#! /usr/bin/env python3 # def prism_jaskowiec_rule_test ( ): #*****************************************************************************80 # ## prism_jaskowiec_rule_test() tests prism_jaskowiec_rule(). # # Licensing: # # This code is distributed under the GNU GPL license. # # Modified: # # 14 July 2023 # # Author: # # John Burkardt. # import platform print ( '' ) print ( 'prism_jaskowiec_rule_test():' ) print ( ' Python version: ' + platform.python_version ( ) ) print ( ' Test prism_jaskowiec_rule().' ) p = 5 prism_jaskowiec_rule_test01 ( p ) p = 5 prism_jaskowiec_rule_test02 ( p ) p_lo = 0 p_hi = 20 prism_jaskowiec_rule_test03 ( p_lo, p_hi ) # # Terminate. # print ( '' ) print ( 'prism_jaskowiec_rule_test():' ) print ( ' Normal end of execution.' ) return def prism_jaskowiec_rule_test01 ( p ): #*****************************************************************************80 # ## prism_jaskowiec_rule_test01() computes a quadrature rule of precision P. # # Licensing: # # This code is distributed under the GNU GPL license. # # Modified: # # 26 May 2023 # # Author: # # John Burkardt. # import numpy as np print ( '' ) print ( 'prism_jaskowiec_rule_test01():' ) print ( ' Quadrature rule for the unit prism,' ) print ( ' Precision p =', p ) # # Retrieve a rule. # n, x, y, z, w = prism_jaskowiec_rule ( p ) # # Print the rule. # print ( '' ) print ( ' I W X Y Z' ) print ( '' ) for i in range ( 0, n ): print ( ' %4d %10.6g %10.6g %10.6g %10.6g' \ % ( i, w[i], x[i], y[i], z[i] ) ) # # Verify that weights sum to 1. # w_sum = np.sum ( w ) print ( '' ) print ( ' Weight Sum ', w_sum ) return def prism_jaskowiec_rule_test02 ( p ): #*****************************************************************************80 # ## prism_jaskowiec_rule_test02() tests a quadrature rule of precision P. # # Licensing: # # This code is distributed under the GNU GPL license. # # Modified: # # 17 May 2023 # # Author: # # John Burkardt. # import numpy as np print ( '' ) print ( 'prism_jaskowiec_rule_test02():' ) print ( ' Test the precision of a quadrature rule for the unit prism.' ) dim_num = 3 # # Retrieve the rule. # n, x, y, z, w = prism_jaskowiec_rule ( p ) # # Pack the x, y, z vectors as rows of an array. # xyz = np.transpose ( np.array ( [ x, y, z ] ) ) print ( '' ) print ( ' Stated precision of rule = ', p ) print ( ' Number of quadrature points = ', n ) print ( '' ) print ( ' Degree Maximum error' ) print ( '' ) for degree in range ( 0, p + 3 ): expon = np.zeros ( dim_num, dtype = int ) more = False h = 0 t = 0 max_error = 0.0 while ( True ): expon, more, h, t = comp_next ( degree, dim_num, expon, more, h, t ) v = monomial_value ( expon, xyz ) quad = prism_unit_volume ( ) * np.dot ( w, v ) exact = prism_unit_monomial_integral ( expon ) quad_error = np.abs ( quad - exact ) max_error = max ( max_error, quad_error ) if ( not more ): break print ( ' %2d %24.16g' % ( degree, max_error ) ) return def prism_jaskowiec_rule_test03 ( p_lo, p_hi ): #*****************************************************************************80 # ## prism_jaskowiec_rule_test03() tests absolute and relative precision. # # Licensing: # # This code is distributed under the GNU GPL license. # # Modified: # # 14 July 2023 # # Author: # # John Burkardt. # # integer p_lo, p_hi: the lowest and highest rules to check. # import numpy as np print ( '' ) print ( 'prism_jaskowiec_rule_test03():' ) print ( ' Test the precision of quadrature rules for the unit prism.' ) print ( ' Check rules of precision p =', p_lo, 'through', p_hi ) print ( ' for error in approximating integrals of monomials.' ) dim_num = 3 print ( '' ) print ( ' maximum maximum' ) print ( ' p absolute relative' ) print ( ' error error' ) print ( '' ) for p in range ( p_lo, p_hi + 1 ): n, x, y, z, w = prism_jaskowiec_rule ( p ) # # Pack the x, y, z vectors as rows of an array. # xyz = np.transpose ( np.array ( [ x, y, z ] ) ) max_abs = 0.0 max_rel = 0.0 for degree in range ( 0, p + 1 ): expon = np.zeros ( dim_num, dtype = int ) more = False h = 0 t = 0 while ( True ): expon, more, h, t = comp_next ( degree, dim_num, expon, more, h, t ) v = monomial_value ( expon, xyz ) quad = prism_unit_volume ( ) * np.dot ( w, v ) exact = prism_unit_monomial_integral ( expon ) quad_error = np.abs ( quad - exact ) max_abs = max ( max_abs, quad_error ) if ( exact != 0 ): max_rel = max ( max_rel, quad_error / abs ( exact ) ) if ( not more ): break print ( ' %2d %24.16g %24.16g' % ( p, max_abs, max_rel ) ) return def comp_next ( n, k, a, more, h, t ): #*****************************************************************************80 # ## comp_next() computes the compositions of the integer N into K parts. # # Discussion: # # A composition of the integer N into K parts is an ordered sequence # of K nonnegative integers which sum to N. The compositions (1,2,1) # and (1,1,2) are considered to be distinct. # # The routine computes one composition on each call until there are no more. # For instance, one composition of 6 into 3 parts is # 3+2+1, another would be 6+0+0. # # On the first call to this routine, set MORE = FALSE. The routine # will compute the first element in the sequence of compositions, and # return it, as well as setting MORE = TRUE. If more compositions # are desired, call again, and again. Each time, the routine will # return with a new composition. # # However, when the LAST composition in the sequence is computed # and returned, the routine will reset MORE to FALSE, signaling that # the end of the sequence has been reached. # # This routine originally used a SAVE statement to maintain the # variables H and T. I have decided that it is safer # to pass these variables as arguments, even though the user should # never alter them. This allows this routine to safely shuffle # between several ongoing calculations. # # There are 28 compositions of 6 into three parts. This routine will # produce those compositions in the following order: # # I A # - --------- # 1 6 0 0 # 2 5 1 0 # 3 4 2 0 # 4 3 3 0 # 5 2 4 0 # 6 1 5 0 # 7 0 6 0 # 8 5 0 1 # 9 4 1 1 # 10 3 2 1 # 11 2 3 1 # 12 1 4 1 # 13 0 5 1 # 14 4 0 2 # 15 3 1 2 # 16 2 2 2 # 17 1 3 2 # 18 0 4 2 # 19 3 0 3 # 20 2 1 3 # 21 1 2 3 # 22 0 3 3 # 23 2 0 4 # 24 1 1 4 # 25 0 2 4 # 26 1 0 5 # 27 0 1 5 # 28 0 0 6 # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 20 May 2015 # # Author: # # John Burkardt. # # Reference: # # Albert Nijenhuis, Herbert Wilf, # Combinatorial Algorithms for Computers and Calculators, # Second Edition, # Academic Press, 1978, # ISBN: 0-12-519260-6, # LC: QA164.N54. # # Input: # # integer N, the integer whose compositions are desired. # # integer K, the number of parts in the composition. # # integer A(K), the previous composition. On the first call, # with MORE = FALSE, set A = []. Thereafter, A should be the # value of A output from the previous call. # # bool MORE. The input value of MORE on the first # call should be FALSE, which tells the program to initialize. # On subsequent calls, MORE should be TRUE, or simply the # output value of MORE from the previous call. # # integer H, T, two internal parameters needed for the # computation. The user may need to initialize these before the # very first call, but these initial values are not important. # The user should not alter these parameters once the computation # begins. # # Output: # # integer A(K), the next composition. # # bool MORE, will be TRUE unless the composition # that is being returned is the final one in the sequence. # # integer H, T, the updated values of the two internal # variables. # if ( k == 0 ): a = [] more = False t = n h = 0 return a, more, h, t if ( not more ): t = n h = 0 a[0] = n for i in range ( 1, k ): a[i] = 0 else: if ( 1 < t ): h = 0 t = a[h] a[h] = 0 a[0] = t - 1 a[h+1] = a[h+1] + 1 h = h + 1 more = ( a[k-1] != n ) return a, more, h, t def monomial_value ( e, x ): #*****************************************************************************80 # ## monomial_value() evaluates a monomial. # # Discussion: # # This routine evaluates a monomial of the form # # product ( 1 <= i <= m ) x(i)^e(i) # # The combination 0.0^0, if encountered, is treated as 1.0. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 15 May 2023 # # Author: # # John Burkardt # # Input: # # integer E(D): the exponents. # # real X(N,D): the point coordinates. # # Output: # # real V(N): the monomial values. # import numpy as np n, d = x.shape v = np.ones ( n ) for j in range ( 0, d ): if ( 0 != e[j] ): v[0:n] = v[0:n] * x[0:n,j] ** e[j] return v def prism_jaskowiec_rule ( p ): #*****************************************************************************80 # ## prism_jaskowiec_rule() returns a prism quadrature rule of given precision. # # Discussion: # # The unit prism is defined as: # # - ( 1 - Z ) <= X <= 1 - Z # - ( 1 - Z ) <= Y <= 1 - Z # 0 <= Z <= 1. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 17 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Input: # # integer p: the precision, 0 <= p <= 20. # # Output: # # integer n: the order of the rule. # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights. # if ( p < 0 ): raise Exception ( 'prism_jaskowiec_rule(): Input p < 0.' ) if ( 20 < p ): raise Exception ( 'prism_jaskowiec_rule(): Input 20 < p.' ) n = rule_order ( p ) if ( p == 0 ): x, y, z, w = rule00 ( ) elif ( p == 1 ): x, y, z, w = rule01 ( ) elif ( p == 2 ): x, y, z, w = rule02 ( ) elif ( p == 3 ): x, y, z, w = rule03 ( ) elif ( p == 4 ): x, y, z, w = rule04 ( ) elif ( p == 5 ): x, y, z, w = rule05 ( ) elif ( p == 6 ): x, y, z, w = rule06 ( ) elif ( p == 7 ): x, y, z, w = rule07 ( ) elif ( p == 8 ): x, y, z, w = rule08 ( ) elif ( p == 9 ): x, y, z, w = rule09 ( ) elif ( p == 10 ): x, y, z, w = rule10 ( ) elif ( p == 11 ): x, y, z, w = rule11 ( ) elif ( p == 12 ): x, y, z, w = rule12 ( ) elif ( p == 13 ): x, y, z, w = rule13 ( ) elif ( p == 14 ): x, y, z, w = rule14 ( ) elif ( p == 15 ): x, y, z, w = rule15 ( ) elif ( p == 16 ): x, y, z, w = rule16 ( ) elif ( p == 17 ): x, y, z, w = rule17 ( ) elif ( p == 18 ): x, y, z, w = rule18 ( ) elif ( p == 19 ): x, y, z, w = rule19 ( ) elif ( p == 20 ): x, y, z, w = rule20 ( ) return n, x, y, z, w def prism_unit_monomial_integral ( expon ): #*****************************************************************************80 # ## prism_unit_monomial_integral(): monomial integral in a unit prism. # # Discussion: # # This routine returns the integral of # # product ( 1 <= I <= 3 ) X(I)^EXPON(I) # # over the unit prism. # # The unit prism is defined as: # # - ( 1 - Z ) <= X <= 1 - Z # - ( 1 - Z ) <= Y <= 1 - Z # 0 <= Z <= 1. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 22 June 2015 # # Author: # # John Burkardt # # Reference: # # Arthur Stroud, # Approximate Calculation of Multiple Integrals, # Prentice Hall, 1971, # ISBN: 0130438936, # LC: QA311.S85. # # Input: # # integer EXPON(3): the exponents. # # Output: # # real VALUE: the integral of the monomial. # from scipy.special import factorial value = factorial ( expon[0] ) \ * factorial ( expon[1] ) \ / ( expon[2] + 1 ) \ / factorial ( expon[0] + expon[1] + 2 ) return value def prism_unit_volume ( ): #*****************************************************************************80 # ## prism_unit_volume() returns the volume of a unit prism. # # Discussion: # # A unit triangular prism has a triangular base in the (x,y) plane, # projected vertically one unit in the z direction. # # The vertices are # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 27 April 2023 # # Author: # # John Burkardt # # Output: # # real value: the volume. # value = 0.5 return value def rule_order ( p ): #*****************************************************************************80 # ## rule_order() returns the order of a prism quadrature rule of given precision. # # Discussion: # # The "order" of a quadrature rule is the number of points involved. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 27 April 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Input: # # integer p: the precision, 0 <= p <= 20. # # Output: # # integer order: the order of the rule. # import numpy as np if ( p < 0 ): raise Exception ( 'rule_order(): Input p < 0.' ) if ( 20 < p ): raise Exception ( 'rule_order(): Input 20 < p.' ) order_vec = np.array ( [ \ 1, \ 1, 5, 8, 11, 16, 28, 35, 46, 59, 84, \ 99, 136, 162, 194, 238, 287, 338, 396, 420, 518 ] ) order = order_vec[p] return order def rule00 ( ): #*****************************************************************************80 # ## rule00() returns the prism rule of precision 0. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333 ] ) y = np.array ( [ \ 0.3333333333333333 ] ) z = np.array ( [ \ 0.5000000000000000 ] ) w = np.array ( [ \ 1.0000000000000000 ] ) return x, y, z, w def rule01 ( ): #*****************************************************************************80 # ## rule01() returns the prism rule of precision 1. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333 ] ) y = np.array ( [ \ 0.3333333333333333 ] ) z = np.array ( [ \ 0.5000000000000000 ] ) w = np.array ( [ \ 1.0000000000000000 ] ) return x, y, z, w def rule02 ( ): #*****************************************************************************80 # ## rule02() returns the prism rule of precision 2. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.1046424703769979, \ 0.1046424703769979, \ 0.7907150592460042 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.1046424703769979, \ 0.7907150592460042, \ 0.1046424703769979 ] ) z = np.array ( [ \ 0.0784174655667168, \ 0.9215825344332832, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000 ] ) w = np.array ( [ \ 0.2344355869392759, \ 0.2344355869392759, \ 0.1770429420404827, \ 0.1770429420404827, \ 0.1770429420404827 ] ) return x, y, z, w def rule03 ( ): #*****************************************************************************80 # ## rule03() returns the prism rule of precision 3. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4890588576053607, \ 0.4890588576053607, \ 0.0218822847892786, \ 0.0778317780273062, \ 0.0778317780273062, \ 0.8443364439453875 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4890588576053607, \ 0.0218822847892786, \ 0.4890588576053607, \ 0.0778317780273062, \ 0.8443364439453875, \ 0.0778317780273062 ] ) z = np.array ( [ \ 0.0192797910055989, \ 0.9807202089944012, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000 ] ) w = np.array ( [ \ 0.1803034341765672, \ 0.1803034341765672, \ 0.1134313729984015, \ 0.1134313729984015, \ 0.1134313729984015, \ 0.0996996708838870, \ 0.0996996708838870, \ 0.0996996708838870 ] ) return x, y, z, w def rule04 ( ): #*****************************************************************************80 # ## rule04() returns the prism rule of precision 4. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4686558098619952, \ 0.4686558098619952, \ 0.0626883802760096, \ 0.1007404057989106, \ 0.1007404057989106, \ 0.7985191884021787, \ 0.1007404057989106, \ 0.1007404057989106, \ 0.7985191884021787 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4686558098619952, \ 0.0626883802760096, \ 0.4686558098619952, \ 0.1007404057989106, \ 0.7985191884021787, \ 0.1007404057989106, \ 0.1007404057989106, \ 0.7985191884021787, \ 0.1007404057989106 ] ) z = np.array ( [ \ 0.0665690129954826, \ 0.9334309870045174, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.8378199118411298, \ 0.8378199118411298, \ 0.8378199118411298, \ 0.1621800881588701, \ 0.1621800881588701, \ 0.1621800881588701 ] ) w = np.array ( [ \ 0.1079119748155355, \ 0.1079119748155355, \ 0.1364146126054776, \ 0.1364146126054776, \ 0.1364146126054776, \ 0.0624887020920827, \ 0.0624887020920827, \ 0.0624887020920827, \ 0.0624887020920827, \ 0.0624887020920827, \ 0.0624887020920827 ] ) return x, y, z, w def rule05 ( ): #*****************************************************************************80 # ## rule05() returns the prism rule of precision 5. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.0517646178271648, \ 0.0517646178271648, \ 0.8964707643456705, \ 0.1663967696311171, \ 0.1663967696311171, \ 0.6672064607377658, \ 0.1663967696311171, \ 0.1663967696311171, \ 0.6672064607377658, \ 0.4976649895838920, \ 0.4976649895838920, \ 0.0046700208322159, \ 0.4976649895838920, \ 0.4976649895838920, \ 0.0046700208322159 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.0517646178271648, \ 0.8964707643456705, \ 0.0517646178271648, \ 0.1663967696311171, \ 0.6672064607377658, \ 0.1663967696311171, \ 0.1663967696311171, \ 0.6672064607377658, \ 0.1663967696311171, \ 0.4976649895838920, \ 0.0046700208322159, \ 0.4976649895838920, \ 0.4976649895838920, \ 0.0046700208322159, \ 0.4976649895838920 ] ) z = np.array ( [ \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.9035817431942219, \ 0.9035817431942219, \ 0.9035817431942219, \ 0.0964182568057780, \ 0.0964182568057780, \ 0.0964182568057780, \ 0.3013691627751696, \ 0.3013691627751696, \ 0.3013691627751696, \ 0.6986308372248304, \ 0.6986308372248304, \ 0.6986308372248304 ] ) w = np.array ( [ \ 0.2071428343483058, \ 0.0380755890309976, \ 0.0380755890309976, \ 0.0380755890309976, \ 0.0763742613922619, \ 0.0763742613922619, \ 0.0763742613922619, \ 0.0763742613922619, \ 0.0763742613922619, \ 0.0763742613922619, \ 0.0367308050341883, \ 0.0367308050341883, \ 0.0367308050341883, \ 0.0367308050341883, \ 0.0367308050341883, \ 0.0367308050341883 ] ) return x, y, z, w def rule06 ( ): #*****************************************************************************80 # ## rule06() returns the prism rule of precision 6. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.0175042446586512, \ 0.0175042446586512, \ 0.9649915106826975, \ 0.1617417813899514, \ 0.1617417813899514, \ 0.6765164372200974, \ 0.4656535513495914, \ 0.4656535513495914, \ 0.0686928973008173, \ 0.4656535513495914, \ 0.4656535513495914, \ 0.0686928973008173, \ 0.0345948698524570, \ 0.2025039451729335, \ 0.0345948698524570, \ 0.7629011849746096, \ 0.2025039451729335, \ 0.7629011849746096, \ 0.0345948698524570, \ 0.2025039451729335, \ 0.0345948698524570, \ 0.7629011849746096, \ 0.2025039451729335, \ 0.7629011849746096 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.0175042446586512, \ 0.9649915106826975, \ 0.0175042446586512, \ 0.1617417813899514, \ 0.6765164372200974, \ 0.1617417813899514, \ 0.4656535513495914, \ 0.0686928973008173, \ 0.4656535513495914, \ 0.4656535513495914, \ 0.0686928973008173, \ 0.4656535513495914, \ 0.2025039451729335, \ 0.0345948698524570, \ 0.7629011849746096, \ 0.0345948698524570, \ 0.7629011849746096, \ 0.2025039451729335, \ 0.2025039451729335, \ 0.0345948698524570, \ 0.7629011849746096, \ 0.0345948698524570, \ 0.7629011849746096, \ 0.2025039451729335 ] ) z = np.array ( [ \ 0.9925192962207799, \ 0.0074807037792201, \ 0.2480010523941030, \ 0.7519989476058970, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.7405713004233492, \ 0.7405713004233492, \ 0.7405713004233492, \ 0.2594286995766508, \ 0.2594286995766508, \ 0.2594286995766508, \ 0.0952682547954233, \ 0.0952682547954233, \ 0.0952682547954233, \ 0.0952682547954233, \ 0.0952682547954233, \ 0.0952682547954233, \ 0.9047317452045767, \ 0.9047317452045767, \ 0.9047317452045767, \ 0.9047317452045767, \ 0.9047317452045767, \ 0.9047317452045767 ] ) w = np.array ( [ \ 0.0361446293820950, \ 0.0361446293820950, \ 0.0554469020242208, \ 0.0554469020242208, \ 0.0116429635765844, \ 0.0116429635765844, \ 0.0116429635765844, \ 0.0768806419265571, \ 0.0768806419265571, \ 0.0768806419265571, \ 0.0496268551496232, \ 0.0496268551496232, \ 0.0496268551496232, \ 0.0496268551496232, \ 0.0496268551496232, \ 0.0496268551496232, \ 0.0211237491483504, \ 0.0211237491483504, \ 0.0211237491483504, \ 0.0211237491483504, \ 0.0211237491483504, \ 0.0211237491483504, \ 0.0211237491483504, \ 0.0211237491483504, \ 0.0211237491483504, \ 0.0211237491483504, \ 0.0211237491483504, \ 0.0211237491483504 ] ) return x, y, z, w def rule07 ( ): #*****************************************************************************80 # ## rule07() returns the prism rule of precision 7. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.0083375954660215, \ 0.0083375954660215, \ 0.9833248090679571, \ 0.4815219753291366, \ 0.4815219753291366, \ 0.0369560493417268, \ 0.4815219753291366, \ 0.4815219753291366, \ 0.0369560493417268, \ 0.0954832483714894, \ 0.0954832483714894, \ 0.8090335032570213, \ 0.0954832483714894, \ 0.0954832483714894, \ 0.8090335032570213, \ 0.7429966820728956, \ 0.0121491315983783, \ 0.7429966820728956, \ 0.2448541863287261, \ 0.0121491315983783, \ 0.2448541863287261, \ 0.1529845984247976, \ 0.3051562164322261, \ 0.1529845984247976, \ 0.5418591851429763, \ 0.3051562164322261, \ 0.5418591851429763, \ 0.1529845984247976, \ 0.3051562164322261, \ 0.1529845984247976, \ 0.5418591851429763, \ 0.3051562164322261, \ 0.5418591851429763 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.0083375954660215, \ 0.9833248090679571, \ 0.0083375954660215, \ 0.4815219753291366, \ 0.0369560493417268, \ 0.4815219753291366, \ 0.4815219753291366, \ 0.0369560493417268, \ 0.4815219753291366, \ 0.0954832483714894, \ 0.8090335032570213, \ 0.0954832483714894, \ 0.0954832483714894, \ 0.8090335032570213, \ 0.0954832483714894, \ 0.0121491315983783, \ 0.7429966820728956, \ 0.2448541863287261, \ 0.7429966820728956, \ 0.2448541863287261, \ 0.0121491315983783, \ 0.3051562164322261, \ 0.1529845984247976, \ 0.5418591851429763, \ 0.1529845984247976, \ 0.5418591851429763, \ 0.3051562164322261, \ 0.3051562164322261, \ 0.1529845984247976, \ 0.5418591851429763, \ 0.1529845984247976, \ 0.5418591851429763, \ 0.3051562164322261 ] ) z = np.array ( [ \ 0.9901140347954458, \ 0.0098859652045542, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.0793063228967370, \ 0.0793063228967370, \ 0.0793063228967370, \ 0.9206936771032630, \ 0.9206936771032630, \ 0.9206936771032630, \ 0.1020754547065085, \ 0.1020754547065085, \ 0.1020754547065085, \ 0.8979245452934915, \ 0.8979245452934915, \ 0.8979245452934915, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.7019672802660549, \ 0.7019672802660549, \ 0.7019672802660549, \ 0.7019672802660549, \ 0.7019672802660549, \ 0.7019672802660549, \ 0.2980327197339451, \ 0.2980327197339451, \ 0.2980327197339451, \ 0.2980327197339451, \ 0.2980327197339451, \ 0.2980327197339451 ] ) w = np.array ( [ \ 0.0284456811126893, \ 0.0284456811126893, \ 0.0061182421730950, \ 0.0061182421730950, \ 0.0061182421730950, \ 0.0215508640862265, \ 0.0215508640862265, \ 0.0215508640862265, \ 0.0215508640862265, \ 0.0215508640862265, \ 0.0215508640862265, \ 0.0291785249020985, \ 0.0291785249020985, \ 0.0291785249020985, \ 0.0291785249020985, \ 0.0291785249020985, \ 0.0291785249020985, \ 0.0255148563351493, \ 0.0255148563351493, \ 0.0255148563351493, \ 0.0255148563351493, \ 0.0255148563351493, \ 0.0255148563351493, \ 0.0389407032762076, \ 0.0389407032762076, \ 0.0389407032762076, \ 0.0389407032762076, \ 0.0389407032762076, \ 0.0389407032762076, \ 0.0389407032762076, \ 0.0389407032762076, \ 0.0389407032762076, \ 0.0389407032762076, \ 0.0389407032762076, \ 0.0389407032762076 ] ) return x, y, z, w def rule08 ( ): #*****************************************************************************80 # ## rule08() returns the prism rule of precision 8. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4600889628137106, \ 0.4600889628137106, \ 0.0798220743725788, \ 0.0534123509369071, \ 0.0534123509369071, \ 0.8931752981261857, \ 0.0472387858397694, \ 0.0472387858397694, \ 0.9055224283204613, \ 0.0472387858397694, \ 0.0472387858397694, \ 0.9055224283204613, \ 0.1740616079243704, \ 0.1740616079243704, \ 0.6518767841512593, \ 0.1740616079243704, \ 0.1740616079243704, \ 0.6518767841512593, \ 0.1597492639425890, \ 0.1597492639425890, \ 0.6805014721148220, \ 0.1597492639425890, \ 0.1597492639425890, \ 0.6805014721148220, \ 0.4585690687909513, \ 0.4585690687909513, \ 0.0828618624180973, \ 0.4585690687909513, \ 0.4585690687909513, \ 0.0828618624180973, \ 0.0085881275077590, \ 0.7285980718010000, \ 0.0085881275077590, \ 0.2628138006912410, \ 0.7285980718010000, \ 0.2628138006912410, \ 0.0085881275077590, \ 0.7285980718010000, \ 0.0085881275077590, \ 0.2628138006912410, \ 0.7285980718010000, \ 0.2628138006912410 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4600889628137106, \ 0.0798220743725788, \ 0.4600889628137106, \ 0.0534123509369071, \ 0.8931752981261857, \ 0.0534123509369071, \ 0.0472387858397694, \ 0.9055224283204613, \ 0.0472387858397694, \ 0.0472387858397694, \ 0.9055224283204613, \ 0.0472387858397694, \ 0.1740616079243704, \ 0.6518767841512593, \ 0.1740616079243704, \ 0.1740616079243704, \ 0.6518767841512593, \ 0.1740616079243704, \ 0.1597492639425890, \ 0.6805014721148220, \ 0.1597492639425890, \ 0.1597492639425890, \ 0.6805014721148220, \ 0.1597492639425890, \ 0.4585690687909513, \ 0.0828618624180973, \ 0.4585690687909513, \ 0.4585690687909513, \ 0.0828618624180973, \ 0.4585690687909513, \ 0.7285980718010000, \ 0.0085881275077590, \ 0.2628138006912410, \ 0.0085881275077590, \ 0.2628138006912410, \ 0.7285980718010000, \ 0.7285980718010000, \ 0.0085881275077590, \ 0.2628138006912410, \ 0.0085881275077590, \ 0.2628138006912410, \ 0.7285980718010000 ] ) z = np.array ( [ \ 0.6122169330529953, \ 0.3877830669470047, \ 0.1590872792145671, \ 0.8409127207854329, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.9144077215293400, \ 0.9144077215293400, \ 0.9144077215293400, \ 0.0855922784706599, \ 0.0855922784706599, \ 0.0855922784706599, \ 0.7030421146772952, \ 0.7030421146772952, \ 0.7030421146772952, \ 0.2969578853227048, \ 0.2969578853227048, \ 0.2969578853227048, \ 0.9829325832703272, \ 0.9829325832703272, \ 0.9829325832703272, \ 0.0170674167296728, \ 0.0170674167296728, \ 0.0170674167296728, \ 0.9380979550001272, \ 0.9380979550001272, \ 0.9380979550001272, \ 0.0619020449998729, \ 0.0619020449998729, \ 0.0619020449998729, \ 0.7886751345948129, \ 0.7886751345948129, \ 0.7886751345948129, \ 0.7886751345948129, \ 0.7886751345948129, \ 0.7886751345948129, \ 0.2113248654051871, \ 0.2113248654051871, \ 0.2113248654051871, \ 0.2113248654051871, \ 0.2113248654051871, \ 0.2113248654051871 ] ) w = np.array ( [ \ 0.0207349366428555, \ 0.0207349366428555, \ 0.0510894271121815, \ 0.0510894271121815, \ 0.0526997449065072, \ 0.0526997449065072, \ 0.0526997449065072, \ 0.0181522487841497, \ 0.0181522487841497, \ 0.0181522487841497, \ 0.0071453577535386, \ 0.0071453577535386, \ 0.0071453577535386, \ 0.0071453577535386, \ 0.0071453577535386, \ 0.0071453577535386, \ 0.0406292626589893, \ 0.0406292626589893, \ 0.0406292626589893, \ 0.0406292626589893, \ 0.0406292626589893, \ 0.0406292626589893, \ 0.0111430273484653, \ 0.0111430273484653, \ 0.0111430273484653, \ 0.0111430273484653, \ 0.0111430273484653, \ 0.0111430273484653, \ 0.0210865092935865, \ 0.0210865092935865, \ 0.0210865092935865, \ 0.0210865092935865, \ 0.0210865092935865, \ 0.0210865092935865, \ 0.0136475290908731, \ 0.0136475290908731, \ 0.0136475290908731, \ 0.0136475290908731, \ 0.0136475290908731, \ 0.0136475290908731, \ 0.0136475290908731, \ 0.0136475290908731, \ 0.0136475290908731, \ 0.0136475290908731, \ 0.0136475290908731, \ 0.0136475290908731 ] ) return x, y, z, w def rule09 ( ): #*****************************************************************************80 # ## rule09() returns the prism rule of precision 9. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4938630638969568, \ 0.4938630638969568, \ 0.0122738722060863, \ 0.2362540169543293, \ 0.2362540169543293, \ 0.5274919660913414, \ 0.2362540169543293, \ 0.2362540169543293, \ 0.5274919660913414, \ 0.0722383394163824, \ 0.0722383394163824, \ 0.8555233211672353, \ 0.0722383394163824, \ 0.0722383394163824, \ 0.8555233211672353, \ 0.4383137607101617, \ 0.4383137607101617, \ 0.1233724785796766, \ 0.4383137607101617, \ 0.4383137607101617, \ 0.1233724785796766, \ 0.0364340164940779, \ 0.0364340164940779, \ 0.9271319670118442, \ 0.0364340164940779, \ 0.0364340164940779, \ 0.9271319670118442, \ 0.4828779929693860, \ 0.4828779929693860, \ 0.0342440140612279, \ 0.4828779929693860, \ 0.4828779929693860, \ 0.0342440140612279, \ 0.1628698857202373, \ 0.1628698857202373, \ 0.6742602285595254, \ 0.1628698857202373, \ 0.1628698857202373, \ 0.6742602285595254, \ 0.8213377527237301, \ 0.1626087609745086, \ 0.8213377527237301, \ 0.0160534863017613, \ 0.1626087609745086, \ 0.0160534863017613, \ 0.0424944495063928, \ 0.2561926710584905, \ 0.0424944495063928, \ 0.7013128794351167, \ 0.2561926710584905, \ 0.7013128794351167, \ 0.0424944495063928, \ 0.2561926710584905, \ 0.0424944495063928, \ 0.7013128794351167, \ 0.2561926710584905, \ 0.7013128794351167 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4938630638969568, \ 0.0122738722060863, \ 0.4938630638969568, \ 0.2362540169543293, \ 0.5274919660913414, \ 0.2362540169543293, \ 0.2362540169543293, \ 0.5274919660913414, \ 0.2362540169543293, \ 0.0722383394163824, \ 0.8555233211672353, \ 0.0722383394163824, \ 0.0722383394163824, \ 0.8555233211672353, \ 0.0722383394163824, \ 0.4383137607101617, \ 0.1233724785796766, \ 0.4383137607101617, \ 0.4383137607101617, \ 0.1233724785796766, \ 0.4383137607101617, \ 0.0364340164940779, \ 0.9271319670118442, \ 0.0364340164940779, \ 0.0364340164940779, \ 0.9271319670118442, \ 0.0364340164940779, \ 0.4828779929693860, \ 0.0342440140612279, \ 0.4828779929693860, \ 0.4828779929693860, \ 0.0342440140612279, \ 0.4828779929693860, \ 0.1628698857202373, \ 0.6742602285595254, \ 0.1628698857202373, \ 0.1628698857202373, \ 0.6742602285595254, \ 0.1628698857202373, \ 0.1626087609745086, \ 0.8213377527237301, \ 0.0160534863017613, \ 0.8213377527237301, \ 0.0160534863017613, \ 0.1626087609745086, \ 0.2561926710584905, \ 0.0424944495063928, \ 0.7013128794351167, \ 0.0424944495063928, \ 0.7013128794351167, \ 0.2561926710584905, \ 0.2561926710584905, \ 0.0424944495063928, \ 0.7013128794351167, \ 0.0424944495063928, \ 0.7013128794351167, \ 0.2561926710584905 ] ) z = np.array ( [ \ 0.3624907904961116, \ 0.6375092095038883, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.9356193788144789, \ 0.9356193788144789, \ 0.9356193788144789, \ 0.0643806211855212, \ 0.0643806211855212, \ 0.0643806211855212, \ 0.9775968701180802, \ 0.9775968701180802, \ 0.9775968701180802, \ 0.0224031298819198, \ 0.0224031298819198, \ 0.0224031298819198, \ 0.7182058908565040, \ 0.7182058908565040, \ 0.7182058908565040, \ 0.2817941091434960, \ 0.2817941091434960, \ 0.2817941091434960, \ 0.7515028180283210, \ 0.7515028180283210, \ 0.7515028180283210, \ 0.2484971819716791, \ 0.2484971819716791, \ 0.2484971819716791, \ 0.0091271384992068, \ 0.0091271384992068, \ 0.0091271384992068, \ 0.9908728615007932, \ 0.9908728615007932, \ 0.9908728615007932, \ 0.4125665631901645, \ 0.4125665631901645, \ 0.4125665631901645, \ 0.5874334368098355, \ 0.5874334368098355, \ 0.5874334368098355, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.8418079279883068, \ 0.8418079279883068, \ 0.8418079279883068, \ 0.8418079279883068, \ 0.8418079279883068, \ 0.8418079279883068, \ 0.1581920720116932, \ 0.1581920720116932, \ 0.1581920720116932, \ 0.1581920720116932, \ 0.1581920720116932, \ 0.1581920720116932 ] ) w = np.array ( [ \ 0.0342834712909237, \ 0.0342834712909237, \ 0.0163927977044536, \ 0.0163927977044536, \ 0.0163927977044536, \ 0.0234717806987032, \ 0.0234717806987032, \ 0.0234717806987032, \ 0.0234717806987032, \ 0.0234717806987032, \ 0.0234717806987032, \ 0.0057618481582789, \ 0.0057618481582789, \ 0.0057618481582789, \ 0.0057618481582789, \ 0.0057618481582789, \ 0.0057618481582789, \ 0.0349247930577802, \ 0.0349247930577802, \ 0.0349247930577802, \ 0.0349247930577802, \ 0.0349247930577802, \ 0.0349247930577802, \ 0.0073738996232123, \ 0.0073738996232123, \ 0.0073738996232123, \ 0.0073738996232123, \ 0.0073738996232123, \ 0.0073738996232123, \ 0.0057224608464489, \ 0.0057224608464489, \ 0.0057224608464489, \ 0.0057224608464489, \ 0.0057224608464489, \ 0.0057224608464489, \ 0.0243387430144453, \ 0.0243387430144453, \ 0.0243387430144453, \ 0.0243387430144453, \ 0.0243387430144453, \ 0.0243387430144453, \ 0.0094129636631352, \ 0.0094129636631352, \ 0.0094129636631352, \ 0.0094129636631352, \ 0.0094129636631352, \ 0.0094129636631352, \ 0.0180179774943973, \ 0.0180179774943973, \ 0.0180179774943973, \ 0.0180179774943973, \ 0.0180179774943973, \ 0.0180179774943973, \ 0.0180179774943973, \ 0.0180179774943973, \ 0.0180179774943973, \ 0.0180179774943973, \ 0.0180179774943973, \ 0.0180179774943973 ] ) return x, y, z, w def rule10 ( ): #*****************************************************************************80 # ## rule10() returns the prism rule of precision 10. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4300104731739727, \ 0.4300104731739727, \ 0.1399790536520547, \ 0.1095696683547513, \ 0.1095696683547513, \ 0.7808606632904974, \ 0.0158148366313868, \ 0.0158148366313868, \ 0.9683703267372263, \ 0.4356018236697312, \ 0.4356018236697312, \ 0.1287963526605377, \ 0.4356018236697312, \ 0.4356018236697312, \ 0.1287963526605377, \ 0.1485976165307029, \ 0.1485976165307029, \ 0.7028047669385943, \ 0.1485976165307029, \ 0.1485976165307029, \ 0.7028047669385943, \ 0.0519664835822357, \ 0.0519664835822357, \ 0.8960670328355285, \ 0.0519664835822357, \ 0.0519664835822357, \ 0.8960670328355285, \ 0.4990219694442680, \ 0.4990219694442680, \ 0.0019560611114641, \ 0.4990219694442680, \ 0.4990219694442680, \ 0.0019560611114641, \ 0.0415525016027702, \ 0.0415525016027702, \ 0.9168949967944596, \ 0.0415525016027702, \ 0.0415525016027702, \ 0.9168949967944596, \ 0.2343945196820784, \ 0.2343945196820784, \ 0.5312109606358433, \ 0.2343945196820784, \ 0.2343945196820784, \ 0.5312109606358433, \ 0.0528115168465621, \ 0.2716521744885937, \ 0.0528115168465621, \ 0.6755363086648442, \ 0.2716521744885937, \ 0.6755363086648442, \ 0.0528115168465621, \ 0.2716521744885937, \ 0.0528115168465621, \ 0.6755363086648442, \ 0.2716521744885937, \ 0.6755363086648442, \ 0.1675738337212976, \ 0.0101912520986929, \ 0.1675738337212976, \ 0.8222349141800095, \ 0.0101912520986929, \ 0.8222349141800095, \ 0.1675738337212976, \ 0.0101912520986929, \ 0.1675738337212976, \ 0.8222349141800095, \ 0.0101912520986929, \ 0.8222349141800095, \ 0.3291869417398026, \ 0.0509081627669518, \ 0.3291869417398026, \ 0.6199048954932456, \ 0.0509081627669518, \ 0.6199048954932456, \ 0.3291869417398026, \ 0.0509081627669518, \ 0.3291869417398026, \ 0.6199048954932456, \ 0.0509081627669518, \ 0.6199048954932456 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4300104731739727, \ 0.1399790536520547, \ 0.4300104731739727, \ 0.1095696683547513, \ 0.7808606632904974, \ 0.1095696683547513, \ 0.0158148366313868, \ 0.9683703267372263, \ 0.0158148366313868, \ 0.4356018236697312, \ 0.1287963526605377, \ 0.4356018236697312, \ 0.4356018236697312, \ 0.1287963526605377, \ 0.4356018236697312, \ 0.1485976165307029, \ 0.7028047669385943, \ 0.1485976165307029, \ 0.1485976165307029, \ 0.7028047669385943, \ 0.1485976165307029, \ 0.0519664835822357, \ 0.8960670328355285, \ 0.0519664835822357, \ 0.0519664835822357, \ 0.8960670328355285, \ 0.0519664835822357, \ 0.4990219694442680, \ 0.0019560611114641, \ 0.4990219694442680, \ 0.4990219694442680, \ 0.0019560611114641, \ 0.4990219694442680, \ 0.0415525016027702, \ 0.9168949967944596, \ 0.0415525016027702, \ 0.0415525016027702, \ 0.9168949967944596, \ 0.0415525016027702, \ 0.2343945196820784, \ 0.5312109606358433, \ 0.2343945196820784, \ 0.2343945196820784, \ 0.5312109606358433, \ 0.2343945196820784, \ 0.2716521744885937, \ 0.0528115168465621, \ 0.6755363086648442, \ 0.0528115168465621, \ 0.6755363086648442, \ 0.2716521744885937, \ 0.2716521744885937, \ 0.0528115168465621, \ 0.6755363086648442, \ 0.0528115168465621, \ 0.6755363086648442, \ 0.2716521744885937, \ 0.0101912520986929, \ 0.1675738337212976, \ 0.8222349141800095, \ 0.1675738337212976, \ 0.8222349141800095, \ 0.0101912520986929, \ 0.0101912520986929, \ 0.1675738337212976, \ 0.8222349141800095, \ 0.1675738337212976, \ 0.8222349141800095, \ 0.0101912520986929, \ 0.0509081627669518, \ 0.3291869417398026, \ 0.6199048954932456, \ 0.3291869417398026, \ 0.6199048954932456, \ 0.0509081627669518, \ 0.0509081627669518, \ 0.3291869417398026, \ 0.6199048954932456, \ 0.3291869417398026, \ 0.6199048954932456, \ 0.0509081627669518 ] ) z = np.array ( [ \ 0.5000000000000000, \ 0.9825433408084738, \ 0.0174566591915262, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.8914517952143866, \ 0.8914517952143866, \ 0.8914517952143866, \ 0.1085482047856134, \ 0.1085482047856134, \ 0.1085482047856134, \ 0.8836628766030707, \ 0.8836628766030707, \ 0.8836628766030707, \ 0.1163371233969292, \ 0.1163371233969292, \ 0.1163371233969292, \ 0.7446456974623019, \ 0.7446456974623019, \ 0.7446456974623019, \ 0.2553543025376981, \ 0.2553543025376981, \ 0.2553543025376981, \ 0.8458358146384862, \ 0.8458358146384862, \ 0.8458358146384862, \ 0.1541641853615138, \ 0.1541641853615138, \ 0.1541641853615138, \ 0.9455005421560339, \ 0.9455005421560339, \ 0.9455005421560339, \ 0.0544994578439661, \ 0.0544994578439661, \ 0.0544994578439661, \ 0.7325671383641421, \ 0.7325671383641421, \ 0.7325671383641421, \ 0.2674328616358579, \ 0.2674328616358579, \ 0.2674328616358579, \ 0.9783107845287782, \ 0.9783107845287782, \ 0.9783107845287782, \ 0.9783107845287782, \ 0.9783107845287782, \ 0.9783107845287782, \ 0.0216892154712218, \ 0.0216892154712218, \ 0.0216892154712218, \ 0.0216892154712218, \ 0.0216892154712218, \ 0.0216892154712218, \ 0.7695898729425880, \ 0.7695898729425880, \ 0.7695898729425880, \ 0.7695898729425880, \ 0.7695898729425880, \ 0.7695898729425880, \ 0.2304101270574120, \ 0.2304101270574120, \ 0.2304101270574120, \ 0.2304101270574120, \ 0.2304101270574120, \ 0.2304101270574120, \ 0.3678576424324543, \ 0.3678576424324543, \ 0.3678576424324543, \ 0.3678576424324543, \ 0.3678576424324543, \ 0.3678576424324543, \ 0.6321423575675457, \ 0.6321423575675457, \ 0.6321423575675457, \ 0.6321423575675457, \ 0.6321423575675457, \ 0.6321423575675457 ] ) w = np.array ( [ \ 0.0321960882383680, \ 0.0132395953233874, \ 0.0132395953233874, \ 0.0166307172641394, \ 0.0166307172641394, \ 0.0166307172641394, \ 0.0223711737567938, \ 0.0223711737567938, \ 0.0223711737567938, \ 0.0029651442680284, \ 0.0029651442680284, \ 0.0029651442680284, \ 0.0204122071267436, \ 0.0204122071267436, \ 0.0204122071267436, \ 0.0204122071267436, \ 0.0204122071267436, \ 0.0204122071267436, \ 0.0141443496976585, \ 0.0141443496976585, \ 0.0141443496976585, \ 0.0141443496976585, \ 0.0141443496976585, \ 0.0141443496976585, \ 0.0035457088242710, \ 0.0035457088242710, \ 0.0035457088242710, \ 0.0035457088242710, \ 0.0035457088242710, \ 0.0035457088242710, \ 0.0060580804829928, \ 0.0060580804829928, \ 0.0060580804829928, \ 0.0060580804829928, \ 0.0060580804829928, \ 0.0060580804829928, \ 0.0035558880399586, \ 0.0035558880399586, \ 0.0035558880399586, \ 0.0035558880399586, \ 0.0035558880399586, \ 0.0035558880399586, \ 0.0303810246652175, \ 0.0303810246652175, \ 0.0303810246652175, \ 0.0303810246652175, \ 0.0303810246652175, \ 0.0303810246652175, \ 0.0062118567570921, \ 0.0062118567570921, \ 0.0062118567570921, \ 0.0062118567570921, \ 0.0062118567570921, \ 0.0062118567570921, \ 0.0062118567570921, \ 0.0062118567570921, \ 0.0062118567570921, \ 0.0062118567570921, \ 0.0062118567570921, \ 0.0062118567570921, \ 0.0066184191661461, \ 0.0066184191661461, \ 0.0066184191661461, \ 0.0066184191661461, \ 0.0066184191661461, \ 0.0066184191661461, \ 0.0066184191661461, \ 0.0066184191661461, \ 0.0066184191661461, \ 0.0066184191661461, \ 0.0066184191661461, \ 0.0066184191661461, \ 0.0160730625956718, \ 0.0160730625956718, \ 0.0160730625956718, \ 0.0160730625956718, \ 0.0160730625956718, \ 0.0160730625956718, \ 0.0160730625956718, \ 0.0160730625956718, \ 0.0160730625956718, \ 0.0160730625956718, \ 0.0160730625956718, \ 0.0160730625956718 ] ) return x, y, z, w def rule11 ( ): #*****************************************************************************80 # ## rule11() returns the prism rule of precision 11. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4269221881685347, \ 0.4269221881685347, \ 0.1461556236629306, \ 0.4396327659106838, \ 0.4396327659106838, \ 0.1207344681786323, \ 0.4396327659106838, \ 0.4396327659106838, \ 0.1207344681786323, \ 0.2009128761138035, \ 0.2009128761138035, \ 0.5981742477723929, \ 0.2009128761138035, \ 0.2009128761138035, \ 0.5981742477723929, \ 0.0215850980431769, \ 0.0215850980431769, \ 0.9568298039136461, \ 0.0215850980431769, \ 0.0215850980431769, \ 0.9568298039136461, \ 0.4941546402080231, \ 0.4941546402080231, \ 0.0116907195839538, \ 0.4941546402080231, \ 0.4941546402080231, \ 0.0116907195839538, \ 0.0599602772654436, \ 0.0599602772654436, \ 0.8800794454691128, \ 0.0599602772654436, \ 0.0599602772654436, \ 0.8800794454691128, \ 0.2328102236807494, \ 0.2328102236807494, \ 0.5343795526385012, \ 0.2328102236807494, \ 0.2328102236807494, \ 0.5343795526385012, \ 0.1213201391549184, \ 0.1213201391549184, \ 0.7573597216901632, \ 0.1213201391549184, \ 0.1213201391549184, \ 0.7573597216901632, \ 0.4987543911906001, \ 0.4987543911906001, \ 0.0024912176187997, \ 0.4987543911906001, \ 0.4987543911906001, \ 0.0024912176187997, \ 0.1046140524481813, \ 0.0165184963342511, \ 0.1046140524481813, \ 0.8788674512175675, \ 0.0165184963342511, \ 0.8788674512175675, \ 0.0719632569755848, \ 0.2979854667459965, \ 0.0719632569755848, \ 0.6300512762784186, \ 0.2979854667459965, \ 0.6300512762784186, \ 0.0719632569755848, \ 0.2979854667459965, \ 0.0719632569755848, \ 0.6300512762784186, \ 0.2979854667459965, \ 0.6300512762784186, \ 0.1998026706474004, \ 0.0172094825510263, \ 0.1998026706474004, \ 0.7829878468015733, \ 0.0172094825510263, \ 0.7829878468015733, \ 0.1998026706474004, \ 0.0172094825510263, \ 0.1998026706474004, \ 0.7829878468015733, \ 0.0172094825510263, \ 0.7829878468015733, \ 0.3197359768880742, \ 0.0462852386525127, \ 0.3197359768880742, \ 0.6339787844594131, \ 0.0462852386525127, \ 0.6339787844594131, \ 0.3197359768880742, \ 0.0462852386525127, \ 0.3197359768880742, \ 0.6339787844594131, \ 0.0462852386525127, \ 0.6339787844594131 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4269221881685347, \ 0.1461556236629306, \ 0.4269221881685347, \ 0.4396327659106838, \ 0.1207344681786323, \ 0.4396327659106838, \ 0.4396327659106838, \ 0.1207344681786323, \ 0.4396327659106838, \ 0.2009128761138035, \ 0.5981742477723929, \ 0.2009128761138035, \ 0.2009128761138035, \ 0.5981742477723929, \ 0.2009128761138035, \ 0.0215850980431769, \ 0.9568298039136461, \ 0.0215850980431769, \ 0.0215850980431769, \ 0.9568298039136461, \ 0.0215850980431769, \ 0.4941546402080231, \ 0.0116907195839538, \ 0.4941546402080231, \ 0.4941546402080231, \ 0.0116907195839538, \ 0.4941546402080231, \ 0.0599602772654436, \ 0.8800794454691128, \ 0.0599602772654436, \ 0.0599602772654436, \ 0.8800794454691128, \ 0.0599602772654436, \ 0.2328102236807494, \ 0.5343795526385012, \ 0.2328102236807494, \ 0.2328102236807494, \ 0.5343795526385012, \ 0.2328102236807494, \ 0.1213201391549184, \ 0.7573597216901632, \ 0.1213201391549184, \ 0.1213201391549184, \ 0.7573597216901632, \ 0.1213201391549184, \ 0.4987543911906001, \ 0.0024912176187997, \ 0.4987543911906001, \ 0.4987543911906001, \ 0.0024912176187997, \ 0.4987543911906001, \ 0.0165184963342511, \ 0.1046140524481813, \ 0.8788674512175675, \ 0.1046140524481813, \ 0.8788674512175675, \ 0.0165184963342511, \ 0.2979854667459965, \ 0.0719632569755848, \ 0.6300512762784186, \ 0.0719632569755848, \ 0.6300512762784186, \ 0.2979854667459965, \ 0.2979854667459965, \ 0.0719632569755848, \ 0.6300512762784186, \ 0.0719632569755848, \ 0.6300512762784186, \ 0.2979854667459965, \ 0.0172094825510263, \ 0.1998026706474004, \ 0.7829878468015733, \ 0.1998026706474004, \ 0.7829878468015733, \ 0.0172094825510263, \ 0.0172094825510263, \ 0.1998026706474004, \ 0.7829878468015733, \ 0.1998026706474004, \ 0.7829878468015733, \ 0.0172094825510263, \ 0.0462852386525127, \ 0.3197359768880742, \ 0.6339787844594131, \ 0.3197359768880742, \ 0.6339787844594131, \ 0.0462852386525127, \ 0.0462852386525127, \ 0.3197359768880742, \ 0.6339787844594131, \ 0.3197359768880742, \ 0.6339787844594131, \ 0.0462852386525127 ] ) z = np.array ( [ \ 0.2225225095750167, \ 0.7774774904249833, \ 0.9852161050240122, \ 0.0147838949759878, \ 0.4588747274117720, \ 0.5411252725882280, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.8379989630425995, \ 0.8379989630425995, \ 0.8379989630425995, \ 0.1620010369574005, \ 0.1620010369574005, \ 0.1620010369574005, \ 0.9180267718000730, \ 0.9180267718000730, \ 0.9180267718000730, \ 0.0819732281999270, \ 0.0819732281999270, \ 0.0819732281999270, \ 0.7930952762440336, \ 0.7930952762440336, \ 0.7930952762440336, \ 0.2069047237559664, \ 0.2069047237559664, \ 0.2069047237559664, \ 0.9873500827745887, \ 0.9873500827745887, \ 0.9873500827745887, \ 0.0126499172254113, \ 0.0126499172254113, \ 0.0126499172254113, \ 0.9725136338749176, \ 0.9725136338749176, \ 0.9725136338749176, \ 0.0274863661250824, \ 0.0274863661250824, \ 0.0274863661250824, \ 0.6317612421960503, \ 0.6317612421960503, \ 0.6317612421960503, \ 0.3682387578039497, \ 0.3682387578039497, \ 0.3682387578039497, \ 0.7219895510260312, \ 0.7219895510260312, \ 0.7219895510260312, \ 0.2780104489739688, \ 0.2780104489739688, \ 0.2780104489739688, \ 0.7875076710403349, \ 0.7875076710403349, \ 0.7875076710403349, \ 0.2124923289596651, \ 0.2124923289596651, \ 0.2124923289596651, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.9685666190309952, \ 0.9685666190309952, \ 0.9685666190309952, \ 0.9685666190309952, \ 0.9685666190309952, \ 0.9685666190309952, \ 0.0314333809690049, \ 0.0314333809690049, \ 0.0314333809690049, \ 0.0314333809690049, \ 0.0314333809690049, \ 0.0314333809690049, \ 0.8651322718570146, \ 0.8651322718570146, \ 0.8651322718570146, \ 0.8651322718570146, \ 0.8651322718570146, \ 0.8651322718570146, \ 0.1348677281429854, \ 0.1348677281429854, \ 0.1348677281429854, \ 0.1348677281429854, \ 0.1348677281429854, \ 0.1348677281429854, \ 0.3827016872275200, \ 0.3827016872275200, \ 0.3827016872275200, \ 0.3827016872275200, \ 0.3827016872275200, \ 0.3827016872275200, \ 0.6172983127724800, \ 0.6172983127724800, \ 0.6172983127724800, \ 0.6172983127724800, \ 0.6172983127724800, \ 0.6172983127724800 ] ) w = np.array ( [ \ 0.0216801440673047, \ 0.0216801440673047, \ 0.0104576079858723, \ 0.0104576079858723, \ 0.0034934625077134, \ 0.0034934625077134, \ 0.0161061041817141, \ 0.0161061041817141, \ 0.0161061041817141, \ 0.0220331028546643, \ 0.0220331028546643, \ 0.0220331028546643, \ 0.0220331028546643, \ 0.0220331028546643, \ 0.0220331028546643, \ 0.0127441869618096, \ 0.0127441869618096, \ 0.0127441869618096, \ 0.0127441869618096, \ 0.0127441869618096, \ 0.0127441869618096, \ 0.0028425261018310, \ 0.0028425261018310, \ 0.0028425261018310, \ 0.0028425261018310, \ 0.0028425261018310, \ 0.0028425261018310, \ 0.0015098577228106, \ 0.0015098577228106, \ 0.0015098577228106, \ 0.0015098577228106, \ 0.0015098577228106, \ 0.0015098577228106, \ 0.0036504270647864, \ 0.0036504270647864, \ 0.0036504270647864, \ 0.0036504270647864, \ 0.0036504270647864, \ 0.0036504270647864, \ 0.0210029067169834, \ 0.0210029067169834, \ 0.0210029067169834, \ 0.0210029067169834, \ 0.0210029067169834, \ 0.0210029067169834, \ 0.0187619585366729, \ 0.0187619585366729, \ 0.0187619585366729, \ 0.0187619585366729, \ 0.0187619585366729, \ 0.0187619585366729, \ 0.0051948191291646, \ 0.0051948191291646, \ 0.0051948191291646, \ 0.0051948191291646, \ 0.0051948191291646, \ 0.0051948191291646, \ 0.0070505879387705, \ 0.0070505879387705, \ 0.0070505879387705, \ 0.0070505879387705, \ 0.0070505879387705, \ 0.0070505879387705, \ 0.0059012699482948, \ 0.0059012699482948, \ 0.0059012699482948, \ 0.0059012699482948, \ 0.0059012699482948, \ 0.0059012699482948, \ 0.0059012699482948, \ 0.0059012699482948, \ 0.0059012699482948, \ 0.0059012699482948, \ 0.0059012699482948, \ 0.0059012699482948, \ 0.0062126654655306, \ 0.0062126654655306, \ 0.0062126654655306, \ 0.0062126654655306, \ 0.0062126654655306, \ 0.0062126654655306, \ 0.0062126654655306, \ 0.0062126654655306, \ 0.0062126654655306, \ 0.0062126654655306, \ 0.0062126654655306, \ 0.0062126654655306, \ 0.0138591496001844, \ 0.0138591496001844, \ 0.0138591496001844, \ 0.0138591496001844, \ 0.0138591496001844, \ 0.0138591496001844, \ 0.0138591496001844, \ 0.0138591496001844, \ 0.0138591496001844, \ 0.0138591496001844, \ 0.0138591496001844, \ 0.0138591496001844 ] ) return x, y, z, w def rule12 ( ): #*****************************************************************************80 # ## rule12() returns the prism rule of precision 12. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.0485337200788840, \ 0.0485337200788840, \ 0.9029325598422321, \ 0.0003340541016131, \ 0.0003340541016131, \ 0.9993318917967738, \ 0.4849074297077183, \ 0.4849074297077183, \ 0.0301851405845634, \ 0.4849074297077183, \ 0.4849074297077183, \ 0.0301851405845634, \ 0.1497566246841529, \ 0.1497566246841529, \ 0.7004867506316943, \ 0.1497566246841529, \ 0.1497566246841529, \ 0.7004867506316943, \ 0.2344536517846724, \ 0.2344536517846724, \ 0.5310926964306552, \ 0.2344536517846724, \ 0.2344536517846724, \ 0.5310926964306552, \ 0.4409030162469282, \ 0.4409030162469282, \ 0.1181939675061436, \ 0.4409030162469282, \ 0.4409030162469282, \ 0.1181939675061436, \ 0.4870412467653055, \ 0.4870412467653055, \ 0.0259175064693890, \ 0.4870412467653055, \ 0.4870412467653055, \ 0.0259175064693890, \ 0.0248638193002129, \ 0.0248638193002129, \ 0.9502723613995742, \ 0.0248638193002129, \ 0.0248638193002129, \ 0.9502723613995742, \ 0.1118542147928236, \ 0.1118542147928236, \ 0.7762915704143528, \ 0.1118542147928236, \ 0.1118542147928236, \ 0.7762915704143528, \ 0.1207867185816364, \ 0.7097236881695401, \ 0.1207867185816364, \ 0.1694895932488235, \ 0.7097236881695401, \ 0.1694895932488235, \ 0.6481099336610571, \ 0.3419306029008594, \ 0.6481099336610571, \ 0.0099594634380836, \ 0.3419306029008594, \ 0.0099594634380836, \ 0.0200903650277176, \ 0.1335404714654308, \ 0.0200903650277176, \ 0.8463691635068515, \ 0.1335404714654308, \ 0.8463691635068515, \ 0.0200903650277176, \ 0.1335404714654308, \ 0.0200903650277176, \ 0.8463691635068515, \ 0.1335404714654308, \ 0.8463691635068515, \ 0.3214917379706315, \ 0.1648190492804087, \ 0.3214917379706315, \ 0.5136892127489598, \ 0.1648190492804087, \ 0.5136892127489598, \ 0.3214917379706315, \ 0.1648190492804087, \ 0.3214917379706315, \ 0.5136892127489598, \ 0.1648190492804087, \ 0.5136892127489598, \ 0.0689956505491457, \ 0.6636404691861656, \ 0.0689956505491457, \ 0.2673638802646887, \ 0.6636404691861656, \ 0.2673638802646887, \ 0.0689956505491457, \ 0.6636404691861656, \ 0.0689956505491457, \ 0.2673638802646887, \ 0.6636404691861656, \ 0.2673638802646887, \ 0.0199626366433414, \ 0.2615529990296567, \ 0.0199626366433414, \ 0.7184843643270019, \ 0.2615529990296567, \ 0.7184843643270019, \ 0.0199626366433414, \ 0.2615529990296567, \ 0.0199626366433414, \ 0.7184843643270019, \ 0.2615529990296567, \ 0.7184843643270019, \ 0.0900297723891655, \ 0.0155218563345725, \ 0.0900297723891655, \ 0.8944483712762620, \ 0.0155218563345725, \ 0.8944483712762620, \ 0.0900297723891655, \ 0.0155218563345725, \ 0.0900297723891655, \ 0.8944483712762620, \ 0.0155218563345725, \ 0.8944483712762620, \ 0.1172214513692467, \ 0.5641016399337317, \ 0.1172214513692467, \ 0.3186769086970216, \ 0.5641016399337317, \ 0.3186769086970216, \ 0.1172214513692467, \ 0.5641016399337317, \ 0.1172214513692467, \ 0.3186769086970216, \ 0.5641016399337317, \ 0.3186769086970216 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.0485337200788840, \ 0.9029325598422321, \ 0.0485337200788840, \ 0.0003340541016131, \ 0.9993318917967738, \ 0.0003340541016131, \ 0.4849074297077183, \ 0.0301851405845634, \ 0.4849074297077183, \ 0.4849074297077183, \ 0.0301851405845634, \ 0.4849074297077183, \ 0.1497566246841529, \ 0.7004867506316943, \ 0.1497566246841529, \ 0.1497566246841529, \ 0.7004867506316943, \ 0.1497566246841529, \ 0.2344536517846724, \ 0.5310926964306552, \ 0.2344536517846724, \ 0.2344536517846724, \ 0.5310926964306552, \ 0.2344536517846724, \ 0.4409030162469282, \ 0.1181939675061436, \ 0.4409030162469282, \ 0.4409030162469282, \ 0.1181939675061436, \ 0.4409030162469282, \ 0.4870412467653055, \ 0.0259175064693890, \ 0.4870412467653055, \ 0.4870412467653055, \ 0.0259175064693890, \ 0.4870412467653055, \ 0.0248638193002129, \ 0.9502723613995742, \ 0.0248638193002129, \ 0.0248638193002129, \ 0.9502723613995742, \ 0.0248638193002129, \ 0.1118542147928236, \ 0.7762915704143528, \ 0.1118542147928236, \ 0.1118542147928236, \ 0.7762915704143528, \ 0.1118542147928236, \ 0.7097236881695401, \ 0.1207867185816364, \ 0.1694895932488235, \ 0.1207867185816364, \ 0.1694895932488235, \ 0.7097236881695401, \ 0.3419306029008594, \ 0.6481099336610571, \ 0.0099594634380836, \ 0.6481099336610571, \ 0.0099594634380836, \ 0.3419306029008594, \ 0.1335404714654308, \ 0.0200903650277176, \ 0.8463691635068515, \ 0.0200903650277176, \ 0.8463691635068515, \ 0.1335404714654308, \ 0.1335404714654308, \ 0.0200903650277176, \ 0.8463691635068515, \ 0.0200903650277176, \ 0.8463691635068515, \ 0.1335404714654308, \ 0.1648190492804087, \ 0.3214917379706315, \ 0.5136892127489598, \ 0.3214917379706315, \ 0.5136892127489598, \ 0.1648190492804087, \ 0.1648190492804087, \ 0.3214917379706315, \ 0.5136892127489598, \ 0.3214917379706315, \ 0.5136892127489598, \ 0.1648190492804087, \ 0.6636404691861656, \ 0.0689956505491457, \ 0.2673638802646887, \ 0.0689956505491457, \ 0.2673638802646887, \ 0.6636404691861656, \ 0.6636404691861656, \ 0.0689956505491457, \ 0.2673638802646887, \ 0.0689956505491457, \ 0.2673638802646887, \ 0.6636404691861656, \ 0.2615529990296567, \ 0.0199626366433414, \ 0.7184843643270019, \ 0.0199626366433414, \ 0.7184843643270019, \ 0.2615529990296567, \ 0.2615529990296567, \ 0.0199626366433414, \ 0.7184843643270019, \ 0.0199626366433414, \ 0.7184843643270019, \ 0.2615529990296567, \ 0.0155218563345725, \ 0.0900297723891655, \ 0.8944483712762620, \ 0.0900297723891655, \ 0.8944483712762620, \ 0.0155218563345725, \ 0.0155218563345725, \ 0.0900297723891655, \ 0.8944483712762620, \ 0.0900297723891655, \ 0.8944483712762620, \ 0.0155218563345725, \ 0.5641016399337317, \ 0.1172214513692467, \ 0.3186769086970216, \ 0.1172214513692467, \ 0.3186769086970216, \ 0.5641016399337317, \ 0.5641016399337317, \ 0.1172214513692467, \ 0.3186769086970216, \ 0.1172214513692467, \ 0.3186769086970216, \ 0.5641016399337317 ] ) z = np.array ( [ \ 0.9624802640887508, \ 0.0375197359112492, \ 0.6587282524060957, \ 0.3412717475939043, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.7793516203235177, \ 0.7793516203235177, \ 0.7793516203235177, \ 0.2206483796764823, \ 0.2206483796764823, \ 0.2206483796764823, \ 0.0004872322680031, \ 0.0004872322680031, \ 0.0004872322680031, \ 0.9995127677319969, \ 0.9995127677319969, \ 0.9995127677319969, \ 0.6654124101220354, \ 0.6654124101220354, \ 0.6654124101220354, \ 0.3345875898779646, \ 0.3345875898779646, \ 0.3345875898779646, \ 0.4636840317697777, \ 0.4636840317697777, \ 0.4636840317697777, \ 0.5363159682302223, \ 0.5363159682302223, \ 0.5363159682302223, \ 0.9686710476965903, \ 0.9686710476965903, \ 0.9686710476965903, \ 0.0313289523034098, \ 0.0313289523034098, \ 0.0313289523034098, \ 0.8407183935395043, \ 0.8407183935395043, \ 0.8407183935395043, \ 0.1592816064604957, \ 0.1592816064604957, \ 0.1592816064604957, \ 0.8773254399543938, \ 0.8773254399543938, \ 0.8773254399543938, \ 0.1226745600456062, \ 0.1226745600456062, \ 0.1226745600456062, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.6944689221677809, \ 0.6944689221677809, \ 0.6944689221677809, \ 0.6944689221677809, \ 0.6944689221677809, \ 0.6944689221677809, \ 0.3055310778322191, \ 0.3055310778322191, \ 0.3055310778322191, \ 0.3055310778322191, \ 0.3055310778322191, \ 0.3055310778322191, \ 0.8559006310012134, \ 0.8559006310012134, \ 0.8559006310012134, \ 0.8559006310012134, \ 0.8559006310012134, \ 0.8559006310012134, \ 0.1440993689987866, \ 0.1440993689987866, \ 0.1440993689987866, \ 0.1440993689987866, \ 0.1440993689987866, \ 0.1440993689987866, \ 0.2888455393681256, \ 0.2888455393681256, \ 0.2888455393681256, \ 0.2888455393681256, \ 0.2888455393681256, \ 0.2888455393681256, \ 0.7111544606318744, \ 0.7111544606318744, \ 0.7111544606318744, \ 0.7111544606318744, \ 0.7111544606318744, \ 0.7111544606318744, \ 0.9221446748533075, \ 0.9221446748533075, \ 0.9221446748533075, \ 0.9221446748533075, \ 0.9221446748533075, \ 0.9221446748533075, \ 0.0778553251466925, \ 0.0778553251466925, \ 0.0778553251466925, \ 0.0778553251466925, \ 0.0778553251466925, \ 0.0778553251466925, \ 0.9762657414250897, \ 0.9762657414250897, \ 0.9762657414250897, \ 0.9762657414250897, \ 0.9762657414250897, \ 0.9762657414250897, \ 0.0237342585749103, \ 0.0237342585749103, \ 0.0237342585749103, \ 0.0237342585749103, \ 0.0237342585749103, \ 0.0237342585749103, \ 0.0278439909861404, \ 0.0278439909861404, \ 0.0278439909861404, \ 0.0278439909861404, \ 0.0278439909861404, \ 0.0278439909861404, \ 0.9721560090138596, \ 0.9721560090138596, \ 0.9721560090138596, \ 0.9721560090138596, \ 0.9721560090138596, \ 0.9721560090138596 ] ) w = np.array ( [ \ 0.0112254009526498, \ 0.0112254009526498, \ 0.0225097870163436, \ 0.0225097870163436, \ 0.0054894240839370, \ 0.0054894240839370, \ 0.0054894240839370, \ 0.0004964406537954, \ 0.0004964406537954, \ 0.0004964406537954, \ 0.0102090400109948, \ 0.0102090400109948, \ 0.0102090400109948, \ 0.0102090400109948, \ 0.0102090400109948, \ 0.0102090400109948, \ 0.0021006495508094, \ 0.0021006495508094, \ 0.0021006495508094, \ 0.0021006495508094, \ 0.0021006495508094, \ 0.0021006495508094, \ 0.0152094206670507, \ 0.0152094206670507, \ 0.0152094206670507, \ 0.0152094206670507, \ 0.0152094206670507, \ 0.0152094206670507, \ 0.0136582892001806, \ 0.0136582892001806, \ 0.0136582892001806, \ 0.0136582892001806, \ 0.0136582892001806, \ 0.0136582892001806, \ 0.0027887393220490, \ 0.0027887393220490, \ 0.0027887393220490, \ 0.0027887393220490, \ 0.0027887393220490, \ 0.0027887393220490, \ 0.0022520950855930, \ 0.0022520950855930, \ 0.0022520950855930, \ 0.0022520950855930, \ 0.0022520950855930, \ 0.0022520950855930, \ 0.0099174482589678, \ 0.0099174482589678, \ 0.0099174482589678, \ 0.0099174482589678, \ 0.0099174482589678, \ 0.0099174482589678, \ 0.0095245941038326, \ 0.0095245941038326, \ 0.0095245941038326, \ 0.0095245941038326, \ 0.0095245941038326, \ 0.0095245941038326, \ 0.0066881354290901, \ 0.0066881354290901, \ 0.0066881354290901, \ 0.0066881354290901, \ 0.0066881354290901, \ 0.0066881354290901, \ 0.0056673224713314, \ 0.0056673224713314, \ 0.0056673224713314, \ 0.0056673224713314, \ 0.0056673224713314, \ 0.0056673224713314, \ 0.0056673224713314, \ 0.0056673224713314, \ 0.0056673224713314, \ 0.0056673224713314, \ 0.0056673224713314, \ 0.0056673224713314, \ 0.0125866454585098, \ 0.0125866454585098, \ 0.0125866454585098, \ 0.0125866454585098, \ 0.0125866454585098, \ 0.0125866454585098, \ 0.0125866454585098, \ 0.0125866454585098, \ 0.0125866454585098, \ 0.0125866454585098, \ 0.0125866454585098, \ 0.0125866454585098, \ 0.0114370118000321, \ 0.0114370118000321, \ 0.0114370118000321, \ 0.0114370118000321, \ 0.0114370118000321, \ 0.0114370118000321, \ 0.0114370118000321, \ 0.0114370118000321, \ 0.0114370118000321, \ 0.0114370118000321, \ 0.0114370118000321, \ 0.0114370118000321, \ 0.0044955975561441, \ 0.0044955975561441, \ 0.0044955975561441, \ 0.0044955975561441, \ 0.0044955975561441, \ 0.0044955975561441, \ 0.0044955975561441, \ 0.0044955975561441, \ 0.0044955975561441, \ 0.0044955975561441, \ 0.0044955975561441, \ 0.0044955975561441, \ 0.0011897740209102, \ 0.0011897740209102, \ 0.0011897740209102, \ 0.0011897740209102, \ 0.0011897740209102, \ 0.0011897740209102, \ 0.0011897740209102, \ 0.0011897740209102, \ 0.0011897740209102, \ 0.0011897740209102, \ 0.0011897740209102, \ 0.0011897740209102, \ 0.0046637786995230, \ 0.0046637786995230, \ 0.0046637786995230, \ 0.0046637786995230, \ 0.0046637786995230, \ 0.0046637786995230, \ 0.0046637786995230, \ 0.0046637786995230, \ 0.0046637786995230, \ 0.0046637786995230, \ 0.0046637786995230, \ 0.0046637786995230 ] ) return x, y, z, w def rule13 ( ): #*****************************************************************************80 # ## rule13() returns the prism rule of precision 13. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.4898439392870481, \ 0.4898439392870481, \ 0.0203121214259039, \ 0.4898439392870481, \ 0.4898439392870481, \ 0.0203121214259039, \ 0.4652402431593082, \ 0.4652402431593082, \ 0.0695195136813836, \ 0.4652402431593082, \ 0.4652402431593082, \ 0.0695195136813836, \ 0.1189161740974999, \ 0.1189161740974999, \ 0.7621676518050002, \ 0.1189161740974999, \ 0.1189161740974999, \ 0.7621676518050002, \ 0.4065656818638698, \ 0.4065656818638698, \ 0.1868686362722604, \ 0.4065656818638698, \ 0.4065656818638698, \ 0.1868686362722604, \ 0.3731243598486834, \ 0.3731243598486834, \ 0.2537512803026333, \ 0.3731243598486834, \ 0.3731243598486834, \ 0.2537512803026333, \ 0.0222257711836755, \ 0.0222257711836755, \ 0.9555484576326490, \ 0.0222257711836755, \ 0.0222257711836755, \ 0.9555484576326490, \ 0.4252710267490136, \ 0.4252710267490136, \ 0.1494579465019728, \ 0.4252710267490136, \ 0.4252710267490136, \ 0.1494579465019728, \ 0.2896929731593648, \ 0.2896929731593648, \ 0.4206140536812703, \ 0.2896929731593648, \ 0.2896929731593648, \ 0.4206140536812703, \ 0.1072868932263340, \ 0.1072868932263340, \ 0.7854262135473320, \ 0.1072868932263340, \ 0.1072868932263340, \ 0.7854262135473320, \ 0.2080125480772502, \ 0.2080125480772502, \ 0.5839749038454997, \ 0.2080125480772502, \ 0.2080125480772502, \ 0.5839749038454997, \ 0.2230030950549982, \ 0.2230030950549982, \ 0.5539938098900037, \ 0.2230030950549982, \ 0.2230030950549982, \ 0.5539938098900037, \ 0.0261743916084927, \ 0.0261743916084927, \ 0.9476512167830147, \ 0.0261743916084927, \ 0.0261743916084927, \ 0.9476512167830147, \ 0.0069141701592508, \ 0.2742659465495736, \ 0.0069141701592508, \ 0.7188198832911755, \ 0.2742659465495736, \ 0.7188198832911755, \ 0.1210652398684976, \ 0.0234237758449634, \ 0.1210652398684976, \ 0.8555109842865389, \ 0.0234237758449634, \ 0.8555109842865389, \ 0.0935360177546358, \ 0.5415308571572887, \ 0.0935360177546358, \ 0.3649331250880756, \ 0.5415308571572887, \ 0.3649331250880756, \ 0.0190809537819619, \ 0.4425451813256520, \ 0.0190809537819619, \ 0.5383738648923861, \ 0.4425451813256520, \ 0.5383738648923861, \ 0.0190809537819619, \ 0.4425451813256520, \ 0.0190809537819619, \ 0.5383738648923861, \ 0.4425451813256520, \ 0.5383738648923861, \ 0.2923396969545124, \ 0.0181776522602859, \ 0.2923396969545124, \ 0.6894826507852017, \ 0.0181776522602859, \ 0.6894826507852017, \ 0.2923396969545124, \ 0.0181776522602859, \ 0.2923396969545124, \ 0.6894826507852017, \ 0.0181776522602859, \ 0.6894826507852017, \ 0.0212309381671597, \ 0.8615902483468726, \ 0.0212309381671597, \ 0.1171788134859677, \ 0.8615902483468726, \ 0.1171788134859677, \ 0.0212309381671597, \ 0.8615902483468726, \ 0.0212309381671597, \ 0.1171788134859677, \ 0.8615902483468726, \ 0.1171788134859677, \ 0.2541452627735283, \ 0.0752550719401233, \ 0.2541452627735283, \ 0.6705996652863484, \ 0.0752550719401233, \ 0.6705996652863484, \ 0.2541452627735283, \ 0.0752550719401233, \ 0.2541452627735283, \ 0.6705996652863484, \ 0.0752550719401233, \ 0.6705996652863484, \ 0.1414034499801619, \ 0.0248530932165772, \ 0.1414034499801619, \ 0.8337434568032609, \ 0.0248530932165772, \ 0.8337434568032609, \ 0.1414034499801619, \ 0.0248530932165772, \ 0.1414034499801619, \ 0.8337434568032609, \ 0.0248530932165772, \ 0.8337434568032609, \ 0.5920519581168684, \ 0.1079442302776815, \ 0.5920519581168684, \ 0.3000038116054501, \ 0.1079442302776815, \ 0.3000038116054501, \ 0.5920519581168684, \ 0.1079442302776815, \ 0.5920519581168684, \ 0.3000038116054501, \ 0.1079442302776815, \ 0.3000038116054501 ] ) y = np.array ( [ \ 0.4898439392870481, \ 0.0203121214259039, \ 0.4898439392870481, \ 0.4898439392870481, \ 0.0203121214259039, \ 0.4898439392870481, \ 0.4652402431593082, \ 0.0695195136813836, \ 0.4652402431593082, \ 0.4652402431593082, \ 0.0695195136813836, \ 0.4652402431593082, \ 0.1189161740974999, \ 0.7621676518050002, \ 0.1189161740974999, \ 0.1189161740974999, \ 0.7621676518050002, \ 0.1189161740974999, \ 0.4065656818638698, \ 0.1868686362722604, \ 0.4065656818638698, \ 0.4065656818638698, \ 0.1868686362722604, \ 0.4065656818638698, \ 0.3731243598486834, \ 0.2537512803026333, \ 0.3731243598486834, \ 0.3731243598486834, \ 0.2537512803026333, \ 0.3731243598486834, \ 0.0222257711836755, \ 0.9555484576326490, \ 0.0222257711836755, \ 0.0222257711836755, \ 0.9555484576326490, \ 0.0222257711836755, \ 0.4252710267490136, \ 0.1494579465019728, \ 0.4252710267490136, \ 0.4252710267490136, \ 0.1494579465019728, \ 0.4252710267490136, \ 0.2896929731593648, \ 0.4206140536812703, \ 0.2896929731593648, \ 0.2896929731593648, \ 0.4206140536812703, \ 0.2896929731593648, \ 0.1072868932263340, \ 0.7854262135473320, \ 0.1072868932263340, \ 0.1072868932263340, \ 0.7854262135473320, \ 0.1072868932263340, \ 0.2080125480772502, \ 0.5839749038454997, \ 0.2080125480772502, \ 0.2080125480772502, \ 0.5839749038454997, \ 0.2080125480772502, \ 0.2230030950549982, \ 0.5539938098900037, \ 0.2230030950549982, \ 0.2230030950549982, \ 0.5539938098900037, \ 0.2230030950549982, \ 0.0261743916084927, \ 0.9476512167830147, \ 0.0261743916084927, \ 0.0261743916084927, \ 0.9476512167830147, \ 0.0261743916084927, \ 0.2742659465495736, \ 0.0069141701592508, \ 0.7188198832911755, \ 0.0069141701592508, \ 0.7188198832911755, \ 0.2742659465495736, \ 0.0234237758449634, \ 0.1210652398684976, \ 0.8555109842865389, \ 0.1210652398684976, \ 0.8555109842865389, \ 0.0234237758449634, \ 0.5415308571572887, \ 0.0935360177546358, \ 0.3649331250880756, \ 0.0935360177546358, \ 0.3649331250880756, \ 0.5415308571572887, \ 0.4425451813256520, \ 0.0190809537819619, \ 0.5383738648923861, \ 0.0190809537819619, \ 0.5383738648923861, \ 0.4425451813256520, \ 0.4425451813256520, \ 0.0190809537819619, \ 0.5383738648923861, \ 0.0190809537819619, \ 0.5383738648923861, \ 0.4425451813256520, \ 0.0181776522602859, \ 0.2923396969545124, \ 0.6894826507852017, \ 0.2923396969545124, \ 0.6894826507852017, \ 0.0181776522602859, \ 0.0181776522602859, \ 0.2923396969545124, \ 0.6894826507852017, \ 0.2923396969545124, \ 0.6894826507852017, \ 0.0181776522602859, \ 0.8615902483468726, \ 0.0212309381671597, \ 0.1171788134859677, \ 0.0212309381671597, \ 0.1171788134859677, \ 0.8615902483468726, \ 0.8615902483468726, \ 0.0212309381671597, \ 0.1171788134859677, \ 0.0212309381671597, \ 0.1171788134859677, \ 0.8615902483468726, \ 0.0752550719401233, \ 0.2541452627735283, \ 0.6705996652863484, \ 0.2541452627735283, \ 0.6705996652863484, \ 0.0752550719401233, \ 0.0752550719401233, \ 0.2541452627735283, \ 0.6705996652863484, \ 0.2541452627735283, \ 0.6705996652863484, \ 0.0752550719401233, \ 0.0248530932165772, \ 0.1414034499801619, \ 0.8337434568032609, \ 0.1414034499801619, \ 0.8337434568032609, \ 0.0248530932165772, \ 0.0248530932165772, \ 0.1414034499801619, \ 0.8337434568032609, \ 0.1414034499801619, \ 0.8337434568032609, \ 0.0248530932165772, \ 0.1079442302776815, \ 0.5920519581168684, \ 0.3000038116054501, \ 0.5920519581168684, \ 0.3000038116054501, \ 0.1079442302776815, \ 0.1079442302776815, \ 0.5920519581168684, \ 0.3000038116054501, \ 0.5920519581168684, \ 0.3000038116054501, \ 0.1079442302776815 ] ) z = np.array ( [ \ 0.9628749136752521, \ 0.9628749136752521, \ 0.9628749136752521, \ 0.0371250863247479, \ 0.0371250863247479, \ 0.0371250863247479, \ 0.1596095438845206, \ 0.1596095438845206, \ 0.1596095438845206, \ 0.8403904561154794, \ 0.8403904561154794, \ 0.8403904561154794, \ 0.9056527042389668, \ 0.9056527042389668, \ 0.9056527042389668, \ 0.0943472957610332, \ 0.0943472957610332, \ 0.0943472957610332, \ 0.5096252745686303, \ 0.5096252745686303, \ 0.5096252745686303, \ 0.4903747254313698, \ 0.4903747254313698, \ 0.4903747254313698, \ 0.9622105990610692, \ 0.9622105990610692, \ 0.9622105990610692, \ 0.0377894009389309, \ 0.0377894009389309, \ 0.0377894009389309, \ 0.6451848165106220, \ 0.6451848165106220, \ 0.6451848165106220, \ 0.3548151834893781, \ 0.3548151834893781, \ 0.3548151834893781, \ 0.8358323923694781, \ 0.8358323923694781, \ 0.8358323923694781, \ 0.1641676076305219, \ 0.1641676076305219, \ 0.1641676076305219, \ 0.6970082735594627, \ 0.6970082735594627, \ 0.6970082735594627, \ 0.3029917264405373, \ 0.3029917264405373, \ 0.3029917264405373, \ 0.5806209783551364, \ 0.5806209783551364, \ 0.5806209783551364, \ 0.4193790216448636, \ 0.4193790216448636, \ 0.4193790216448636, \ 0.6310517191712730, \ 0.6310517191712730, \ 0.6310517191712730, \ 0.3689482808287270, \ 0.3689482808287270, \ 0.3689482808287270, \ 0.8758646517044397, \ 0.8758646517044397, \ 0.8758646517044397, \ 0.1241353482955603, \ 0.1241353482955603, \ 0.1241353482955603, \ 0.9265272967069804, \ 0.9265272967069804, \ 0.9265272967069804, \ 0.0734727032930196, \ 0.0734727032930196, \ 0.0734727032930196, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.6831055612129073, \ 0.6831055612129073, \ 0.6831055612129073, \ 0.6831055612129073, \ 0.6831055612129073, \ 0.6831055612129073, \ 0.3168944387870927, \ 0.3168944387870927, \ 0.3168944387870927, \ 0.3168944387870927, \ 0.3168944387870927, \ 0.3168944387870927, \ 0.8946991133344783, \ 0.8946991133344783, \ 0.8946991133344783, \ 0.8946991133344783, \ 0.8946991133344783, \ 0.8946991133344783, \ 0.1053008866655217, \ 0.1053008866655217, \ 0.1053008866655217, \ 0.1053008866655217, \ 0.1053008866655217, \ 0.1053008866655217, \ 0.7783161730088111, \ 0.7783161730088111, \ 0.7783161730088111, \ 0.7783161730088111, \ 0.7783161730088111, \ 0.7783161730088111, \ 0.2216838269911889, \ 0.2216838269911889, \ 0.2216838269911889, \ 0.2216838269911889, \ 0.2216838269911889, \ 0.2216838269911889, \ 0.2838393063545257, \ 0.2838393063545257, \ 0.2838393063545257, \ 0.2838393063545257, \ 0.2838393063545257, \ 0.2838393063545257, \ 0.7161606936454743, \ 0.7161606936454743, \ 0.7161606936454743, \ 0.7161606936454743, \ 0.7161606936454743, \ 0.7161606936454743, \ 0.9877489005025584, \ 0.9877489005025584, \ 0.9877489005025584, \ 0.9877489005025584, \ 0.9877489005025584, \ 0.9877489005025584, \ 0.0122510994974416, \ 0.0122510994974416, \ 0.0122510994974416, \ 0.0122510994974416, \ 0.0122510994974416, \ 0.0122510994974416, \ 0.9801022671238835, \ 0.9801022671238835, \ 0.9801022671238835, \ 0.9801022671238835, \ 0.9801022671238835, \ 0.9801022671238835, \ 0.0198977328761166, \ 0.0198977328761166, \ 0.0198977328761166, \ 0.0198977328761166, \ 0.0198977328761166, \ 0.0198977328761166 ] ) w = np.array ( [ \ 0.0026290597067936, \ 0.0026290597067936, \ 0.0026290597067936, \ 0.0026290597067936, \ 0.0026290597067936, \ 0.0026290597067936, \ 0.0059734237493126, \ 0.0059734237493126, \ 0.0059734237493126, \ 0.0059734237493126, \ 0.0059734237493126, \ 0.0059734237493126, \ 0.0080935494241557, \ 0.0080935494241557, \ 0.0080935494241557, \ 0.0080935494241557, \ 0.0080935494241557, \ 0.0080935494241557, \ 0.0068191162590357, \ 0.0068191162590357, \ 0.0068191162590357, \ 0.0068191162590357, \ 0.0068191162590357, \ 0.0068191162590357, \ 0.0049334513687730, \ 0.0049334513687730, \ 0.0049334513687730, \ 0.0049334513687730, \ 0.0049334513687730, \ 0.0049334513687730, \ 0.0020147080132884, \ 0.0020147080132884, \ 0.0020147080132884, \ 0.0020147080132884, \ 0.0020147080132884, \ 0.0020147080132884, \ 0.0123357707370753, \ 0.0123357707370753, \ 0.0123357707370753, \ 0.0123357707370753, \ 0.0123357707370753, \ 0.0123357707370753, \ 0.0113647943376711, \ 0.0113647943376711, \ 0.0113647943376711, \ 0.0113647943376711, \ 0.0113647943376711, \ 0.0113647943376711, \ 0.0066668649606825, \ 0.0066668649606825, \ 0.0066668649606825, \ 0.0066668649606825, \ 0.0066668649606825, \ 0.0066668649606825, \ 0.0119452201363976, \ 0.0119452201363976, \ 0.0119452201363976, \ 0.0119452201363976, \ 0.0119452201363976, \ 0.0119452201363976, \ 0.0106878809406112, \ 0.0106878809406112, \ 0.0106878809406112, \ 0.0106878809406112, \ 0.0106878809406112, \ 0.0106878809406112, \ 0.0016663443274632, \ 0.0016663443274632, \ 0.0016663443274632, \ 0.0016663443274632, \ 0.0016663443274632, \ 0.0016663443274632, \ 0.0045120516170652, \ 0.0045120516170652, \ 0.0045120516170652, \ 0.0045120516170652, \ 0.0045120516170652, \ 0.0045120516170652, \ 0.0034887068394433, \ 0.0034887068394433, \ 0.0034887068394433, \ 0.0034887068394433, \ 0.0034887068394433, \ 0.0034887068394433, \ 0.0111803506944158, \ 0.0111803506944158, \ 0.0111803506944158, \ 0.0111803506944158, \ 0.0111803506944158, \ 0.0111803506944158, \ 0.0042052602324505, \ 0.0042052602324505, \ 0.0042052602324505, \ 0.0042052602324505, \ 0.0042052602324505, \ 0.0042052602324505, \ 0.0042052602324505, \ 0.0042052602324505, \ 0.0042052602324505, \ 0.0042052602324505, \ 0.0042052602324505, \ 0.0042052602324505, \ 0.0043767917497427, \ 0.0043767917497427, \ 0.0043767917497427, \ 0.0043767917497427, \ 0.0043767917497427, \ 0.0043767917497427, \ 0.0043767917497427, \ 0.0043767917497427, \ 0.0043767917497427, \ 0.0043767917497427, \ 0.0043767917497427, \ 0.0043767917497427, \ 0.0044067550419370, \ 0.0044067550419370, \ 0.0044067550419370, \ 0.0044067550419370, \ 0.0044067550419370, \ 0.0044067550419370, \ 0.0044067550419370, \ 0.0044067550419370, \ 0.0044067550419370, \ 0.0044067550419370, \ 0.0044067550419370, \ 0.0044067550419370, \ 0.0120350326772681, \ 0.0120350326772681, \ 0.0120350326772681, \ 0.0120350326772681, \ 0.0120350326772681, \ 0.0120350326772681, \ 0.0120350326772681, \ 0.0120350326772681, \ 0.0120350326772681, \ 0.0120350326772681, \ 0.0120350326772681, \ 0.0120350326772681, \ 0.0013986921884649, \ 0.0013986921884649, \ 0.0013986921884649, \ 0.0013986921884649, \ 0.0013986921884649, \ 0.0013986921884649, \ 0.0013986921884649, \ 0.0013986921884649, \ 0.0013986921884649, \ 0.0013986921884649, \ 0.0013986921884649, \ 0.0013986921884649, \ 0.0047551548873781, \ 0.0047551548873781, \ 0.0047551548873781, \ 0.0047551548873781, \ 0.0047551548873781, \ 0.0047551548873781, \ 0.0047551548873781, \ 0.0047551548873781, \ 0.0047551548873781, \ 0.0047551548873781, \ 0.0047551548873781, \ 0.0047551548873781 ] ) return x, y, z, w def rule14 ( ): #*****************************************************************************80 # ## rule14() returns the prism rule of precision 14. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.1531409494078462, \ 0.1531409494078462, \ 0.6937181011843075, \ 0.1094642266754876, \ 0.1094642266754876, \ 0.7810715466490248, \ 0.4852139749601931, \ 0.4852139749601931, \ 0.0295720500796137, \ 0.4852139749601931, \ 0.4852139749601931, \ 0.0295720500796137, \ 0.4811857851081537, \ 0.4811857851081537, \ 0.0376284297836926, \ 0.4811857851081537, \ 0.4811857851081537, \ 0.0376284297836926, \ 0.0854252549169795, \ 0.0854252549169795, \ 0.8291494901660410, \ 0.0854252549169795, \ 0.0854252549169795, \ 0.8291494901660410, \ 0.4949498254464672, \ 0.4949498254464672, \ 0.0101003491070657, \ 0.4949498254464672, \ 0.4949498254464672, \ 0.0101003491070657, \ 0.2006722689888837, \ 0.2006722689888837, \ 0.5986554620222326, \ 0.2006722689888837, \ 0.2006722689888837, \ 0.5986554620222326, \ 0.0134631402389198, \ 0.0134631402389198, \ 0.9730737195221603, \ 0.0134631402389198, \ 0.0134631402389198, \ 0.9730737195221603, \ 0.3888073656098609, \ 0.3888073656098609, \ 0.2223852687802782, \ 0.3888073656098609, \ 0.3888073656098609, \ 0.2223852687802782, \ 0.2589597694248802, \ 0.2589597694248802, \ 0.4820804611502396, \ 0.2589597694248802, \ 0.2589597694248802, \ 0.4820804611502396, \ 0.0656674639158517, \ 0.0656674639158517, \ 0.8686650721682967, \ 0.0656674639158517, \ 0.0656674639158517, \ 0.8686650721682967, \ 0.4462504387663027, \ 0.4462504387663027, \ 0.1074991224673947, \ 0.4462504387663027, \ 0.4462504387663027, \ 0.1074991224673947, \ 0.1155506883468244, \ 0.1155506883468244, \ 0.7688986233063512, \ 0.1155506883468244, \ 0.1155506883468244, \ 0.7688986233063512, \ 0.0257939798379175, \ 0.0257939798379175, \ 0.9484120403241649, \ 0.0257939798379175, \ 0.0257939798379175, \ 0.9484120403241649, \ 0.0231313365079955, \ 0.3737411598567404, \ 0.0231313365079955, \ 0.6031275036352640, \ 0.3737411598567404, \ 0.6031275036352640, \ 0.0780837656724579, \ 0.0087167837437324, \ 0.0780837656724579, \ 0.9131994505838098, \ 0.0087167837437324, \ 0.9131994505838098, \ 0.1932090397525293, \ 0.4408189862889988, \ 0.1932090397525293, \ 0.3659719739584719, \ 0.4408189862889988, \ 0.3659719739584719, \ 0.0110695610243579, \ 0.3495104935335414, \ 0.0110695610243579, \ 0.6394199454421007, \ 0.3495104935335414, \ 0.6394199454421007, \ 0.0110695610243579, \ 0.3495104935335414, \ 0.0110695610243579, \ 0.6394199454421007, \ 0.3495104935335414, \ 0.6394199454421007, \ 0.2813786819004687, \ 0.0725240717490409, \ 0.2813786819004687, \ 0.6460972463504904, \ 0.0725240717490409, \ 0.6460972463504904, \ 0.2813786819004687, \ 0.0725240717490409, \ 0.2813786819004687, \ 0.6460972463504904, \ 0.0725240717490409, \ 0.6460972463504904, \ 0.0136067459730556, \ 0.7694543610610796, \ 0.0136067459730556, \ 0.2169388929658648, \ 0.7694543610610796, \ 0.2169388929658648, \ 0.0136067459730556, \ 0.7694543610610796, \ 0.0136067459730556, \ 0.2169388929658648, \ 0.7694543610610796, \ 0.2169388929658648, \ 0.3160642637515643, \ 0.0980612921350026, \ 0.3160642637515643, \ 0.5858744441134331, \ 0.0980612921350026, \ 0.5858744441134331, \ 0.3160642637515643, \ 0.0980612921350026, \ 0.3160642637515643, \ 0.5858744441134331, \ 0.0980612921350026, \ 0.5858744441134331, \ 0.2010219519914552, \ 0.0101533805746944, \ 0.2010219519914552, \ 0.7888246674338504, \ 0.0101533805746944, \ 0.7888246674338504, \ 0.2010219519914552, \ 0.0101533805746944, \ 0.2010219519914552, \ 0.7888246674338504, \ 0.0101533805746944, \ 0.7888246674338504, \ 0.5404862370418329, \ 0.1675040220811526, \ 0.5404862370418329, \ 0.2920097408770145, \ 0.1675040220811526, \ 0.2920097408770145, \ 0.5404862370418329, \ 0.1675040220811526, \ 0.5404862370418329, \ 0.2920097408770145, \ 0.1675040220811526, \ 0.2920097408770145, \ 0.1016331118955575, \ 0.0168250918958854, \ 0.1016331118955575, \ 0.8815417962085570, \ 0.0168250918958854, \ 0.8815417962085570, \ 0.1016331118955575, \ 0.0168250918958854, \ 0.1016331118955575, \ 0.8815417962085570, \ 0.0168250918958854, \ 0.8815417962085570, \ 0.7553391402298588, \ 0.0552850680433709, \ 0.7553391402298588, \ 0.1893757917267703, \ 0.0552850680433709, \ 0.1893757917267703, \ 0.7553391402298588, \ 0.0552850680433709, \ 0.7553391402298588, \ 0.1893757917267703, \ 0.0552850680433709, \ 0.1893757917267703 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.1531409494078462, \ 0.6937181011843075, \ 0.1531409494078462, \ 0.1094642266754876, \ 0.7810715466490248, \ 0.1094642266754876, \ 0.4852139749601931, \ 0.0295720500796137, \ 0.4852139749601931, \ 0.4852139749601931, \ 0.0295720500796137, \ 0.4852139749601931, \ 0.4811857851081537, \ 0.0376284297836926, \ 0.4811857851081537, \ 0.4811857851081537, \ 0.0376284297836926, \ 0.4811857851081537, \ 0.0854252549169795, \ 0.8291494901660410, \ 0.0854252549169795, \ 0.0854252549169795, \ 0.8291494901660410, \ 0.0854252549169795, \ 0.4949498254464672, \ 0.0101003491070657, \ 0.4949498254464672, \ 0.4949498254464672, \ 0.0101003491070657, \ 0.4949498254464672, \ 0.2006722689888837, \ 0.5986554620222326, \ 0.2006722689888837, \ 0.2006722689888837, \ 0.5986554620222326, \ 0.2006722689888837, \ 0.0134631402389198, \ 0.9730737195221603, \ 0.0134631402389198, \ 0.0134631402389198, \ 0.9730737195221603, \ 0.0134631402389198, \ 0.3888073656098609, \ 0.2223852687802782, \ 0.3888073656098609, \ 0.3888073656098609, \ 0.2223852687802782, \ 0.3888073656098609, \ 0.2589597694248802, \ 0.4820804611502396, \ 0.2589597694248802, \ 0.2589597694248802, \ 0.4820804611502396, \ 0.2589597694248802, \ 0.0656674639158517, \ 0.8686650721682967, \ 0.0656674639158517, \ 0.0656674639158517, \ 0.8686650721682967, \ 0.0656674639158517, \ 0.4462504387663027, \ 0.1074991224673947, \ 0.4462504387663027, \ 0.4462504387663027, \ 0.1074991224673947, \ 0.4462504387663027, \ 0.1155506883468244, \ 0.7688986233063512, \ 0.1155506883468244, \ 0.1155506883468244, \ 0.7688986233063512, \ 0.1155506883468244, \ 0.0257939798379175, \ 0.9484120403241649, \ 0.0257939798379175, \ 0.0257939798379175, \ 0.9484120403241649, \ 0.0257939798379175, \ 0.3737411598567404, \ 0.0231313365079955, \ 0.6031275036352640, \ 0.0231313365079955, \ 0.6031275036352640, \ 0.3737411598567404, \ 0.0087167837437324, \ 0.0780837656724579, \ 0.9131994505838098, \ 0.0780837656724579, \ 0.9131994505838098, \ 0.0087167837437324, \ 0.4408189862889988, \ 0.1932090397525293, \ 0.3659719739584719, \ 0.1932090397525293, \ 0.3659719739584719, \ 0.4408189862889988, \ 0.3495104935335414, \ 0.0110695610243579, \ 0.6394199454421007, \ 0.0110695610243579, \ 0.6394199454421007, \ 0.3495104935335414, \ 0.3495104935335414, \ 0.0110695610243579, \ 0.6394199454421007, \ 0.0110695610243579, \ 0.6394199454421007, \ 0.3495104935335414, \ 0.0725240717490409, \ 0.2813786819004687, \ 0.6460972463504904, \ 0.2813786819004687, \ 0.6460972463504904, \ 0.0725240717490409, \ 0.0725240717490409, \ 0.2813786819004687, \ 0.6460972463504904, \ 0.2813786819004687, \ 0.6460972463504904, \ 0.0725240717490409, \ 0.7694543610610796, \ 0.0136067459730556, \ 0.2169388929658648, \ 0.0136067459730556, \ 0.2169388929658648, \ 0.7694543610610796, \ 0.7694543610610796, \ 0.0136067459730556, \ 0.2169388929658648, \ 0.0136067459730556, \ 0.2169388929658648, \ 0.7694543610610796, \ 0.0980612921350026, \ 0.3160642637515643, \ 0.5858744441134331, \ 0.3160642637515643, \ 0.5858744441134331, \ 0.0980612921350026, \ 0.0980612921350026, \ 0.3160642637515643, \ 0.5858744441134331, \ 0.3160642637515643, \ 0.5858744441134331, \ 0.0980612921350026, \ 0.0101533805746944, \ 0.2010219519914552, \ 0.7888246674338504, \ 0.2010219519914552, \ 0.7888246674338504, \ 0.0101533805746944, \ 0.0101533805746944, \ 0.2010219519914552, \ 0.7888246674338504, \ 0.2010219519914552, \ 0.7888246674338504, \ 0.0101533805746944, \ 0.1675040220811526, \ 0.5404862370418329, \ 0.2920097408770145, \ 0.5404862370418329, \ 0.2920097408770145, \ 0.1675040220811526, \ 0.1675040220811526, \ 0.5404862370418329, \ 0.2920097408770145, \ 0.5404862370418329, \ 0.2920097408770145, \ 0.1675040220811526, \ 0.0168250918958854, \ 0.1016331118955575, \ 0.8815417962085570, \ 0.1016331118955575, \ 0.8815417962085570, \ 0.0168250918958854, \ 0.0168250918958854, \ 0.1016331118955575, \ 0.8815417962085570, \ 0.1016331118955575, \ 0.8815417962085570, \ 0.0168250918958854, \ 0.0552850680433709, \ 0.7553391402298588, \ 0.1893757917267703, \ 0.7553391402298588, \ 0.1893757917267703, \ 0.0552850680433709, \ 0.0552850680433709, \ 0.7553391402298588, \ 0.1893757917267703, \ 0.7553391402298588, \ 0.1893757917267703, \ 0.0552850680433709 ] ) z = np.array ( [ \ 0.2284245901167349, \ 0.7715754098832651, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.9749706927792001, \ 0.9749706927792001, \ 0.9749706927792001, \ 0.0250293072207999, \ 0.0250293072207999, \ 0.0250293072207999, \ 0.3291597905587964, \ 0.3291597905587964, \ 0.3291597905587964, \ 0.6708402094412036, \ 0.6708402094412036, \ 0.6708402094412036, \ 0.9996712623901741, \ 0.9996712623901741, \ 0.9996712623901741, \ 0.0003287376098259, \ 0.0003287376098259, \ 0.0003287376098259, \ 0.7021882096439400, \ 0.7021882096439400, \ 0.7021882096439400, \ 0.2978117903560600, \ 0.2978117903560600, \ 0.2978117903560600, \ 0.8352161845608261, \ 0.8352161845608261, \ 0.8352161845608261, \ 0.1647838154391739, \ 0.1647838154391739, \ 0.1647838154391739, \ 0.7201109571464277, \ 0.7201109571464277, \ 0.7201109571464277, \ 0.2798890428535724, \ 0.2798890428535724, \ 0.2798890428535724, \ 0.9332471062622827, \ 0.9332471062622827, \ 0.9332471062622827, \ 0.0667528937377173, \ 0.0667528937377173, \ 0.0667528937377173, \ 0.3097333917149071, \ 0.3097333917149071, \ 0.3097333917149071, \ 0.6902666082850929, \ 0.6902666082850929, \ 0.6902666082850929, \ 0.6590250213320482, \ 0.6590250213320482, \ 0.6590250213320482, \ 0.3409749786679518, \ 0.3409749786679518, \ 0.3409749786679518, \ 0.8368689390766315, \ 0.8368689390766315, \ 0.8368689390766315, \ 0.1631310609233684, \ 0.1631310609233684, \ 0.1631310609233684, \ 0.9263853262636454, \ 0.9263853262636454, \ 0.9263853262636454, \ 0.0736146737363546, \ 0.0736146737363546, \ 0.0736146737363546, \ 0.9586297434779142, \ 0.9586297434779142, \ 0.9586297434779142, \ 0.0413702565220858, \ 0.0413702565220858, \ 0.0413702565220858, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.8444056700436123, \ 0.8444056700436123, \ 0.8444056700436123, \ 0.8444056700436123, \ 0.8444056700436123, \ 0.8444056700436123, \ 0.1555943299563877, \ 0.1555943299563877, \ 0.1555943299563877, \ 0.1555943299563877, \ 0.1555943299563877, \ 0.1555943299563877, \ 0.9409140558660263, \ 0.9409140558660263, \ 0.9409140558660263, \ 0.9409140558660263, \ 0.9409140558660263, \ 0.9409140558660263, \ 0.0590859441339738, \ 0.0590859441339738, \ 0.0590859441339738, \ 0.0590859441339738, \ 0.0590859441339738, \ 0.0590859441339738, \ 0.6041114898852048, \ 0.6041114898852048, \ 0.6041114898852048, \ 0.6041114898852048, \ 0.6041114898852048, \ 0.6041114898852048, \ 0.3958885101147953, \ 0.3958885101147953, \ 0.3958885101147953, \ 0.3958885101147953, \ 0.3958885101147953, \ 0.3958885101147953, \ 0.3519734130158997, \ 0.3519734130158997, \ 0.3519734130158997, \ 0.3519734130158997, \ 0.3519734130158997, \ 0.3519734130158997, \ 0.6480265869841003, \ 0.6480265869841003, \ 0.6480265869841003, \ 0.6480265869841003, \ 0.6480265869841003, \ 0.6480265869841003, \ 0.9783845500663495, \ 0.9783845500663495, \ 0.9783845500663495, \ 0.9783845500663495, \ 0.9783845500663495, \ 0.9783845500663495, \ 0.0216154499336504, \ 0.0216154499336504, \ 0.0216154499336504, \ 0.0216154499336504, \ 0.0216154499336504, \ 0.0216154499336504, \ 0.9921911717290701, \ 0.9921911717290701, \ 0.9921911717290701, \ 0.9921911717290701, \ 0.9921911717290701, \ 0.9921911717290701, \ 0.0078088282709299, \ 0.0078088282709299, \ 0.0078088282709299, \ 0.0078088282709299, \ 0.0078088282709299, \ 0.0078088282709299, \ 0.8588015965488265, \ 0.8588015965488265, \ 0.8588015965488265, \ 0.8588015965488265, \ 0.8588015965488265, \ 0.8588015965488265, \ 0.1411984034511735, \ 0.1411984034511735, \ 0.1411984034511735, \ 0.1411984034511735, \ 0.1411984034511735, \ 0.1411984034511735, \ 0.2510713639088549, \ 0.2510713639088549, \ 0.2510713639088549, \ 0.2510713639088549, \ 0.2510713639088549, \ 0.2510713639088549, \ 0.7489286360911451, \ 0.7489286360911451, \ 0.7489286360911451, \ 0.7489286360911451, \ 0.7489286360911451, \ 0.7489286360911451 ] ) w = np.array ( [ \ 0.0117723371203579, \ 0.0117723371203579, \ 0.0135361913058678, \ 0.0135361913058678, \ 0.0135361913058678, \ 0.0028636244590190, \ 0.0028636244590190, \ 0.0028636244590190, \ 0.0028317101953144, \ 0.0028317101953144, \ 0.0028317101953144, \ 0.0028317101953144, \ 0.0028317101953144, \ 0.0028317101953144, \ 0.0043546495593520, \ 0.0043546495593520, \ 0.0043546495593520, \ 0.0043546495593520, \ 0.0043546495593520, \ 0.0043546495593520, \ 0.0009145278212111, \ 0.0009145278212111, \ 0.0009145278212111, \ 0.0009145278212111, \ 0.0009145278212111, \ 0.0009145278212111, \ 0.0016406658653518, \ 0.0016406658653518, \ 0.0016406658653518, \ 0.0016406658653518, \ 0.0016406658653518, \ 0.0016406658653518, \ 0.0121732957893404, \ 0.0121732957893404, \ 0.0121732957893404, \ 0.0121732957893404, \ 0.0121732957893404, \ 0.0121732957893404, \ 0.0010621280231715, \ 0.0010621280231715, \ 0.0010621280231715, \ 0.0010621280231715, \ 0.0010621280231715, \ 0.0010621280231715, \ 0.0081462190313536, \ 0.0081462190313536, \ 0.0081462190313536, \ 0.0081462190313536, \ 0.0081462190313536, \ 0.0081462190313536, \ 0.0118819825086521, \ 0.0118819825086521, \ 0.0118819825086521, \ 0.0118819825086521, \ 0.0118819825086521, \ 0.0118819825086521, \ 0.0045106060735437, \ 0.0045106060735437, \ 0.0045106060735437, \ 0.0045106060735437, \ 0.0045106060735437, \ 0.0045106060735437, \ 0.0120105850348785, \ 0.0120105850348785, \ 0.0120105850348785, \ 0.0120105850348785, \ 0.0120105850348785, \ 0.0120105850348785, \ 0.0047664383021567, \ 0.0047664383021567, \ 0.0047664383021567, \ 0.0047664383021567, \ 0.0047664383021567, \ 0.0047664383021567, \ 0.0009777024948606, \ 0.0009777024948606, \ 0.0009777024948606, \ 0.0009777024948606, \ 0.0009777024948606, \ 0.0009777024948606, \ 0.0051118807026059, \ 0.0051118807026059, \ 0.0051118807026059, \ 0.0051118807026059, \ 0.0051118807026059, \ 0.0051118807026059, \ 0.0021717621070222, \ 0.0021717621070222, \ 0.0021717621070222, \ 0.0021717621070222, \ 0.0021717621070222, \ 0.0021717621070222, \ 0.0104022745738753, \ 0.0104022745738753, \ 0.0104022745738753, \ 0.0104022745738753, \ 0.0104022745738753, \ 0.0104022745738753, \ 0.0035903626892995, \ 0.0035903626892995, \ 0.0035903626892995, \ 0.0035903626892995, \ 0.0035903626892995, \ 0.0035903626892995, \ 0.0035903626892995, \ 0.0035903626892995, \ 0.0035903626892995, \ 0.0035903626892995, \ 0.0035903626892995, \ 0.0035903626892995, \ 0.0052108482781140, \ 0.0052108482781140, \ 0.0052108482781140, \ 0.0052108482781140, \ 0.0052108482781140, \ 0.0052108482781140, \ 0.0052108482781140, \ 0.0052108482781140, \ 0.0052108482781140, \ 0.0052108482781140, \ 0.0052108482781140, \ 0.0052108482781140, \ 0.0029024210927214, \ 0.0029024210927214, \ 0.0029024210927214, \ 0.0029024210927214, \ 0.0029024210927214, \ 0.0029024210927214, \ 0.0029024210927214, \ 0.0029024210927214, \ 0.0029024210927214, \ 0.0029024210927214, \ 0.0029024210927214, \ 0.0029024210927214, \ 0.0109370973987031, \ 0.0109370973987031, \ 0.0109370973987031, \ 0.0109370973987031, \ 0.0109370973987031, \ 0.0109370973987031, \ 0.0109370973987031, \ 0.0109370973987031, \ 0.0109370973987031, \ 0.0109370973987031, \ 0.0109370973987031, \ 0.0109370973987031, \ 0.0010706231542004, \ 0.0010706231542004, \ 0.0010706231542004, \ 0.0010706231542004, \ 0.0010706231542004, \ 0.0010706231542004, \ 0.0010706231542004, \ 0.0010706231542004, \ 0.0010706231542004, \ 0.0010706231542004, \ 0.0010706231542004, \ 0.0010706231542004, \ 0.0023660542319623, \ 0.0023660542319623, \ 0.0023660542319623, \ 0.0023660542319623, \ 0.0023660542319623, \ 0.0023660542319623, \ 0.0023660542319623, \ 0.0023660542319623, \ 0.0023660542319623, \ 0.0023660542319623, \ 0.0023660542319623, \ 0.0023660542319623, \ 0.0026196720900109, \ 0.0026196720900109, \ 0.0026196720900109, \ 0.0026196720900109, \ 0.0026196720900109, \ 0.0026196720900109, \ 0.0026196720900109, \ 0.0026196720900109, \ 0.0026196720900109, \ 0.0026196720900109, \ 0.0026196720900109, \ 0.0026196720900109, \ 0.0070960302290289, \ 0.0070960302290289, \ 0.0070960302290289, \ 0.0070960302290289, \ 0.0070960302290289, \ 0.0070960302290289, \ 0.0070960302290289, \ 0.0070960302290289, \ 0.0070960302290289, \ 0.0070960302290289, \ 0.0070960302290289, \ 0.0070960302290289 ] ) return x, y, z, w def rule15 ( ): #*****************************************************************************80 # ## rule15() returns the prism rule of precision 15. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.2180179116942069, \ 0.2180179116942069, \ 0.5639641766115862, \ 0.1819996803717089, \ 0.1819996803717089, \ 0.6360006392565821, \ 0.0174014011631261, \ 0.0174014011631261, \ 0.9651971976737478, \ 0.0542711501859966, \ 0.0542711501859966, \ 0.8914576996280067, \ 0.3702896314191779, \ 0.3702896314191779, \ 0.2594207371616443, \ 0.3702896314191779, \ 0.3702896314191779, \ 0.2594207371616443, \ 0.0683792324219036, \ 0.0683792324219036, \ 0.8632415351561927, \ 0.0683792324219036, \ 0.0683792324219036, \ 0.8632415351561927, \ 0.4041029865296590, \ 0.4041029865296590, \ 0.1917940269406819, \ 0.4041029865296590, \ 0.4041029865296590, \ 0.1917940269406819, \ 0.4924009437532922, \ 0.4924009437532922, \ 0.0151981124934156, \ 0.4924009437532922, \ 0.4924009437532922, \ 0.0151981124934156, \ 0.4679592294726652, \ 0.4679592294726652, \ 0.0640815410546695, \ 0.4679592294726652, \ 0.4679592294726652, \ 0.0640815410546695, \ 0.2538613877625208, \ 0.2538613877625208, \ 0.4922772244749584, \ 0.2538613877625208, \ 0.2538613877625208, \ 0.4922772244749584, \ 0.1490233678882002, \ 0.1490233678882002, \ 0.7019532642235996, \ 0.1490233678882002, \ 0.1490233678882002, \ 0.7019532642235996, \ 0.4943016520035221, \ 0.4943016520035221, \ 0.0113966959929558, \ 0.4943016520035221, \ 0.4943016520035221, \ 0.0113966959929558, \ 0.2380227290434666, \ 0.0191912097570085, \ 0.2380227290434666, \ 0.7427860611995248, \ 0.0191912097570085, \ 0.7427860611995248, \ 0.2658819381343187, \ 0.0947024276247834, \ 0.2658819381343187, \ 0.6394156342408979, \ 0.0947024276247834, \ 0.6394156342408979, \ 0.2658819381343187, \ 0.0947024276247834, \ 0.2658819381343187, \ 0.6394156342408979, \ 0.0947024276247834, \ 0.6394156342408979, \ 0.0282488031371648, \ 0.0067976462534296, \ 0.0282488031371648, \ 0.9649535506094056, \ 0.0067976462534296, \ 0.9649535506094056, \ 0.0282488031371648, \ 0.0067976462534296, \ 0.0282488031371648, \ 0.9649535506094056, \ 0.0067976462534296, \ 0.9649535506094056, \ 0.6410939576871652, \ 0.0151121945323486, \ 0.6410939576871652, \ 0.3437938477804863, \ 0.0151121945323486, \ 0.3437938477804863, \ 0.6410939576871652, \ 0.0151121945323486, \ 0.6410939576871652, \ 0.3437938477804863, \ 0.0151121945323486, \ 0.3437938477804863, \ 0.1570345607673937, \ 0.0792048851268470, \ 0.1570345607673937, \ 0.7637605541057593, \ 0.0792048851268470, \ 0.7637605541057593, \ 0.1570345607673937, \ 0.0792048851268470, \ 0.1570345607673937, \ 0.7637605541057593, \ 0.0792048851268470, \ 0.7637605541057593, \ 0.2618681572601245, \ 0.1978262294209248, \ 0.2618681572601245, \ 0.5403056133189508, \ 0.1978262294209248, \ 0.5403056133189508, \ 0.2618681572601245, \ 0.1978262294209248, \ 0.2618681572601245, \ 0.5403056133189508, \ 0.1978262294209248, \ 0.5403056133189508, \ 0.0065324588746805, \ 0.0486805622526683, \ 0.0065324588746805, \ 0.9447869788726512, \ 0.0486805622526683, \ 0.9447869788726512, \ 0.0065324588746805, \ 0.0486805622526683, \ 0.0065324588746805, \ 0.9447869788726512, \ 0.0486805622526683, \ 0.9447869788726512, \ 0.3702624031957258, \ 0.0912977133028943, \ 0.3702624031957258, \ 0.5384398835013798, \ 0.0912977133028943, \ 0.5384398835013798, \ 0.3702624031957258, \ 0.0912977133028943, \ 0.3702624031957258, \ 0.5384398835013798, \ 0.0912977133028943, \ 0.5384398835013798, \ 0.1377079768509805, \ 0.0373008369533812, \ 0.1377079768509805, \ 0.8249911861956383, \ 0.0373008369533812, \ 0.8249911861956383, \ 0.1377079768509805, \ 0.0373008369533812, \ 0.1377079768509805, \ 0.8249911861956383, \ 0.0373008369533812, \ 0.8249911861956383, \ 0.4784950101267365, \ 0.1635654688089934, \ 0.4784950101267365, \ 0.3579395210642701, \ 0.1635654688089934, \ 0.3579395210642701, \ 0.4784950101267365, \ 0.1635654688089934, \ 0.4784950101267365, \ 0.3579395210642701, \ 0.1635654688089934, \ 0.3579395210642701, \ 0.3278240979071815, \ 0.0249615816629217, \ 0.3278240979071815, \ 0.6472143204298968, \ 0.0249615816629217, \ 0.6472143204298968, \ 0.3278240979071815, \ 0.0249615816629217, \ 0.3278240979071815, \ 0.6472143204298968, \ 0.0249615816629217, \ 0.6472143204298968, \ 0.3475735063324100, \ 0.0871302288391398, \ 0.3475735063324100, \ 0.5652962648284502, \ 0.0871302288391398, \ 0.5652962648284502, \ 0.3475735063324100, \ 0.0871302288391398, \ 0.3475735063324100, \ 0.5652962648284502, \ 0.0871302288391398, \ 0.5652962648284502, \ 0.1096327891311607, \ 0.0128395677390919, \ 0.1096327891311607, \ 0.8775276431297473, \ 0.0128395677390919, \ 0.8775276431297473, \ 0.1096327891311607, \ 0.0128395677390919, \ 0.1096327891311607, \ 0.8775276431297473, \ 0.0128395677390919, \ 0.8775276431297473, \ 0.2078784801536998, \ 0.0849045675529205, \ 0.2078784801536998, \ 0.7072169522933797, \ 0.0849045675529205, \ 0.7072169522933797, \ 0.2078784801536998, \ 0.0849045675529205, \ 0.2078784801536998, \ 0.7072169522933797, \ 0.0849045675529205, \ 0.7072169522933797, \ 0.1953330193360589, \ 0.0143045787175747, \ 0.1953330193360589, \ 0.7903624019463664, \ 0.0143045787175747, \ 0.7903624019463664, \ 0.1953330193360589, \ 0.0143045787175747, \ 0.1953330193360589, \ 0.7903624019463664, \ 0.0143045787175747, \ 0.7903624019463664 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.2180179116942069, \ 0.5639641766115862, \ 0.2180179116942069, \ 0.1819996803717089, \ 0.6360006392565821, \ 0.1819996803717089, \ 0.0174014011631261, \ 0.9651971976737478, \ 0.0174014011631261, \ 0.0542711501859966, \ 0.8914576996280067, \ 0.0542711501859966, \ 0.3702896314191779, \ 0.2594207371616443, \ 0.3702896314191779, \ 0.3702896314191779, \ 0.2594207371616443, \ 0.3702896314191779, \ 0.0683792324219036, \ 0.8632415351561927, \ 0.0683792324219036, \ 0.0683792324219036, \ 0.8632415351561927, \ 0.0683792324219036, \ 0.4041029865296590, \ 0.1917940269406819, \ 0.4041029865296590, \ 0.4041029865296590, \ 0.1917940269406819, \ 0.4041029865296590, \ 0.4924009437532922, \ 0.0151981124934156, \ 0.4924009437532922, \ 0.4924009437532922, \ 0.0151981124934156, \ 0.4924009437532922, \ 0.4679592294726652, \ 0.0640815410546695, \ 0.4679592294726652, \ 0.4679592294726652, \ 0.0640815410546695, \ 0.4679592294726652, \ 0.2538613877625208, \ 0.4922772244749584, \ 0.2538613877625208, \ 0.2538613877625208, \ 0.4922772244749584, \ 0.2538613877625208, \ 0.1490233678882002, \ 0.7019532642235996, \ 0.1490233678882002, \ 0.1490233678882002, \ 0.7019532642235996, \ 0.1490233678882002, \ 0.4943016520035221, \ 0.0113966959929558, \ 0.4943016520035221, \ 0.4943016520035221, \ 0.0113966959929558, \ 0.4943016520035221, \ 0.0191912097570085, \ 0.2380227290434666, \ 0.7427860611995248, \ 0.2380227290434666, \ 0.7427860611995248, \ 0.0191912097570085, \ 0.0947024276247834, \ 0.2658819381343187, \ 0.6394156342408979, \ 0.2658819381343187, \ 0.6394156342408979, \ 0.0947024276247834, \ 0.0947024276247834, \ 0.2658819381343187, \ 0.6394156342408979, \ 0.2658819381343187, \ 0.6394156342408979, \ 0.0947024276247834, \ 0.0067976462534296, \ 0.0282488031371648, \ 0.9649535506094056, \ 0.0282488031371648, \ 0.9649535506094056, \ 0.0067976462534296, \ 0.0067976462534296, \ 0.0282488031371648, \ 0.9649535506094056, \ 0.0282488031371648, \ 0.9649535506094056, \ 0.0067976462534296, \ 0.0151121945323486, \ 0.6410939576871652, \ 0.3437938477804863, \ 0.6410939576871652, \ 0.3437938477804863, \ 0.0151121945323486, \ 0.0151121945323486, \ 0.6410939576871652, \ 0.3437938477804863, \ 0.6410939576871652, \ 0.3437938477804863, \ 0.0151121945323486, \ 0.0792048851268470, \ 0.1570345607673937, \ 0.7637605541057593, \ 0.1570345607673937, \ 0.7637605541057593, \ 0.0792048851268470, \ 0.0792048851268470, \ 0.1570345607673937, \ 0.7637605541057593, \ 0.1570345607673937, \ 0.7637605541057593, \ 0.0792048851268470, \ 0.1978262294209248, \ 0.2618681572601245, \ 0.5403056133189508, \ 0.2618681572601245, \ 0.5403056133189508, \ 0.1978262294209248, \ 0.1978262294209248, \ 0.2618681572601245, \ 0.5403056133189508, \ 0.2618681572601245, \ 0.5403056133189508, \ 0.1978262294209248, \ 0.0486805622526683, \ 0.0065324588746805, \ 0.9447869788726512, \ 0.0065324588746805, \ 0.9447869788726512, \ 0.0486805622526683, \ 0.0486805622526683, \ 0.0065324588746805, \ 0.9447869788726512, \ 0.0065324588746805, \ 0.9447869788726512, \ 0.0486805622526683, \ 0.0912977133028943, \ 0.3702624031957258, \ 0.5384398835013798, \ 0.3702624031957258, \ 0.5384398835013798, \ 0.0912977133028943, \ 0.0912977133028943, \ 0.3702624031957258, \ 0.5384398835013798, \ 0.3702624031957258, \ 0.5384398835013798, \ 0.0912977133028943, \ 0.0373008369533812, \ 0.1377079768509805, \ 0.8249911861956383, \ 0.1377079768509805, \ 0.8249911861956383, \ 0.0373008369533812, \ 0.0373008369533812, \ 0.1377079768509805, \ 0.8249911861956383, \ 0.1377079768509805, \ 0.8249911861956383, \ 0.0373008369533812, \ 0.1635654688089934, \ 0.4784950101267365, \ 0.3579395210642701, \ 0.4784950101267365, \ 0.3579395210642701, \ 0.1635654688089934, \ 0.1635654688089934, \ 0.4784950101267365, \ 0.3579395210642701, \ 0.4784950101267365, \ 0.3579395210642701, \ 0.1635654688089934, \ 0.0249615816629217, \ 0.3278240979071815, \ 0.6472143204298968, \ 0.3278240979071815, \ 0.6472143204298968, \ 0.0249615816629217, \ 0.0249615816629217, \ 0.3278240979071815, \ 0.6472143204298968, \ 0.3278240979071815, \ 0.6472143204298968, \ 0.0249615816629217, \ 0.0871302288391398, \ 0.3475735063324100, \ 0.5652962648284502, \ 0.3475735063324100, \ 0.5652962648284502, \ 0.0871302288391398, \ 0.0871302288391398, \ 0.3475735063324100, \ 0.5652962648284502, \ 0.3475735063324100, \ 0.5652962648284502, \ 0.0871302288391398, \ 0.0128395677390919, \ 0.1096327891311607, \ 0.8775276431297473, \ 0.1096327891311607, \ 0.8775276431297473, \ 0.0128395677390919, \ 0.0128395677390919, \ 0.1096327891311607, \ 0.8775276431297473, \ 0.1096327891311607, \ 0.8775276431297473, \ 0.0128395677390919, \ 0.0849045675529205, \ 0.2078784801536998, \ 0.7072169522933797, \ 0.2078784801536998, \ 0.7072169522933797, \ 0.0849045675529205, \ 0.0849045675529205, \ 0.2078784801536998, \ 0.7072169522933797, \ 0.2078784801536998, \ 0.7072169522933797, \ 0.0849045675529205, \ 0.0143045787175747, \ 0.1953330193360589, \ 0.7903624019463664, \ 0.1953330193360589, \ 0.7903624019463664, \ 0.0143045787175747, \ 0.0143045787175747, \ 0.1953330193360589, \ 0.7903624019463664, \ 0.1953330193360589, \ 0.7903624019463664, \ 0.0143045787175747 ] ) z = np.array ( [ \ 0.8401109114804578, \ 0.1598890885195422, \ 0.9524833096339549, \ 0.0475166903660451, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5508863349912149, \ 0.5508863349912149, \ 0.5508863349912149, \ 0.4491136650087851, \ 0.4491136650087851, \ 0.4491136650087851, \ 0.8664406024867690, \ 0.8664406024867690, \ 0.8664406024867690, \ 0.1335593975132310, \ 0.1335593975132310, \ 0.1335593975132310, \ 0.3291089745865182, \ 0.3291089745865182, \ 0.3291089745865182, \ 0.6708910254134818, \ 0.6708910254134818, \ 0.6708910254134818, \ 0.4287046442912852, \ 0.4287046442912852, \ 0.4287046442912852, \ 0.5712953557087147, \ 0.5712953557087147, \ 0.5712953557087147, \ 0.8306396134186287, \ 0.8306396134186287, \ 0.8306396134186287, \ 0.1693603865813713, \ 0.1693603865813713, \ 0.1693603865813713, \ 0.9826991514876333, \ 0.9826991514876333, \ 0.9826991514876333, \ 0.0173008485123666, \ 0.0173008485123666, \ 0.0173008485123666, \ 0.9603593395589844, \ 0.9603593395589844, \ 0.9603593395589844, \ 0.0396406604410156, \ 0.0396406604410156, \ 0.0396406604410156, \ 0.9264398795821308, \ 0.9264398795821308, \ 0.9264398795821308, \ 0.0735601204178692, \ 0.0735601204178692, \ 0.0735601204178692, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.9025925504145782, \ 0.9025925504145782, \ 0.9025925504145782, \ 0.9025925504145782, \ 0.9025925504145782, \ 0.9025925504145782, \ 0.0974074495854217, \ 0.0974074495854217, \ 0.0974074495854217, \ 0.0974074495854217, \ 0.0974074495854217, \ 0.0974074495854217, \ 0.7775063050268207, \ 0.7775063050268207, \ 0.7775063050268207, \ 0.7775063050268207, \ 0.7775063050268207, \ 0.7775063050268207, \ 0.2224936949731793, \ 0.2224936949731793, \ 0.2224936949731793, \ 0.2224936949731793, \ 0.2224936949731793, \ 0.2224936949731793, \ 0.7482800508722226, \ 0.7482800508722226, \ 0.7482800508722226, \ 0.7482800508722226, \ 0.7482800508722226, \ 0.7482800508722226, \ 0.2517199491277774, \ 0.2517199491277774, \ 0.2517199491277774, \ 0.2517199491277774, \ 0.2517199491277774, \ 0.2517199491277774, \ 0.6403701196964353, \ 0.6403701196964353, \ 0.6403701196964353, \ 0.6403701196964353, \ 0.6403701196964353, \ 0.6403701196964353, \ 0.3596298803035647, \ 0.3596298803035647, \ 0.3596298803035647, \ 0.3596298803035647, \ 0.3596298803035647, \ 0.3596298803035647, \ 0.7769630399535563, \ 0.7769630399535563, \ 0.7769630399535563, \ 0.7769630399535563, \ 0.7769630399535563, \ 0.7769630399535563, \ 0.2230369600464437, \ 0.2230369600464437, \ 0.2230369600464437, \ 0.2230369600464437, \ 0.2230369600464437, \ 0.2230369600464437, \ 0.9565498865251021, \ 0.9565498865251021, \ 0.9565498865251021, \ 0.9565498865251021, \ 0.9565498865251021, \ 0.9565498865251021, \ 0.0434501134748980, \ 0.0434501134748980, \ 0.0434501134748980, \ 0.0434501134748980, \ 0.0434501134748980, \ 0.0434501134748980, \ 0.9928793309605464, \ 0.9928793309605464, \ 0.9928793309605464, \ 0.9928793309605464, \ 0.9928793309605464, \ 0.9928793309605464, \ 0.0071206690394536, \ 0.0071206690394536, \ 0.0071206690394536, \ 0.0071206690394536, \ 0.0071206690394536, \ 0.0071206690394536, \ 0.9880631245383409, \ 0.9880631245383409, \ 0.9880631245383409, \ 0.9880631245383409, \ 0.9880631245383409, \ 0.9880631245383409, \ 0.0119368754616592, \ 0.0119368754616592, \ 0.0119368754616592, \ 0.0119368754616592, \ 0.0119368754616592, \ 0.0119368754616592, \ 0.9154311485206492, \ 0.9154311485206492, \ 0.9154311485206492, \ 0.9154311485206492, \ 0.9154311485206492, \ 0.9154311485206492, \ 0.0845688514793507, \ 0.0845688514793507, \ 0.0845688514793507, \ 0.0845688514793507, \ 0.0845688514793507, \ 0.0845688514793507, \ 0.9705250000766562, \ 0.9705250000766562, \ 0.9705250000766562, \ 0.9705250000766562, \ 0.9705250000766562, \ 0.9705250000766562, \ 0.0294749999233438, \ 0.0294749999233438, \ 0.0294749999233438, \ 0.0294749999233438, \ 0.0294749999233438, \ 0.0294749999233438, \ 0.3853017195325944, \ 0.3853017195325944, \ 0.3853017195325944, \ 0.3853017195325944, \ 0.3853017195325944, \ 0.3853017195325944, \ 0.6146982804674056, \ 0.6146982804674056, \ 0.6146982804674056, \ 0.6146982804674056, \ 0.6146982804674056, \ 0.6146982804674056, \ 0.6900924303976952, \ 0.6900924303976952, \ 0.6900924303976952, \ 0.6900924303976952, \ 0.6900924303976952, \ 0.6900924303976952, \ 0.3099075696023049, \ 0.3099075696023049, \ 0.3099075696023049, \ 0.3099075696023049, \ 0.3099075696023049, \ 0.3099075696023049, \ 0.7816502945768592, \ 0.7816502945768592, \ 0.7816502945768592, \ 0.7816502945768592, \ 0.7816502945768592, \ 0.7816502945768592, \ 0.2183497054231408, \ 0.2183497054231408, \ 0.2183497054231408, \ 0.2183497054231408, \ 0.2183497054231408, \ 0.2183497054231408, \ 0.8883851866879513, \ 0.8883851866879513, \ 0.8883851866879513, \ 0.8883851866879513, \ 0.8883851866879513, \ 0.8883851866879513, \ 0.1116148133120487, \ 0.1116148133120487, \ 0.1116148133120487, \ 0.1116148133120487, \ 0.1116148133120487, \ 0.1116148133120487 ] ) w = np.array ( [ \ 0.0109734885358618, \ 0.0109734885358618, \ 0.0032613958307349, \ 0.0032613958307349, \ 0.0093941405464380, \ 0.0093941405464380, \ 0.0093941405464380, \ 0.0074721925384407, \ 0.0074721925384407, \ 0.0074721925384407, \ 0.0011251486610045, \ 0.0011251486610045, \ 0.0011251486610045, \ 0.0040278942594456, \ 0.0040278942594456, \ 0.0040278942594456, \ 0.0046987405274126, \ 0.0046987405274126, \ 0.0046987405274126, \ 0.0046987405274126, \ 0.0046987405274126, \ 0.0046987405274126, \ 0.0044231068731859, \ 0.0044231068731859, \ 0.0044231068731859, \ 0.0044231068731859, \ 0.0044231068731859, \ 0.0044231068731859, \ 0.0119087119167916, \ 0.0119087119167916, \ 0.0119087119167916, \ 0.0119087119167916, \ 0.0119087119167916, \ 0.0119087119167916, \ 0.0034190583552652, \ 0.0034190583552652, \ 0.0034190583552652, \ 0.0034190583552652, \ 0.0034190583552652, \ 0.0034190583552652, \ 0.0076078695830692, \ 0.0076078695830692, \ 0.0076078695830692, \ 0.0076078695830692, \ 0.0076078695830692, \ 0.0076078695830692, \ 0.0034877218143443, \ 0.0034877218143443, \ 0.0034877218143443, \ 0.0034877218143443, \ 0.0034877218143443, \ 0.0034877218143443, \ 0.0041265711237682, \ 0.0041265711237682, \ 0.0041265711237682, \ 0.0041265711237682, \ 0.0041265711237682, \ 0.0041265711237682, \ 0.0020552290439392, \ 0.0020552290439392, \ 0.0020552290439392, \ 0.0020552290439392, \ 0.0020552290439392, \ 0.0020552290439392, \ 0.0053334369811175, \ 0.0053334369811175, \ 0.0053334369811175, \ 0.0053334369811175, \ 0.0053334369811175, \ 0.0053334369811175, \ 0.0045658362940629, \ 0.0045658362940629, \ 0.0045658362940629, \ 0.0045658362940629, \ 0.0045658362940629, \ 0.0045658362940629, \ 0.0045658362940629, \ 0.0045658362940629, \ 0.0045658362940629, \ 0.0045658362940629, \ 0.0045658362940629, \ 0.0045658362940629, \ 0.0005459132755231, \ 0.0005459132755231, \ 0.0005459132755231, \ 0.0005459132755231, \ 0.0005459132755231, \ 0.0005459132755231, \ 0.0005459132755231, \ 0.0005459132755231, \ 0.0005459132755231, \ 0.0005459132755231, \ 0.0005459132755231, \ 0.0005459132755231, \ 0.0043191948291063, \ 0.0043191948291063, \ 0.0043191948291063, \ 0.0043191948291063, \ 0.0043191948291063, \ 0.0043191948291063, \ 0.0043191948291063, \ 0.0043191948291063, \ 0.0043191948291063, \ 0.0043191948291063, \ 0.0043191948291063, \ 0.0043191948291063, \ 0.0053774143502532, \ 0.0053774143502532, \ 0.0053774143502532, \ 0.0053774143502532, \ 0.0053774143502532, \ 0.0053774143502532, \ 0.0053774143502532, \ 0.0053774143502532, \ 0.0053774143502532, \ 0.0053774143502532, \ 0.0053774143502532, \ 0.0053774143502532, \ 0.0071616945998409, \ 0.0071616945998409, \ 0.0071616945998409, \ 0.0071616945998409, \ 0.0071616945998409, \ 0.0071616945998409, \ 0.0071616945998409, \ 0.0071616945998409, \ 0.0071616945998409, \ 0.0071616945998409, \ 0.0071616945998409, \ 0.0071616945998409, \ 0.0005425741614460, \ 0.0005425741614460, \ 0.0005425741614460, \ 0.0005425741614460, \ 0.0005425741614460, \ 0.0005425741614460, \ 0.0005425741614460, \ 0.0005425741614460, \ 0.0005425741614460, \ 0.0005425741614460, \ 0.0005425741614460, \ 0.0005425741614460, \ 0.0014376691849096, \ 0.0014376691849096, \ 0.0014376691849096, \ 0.0014376691849096, \ 0.0014376691849096, \ 0.0014376691849096, \ 0.0014376691849096, \ 0.0014376691849096, \ 0.0014376691849096, \ 0.0014376691849096, \ 0.0014376691849096, \ 0.0014376691849096, \ 0.0011160390387250, \ 0.0011160390387250, \ 0.0011160390387250, \ 0.0011160390387250, \ 0.0011160390387250, \ 0.0011160390387250, \ 0.0011160390387250, \ 0.0011160390387250, \ 0.0011160390387250, \ 0.0011160390387250, \ 0.0011160390387250, \ 0.0011160390387250, \ 0.0048559812857500, \ 0.0048559812857500, \ 0.0048559812857500, \ 0.0048559812857500, \ 0.0048559812857500, \ 0.0048559812857500, \ 0.0048559812857500, \ 0.0048559812857500, \ 0.0048559812857500, \ 0.0048559812857500, \ 0.0048559812857500, \ 0.0048559812857500, \ 0.0017971531168555, \ 0.0017971531168555, \ 0.0017971531168555, \ 0.0017971531168555, \ 0.0017971531168555, \ 0.0017971531168555, \ 0.0017971531168555, \ 0.0017971531168555, \ 0.0017971531168555, \ 0.0017971531168555, \ 0.0017971531168555, \ 0.0017971531168555, \ 0.0096050585986694, \ 0.0096050585986694, \ 0.0096050585986694, \ 0.0096050585986694, \ 0.0096050585986694, \ 0.0096050585986694, \ 0.0096050585986694, \ 0.0096050585986694, \ 0.0096050585986694, \ 0.0096050585986694, \ 0.0096050585986694, \ 0.0096050585986694, \ 0.0027021277312842, \ 0.0027021277312842, \ 0.0027021277312842, \ 0.0027021277312842, \ 0.0027021277312842, \ 0.0027021277312842, \ 0.0027021277312842, \ 0.0027021277312842, \ 0.0027021277312842, \ 0.0027021277312842, \ 0.0027021277312842, \ 0.0027021277312842, \ 0.0055044460347859, \ 0.0055044460347859, \ 0.0055044460347859, \ 0.0055044460347859, \ 0.0055044460347859, \ 0.0055044460347859, \ 0.0055044460347859, \ 0.0055044460347859, \ 0.0055044460347859, \ 0.0055044460347859, \ 0.0055044460347859, \ 0.0055044460347859, \ 0.0023946829935762, \ 0.0023946829935762, \ 0.0023946829935762, \ 0.0023946829935762, \ 0.0023946829935762, \ 0.0023946829935762, \ 0.0023946829935762, \ 0.0023946829935762, \ 0.0023946829935762, \ 0.0023946829935762, \ 0.0023946829935762, \ 0.0023946829935762 ] ) return x, y, z, w def rule16 ( ): #*****************************************************************************80 # ## rule16() returns the prism rule of precision 16. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.0089540840312146, \ 0.0089540840312146, \ 0.9820918319375707, \ 0.0553908100440216, \ 0.0553908100440216, \ 0.8892183799119567, \ 0.3916317627335951, \ 0.3916317627335951, \ 0.2167364745328097, \ 0.0561000778858347, \ 0.0561000778858347, \ 0.8877998442283307, \ 0.0561000778858347, \ 0.0561000778858347, \ 0.8877998442283307, \ 0.2669967593271997, \ 0.2669967593271997, \ 0.4660064813456005, \ 0.2669967593271997, \ 0.2669967593271997, \ 0.4660064813456005, \ 0.1619375672046858, \ 0.1619375672046858, \ 0.6761248655906285, \ 0.1619375672046858, \ 0.1619375672046858, \ 0.6761248655906285, \ 0.4259891438745869, \ 0.4259891438745869, \ 0.1480217122508263, \ 0.4259891438745869, \ 0.4259891438745869, \ 0.1480217122508263, \ 0.4943682456985981, \ 0.4943682456985981, \ 0.0112635086028039, \ 0.4943682456985981, \ 0.4943682456985981, \ 0.0112635086028039, \ 0.4745817664118026, \ 0.4745817664118026, \ 0.0508364671763949, \ 0.4745817664118026, \ 0.4745817664118026, \ 0.0508364671763949, \ 0.2785739189962955, \ 0.2785739189962955, \ 0.4428521620074090, \ 0.2785739189962955, \ 0.2785739189962955, \ 0.4428521620074090, \ 0.1564426776295505, \ 0.1564426776295505, \ 0.6871146447408989, \ 0.1564426776295505, \ 0.1564426776295505, \ 0.6871146447408989, \ 0.4933597406415689, \ 0.4933597406415689, \ 0.0132805187168622, \ 0.4933597406415689, \ 0.4933597406415689, \ 0.0132805187168622, \ 0.5254267651000835, \ 0.0993149454903146, \ 0.5254267651000835, \ 0.3752582894096019, \ 0.0993149454903146, \ 0.3752582894096019, \ 0.3482743570815052, \ 0.0212715561472286, \ 0.3482743570815052, \ 0.6304540867712661, \ 0.0212715561472286, \ 0.6304540867712661, \ 0.1654756461617898, \ 0.0860850979983652, \ 0.1654756461617898, \ 0.7484392558398449, \ 0.0860850979983652, \ 0.7484392558398449, \ 0.1639763763638563, \ 0.0551722290870520, \ 0.1639763763638563, \ 0.7808513945490917, \ 0.0551722290870520, \ 0.7808513945490917, \ 0.1639763763638563, \ 0.0551722290870520, \ 0.1639763763638563, \ 0.7808513945490917, \ 0.0551722290870520, \ 0.7808513945490917, \ 0.0370382522819345, \ 0.0105036665914126, \ 0.0370382522819345, \ 0.9524580811266530, \ 0.0105036665914126, \ 0.9524580811266530, \ 0.0370382522819345, \ 0.0105036665914126, \ 0.0370382522819345, \ 0.9524580811266530, \ 0.0105036665914126, \ 0.9524580811266530, \ 0.6350080118967869, \ 0.0055639594221052, \ 0.6350080118967869, \ 0.3594280286811078, \ 0.0055639594221052, \ 0.3594280286811078, \ 0.6350080118967869, \ 0.0055639594221052, \ 0.6350080118967869, \ 0.3594280286811078, \ 0.0055639594221052, \ 0.3594280286811078, \ 0.1297894012222233, \ 0.0724868197207572, \ 0.1297894012222233, \ 0.7977237790570195, \ 0.0724868197207572, \ 0.7977237790570195, \ 0.1297894012222233, \ 0.0724868197207572, \ 0.1297894012222233, \ 0.7977237790570195, \ 0.0724868197207572, \ 0.7977237790570195, \ 0.3059321030072774, \ 0.0621201882612809, \ 0.3059321030072774, \ 0.6319477087314417, \ 0.0621201882612809, \ 0.6319477087314417, \ 0.3059321030072774, \ 0.0621201882612809, \ 0.3059321030072774, \ 0.6319477087314417, \ 0.0621201882612809, \ 0.6319477087314417, \ 0.1677554820715698, \ 0.2612303617350463, \ 0.1677554820715698, \ 0.5710141561933839, \ 0.2612303617350463, \ 0.5710141561933839, \ 0.1677554820715698, \ 0.2612303617350463, \ 0.1677554820715698, \ 0.5710141561933839, \ 0.2612303617350463, \ 0.5710141561933839, \ 0.0015296219296330, \ 0.0337686922990924, \ 0.0015296219296330, \ 0.9647016857712746, \ 0.0337686922990924, \ 0.9647016857712746, \ 0.0015296219296330, \ 0.0337686922990924, \ 0.0015296219296330, \ 0.9647016857712746, \ 0.0337686922990924, \ 0.9647016857712746, \ 0.3532796606132861, \ 0.0949242221588698, \ 0.3532796606132861, \ 0.5517961172278441, \ 0.0949242221588698, \ 0.5517961172278441, \ 0.3532796606132861, \ 0.0949242221588698, \ 0.3532796606132861, \ 0.5517961172278441, \ 0.0949242221588698, \ 0.5517961172278441, \ 0.0959262325920420, \ 0.0248817170289358, \ 0.0959262325920420, \ 0.8791920503790223, \ 0.0248817170289358, \ 0.8791920503790223, \ 0.0959262325920420, \ 0.0248817170289358, \ 0.0959262325920420, \ 0.8791920503790223, \ 0.0248817170289358, \ 0.8791920503790223, \ 0.4963305263335406, \ 0.1757029526928022, \ 0.4963305263335406, \ 0.3279665209736572, \ 0.1757029526928022, \ 0.3279665209736572, \ 0.4963305263335406, \ 0.1757029526928022, \ 0.4963305263335406, \ 0.3279665209736572, \ 0.1757029526928022, \ 0.3279665209736572, \ 0.2738501035919119, \ 0.0140786122092260, \ 0.2738501035919119, \ 0.7120712841988621, \ 0.0140786122092260, \ 0.7120712841988621, \ 0.2738501035919119, \ 0.0140786122092260, \ 0.2738501035919119, \ 0.7120712841988621, \ 0.0140786122092260, \ 0.7120712841988621, \ 0.3828844998895772, \ 0.0565171623200343, \ 0.3828844998895772, \ 0.5605983377903885, \ 0.0565171623200343, \ 0.5605983377903885, \ 0.3828844998895772, \ 0.0565171623200343, \ 0.3828844998895772, \ 0.5605983377903885, \ 0.0565171623200343, \ 0.5605983377903885, \ 0.0899365323953004, \ 0.0192412570201629, \ 0.0899365323953004, \ 0.8908222105845367, \ 0.0192412570201629, \ 0.8908222105845367, \ 0.0899365323953004, \ 0.0192412570201629, \ 0.0899365323953004, \ 0.8908222105845367, \ 0.0192412570201629, \ 0.8908222105845367, \ 0.2516074150213084, \ 0.0611370184542401, \ 0.2516074150213084, \ 0.6872555665244514, \ 0.0611370184542401, \ 0.6872555665244514, \ 0.2516074150213084, \ 0.0611370184542401, \ 0.2516074150213084, \ 0.6872555665244514, \ 0.0611370184542401, \ 0.6872555665244514, \ 0.1524244609819593, \ 0.0050182699561948, \ 0.1524244609819593, \ 0.8425572690618459, \ 0.0050182699561948, \ 0.8425572690618459, \ 0.1524244609819593, \ 0.0050182699561948, \ 0.1524244609819593, \ 0.8425572690618459, \ 0.0050182699561948, \ 0.8425572690618459, \ 0.2475011786624472, \ 0.1365910574355268, \ 0.2475011786624472, \ 0.6159077639020261, \ 0.1365910574355268, \ 0.6159077639020261, \ 0.2475011786624472, \ 0.1365910574355268, \ 0.2475011786624472, \ 0.6159077639020261, \ 0.1365910574355268, \ 0.6159077639020261, \ 0.7591852986956938, \ 0.0173543718109579, \ 0.7591852986956938, \ 0.2234603294933483, \ 0.0173543718109579, \ 0.2234603294933483, \ 0.7591852986956938, \ 0.0173543718109579, \ 0.7591852986956938, \ 0.2234603294933483, \ 0.0173543718109579, \ 0.2234603294933483 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.0089540840312146, \ 0.9820918319375707, \ 0.0089540840312146, \ 0.0553908100440216, \ 0.8892183799119567, \ 0.0553908100440216, \ 0.3916317627335951, \ 0.2167364745328097, \ 0.3916317627335951, \ 0.0561000778858347, \ 0.8877998442283307, \ 0.0561000778858347, \ 0.0561000778858347, \ 0.8877998442283307, \ 0.0561000778858347, \ 0.2669967593271997, \ 0.4660064813456005, \ 0.2669967593271997, \ 0.2669967593271997, \ 0.4660064813456005, \ 0.2669967593271997, \ 0.1619375672046858, \ 0.6761248655906285, \ 0.1619375672046858, \ 0.1619375672046858, \ 0.6761248655906285, \ 0.1619375672046858, \ 0.4259891438745869, \ 0.1480217122508263, \ 0.4259891438745869, \ 0.4259891438745869, \ 0.1480217122508263, \ 0.4259891438745869, \ 0.4943682456985981, \ 0.0112635086028039, \ 0.4943682456985981, \ 0.4943682456985981, \ 0.0112635086028039, \ 0.4943682456985981, \ 0.4745817664118026, \ 0.0508364671763949, \ 0.4745817664118026, \ 0.4745817664118026, \ 0.0508364671763949, \ 0.4745817664118026, \ 0.2785739189962955, \ 0.4428521620074090, \ 0.2785739189962955, \ 0.2785739189962955, \ 0.4428521620074090, \ 0.2785739189962955, \ 0.1564426776295505, \ 0.6871146447408989, \ 0.1564426776295505, \ 0.1564426776295505, \ 0.6871146447408989, \ 0.1564426776295505, \ 0.4933597406415689, \ 0.0132805187168622, \ 0.4933597406415689, \ 0.4933597406415689, \ 0.0132805187168622, \ 0.4933597406415689, \ 0.0993149454903146, \ 0.5254267651000835, \ 0.3752582894096019, \ 0.5254267651000835, \ 0.3752582894096019, \ 0.0993149454903146, \ 0.0212715561472286, \ 0.3482743570815052, \ 0.6304540867712661, \ 0.3482743570815052, \ 0.6304540867712661, \ 0.0212715561472286, \ 0.0860850979983652, \ 0.1654756461617898, \ 0.7484392558398449, \ 0.1654756461617898, \ 0.7484392558398449, \ 0.0860850979983652, \ 0.0551722290870520, \ 0.1639763763638563, \ 0.7808513945490917, \ 0.1639763763638563, \ 0.7808513945490917, \ 0.0551722290870520, \ 0.0551722290870520, \ 0.1639763763638563, \ 0.7808513945490917, \ 0.1639763763638563, \ 0.7808513945490917, \ 0.0551722290870520, \ 0.0105036665914126, \ 0.0370382522819345, \ 0.9524580811266530, \ 0.0370382522819345, \ 0.9524580811266530, \ 0.0105036665914126, \ 0.0105036665914126, \ 0.0370382522819345, \ 0.9524580811266530, \ 0.0370382522819345, \ 0.9524580811266530, \ 0.0105036665914126, \ 0.0055639594221052, \ 0.6350080118967869, \ 0.3594280286811078, \ 0.6350080118967869, \ 0.3594280286811078, \ 0.0055639594221052, \ 0.0055639594221052, \ 0.6350080118967869, \ 0.3594280286811078, \ 0.6350080118967869, \ 0.3594280286811078, \ 0.0055639594221052, \ 0.0724868197207572, \ 0.1297894012222233, \ 0.7977237790570195, \ 0.1297894012222233, \ 0.7977237790570195, \ 0.0724868197207572, \ 0.0724868197207572, \ 0.1297894012222233, \ 0.7977237790570195, \ 0.1297894012222233, \ 0.7977237790570195, \ 0.0724868197207572, \ 0.0621201882612809, \ 0.3059321030072774, \ 0.6319477087314417, \ 0.3059321030072774, \ 0.6319477087314417, \ 0.0621201882612809, \ 0.0621201882612809, \ 0.3059321030072774, \ 0.6319477087314417, \ 0.3059321030072774, \ 0.6319477087314417, \ 0.0621201882612809, \ 0.2612303617350463, \ 0.1677554820715698, \ 0.5710141561933839, \ 0.1677554820715698, \ 0.5710141561933839, \ 0.2612303617350463, \ 0.2612303617350463, \ 0.1677554820715698, \ 0.5710141561933839, \ 0.1677554820715698, \ 0.5710141561933839, \ 0.2612303617350463, \ 0.0337686922990924, \ 0.0015296219296330, \ 0.9647016857712746, \ 0.0015296219296330, \ 0.9647016857712746, \ 0.0337686922990924, \ 0.0337686922990924, \ 0.0015296219296330, \ 0.9647016857712746, \ 0.0015296219296330, \ 0.9647016857712746, \ 0.0337686922990924, \ 0.0949242221588698, \ 0.3532796606132861, \ 0.5517961172278441, \ 0.3532796606132861, \ 0.5517961172278441, \ 0.0949242221588698, \ 0.0949242221588698, \ 0.3532796606132861, \ 0.5517961172278441, \ 0.3532796606132861, \ 0.5517961172278441, \ 0.0949242221588698, \ 0.0248817170289358, \ 0.0959262325920420, \ 0.8791920503790223, \ 0.0959262325920420, \ 0.8791920503790223, \ 0.0248817170289358, \ 0.0248817170289358, \ 0.0959262325920420, \ 0.8791920503790223, \ 0.0959262325920420, \ 0.8791920503790223, \ 0.0248817170289358, \ 0.1757029526928022, \ 0.4963305263335406, \ 0.3279665209736572, \ 0.4963305263335406, \ 0.3279665209736572, \ 0.1757029526928022, \ 0.1757029526928022, \ 0.4963305263335406, \ 0.3279665209736572, \ 0.4963305263335406, \ 0.3279665209736572, \ 0.1757029526928022, \ 0.0140786122092260, \ 0.2738501035919119, \ 0.7120712841988621, \ 0.2738501035919119, \ 0.7120712841988621, \ 0.0140786122092260, \ 0.0140786122092260, \ 0.2738501035919119, \ 0.7120712841988621, \ 0.2738501035919119, \ 0.7120712841988621, \ 0.0140786122092260, \ 0.0565171623200343, \ 0.3828844998895772, \ 0.5605983377903885, \ 0.3828844998895772, \ 0.5605983377903885, \ 0.0565171623200343, \ 0.0565171623200343, \ 0.3828844998895772, \ 0.5605983377903885, \ 0.3828844998895772, \ 0.5605983377903885, \ 0.0565171623200343, \ 0.0192412570201629, \ 0.0899365323953004, \ 0.8908222105845367, \ 0.0899365323953004, \ 0.8908222105845367, \ 0.0192412570201629, \ 0.0192412570201629, \ 0.0899365323953004, \ 0.8908222105845367, \ 0.0899365323953004, \ 0.8908222105845367, \ 0.0192412570201629, \ 0.0611370184542401, \ 0.2516074150213084, \ 0.6872555665244514, \ 0.2516074150213084, \ 0.6872555665244514, \ 0.0611370184542401, \ 0.0611370184542401, \ 0.2516074150213084, \ 0.6872555665244514, \ 0.2516074150213084, \ 0.6872555665244514, \ 0.0611370184542401, \ 0.0050182699561948, \ 0.1524244609819593, \ 0.8425572690618459, \ 0.1524244609819593, \ 0.8425572690618459, \ 0.0050182699561948, \ 0.0050182699561948, \ 0.1524244609819593, \ 0.8425572690618459, \ 0.1524244609819593, \ 0.8425572690618459, \ 0.0050182699561948, \ 0.1365910574355268, \ 0.2475011786624472, \ 0.6159077639020261, \ 0.2475011786624472, \ 0.6159077639020261, \ 0.1365910574355268, \ 0.1365910574355268, \ 0.2475011786624472, \ 0.6159077639020261, \ 0.2475011786624472, \ 0.6159077639020261, \ 0.1365910574355268, \ 0.0173543718109579, \ 0.7591852986956938, \ 0.2234603294933483, \ 0.7591852986956938, \ 0.2234603294933483, \ 0.0173543718109579, \ 0.0173543718109579, \ 0.7591852986956938, \ 0.2234603294933483, \ 0.7591852986956938, \ 0.2234603294933483, \ 0.0173543718109579 ] ) z = np.array ( [ \ 0.8403117926958672, \ 0.1596882073041329, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.8842725336194912, \ 0.8842725336194912, \ 0.8842725336194912, \ 0.1157274663805087, \ 0.1157274663805087, \ 0.1157274663805087, \ 0.6593583359504520, \ 0.6593583359504520, \ 0.6593583359504520, \ 0.3406416640495479, \ 0.3406416640495479, \ 0.3406416640495479, \ 0.8712606670369063, \ 0.8712606670369063, \ 0.8712606670369063, \ 0.1287393329630936, \ 0.1287393329630936, \ 0.1287393329630936, \ 0.2304229580101690, \ 0.2304229580101690, \ 0.2304229580101690, \ 0.7695770419898310, \ 0.7695770419898310, \ 0.7695770419898310, \ 0.6455038513606328, \ 0.6455038513606328, \ 0.6455038513606328, \ 0.3544961486393671, \ 0.3544961486393671, \ 0.3544961486393671, \ 0.8851421910096543, \ 0.8851421910096543, \ 0.8851421910096543, \ 0.1148578089903457, \ 0.1148578089903457, \ 0.1148578089903457, \ 0.9818829756657210, \ 0.9818829756657210, \ 0.9818829756657210, \ 0.0181170243342790, \ 0.0181170243342790, \ 0.0181170243342790, \ 0.9850118189004804, \ 0.9850118189004804, \ 0.9850118189004804, \ 0.0149881810995195, \ 0.0149881810995195, \ 0.0149881810995195, \ 0.9701315317214787, \ 0.9701315317214787, \ 0.9701315317214787, \ 0.0298684682785212, \ 0.0298684682785212, \ 0.0298684682785212, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.9387595239709540, \ 0.9387595239709540, \ 0.9387595239709540, \ 0.9387595239709540, \ 0.9387595239709540, \ 0.9387595239709540, \ 0.0612404760290460, \ 0.0612404760290460, \ 0.0612404760290460, \ 0.0612404760290460, \ 0.0612404760290460, \ 0.0612404760290460, \ 0.7620680922209941, \ 0.7620680922209941, \ 0.7620680922209941, \ 0.7620680922209941, \ 0.7620680922209941, \ 0.7620680922209941, \ 0.2379319077790059, \ 0.2379319077790059, \ 0.2379319077790059, \ 0.2379319077790059, \ 0.2379319077790059, \ 0.2379319077790059, \ 0.8273450356745347, \ 0.8273450356745347, \ 0.8273450356745347, \ 0.8273450356745347, \ 0.8273450356745347, \ 0.8273450356745347, \ 0.1726549643254653, \ 0.1726549643254653, \ 0.1726549643254653, \ 0.1726549643254653, \ 0.1726549643254653, \ 0.1726549643254653, \ 0.7401886228316812, \ 0.7401886228316812, \ 0.7401886228316812, \ 0.7401886228316812, \ 0.7401886228316812, \ 0.7401886228316812, \ 0.2598113771683188, \ 0.2598113771683188, \ 0.2598113771683188, \ 0.2598113771683188, \ 0.2598113771683188, \ 0.2598113771683188, \ 0.9210357305129615, \ 0.9210357305129615, \ 0.9210357305129615, \ 0.9210357305129615, \ 0.9210357305129615, \ 0.9210357305129615, \ 0.0789642694870386, \ 0.0789642694870386, \ 0.0789642694870386, \ 0.0789642694870386, \ 0.0789642694870386, \ 0.0789642694870386, \ 0.8233326926574609, \ 0.8233326926574609, \ 0.8233326926574609, \ 0.8233326926574609, \ 0.8233326926574609, \ 0.8233326926574609, \ 0.1766673073425390, \ 0.1766673073425390, \ 0.1766673073425390, \ 0.1766673073425390, \ 0.1766673073425390, \ 0.1766673073425390, \ 0.9478854137232893, \ 0.9478854137232893, \ 0.9478854137232893, \ 0.9478854137232893, \ 0.9478854137232893, \ 0.9478854137232893, \ 0.0521145862767107, \ 0.0521145862767107, \ 0.0521145862767107, \ 0.0521145862767107, \ 0.0521145862767107, \ 0.0521145862767107, \ 0.9916391512107949, \ 0.9916391512107949, \ 0.9916391512107949, \ 0.9916391512107949, \ 0.9916391512107949, \ 0.9916391512107949, \ 0.0083608487892050, \ 0.0083608487892050, \ 0.0083608487892050, \ 0.0083608487892050, \ 0.0083608487892050, \ 0.0083608487892050, \ 0.9898679390727328, \ 0.9898679390727328, \ 0.9898679390727328, \ 0.9898679390727328, \ 0.9898679390727328, \ 0.9898679390727328, \ 0.0101320609272672, \ 0.0101320609272672, \ 0.0101320609272672, \ 0.0101320609272672, \ 0.0101320609272672, \ 0.0101320609272672, \ 0.9305712292308971, \ 0.9305712292308971, \ 0.9305712292308971, \ 0.9305712292308971, \ 0.9305712292308971, \ 0.9305712292308971, \ 0.0694287707691029, \ 0.0694287707691029, \ 0.0694287707691029, \ 0.0694287707691029, \ 0.0694287707691029, \ 0.0694287707691029, \ 0.9758011913142881, \ 0.9758011913142881, \ 0.9758011913142881, \ 0.9758011913142881, \ 0.9758011913142881, \ 0.9758011913142881, \ 0.0241988086857118, \ 0.0241988086857118, \ 0.0241988086857118, \ 0.0241988086857118, \ 0.0241988086857118, \ 0.0241988086857118, \ 0.3036894659036268, \ 0.3036894659036268, \ 0.3036894659036268, \ 0.3036894659036268, \ 0.3036894659036268, \ 0.3036894659036268, \ 0.6963105340963731, \ 0.6963105340963731, \ 0.6963105340963731, \ 0.6963105340963731, \ 0.6963105340963731, \ 0.6963105340963731, \ 0.6138684175664644, \ 0.6138684175664644, \ 0.6138684175664644, \ 0.6138684175664644, \ 0.6138684175664644, \ 0.6138684175664644, \ 0.3861315824335356, \ 0.3861315824335356, \ 0.3861315824335356, \ 0.3861315824335356, \ 0.3861315824335356, \ 0.3861315824335356, \ 0.7885638236910046, \ 0.7885638236910046, \ 0.7885638236910046, \ 0.7885638236910046, \ 0.7885638236910046, \ 0.7885638236910046, \ 0.2114361763089954, \ 0.2114361763089954, \ 0.2114361763089954, \ 0.2114361763089954, \ 0.2114361763089954, \ 0.2114361763089954, \ 0.8514618541948253, \ 0.8514618541948253, \ 0.8514618541948253, \ 0.8514618541948253, \ 0.8514618541948253, \ 0.8514618541948253, \ 0.1485381458051747, \ 0.1485381458051747, \ 0.1485381458051747, \ 0.1485381458051747, \ 0.1485381458051747, \ 0.1485381458051747, \ 0.3722497735842279, \ 0.3722497735842279, \ 0.3722497735842279, \ 0.3722497735842279, \ 0.3722497735842279, \ 0.3722497735842279, \ 0.6277502264157721, \ 0.6277502264157721, \ 0.6277502264157721, \ 0.6277502264157721, \ 0.6277502264157721, \ 0.6277502264157721, \ 0.6329858368666343, \ 0.6329858368666343, \ 0.6329858368666343, \ 0.6329858368666343, \ 0.6329858368666343, \ 0.6329858368666343, \ 0.3670141631333657, \ 0.3670141631333657, \ 0.3670141631333657, \ 0.3670141631333657, \ 0.3670141631333657, \ 0.3670141631333657 ] ) w = np.array ( [ \ 0.0126082800938750, \ 0.0126082800938750, \ 0.0005804416586274, \ 0.0005804416586274, \ 0.0005804416586274, \ 0.0014515275277035, \ 0.0014515275277035, \ 0.0014515275277035, \ 0.0110392519457591, \ 0.0110392519457591, \ 0.0110392519457591, \ 0.0023896607884456, \ 0.0023896607884456, \ 0.0023896607884456, \ 0.0023896607884456, \ 0.0023896607884456, \ 0.0023896607884456, \ 0.0111852959227564, \ 0.0111852959227564, \ 0.0111852959227564, \ 0.0111852959227564, \ 0.0111852959227564, \ 0.0111852959227564, \ 0.0044527106261489, \ 0.0044527106261489, \ 0.0044527106261489, \ 0.0044527106261489, \ 0.0044527106261489, \ 0.0044527106261489, \ 0.0099310429205233, \ 0.0099310429205233, \ 0.0099310429205233, \ 0.0099310429205233, \ 0.0099310429205233, \ 0.0099310429205233, \ 0.0027983315098658, \ 0.0027983315098658, \ 0.0027983315098658, \ 0.0027983315098658, \ 0.0027983315098658, \ 0.0027983315098658, \ 0.0045950791440008, \ 0.0045950791440008, \ 0.0045950791440008, \ 0.0045950791440008, \ 0.0045950791440008, \ 0.0045950791440008, \ 0.0026497419827358, \ 0.0026497419827358, \ 0.0026497419827358, \ 0.0026497419827358, \ 0.0026497419827358, \ 0.0026497419827358, \ 0.0023791749450044, \ 0.0023791749450044, \ 0.0023791749450044, \ 0.0023791749450044, \ 0.0023791749450044, \ 0.0023791749450044, \ 0.0012902115379941, \ 0.0012902115379941, \ 0.0012902115379941, \ 0.0012902115379941, \ 0.0012902115379941, \ 0.0012902115379941, \ 0.0071749704457678, \ 0.0071749704457678, \ 0.0071749704457678, \ 0.0071749704457678, \ 0.0071749704457678, \ 0.0071749704457678, \ 0.0039041969129065, \ 0.0039041969129065, \ 0.0039041969129065, \ 0.0039041969129065, \ 0.0039041969129065, \ 0.0039041969129065, \ 0.0058968479213494, \ 0.0058968479213494, \ 0.0058968479213494, \ 0.0058968479213494, \ 0.0058968479213494, \ 0.0058968479213494, \ 0.0024349402323106, \ 0.0024349402323106, \ 0.0024349402323106, \ 0.0024349402323106, \ 0.0024349402323106, \ 0.0024349402323106, \ 0.0024349402323106, \ 0.0024349402323106, \ 0.0024349402323106, \ 0.0024349402323106, \ 0.0024349402323106, \ 0.0024349402323106, \ 0.0007867926281754, \ 0.0007867926281754, \ 0.0007867926281754, \ 0.0007867926281754, \ 0.0007867926281754, \ 0.0007867926281754, \ 0.0007867926281754, \ 0.0007867926281754, \ 0.0007867926281754, \ 0.0007867926281754, \ 0.0007867926281754, \ 0.0007867926281754, \ 0.0019733458950933, \ 0.0019733458950933, \ 0.0019733458950933, \ 0.0019733458950933, \ 0.0019733458950933, \ 0.0019733458950933, \ 0.0019733458950933, \ 0.0019733458950933, \ 0.0019733458950933, \ 0.0019733458950933, \ 0.0019733458950933, \ 0.0019733458950933, \ 0.0039028044764867, \ 0.0039028044764867, \ 0.0039028044764867, \ 0.0039028044764867, \ 0.0039028044764867, \ 0.0039028044764867, \ 0.0039028044764867, \ 0.0039028044764867, \ 0.0039028044764867, \ 0.0039028044764867, \ 0.0039028044764867, \ 0.0039028044764867, \ 0.0028651948489933, \ 0.0028651948489933, \ 0.0028651948489933, \ 0.0028651948489933, \ 0.0028651948489933, \ 0.0028651948489933, \ 0.0028651948489933, \ 0.0028651948489933, \ 0.0028651948489933, \ 0.0028651948489933, \ 0.0028651948489933, \ 0.0028651948489933, \ 0.0042658629737291, \ 0.0042658629737291, \ 0.0042658629737291, \ 0.0042658629737291, \ 0.0042658629737291, \ 0.0042658629737291, \ 0.0042658629737291, \ 0.0042658629737291, \ 0.0042658629737291, \ 0.0042658629737291, \ 0.0042658629737291, \ 0.0042658629737291, \ 0.0002763846307136, \ 0.0002763846307136, \ 0.0002763846307136, \ 0.0002763846307136, \ 0.0002763846307136, \ 0.0002763846307136, \ 0.0002763846307136, \ 0.0002763846307136, \ 0.0002763846307136, \ 0.0002763846307136, \ 0.0002763846307136, \ 0.0002763846307136, \ 0.0015947204945141, \ 0.0015947204945141, \ 0.0015947204945141, \ 0.0015947204945141, \ 0.0015947204945141, \ 0.0015947204945141, \ 0.0015947204945141, \ 0.0015947204945141, \ 0.0015947204945141, \ 0.0015947204945141, \ 0.0015947204945141, \ 0.0015947204945141, \ 0.0005967018601525, \ 0.0005967018601525, \ 0.0005967018601525, \ 0.0005967018601525, \ 0.0005967018601525, \ 0.0005967018601525, \ 0.0005967018601525, \ 0.0005967018601525, \ 0.0005967018601525, \ 0.0005967018601525, \ 0.0005967018601525, \ 0.0005967018601525, \ 0.0050890108573681, \ 0.0050890108573681, \ 0.0050890108573681, \ 0.0050890108573681, \ 0.0050890108573681, \ 0.0050890108573681, \ 0.0050890108573681, \ 0.0050890108573681, \ 0.0050890108573681, \ 0.0050890108573681, \ 0.0050890108573681, \ 0.0050890108573681, \ 0.0010232527285653, \ 0.0010232527285653, \ 0.0010232527285653, \ 0.0010232527285653, \ 0.0010232527285653, \ 0.0010232527285653, \ 0.0010232527285653, \ 0.0010232527285653, \ 0.0010232527285653, \ 0.0010232527285653, \ 0.0010232527285653, \ 0.0010232527285653, \ 0.0048336118379361, \ 0.0048336118379361, \ 0.0048336118379361, \ 0.0048336118379361, \ 0.0048336118379361, \ 0.0048336118379361, \ 0.0048336118379361, \ 0.0048336118379361, \ 0.0048336118379361, \ 0.0048336118379361, \ 0.0048336118379361, \ 0.0048336118379361, \ 0.0022565704494986, \ 0.0022565704494986, \ 0.0022565704494986, \ 0.0022565704494986, \ 0.0022565704494986, \ 0.0022565704494986, \ 0.0022565704494986, \ 0.0022565704494986, \ 0.0022565704494986, \ 0.0022565704494986, \ 0.0022565704494986, \ 0.0022565704494986, \ 0.0043942174418955, \ 0.0043942174418955, \ 0.0043942174418955, \ 0.0043942174418955, \ 0.0043942174418955, \ 0.0043942174418955, \ 0.0043942174418955, \ 0.0043942174418955, \ 0.0043942174418955, \ 0.0043942174418955, \ 0.0043942174418955, \ 0.0043942174418955, \ 0.0014412804208529, \ 0.0014412804208529, \ 0.0014412804208529, \ 0.0014412804208529, \ 0.0014412804208529, \ 0.0014412804208529, \ 0.0014412804208529, \ 0.0014412804208529, \ 0.0014412804208529, \ 0.0014412804208529, \ 0.0014412804208529, \ 0.0014412804208529, \ 0.0076389038664824, \ 0.0076389038664824, \ 0.0076389038664824, \ 0.0076389038664824, \ 0.0076389038664824, \ 0.0076389038664824, \ 0.0076389038664824, \ 0.0076389038664824, \ 0.0076389038664824, \ 0.0076389038664824, \ 0.0076389038664824, \ 0.0076389038664824, \ 0.0032669200631480, \ 0.0032669200631480, \ 0.0032669200631480, \ 0.0032669200631480, \ 0.0032669200631480, \ 0.0032669200631480, \ 0.0032669200631480, \ 0.0032669200631480, \ 0.0032669200631480, \ 0.0032669200631480, \ 0.0032669200631480, \ 0.0032669200631480 ] ) return x, y, z, w def rule17 ( ): #*****************************************************************************80 # ## rule17() returns the prism rule of precision 17. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4665556153120432, \ 0.4665556153120432, \ 0.0668887693759136, \ 0.4665556153120432, \ 0.4665556153120432, \ 0.0668887693759136, \ 0.4923809674098066, \ 0.4923809674098066, \ 0.0152380651803867, \ 0.4923809674098066, \ 0.4923809674098066, \ 0.0152380651803867, \ 0.4466963397682847, \ 0.4466963397682847, \ 0.1066073204634306, \ 0.4466963397682847, \ 0.4466963397682847, \ 0.1066073204634306, \ 0.2058899494758731, \ 0.2058899494758731, \ 0.5882201010482538, \ 0.2058899494758731, \ 0.2058899494758731, \ 0.5882201010482538, \ 0.2305779360323416, \ 0.2305779360323416, \ 0.5388441279353168, \ 0.2305779360323416, \ 0.2305779360323416, \ 0.5388441279353168, \ 0.0070726502342596, \ 0.0070726502342596, \ 0.9858546995314807, \ 0.0070726502342596, \ 0.0070726502342596, \ 0.9858546995314807, \ 0.0524838545504923, \ 0.0524838545504923, \ 0.8950322908990154, \ 0.0524838545504923, \ 0.0524838545504923, \ 0.8950322908990154, \ 0.2780743897936196, \ 0.2780743897936196, \ 0.4438512204127608, \ 0.2780743897936196, \ 0.2780743897936196, \ 0.4438512204127608, \ 0.3893995813144771, \ 0.3893995813144771, \ 0.2212008373710459, \ 0.3893995813144771, \ 0.3893995813144771, \ 0.2212008373710459, \ 0.1132999955833483, \ 0.1132999955833483, \ 0.7734000088333033, \ 0.1132999955833483, \ 0.1132999955833483, \ 0.7734000088333033, \ 0.4687774749283292, \ 0.4687774749283292, \ 0.0624450501433417, \ 0.4687774749283292, \ 0.4687774749283292, \ 0.0624450501433417, \ 0.4942337889964747, \ 0.4942337889964747, \ 0.0115324220070506, \ 0.4942337889964747, \ 0.4942337889964747, \ 0.0115324220070506, \ 0.2906999098818323, \ 0.2906999098818323, \ 0.4186001802363355, \ 0.2906999098818323, \ 0.2906999098818323, \ 0.4186001802363355, \ 0.0937693039918497, \ 0.0937693039918497, \ 0.8124613920163005, \ 0.0937693039918497, \ 0.0937693039918497, \ 0.8124613920163005, \ 0.1632812980104028, \ 0.1632812980104028, \ 0.6734374039791945, \ 0.1632812980104028, \ 0.1632812980104028, \ 0.6734374039791945, \ 0.1774626418569745, \ 0.1774626418569745, \ 0.6450747162860511, \ 0.1774626418569745, \ 0.1774626418569745, \ 0.6450747162860511, \ 0.3930786561208929, \ 0.3930786561208929, \ 0.2138426877582142, \ 0.3930786561208929, \ 0.3930786561208929, \ 0.2138426877582142, \ 0.2711153947647855, \ 0.2711153947647855, \ 0.4577692104704291, \ 0.2711153947647855, \ 0.2711153947647855, \ 0.4577692104704291, \ 0.0243062210015648, \ 0.0243062210015648, \ 0.9513875579968704, \ 0.0243062210015648, \ 0.0243062210015648, \ 0.9513875579968704, \ 0.1273859447314594, \ 0.2765687474993688, \ 0.1273859447314594, \ 0.5960453077691718, \ 0.2765687474993688, \ 0.5960453077691718, \ 0.2450510833431427, \ 0.0852468501828055, \ 0.2450510833431427, \ 0.6697020664740517, \ 0.0852468501828055, \ 0.6697020664740517, \ 0.2450510833431427, \ 0.0852468501828055, \ 0.2450510833431427, \ 0.6697020664740517, \ 0.0852468501828055, \ 0.6697020664740517, \ 0.3192911229101512, \ 0.0276449852112552, \ 0.3192911229101512, \ 0.6530638918785936, \ 0.0276449852112552, \ 0.6530638918785936, \ 0.3192911229101512, \ 0.0276449852112552, \ 0.3192911229101512, \ 0.6530638918785936, \ 0.0276449852112552, \ 0.6530638918785936, \ 0.4577741357714497, \ 0.1385428807092584, \ 0.4577741357714497, \ 0.4036829835192919, \ 0.1385428807092584, \ 0.4036829835192919, \ 0.4577741357714497, \ 0.1385428807092584, \ 0.4577741357714497, \ 0.4036829835192919, \ 0.1385428807092584, \ 0.4036829835192919, \ 0.2421105196372634, \ 0.0124568391026713, \ 0.2421105196372634, \ 0.7454326412600653, \ 0.0124568391026713, \ 0.7454326412600653, \ 0.2421105196372634, \ 0.0124568391026713, \ 0.2421105196372634, \ 0.7454326412600653, \ 0.0124568391026713, \ 0.7454326412600653, \ 0.7269945541994681, \ 0.0888093024099239, \ 0.7269945541994681, \ 0.1841961433906080, \ 0.0888093024099239, \ 0.1841961433906080, \ 0.7269945541994681, \ 0.0888093024099239, \ 0.7269945541994681, \ 0.1841961433906080, \ 0.0888093024099239, \ 0.1841961433906080, \ 0.4870517086755036, \ 0.1725779658472295, \ 0.4870517086755036, \ 0.3403703254772670, \ 0.1725779658472295, \ 0.3403703254772670, \ 0.4870517086755036, \ 0.1725779658472295, \ 0.4870517086755036, \ 0.3403703254772670, \ 0.1725779658472295, \ 0.3403703254772670, \ 0.1582123188291182, \ 0.3108975273704328, \ 0.1582123188291182, \ 0.5308901538004490, \ 0.3108975273704328, \ 0.5308901538004490, \ 0.1582123188291182, \ 0.3108975273704328, \ 0.1582123188291182, \ 0.5308901538004490, \ 0.3108975273704328, \ 0.5308901538004490, \ 0.5973833093262283, \ 0.0432967856355362, \ 0.5973833093262283, \ 0.3593199050382355, \ 0.0432967856355362, \ 0.3593199050382355, \ 0.5973833093262283, \ 0.0432967856355362, \ 0.5973833093262283, \ 0.3593199050382355, \ 0.0432967856355362, \ 0.3593199050382355, \ 0.1111929938785087, \ 0.0368260832055792, \ 0.1111929938785087, \ 0.8519809229159122, \ 0.0368260832055792, \ 0.8519809229159122, \ 0.1111929938785087, \ 0.0368260832055792, \ 0.1111929938785087, \ 0.8519809229159122, \ 0.0368260832055792, \ 0.8519809229159122, \ 0.0387865006290567, \ 0.0070454102530681, \ 0.0387865006290567, \ 0.9541680891178752, \ 0.0070454102530681, \ 0.9541680891178752, \ 0.0387865006290567, \ 0.0070454102530681, \ 0.0387865006290567, \ 0.9541680891178752, \ 0.0070454102530681, \ 0.9541680891178752, \ 0.1417441449927015, \ 0.0066561350058640, \ 0.1417441449927015, \ 0.8515997200014345, \ 0.0066561350058640, \ 0.8515997200014345, \ 0.1417441449927015, \ 0.0066561350058640, \ 0.1417441449927015, \ 0.8515997200014345, \ 0.0066561350058640, \ 0.8515997200014345, \ 0.1343472140817544, \ 0.0269814183822196, \ 0.1343472140817544, \ 0.8386713675360260, \ 0.0269814183822196, \ 0.8386713675360260, \ 0.1343472140817544, \ 0.0269814183822196, \ 0.1343472140817544, \ 0.8386713675360260, \ 0.0269814183822196, \ 0.8386713675360260, \ 0.3195031824862086, \ 0.0813440561372205, \ 0.3195031824862086, \ 0.5991527613765709, \ 0.0813440561372205, \ 0.5991527613765709, \ 0.3195031824862086, \ 0.0813440561372205, \ 0.3195031824862086, \ 0.5991527613765709, \ 0.0813440561372205, \ 0.5991527613765709, \ 0.0826308438588340, \ 0.0102635465039764, \ 0.0826308438588340, \ 0.9071056096371896, \ 0.0102635465039764, \ 0.9071056096371896, \ 0.0826308438588340, \ 0.0102635465039764, \ 0.0826308438588340, \ 0.9071056096371896, \ 0.0102635465039764, \ 0.9071056096371896, \ 0.3827807429336583, \ 0.0067490134150570, \ 0.3827807429336583, \ 0.6104702436512848, \ 0.0067490134150570, \ 0.6104702436512848, \ 0.3827807429336583, \ 0.0067490134150570, \ 0.3827807429336583, \ 0.6104702436512848, \ 0.0067490134150570, \ 0.6104702436512848, \ 0.2446543644502634, \ 0.0056501992086932, \ 0.2446543644502634, \ 0.7496954363410434, \ 0.0056501992086932, \ 0.7496954363410434, \ 0.2446543644502634, \ 0.0056501992086932, \ 0.2446543644502634, \ 0.7496954363410434, \ 0.0056501992086932, \ 0.7496954363410434, \ 0.0263707136798594, \ 0.6264598409479015, \ 0.0263707136798594, \ 0.3471694453722390, \ 0.6264598409479015, \ 0.3471694453722390, \ 0.0263707136798594, \ 0.6264598409479015, \ 0.0263707136798594, \ 0.3471694453722390, \ 0.6264598409479015, \ 0.3471694453722390, \ 0.0434074542015924, \ 0.1947680793017931, \ 0.0434074542015924, \ 0.7618244664966145, \ 0.1947680793017931, \ 0.7618244664966145, \ 0.0434074542015924, \ 0.1947680793017931, \ 0.0434074542015924, \ 0.7618244664966145, \ 0.1947680793017931, \ 0.7618244664966145 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4665556153120432, \ 0.0668887693759136, \ 0.4665556153120432, \ 0.4665556153120432, \ 0.0668887693759136, \ 0.4665556153120432, \ 0.4923809674098066, \ 0.0152380651803867, \ 0.4923809674098066, \ 0.4923809674098066, \ 0.0152380651803867, \ 0.4923809674098066, \ 0.4466963397682847, \ 0.1066073204634306, \ 0.4466963397682847, \ 0.4466963397682847, \ 0.1066073204634306, \ 0.4466963397682847, \ 0.2058899494758731, \ 0.5882201010482538, \ 0.2058899494758731, \ 0.2058899494758731, \ 0.5882201010482538, \ 0.2058899494758731, \ 0.2305779360323416, \ 0.5388441279353168, \ 0.2305779360323416, \ 0.2305779360323416, \ 0.5388441279353168, \ 0.2305779360323416, \ 0.0070726502342596, \ 0.9858546995314807, \ 0.0070726502342596, \ 0.0070726502342596, \ 0.9858546995314807, \ 0.0070726502342596, \ 0.0524838545504923, \ 0.8950322908990154, \ 0.0524838545504923, \ 0.0524838545504923, \ 0.8950322908990154, \ 0.0524838545504923, \ 0.2780743897936196, \ 0.4438512204127608, \ 0.2780743897936196, \ 0.2780743897936196, \ 0.4438512204127608, \ 0.2780743897936196, \ 0.3893995813144771, \ 0.2212008373710459, \ 0.3893995813144771, \ 0.3893995813144771, \ 0.2212008373710459, \ 0.3893995813144771, \ 0.1132999955833483, \ 0.7734000088333033, \ 0.1132999955833483, \ 0.1132999955833483, \ 0.7734000088333033, \ 0.1132999955833483, \ 0.4687774749283292, \ 0.0624450501433417, \ 0.4687774749283292, \ 0.4687774749283292, \ 0.0624450501433417, \ 0.4687774749283292, \ 0.4942337889964747, \ 0.0115324220070506, \ 0.4942337889964747, \ 0.4942337889964747, \ 0.0115324220070506, \ 0.4942337889964747, \ 0.2906999098818323, \ 0.4186001802363355, \ 0.2906999098818323, \ 0.2906999098818323, \ 0.4186001802363355, \ 0.2906999098818323, \ 0.0937693039918497, \ 0.8124613920163005, \ 0.0937693039918497, \ 0.0937693039918497, \ 0.8124613920163005, \ 0.0937693039918497, \ 0.1632812980104028, \ 0.6734374039791945, \ 0.1632812980104028, \ 0.1632812980104028, \ 0.6734374039791945, \ 0.1632812980104028, \ 0.1774626418569745, \ 0.6450747162860511, \ 0.1774626418569745, \ 0.1774626418569745, \ 0.6450747162860511, \ 0.1774626418569745, \ 0.3930786561208929, \ 0.2138426877582142, \ 0.3930786561208929, \ 0.3930786561208929, \ 0.2138426877582142, \ 0.3930786561208929, \ 0.2711153947647855, \ 0.4577692104704291, \ 0.2711153947647855, \ 0.2711153947647855, \ 0.4577692104704291, \ 0.2711153947647855, \ 0.0243062210015648, \ 0.9513875579968704, \ 0.0243062210015648, \ 0.0243062210015648, \ 0.9513875579968704, \ 0.0243062210015648, \ 0.2765687474993688, \ 0.1273859447314594, \ 0.5960453077691718, \ 0.1273859447314594, \ 0.5960453077691718, \ 0.2765687474993688, \ 0.0852468501828055, \ 0.2450510833431427, \ 0.6697020664740517, \ 0.2450510833431427, \ 0.6697020664740517, \ 0.0852468501828055, \ 0.0852468501828055, \ 0.2450510833431427, \ 0.6697020664740517, \ 0.2450510833431427, \ 0.6697020664740517, \ 0.0852468501828055, \ 0.0276449852112552, \ 0.3192911229101512, \ 0.6530638918785936, \ 0.3192911229101512, \ 0.6530638918785936, \ 0.0276449852112552, \ 0.0276449852112552, \ 0.3192911229101512, \ 0.6530638918785936, \ 0.3192911229101512, \ 0.6530638918785936, \ 0.0276449852112552, \ 0.1385428807092584, \ 0.4577741357714497, \ 0.4036829835192919, \ 0.4577741357714497, \ 0.4036829835192919, \ 0.1385428807092584, \ 0.1385428807092584, \ 0.4577741357714497, \ 0.4036829835192919, \ 0.4577741357714497, \ 0.4036829835192919, \ 0.1385428807092584, \ 0.0124568391026713, \ 0.2421105196372634, \ 0.7454326412600653, \ 0.2421105196372634, \ 0.7454326412600653, \ 0.0124568391026713, \ 0.0124568391026713, \ 0.2421105196372634, \ 0.7454326412600653, \ 0.2421105196372634, \ 0.7454326412600653, \ 0.0124568391026713, \ 0.0888093024099239, \ 0.7269945541994681, \ 0.1841961433906080, \ 0.7269945541994681, \ 0.1841961433906080, \ 0.0888093024099239, \ 0.0888093024099239, \ 0.7269945541994681, \ 0.1841961433906080, \ 0.7269945541994681, \ 0.1841961433906080, \ 0.0888093024099239, \ 0.1725779658472295, \ 0.4870517086755036, \ 0.3403703254772670, \ 0.4870517086755036, \ 0.3403703254772670, \ 0.1725779658472295, \ 0.1725779658472295, \ 0.4870517086755036, \ 0.3403703254772670, \ 0.4870517086755036, \ 0.3403703254772670, \ 0.1725779658472295, \ 0.3108975273704328, \ 0.1582123188291182, \ 0.5308901538004490, \ 0.1582123188291182, \ 0.5308901538004490, \ 0.3108975273704328, \ 0.3108975273704328, \ 0.1582123188291182, \ 0.5308901538004490, \ 0.1582123188291182, \ 0.5308901538004490, \ 0.3108975273704328, \ 0.0432967856355362, \ 0.5973833093262283, \ 0.3593199050382355, \ 0.5973833093262283, \ 0.3593199050382355, \ 0.0432967856355362, \ 0.0432967856355362, \ 0.5973833093262283, \ 0.3593199050382355, \ 0.5973833093262283, \ 0.3593199050382355, \ 0.0432967856355362, \ 0.0368260832055792, \ 0.1111929938785087, \ 0.8519809229159122, \ 0.1111929938785087, \ 0.8519809229159122, \ 0.0368260832055792, \ 0.0368260832055792, \ 0.1111929938785087, \ 0.8519809229159122, \ 0.1111929938785087, \ 0.8519809229159122, \ 0.0368260832055792, \ 0.0070454102530681, \ 0.0387865006290567, \ 0.9541680891178752, \ 0.0387865006290567, \ 0.9541680891178752, \ 0.0070454102530681, \ 0.0070454102530681, \ 0.0387865006290567, \ 0.9541680891178752, \ 0.0387865006290567, \ 0.9541680891178752, \ 0.0070454102530681, \ 0.0066561350058640, \ 0.1417441449927015, \ 0.8515997200014345, \ 0.1417441449927015, \ 0.8515997200014345, \ 0.0066561350058640, \ 0.0066561350058640, \ 0.1417441449927015, \ 0.8515997200014345, \ 0.1417441449927015, \ 0.8515997200014345, \ 0.0066561350058640, \ 0.0269814183822196, \ 0.1343472140817544, \ 0.8386713675360260, \ 0.1343472140817544, \ 0.8386713675360260, \ 0.0269814183822196, \ 0.0269814183822196, \ 0.1343472140817544, \ 0.8386713675360260, \ 0.1343472140817544, \ 0.8386713675360260, \ 0.0269814183822196, \ 0.0813440561372205, \ 0.3195031824862086, \ 0.5991527613765709, \ 0.3195031824862086, \ 0.5991527613765709, \ 0.0813440561372205, \ 0.0813440561372205, \ 0.3195031824862086, \ 0.5991527613765709, \ 0.3195031824862086, \ 0.5991527613765709, \ 0.0813440561372205, \ 0.0102635465039764, \ 0.0826308438588340, \ 0.9071056096371896, \ 0.0826308438588340, \ 0.9071056096371896, \ 0.0102635465039764, \ 0.0102635465039764, \ 0.0826308438588340, \ 0.9071056096371896, \ 0.0826308438588340, \ 0.9071056096371896, \ 0.0102635465039764, \ 0.0067490134150570, \ 0.3827807429336583, \ 0.6104702436512848, \ 0.3827807429336583, \ 0.6104702436512848, \ 0.0067490134150570, \ 0.0067490134150570, \ 0.3827807429336583, \ 0.6104702436512848, \ 0.3827807429336583, \ 0.6104702436512848, \ 0.0067490134150570, \ 0.0056501992086932, \ 0.2446543644502634, \ 0.7496954363410434, \ 0.2446543644502634, \ 0.7496954363410434, \ 0.0056501992086932, \ 0.0056501992086932, \ 0.2446543644502634, \ 0.7496954363410434, \ 0.2446543644502634, \ 0.7496954363410434, \ 0.0056501992086932, \ 0.6264598409479015, \ 0.0263707136798594, \ 0.3471694453722390, \ 0.0263707136798594, \ 0.3471694453722390, \ 0.6264598409479015, \ 0.6264598409479015, \ 0.0263707136798594, \ 0.3471694453722390, \ 0.0263707136798594, \ 0.3471694453722390, \ 0.6264598409479015, \ 0.1947680793017931, \ 0.0434074542015924, \ 0.7618244664966145, \ 0.0434074542015924, \ 0.7618244664966145, \ 0.1947680793017931, \ 0.1947680793017931, \ 0.0434074542015924, \ 0.7618244664966145, \ 0.0434074542015924, \ 0.7618244664966145, \ 0.1947680793017931 ] ) z = np.array ( [ \ 0.8666490488529237, \ 0.1333509511470762, \ 0.8703524297116281, \ 0.8703524297116281, \ 0.8703524297116281, \ 0.1296475702883719, \ 0.1296475702883719, \ 0.1296475702883719, \ 0.5962892344246855, \ 0.5962892344246855, \ 0.5962892344246855, \ 0.4037107655753145, \ 0.4037107655753145, \ 0.4037107655753145, \ 0.9862457308604132, \ 0.9862457308604132, \ 0.9862457308604132, \ 0.0137542691395868, \ 0.0137542691395868, \ 0.0137542691395868, \ 0.6193405024389006, \ 0.6193405024389006, \ 0.6193405024389006, \ 0.3806594975610994, \ 0.3806594975610994, \ 0.3806594975610994, \ 0.6506114926978297, \ 0.6506114926978297, \ 0.6506114926978297, \ 0.3493885073021704, \ 0.3493885073021704, \ 0.3493885073021704, \ 0.8258627018814777, \ 0.8258627018814777, \ 0.8258627018814777, \ 0.1741372981185223, \ 0.1741372981185223, \ 0.1741372981185223, \ 0.2477808238551293, \ 0.2477808238551293, \ 0.2477808238551293, \ 0.7522191761448707, \ 0.7522191761448707, \ 0.7522191761448707, \ 0.2921358009316097, \ 0.2921358009316097, \ 0.2921358009316097, \ 0.7078641990683903, \ 0.7078641990683903, \ 0.7078641990683903, \ 0.0981873258283363, \ 0.0981873258283363, \ 0.0981873258283363, \ 0.9018126741716637, \ 0.9018126741716637, \ 0.9018126741716637, \ 0.7125198639059245, \ 0.7125198639059245, \ 0.7125198639059245, \ 0.2874801360940755, \ 0.2874801360940755, \ 0.2874801360940755, \ 0.3516395946331653, \ 0.3516395946331653, \ 0.3516395946331653, \ 0.6483604053668347, \ 0.6483604053668347, \ 0.6483604053668347, \ 0.9459742145088597, \ 0.9459742145088597, \ 0.9459742145088597, \ 0.0540257854911404, \ 0.0540257854911404, \ 0.0540257854911404, \ 0.5658612299497008, \ 0.5658612299497008, \ 0.5658612299497008, \ 0.4341387700502993, \ 0.4341387700502993, \ 0.4341387700502993, \ 0.9273724754855923, \ 0.9273724754855923, \ 0.9273724754855923, \ 0.0726275245144077, \ 0.0726275245144077, \ 0.0726275245144077, \ 0.9989623565279326, \ 0.9989623565279326, \ 0.9989623565279326, \ 0.0010376434720673, \ 0.0010376434720673, \ 0.0010376434720673, \ 0.1604307070751930, \ 0.1604307070751930, \ 0.1604307070751930, \ 0.8395692929248070, \ 0.8395692929248070, \ 0.8395692929248070, \ 0.6345291354152611, \ 0.6345291354152611, \ 0.6345291354152611, \ 0.3654708645847389, \ 0.3654708645847389, \ 0.3654708645847389, \ 0.9811993443803730, \ 0.9811993443803730, \ 0.9811993443803730, \ 0.0188006556196270, \ 0.0188006556196270, \ 0.0188006556196270, \ 0.9712793580501632, \ 0.9712793580501632, \ 0.9712793580501632, \ 0.0287206419498368, \ 0.0287206419498368, \ 0.0287206419498368, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.9553022983267381, \ 0.9553022983267381, \ 0.9553022983267381, \ 0.9553022983267381, \ 0.9553022983267381, \ 0.9553022983267381, \ 0.0446977016732619, \ 0.0446977016732619, \ 0.0446977016732619, \ 0.0446977016732619, \ 0.0446977016732619, \ 0.0446977016732619, \ 0.5287768623754378, \ 0.5287768623754378, \ 0.5287768623754378, \ 0.5287768623754378, \ 0.5287768623754378, \ 0.5287768623754378, \ 0.4712231376245622, \ 0.4712231376245622, \ 0.4712231376245622, \ 0.4712231376245622, \ 0.4712231376245622, \ 0.4712231376245622, \ 0.4538216112941401, \ 0.4538216112941401, \ 0.4538216112941401, \ 0.4538216112941401, \ 0.4538216112941401, \ 0.4538216112941401, \ 0.5461783887058599, \ 0.5461783887058599, \ 0.5461783887058599, \ 0.5461783887058599, \ 0.5461783887058599, \ 0.5461783887058599, \ 0.3624591754389214, \ 0.3624591754389214, \ 0.3624591754389214, \ 0.3624591754389214, \ 0.3624591754389214, \ 0.3624591754389214, \ 0.6375408245610786, \ 0.6375408245610786, \ 0.6375408245610786, \ 0.6375408245610786, \ 0.6375408245610786, \ 0.6375408245610786, \ 0.6176517353683332, \ 0.6176517353683332, \ 0.6176517353683332, \ 0.6176517353683332, \ 0.6176517353683332, \ 0.6176517353683332, \ 0.3823482646316668, \ 0.3823482646316668, \ 0.3823482646316668, \ 0.3823482646316668, \ 0.3823482646316668, \ 0.3823482646316668, \ 0.2130333812420883, \ 0.2130333812420883, \ 0.2130333812420883, \ 0.2130333812420883, \ 0.2130333812420883, \ 0.2130333812420883, \ 0.7869666187579117, \ 0.7869666187579117, \ 0.7869666187579117, \ 0.7869666187579117, \ 0.7869666187579117, \ 0.7869666187579117, \ 0.9281160730015593, \ 0.9281160730015593, \ 0.9281160730015593, \ 0.9281160730015593, \ 0.9281160730015593, \ 0.9281160730015593, \ 0.0718839269984407, \ 0.0718839269984407, \ 0.0718839269984407, \ 0.0718839269984407, \ 0.0718839269984407, \ 0.0718839269984407, \ 0.8760469217983575, \ 0.8760469217983575, \ 0.8760469217983575, \ 0.8760469217983575, \ 0.8760469217983575, \ 0.8760469217983575, \ 0.1239530782016424, \ 0.1239530782016424, \ 0.1239530782016424, \ 0.1239530782016424, \ 0.1239530782016424, \ 0.1239530782016424, \ 0.5105770631364416, \ 0.5105770631364416, \ 0.5105770631364416, \ 0.5105770631364416, \ 0.5105770631364416, \ 0.5105770631364416, \ 0.4894229368635584, \ 0.4894229368635584, \ 0.4894229368635584, \ 0.4894229368635584, \ 0.4894229368635584, \ 0.4894229368635584, \ 0.6124742043718856, \ 0.6124742043718856, \ 0.6124742043718856, \ 0.6124742043718856, \ 0.6124742043718856, \ 0.6124742043718856, \ 0.3875257956281143, \ 0.3875257956281143, \ 0.3875257956281143, \ 0.3875257956281143, \ 0.3875257956281143, \ 0.3875257956281143, \ 0.6912097488491545, \ 0.6912097488491545, \ 0.6912097488491545, \ 0.6912097488491545, \ 0.6912097488491545, \ 0.6912097488491545, \ 0.3087902511508455, \ 0.3087902511508455, \ 0.3087902511508455, \ 0.3087902511508455, \ 0.3087902511508455, \ 0.3087902511508455, \ 0.9870924037489929, \ 0.9870924037489929, \ 0.9870924037489929, \ 0.9870924037489929, \ 0.9870924037489929, \ 0.9870924037489929, \ 0.0129075962510071, \ 0.0129075962510071, \ 0.0129075962510071, \ 0.0129075962510071, \ 0.0129075962510071, \ 0.0129075962510071, \ 0.7190879938703654, \ 0.7190879938703654, \ 0.7190879938703654, \ 0.7190879938703654, \ 0.7190879938703654, \ 0.7190879938703654, \ 0.2809120061296346, \ 0.2809120061296346, \ 0.2809120061296346, \ 0.2809120061296346, \ 0.2809120061296346, \ 0.2809120061296346, \ 0.8842315513280089, \ 0.8842315513280089, \ 0.8842315513280089, \ 0.8842315513280089, \ 0.8842315513280089, \ 0.8842315513280089, \ 0.1157684486719911, \ 0.1157684486719911, \ 0.1157684486719911, \ 0.1157684486719911, \ 0.1157684486719911, \ 0.1157684486719911, \ 0.7732864296384829, \ 0.7732864296384829, \ 0.7732864296384829, \ 0.7732864296384829, \ 0.7732864296384829, \ 0.7732864296384829, \ 0.2267135703615171, \ 0.2267135703615171, \ 0.2267135703615171, \ 0.2267135703615171, \ 0.2267135703615171, \ 0.2267135703615171, \ 0.9276833503490310, \ 0.9276833503490310, \ 0.9276833503490310, \ 0.9276833503490310, \ 0.9276833503490310, \ 0.9276833503490310, \ 0.0723166496509690, \ 0.0723166496509690, \ 0.0723166496509690, \ 0.0723166496509690, \ 0.0723166496509690, \ 0.0723166496509690, \ 0.9877761641032086, \ 0.9877761641032086, \ 0.9877761641032086, \ 0.9877761641032086, \ 0.9877761641032086, \ 0.9877761641032086, \ 0.0122238358967914, \ 0.0122238358967914, \ 0.0122238358967914, \ 0.0122238358967914, \ 0.0122238358967914, \ 0.0122238358967914, \ 0.8232871808501311, \ 0.8232871808501311, \ 0.8232871808501311, \ 0.8232871808501311, \ 0.8232871808501311, \ 0.8232871808501311, \ 0.1767128191498689, \ 0.1767128191498689, \ 0.1767128191498689, \ 0.1767128191498689, \ 0.1767128191498689, \ 0.1767128191498689 ] ) w = np.array ( [ \ 0.0075216525929737, \ 0.0075216525929737, \ 0.0039523401420815, \ 0.0039523401420815, \ 0.0039523401420815, \ 0.0039523401420815, \ 0.0039523401420815, \ 0.0039523401420815, \ 0.0026587971023890, \ 0.0026587971023890, \ 0.0026587971023890, \ 0.0026587971023890, \ 0.0026587971023890, \ 0.0026587971023890, \ 0.0023032616801324, \ 0.0023032616801324, \ 0.0023032616801324, \ 0.0023032616801324, \ 0.0023032616801324, \ 0.0023032616801324, \ 0.0043006096205643, \ 0.0043006096205643, \ 0.0043006096205643, \ 0.0043006096205643, \ 0.0043006096205643, \ 0.0043006096205643, \ 0.0030221060153018, \ 0.0030221060153018, \ 0.0030221060153018, \ 0.0030221060153018, \ 0.0030221060153018, \ 0.0030221060153018, \ 0.0003076949950084, \ 0.0003076949950084, \ 0.0003076949950084, \ 0.0003076949950084, \ 0.0003076949950084, \ 0.0003076949950084, \ 0.0030734403191510, \ 0.0030734403191510, \ 0.0030734403191510, \ 0.0030734403191510, \ 0.0030734403191510, \ 0.0030734403191510, \ 0.0049055295500793, \ 0.0049055295500793, \ 0.0049055295500793, \ 0.0049055295500793, \ 0.0049055295500793, \ 0.0049055295500793, \ 0.0027066696484092, \ 0.0027066696484092, \ 0.0027066696484092, \ 0.0027066696484092, \ 0.0027066696484092, \ 0.0027066696484092, \ 0.0034577749276135, \ 0.0034577749276135, \ 0.0034577749276135, \ 0.0034577749276135, \ 0.0034577749276135, \ 0.0034577749276135, \ 0.0051705174541971, \ 0.0051705174541971, \ 0.0051705174541971, \ 0.0051705174541971, \ 0.0051705174541971, \ 0.0051705174541971, \ 0.0014022583115306, \ 0.0014022583115306, \ 0.0014022583115306, \ 0.0014022583115306, \ 0.0014022583115306, \ 0.0014022583115306, \ 0.0042401026043811, \ 0.0042401026043811, \ 0.0042401026043811, \ 0.0042401026043811, \ 0.0042401026043811, \ 0.0042401026043811, \ 0.0032841487411702, \ 0.0032841487411702, \ 0.0032841487411702, \ 0.0032841487411702, \ 0.0032841487411702, \ 0.0032841487411702, \ 0.0011342259853124, \ 0.0011342259853124, \ 0.0011342259853124, \ 0.0011342259853124, \ 0.0011342259853124, \ 0.0011342259853124, \ 0.0074708548163608, \ 0.0074708548163608, \ 0.0074708548163608, \ 0.0074708548163608, \ 0.0074708548163608, \ 0.0074708548163608, \ 0.0034340384613521, \ 0.0034340384613521, \ 0.0034340384613521, \ 0.0034340384613521, \ 0.0034340384613521, \ 0.0034340384613521, \ 0.0031462609694126, \ 0.0031462609694126, \ 0.0031462609694126, \ 0.0031462609694126, \ 0.0031462609694126, \ 0.0031462609694126, \ 0.0005752613942102, \ 0.0005752613942102, \ 0.0005752613942102, \ 0.0005752613942102, \ 0.0005752613942102, \ 0.0005752613942102, \ 0.0065552795530324, \ 0.0065552795530324, \ 0.0065552795530324, \ 0.0065552795530324, \ 0.0065552795530324, \ 0.0065552795530324, \ 0.0029454095619872, \ 0.0029454095619872, \ 0.0029454095619872, \ 0.0029454095619872, \ 0.0029454095619872, \ 0.0029454095619872, \ 0.0029454095619872, \ 0.0029454095619872, \ 0.0029454095619872, \ 0.0029454095619872, \ 0.0029454095619872, \ 0.0029454095619872, \ 0.0024530630415632, \ 0.0024530630415632, \ 0.0024530630415632, \ 0.0024530630415632, \ 0.0024530630415632, \ 0.0024530630415632, \ 0.0024530630415632, \ 0.0024530630415632, \ 0.0024530630415632, \ 0.0024530630415632, \ 0.0024530630415632, \ 0.0024530630415632, \ 0.0028703966404990, \ 0.0028703966404990, \ 0.0028703966404990, \ 0.0028703966404990, \ 0.0028703966404990, \ 0.0028703966404990, \ 0.0028703966404990, \ 0.0028703966404990, \ 0.0028703966404990, \ 0.0028703966404990, \ 0.0028703966404990, \ 0.0028703966404990, \ 0.0018268277900660, \ 0.0018268277900660, \ 0.0018268277900660, \ 0.0018268277900660, \ 0.0018268277900660, \ 0.0018268277900660, \ 0.0018268277900660, \ 0.0018268277900660, \ 0.0018268277900660, \ 0.0018268277900660, \ 0.0018268277900660, \ 0.0018268277900660, \ 0.0046842877207978, \ 0.0046842877207978, \ 0.0046842877207978, \ 0.0046842877207978, \ 0.0046842877207978, \ 0.0046842877207978, \ 0.0046842877207978, \ 0.0046842877207978, \ 0.0046842877207978, \ 0.0046842877207978, \ 0.0046842877207978, \ 0.0046842877207978, \ 0.0061987832367774, \ 0.0061987832367774, \ 0.0061987832367774, \ 0.0061987832367774, \ 0.0061987832367774, \ 0.0061987832367774, \ 0.0061987832367774, \ 0.0061987832367774, \ 0.0061987832367774, \ 0.0061987832367774, \ 0.0061987832367774, \ 0.0061987832367774, \ 0.0041978630490507, \ 0.0041978630490507, \ 0.0041978630490507, \ 0.0041978630490507, \ 0.0041978630490507, \ 0.0041978630490507, \ 0.0041978630490507, \ 0.0041978630490507, \ 0.0041978630490507, \ 0.0041978630490507, \ 0.0041978630490507, \ 0.0041978630490507, \ 0.0025528142014630, \ 0.0025528142014630, \ 0.0025528142014630, \ 0.0025528142014630, \ 0.0025528142014630, \ 0.0025528142014630, \ 0.0025528142014630, \ 0.0025528142014630, \ 0.0025528142014630, \ 0.0025528142014630, \ 0.0025528142014630, \ 0.0025528142014630, \ 0.0020901151709172, \ 0.0020901151709172, \ 0.0020901151709172, \ 0.0020901151709172, \ 0.0020901151709172, \ 0.0020901151709172, \ 0.0020901151709172, \ 0.0020901151709172, \ 0.0020901151709172, \ 0.0020901151709172, \ 0.0020901151709172, \ 0.0020901151709172, \ 0.0007568671403396, \ 0.0007568671403396, \ 0.0007568671403396, \ 0.0007568671403396, \ 0.0007568671403396, \ 0.0007568671403396, \ 0.0007568671403396, \ 0.0007568671403396, \ 0.0007568671403396, \ 0.0007568671403396, \ 0.0007568671403396, \ 0.0007568671403396, \ 0.0013000657288395, \ 0.0013000657288395, \ 0.0013000657288395, \ 0.0013000657288395, \ 0.0013000657288395, \ 0.0013000657288395, \ 0.0013000657288395, \ 0.0013000657288395, \ 0.0013000657288395, \ 0.0013000657288395, \ 0.0013000657288395, \ 0.0013000657288395, \ 0.0008479873802968, \ 0.0008479873802968, \ 0.0008479873802968, \ 0.0008479873802968, \ 0.0008479873802968, \ 0.0008479873802968, \ 0.0008479873802968, \ 0.0008479873802968, \ 0.0008479873802968, \ 0.0008479873802968, \ 0.0008479873802968, \ 0.0008479873802968, \ 0.0064046130401019, \ 0.0064046130401019, \ 0.0064046130401019, \ 0.0064046130401019, \ 0.0064046130401019, \ 0.0064046130401019, \ 0.0064046130401019, \ 0.0064046130401019, \ 0.0064046130401019, \ 0.0064046130401019, \ 0.0064046130401019, \ 0.0064046130401019, \ 0.0011323520941966, \ 0.0011323520941966, \ 0.0011323520941966, \ 0.0011323520941966, \ 0.0011323520941966, \ 0.0011323520941966, \ 0.0011323520941966, \ 0.0011323520941966, \ 0.0011323520941966, \ 0.0011323520941966, \ 0.0011323520941966, \ 0.0011323520941966, \ 0.0020644102531444, \ 0.0020644102531444, \ 0.0020644102531444, \ 0.0020644102531444, \ 0.0020644102531444, \ 0.0020644102531444, \ 0.0020644102531444, \ 0.0020644102531444, \ 0.0020644102531444, \ 0.0020644102531444, \ 0.0020644102531444, \ 0.0020644102531444, \ 0.0010481934583324, \ 0.0010481934583324, \ 0.0010481934583324, \ 0.0010481934583324, \ 0.0010481934583324, \ 0.0010481934583324, \ 0.0010481934583324, \ 0.0010481934583324, \ 0.0010481934583324, \ 0.0010481934583324, \ 0.0010481934583324, \ 0.0010481934583324, \ 0.0009390285812953, \ 0.0009390285812953, \ 0.0009390285812953, \ 0.0009390285812953, \ 0.0009390285812953, \ 0.0009390285812953, \ 0.0009390285812953, \ 0.0009390285812953, \ 0.0009390285812953, \ 0.0009390285812953, \ 0.0009390285812953, \ 0.0009390285812953, \ 0.0042160603323249, \ 0.0042160603323249, \ 0.0042160603323249, \ 0.0042160603323249, \ 0.0042160603323249, \ 0.0042160603323249, \ 0.0042160603323249, \ 0.0042160603323249, \ 0.0042160603323249, \ 0.0042160603323249, \ 0.0042160603323249, \ 0.0042160603323249 ] ) return x, y, z, w def rule18 ( ): #*****************************************************************************80 # ## rule18() returns the prism rule of precision 18. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.4728631477307805, \ 0.4728631477307805, \ 0.0542737045384390, \ 0.4728631477307805, \ 0.4728631477307805, \ 0.0542737045384390, \ 0.1090318158369727, \ 0.1090318158369727, \ 0.7819363683260546, \ 0.1090318158369727, \ 0.1090318158369727, \ 0.7819363683260546, \ 0.2141954002982068, \ 0.2141954002982068, \ 0.5716091994035865, \ 0.2141954002982068, \ 0.2141954002982068, \ 0.5716091994035865, \ 0.0084150887533426, \ 0.0084150887533426, \ 0.9831698224933149, \ 0.0084150887533426, \ 0.0084150887533426, \ 0.9831698224933149, \ 0.0494156440294096, \ 0.0494156440294096, \ 0.9011687119411808, \ 0.0494156440294096, \ 0.0494156440294096, \ 0.9011687119411808, \ 0.4574258170355346, \ 0.4574258170355346, \ 0.0851483659289308, \ 0.4574258170355346, \ 0.4574258170355346, \ 0.0851483659289308, \ 0.2945570490820424, \ 0.2945570490820424, \ 0.4108859018359152, \ 0.2945570490820424, \ 0.2945570490820424, \ 0.4108859018359152, \ 0.4148583469418042, \ 0.4148583469418042, \ 0.1702833061163916, \ 0.4148583469418042, \ 0.4148583469418042, \ 0.1702833061163916, \ 0.1801567120271234, \ 0.1801567120271234, \ 0.6396865759457532, \ 0.1801567120271234, \ 0.1801567120271234, \ 0.6396865759457532, \ 0.4092592655375933, \ 0.4092592655375933, \ 0.1814814689248134, \ 0.4092592655375933, \ 0.4092592655375933, \ 0.1814814689248134, \ 0.4106876572727113, \ 0.4106876572727113, \ 0.1786246854545774, \ 0.4106876572727113, \ 0.4106876572727113, \ 0.1786246854545774, \ 0.4942616172602019, \ 0.4942616172602019, \ 0.0114767654795962, \ 0.4942616172602019, \ 0.4942616172602019, \ 0.0114767654795962, \ 0.1455821883151533, \ 0.1455821883151533, \ 0.7088356233696933, \ 0.1455821883151533, \ 0.1455821883151533, \ 0.7088356233696933, \ 0.0775928696462990, \ 0.0775928696462990, \ 0.8448142607074021, \ 0.0775928696462990, \ 0.0775928696462990, \ 0.8448142607074021, \ 0.1688466480848478, \ 0.1688466480848478, \ 0.6623067038303044, \ 0.1688466480848478, \ 0.1688466480848478, \ 0.6623067038303044, \ 0.2737821968608845, \ 0.2737821968608845, \ 0.4524356062782311, \ 0.2737821968608845, \ 0.2737821968608845, \ 0.4524356062782311, \ 0.3193703110432156, \ 0.3193703110432156, \ 0.3612593779135688, \ 0.3193703110432156, \ 0.3193703110432156, \ 0.3612593779135688, \ 0.2789278090499041, \ 0.2789278090499041, \ 0.4421443819001918, \ 0.2789278090499041, \ 0.2789278090499041, \ 0.4421443819001918, \ 0.0147343308678136, \ 0.0147343308678136, \ 0.9705313382643729, \ 0.0147343308678136, \ 0.0147343308678136, \ 0.9705313382643729, \ 0.2803931989784251, \ 0.1730708483947652, \ 0.2803931989784251, \ 0.5465359526268097, \ 0.1730708483947652, \ 0.5465359526268097, \ 0.3067402740370590, \ 0.0472583316199373, \ 0.3067402740370590, \ 0.6460013943430037, \ 0.0472583316199373, \ 0.6460013943430037, \ 0.3067402740370590, \ 0.0472583316199373, \ 0.3067402740370590, \ 0.6460013943430037, \ 0.0472583316199373, \ 0.6460013943430037, \ 0.5587472044782793, \ 0.0637486493547561, \ 0.5587472044782793, \ 0.3775041461669645, \ 0.0637486493547561, \ 0.3775041461669645, \ 0.5587472044782793, \ 0.0637486493547561, \ 0.5587472044782793, \ 0.3775041461669645, \ 0.0637486493547561, \ 0.3775041461669645, \ 0.1295467219204834, \ 0.2834002542935580, \ 0.1295467219204834, \ 0.5870530237859586, \ 0.2834002542935580, \ 0.5870530237859586, \ 0.1295467219204834, \ 0.2834002542935580, \ 0.1295467219204834, \ 0.5870530237859586, \ 0.2834002542935580, \ 0.5870530237859586, \ 0.2943165499688927, \ 0.0323606160924521, \ 0.2943165499688927, \ 0.6733228339386552, \ 0.0323606160924521, \ 0.6733228339386552, \ 0.2943165499688927, \ 0.0323606160924521, \ 0.2943165499688927, \ 0.6733228339386552, \ 0.0323606160924521, \ 0.6733228339386552, \ 0.7045618883179275, \ 0.0682727115900655, \ 0.7045618883179275, \ 0.2271654000920070, \ 0.0682727115900655, \ 0.2271654000920070, \ 0.7045618883179275, \ 0.0682727115900655, \ 0.7045618883179275, \ 0.2271654000920070, \ 0.0682727115900655, \ 0.2271654000920070, \ 0.5336092397180819, \ 0.0823227847933820, \ 0.5336092397180819, \ 0.3840679754885361, \ 0.0823227847933820, \ 0.3840679754885361, \ 0.5336092397180819, \ 0.0823227847933820, \ 0.5336092397180819, \ 0.3840679754885361, \ 0.0823227847933820, \ 0.3840679754885361, \ 0.1659343236267790, \ 0.2958566157742775, \ 0.1659343236267790, \ 0.5382090605989435, \ 0.2958566157742775, \ 0.5382090605989435, \ 0.1659343236267790, \ 0.2958566157742775, \ 0.1659343236267790, \ 0.5382090605989435, \ 0.2958566157742775, \ 0.5382090605989435, \ 0.5873912217089957, \ 0.0091538552254334, \ 0.5873912217089957, \ 0.4034549230655710, \ 0.0091538552254334, \ 0.4034549230655710, \ 0.5873912217089957, \ 0.0091538552254334, \ 0.5873912217089957, \ 0.4034549230655710, \ 0.0091538552254334, \ 0.4034549230655710, \ 0.1216404492275597, \ 0.0492455683577680, \ 0.1216404492275597, \ 0.8291139824146723, \ 0.0492455683577680, \ 0.8291139824146723, \ 0.1216404492275597, \ 0.0492455683577680, \ 0.1216404492275597, \ 0.8291139824146723, \ 0.0492455683577680, \ 0.8291139824146723, \ 0.0420809533226036, \ 0.0077537345196640, \ 0.0420809533226036, \ 0.9501653121577324, \ 0.0077537345196640, \ 0.9501653121577324, \ 0.0420809533226036, \ 0.0077537345196640, \ 0.0420809533226036, \ 0.9501653121577324, \ 0.0077537345196640, \ 0.9501653121577324, \ 0.4516351625773616, \ 0.0228219169717125, \ 0.4516351625773616, \ 0.5255429204509259, \ 0.0228219169717125, \ 0.5255429204509259, \ 0.4516351625773616, \ 0.0228219169717125, \ 0.4516351625773616, \ 0.5255429204509259, \ 0.0228219169717125, \ 0.5255429204509259, \ 0.1252580641579170, \ 0.0073963274047276, \ 0.1252580641579170, \ 0.8673456084373554, \ 0.0073963274047276, \ 0.8673456084373554, \ 0.1252580641579170, \ 0.0073963274047276, \ 0.1252580641579170, \ 0.8673456084373554, \ 0.0073963274047276, \ 0.8673456084373554, \ 0.1765722195894258, \ 0.0653002115725378, \ 0.1765722195894258, \ 0.7581275688380364, \ 0.0653002115725378, \ 0.7581275688380364, \ 0.1765722195894258, \ 0.0653002115725378, \ 0.1765722195894258, \ 0.7581275688380364, \ 0.0653002115725378, \ 0.7581275688380364, \ 0.2404267015692479, \ 0.0085879015506819, \ 0.2404267015692479, \ 0.7509853968800702, \ 0.0085879015506819, \ 0.7509853968800702, \ 0.2404267015692479, \ 0.0085879015506819, \ 0.2404267015692479, \ 0.7509853968800702, \ 0.0085879015506819, \ 0.7509853968800702, \ 0.0644058552263157, \ 0.0122678034366560, \ 0.0644058552263157, \ 0.9233263413370284, \ 0.0122678034366560, \ 0.9233263413370284, \ 0.0644058552263157, \ 0.0122678034366560, \ 0.0644058552263157, \ 0.9233263413370284, \ 0.0122678034366560, \ 0.9233263413370284, \ 0.2681069831840066, \ 0.0044793265613672, \ 0.2681069831840066, \ 0.7274136902546262, \ 0.0044793265613672, \ 0.7274136902546262, \ 0.2681069831840066, \ 0.0044793265613672, \ 0.2681069831840066, \ 0.7274136902546262, \ 0.0044793265613672, \ 0.7274136902546262, \ 0.1727205252244992, \ 0.0117334535244670, \ 0.1727205252244992, \ 0.8155460212510338, \ 0.0117334535244670, \ 0.8155460212510338, \ 0.1727205252244992, \ 0.0117334535244670, \ 0.1727205252244992, \ 0.8155460212510338, \ 0.0117334535244670, \ 0.8155460212510338, \ 0.1118329953359960, \ 0.5446775610503505, \ 0.1118329953359960, \ 0.3434894436136536, \ 0.5446775610503505, \ 0.3434894436136536, \ 0.1118329953359960, \ 0.5446775610503505, \ 0.1118329953359960, \ 0.3434894436136536, \ 0.5446775610503505, \ 0.3434894436136536, \ 0.5899536365475773, \ 0.0080175874177025, \ 0.5899536365475773, \ 0.4020287760347202, \ 0.0080175874177025, \ 0.4020287760347202, \ 0.5899536365475773, \ 0.0080175874177025, \ 0.5899536365475773, \ 0.4020287760347202, \ 0.0080175874177025, \ 0.4020287760347202, \ 0.0961540597861764, \ 0.2067498834662211, \ 0.0961540597861764, \ 0.6970960567476024, \ 0.2067498834662211, \ 0.6970960567476024, \ 0.0961540597861764, \ 0.2067498834662211, \ 0.0961540597861764, \ 0.6970960567476024, \ 0.2067498834662211, \ 0.6970960567476024, \ 0.3068968355307726, \ 0.0117746553239110, \ 0.3068968355307726, \ 0.6813285091453164, \ 0.0117746553239110, \ 0.6813285091453164, \ 0.3068968355307726, \ 0.0117746553239110, \ 0.3068968355307726, \ 0.6813285091453164, \ 0.0117746553239110, \ 0.6813285091453164, \ 0.0798486447852760, \ 0.0174063356195718, \ 0.0798486447852760, \ 0.9027450195951523, \ 0.0174063356195718, \ 0.9027450195951523, \ 0.0798486447852760, \ 0.0174063356195718, \ 0.0798486447852760, \ 0.9027450195951523, \ 0.0174063356195718, \ 0.9027450195951523, \ 0.0360777790627768, \ 0.1618562004775664, \ 0.0360777790627768, \ 0.8020660204596567, \ 0.1618562004775664, \ 0.8020660204596567, \ 0.0360777790627768, \ 0.1618562004775664, \ 0.0360777790627768, \ 0.8020660204596567, \ 0.1618562004775664, \ 0.8020660204596567 ] ) y = np.array ( [ \ 0.4728631477307805, \ 0.0542737045384390, \ 0.4728631477307805, \ 0.4728631477307805, \ 0.0542737045384390, \ 0.4728631477307805, \ 0.1090318158369727, \ 0.7819363683260546, \ 0.1090318158369727, \ 0.1090318158369727, \ 0.7819363683260546, \ 0.1090318158369727, \ 0.2141954002982068, \ 0.5716091994035865, \ 0.2141954002982068, \ 0.2141954002982068, \ 0.5716091994035865, \ 0.2141954002982068, \ 0.0084150887533426, \ 0.9831698224933149, \ 0.0084150887533426, \ 0.0084150887533426, \ 0.9831698224933149, \ 0.0084150887533426, \ 0.0494156440294096, \ 0.9011687119411808, \ 0.0494156440294096, \ 0.0494156440294096, \ 0.9011687119411808, \ 0.0494156440294096, \ 0.4574258170355346, \ 0.0851483659289308, \ 0.4574258170355346, \ 0.4574258170355346, \ 0.0851483659289308, \ 0.4574258170355346, \ 0.2945570490820424, \ 0.4108859018359152, \ 0.2945570490820424, \ 0.2945570490820424, \ 0.4108859018359152, \ 0.2945570490820424, \ 0.4148583469418042, \ 0.1702833061163916, \ 0.4148583469418042, \ 0.4148583469418042, \ 0.1702833061163916, \ 0.4148583469418042, \ 0.1801567120271234, \ 0.6396865759457532, \ 0.1801567120271234, \ 0.1801567120271234, \ 0.6396865759457532, \ 0.1801567120271234, \ 0.4092592655375933, \ 0.1814814689248134, \ 0.4092592655375933, \ 0.4092592655375933, \ 0.1814814689248134, \ 0.4092592655375933, \ 0.4106876572727113, \ 0.1786246854545774, \ 0.4106876572727113, \ 0.4106876572727113, \ 0.1786246854545774, \ 0.4106876572727113, \ 0.4942616172602019, \ 0.0114767654795962, \ 0.4942616172602019, \ 0.4942616172602019, \ 0.0114767654795962, \ 0.4942616172602019, \ 0.1455821883151533, \ 0.7088356233696933, \ 0.1455821883151533, \ 0.1455821883151533, \ 0.7088356233696933, \ 0.1455821883151533, \ 0.0775928696462990, \ 0.8448142607074021, \ 0.0775928696462990, \ 0.0775928696462990, \ 0.8448142607074021, \ 0.0775928696462990, \ 0.1688466480848478, \ 0.6623067038303044, \ 0.1688466480848478, \ 0.1688466480848478, \ 0.6623067038303044, \ 0.1688466480848478, \ 0.2737821968608845, \ 0.4524356062782311, \ 0.2737821968608845, \ 0.2737821968608845, \ 0.4524356062782311, \ 0.2737821968608845, \ 0.3193703110432156, \ 0.3612593779135688, \ 0.3193703110432156, \ 0.3193703110432156, \ 0.3612593779135688, \ 0.3193703110432156, \ 0.2789278090499041, \ 0.4421443819001918, \ 0.2789278090499041, \ 0.2789278090499041, \ 0.4421443819001918, \ 0.2789278090499041, \ 0.0147343308678136, \ 0.9705313382643729, \ 0.0147343308678136, \ 0.0147343308678136, \ 0.9705313382643729, \ 0.0147343308678136, \ 0.1730708483947652, \ 0.2803931989784251, \ 0.5465359526268097, \ 0.2803931989784251, \ 0.5465359526268097, \ 0.1730708483947652, \ 0.0472583316199373, \ 0.3067402740370590, \ 0.6460013943430037, \ 0.3067402740370590, \ 0.6460013943430037, \ 0.0472583316199373, \ 0.0472583316199373, \ 0.3067402740370590, \ 0.6460013943430037, \ 0.3067402740370590, \ 0.6460013943430037, \ 0.0472583316199373, \ 0.0637486493547561, \ 0.5587472044782793, \ 0.3775041461669645, \ 0.5587472044782793, \ 0.3775041461669645, \ 0.0637486493547561, \ 0.0637486493547561, \ 0.5587472044782793, \ 0.3775041461669645, \ 0.5587472044782793, \ 0.3775041461669645, \ 0.0637486493547561, \ 0.2834002542935580, \ 0.1295467219204834, \ 0.5870530237859586, \ 0.1295467219204834, \ 0.5870530237859586, \ 0.2834002542935580, \ 0.2834002542935580, \ 0.1295467219204834, \ 0.5870530237859586, \ 0.1295467219204834, \ 0.5870530237859586, \ 0.2834002542935580, \ 0.0323606160924521, \ 0.2943165499688927, \ 0.6733228339386552, \ 0.2943165499688927, \ 0.6733228339386552, \ 0.0323606160924521, \ 0.0323606160924521, \ 0.2943165499688927, \ 0.6733228339386552, \ 0.2943165499688927, \ 0.6733228339386552, \ 0.0323606160924521, \ 0.0682727115900655, \ 0.7045618883179275, \ 0.2271654000920070, \ 0.7045618883179275, \ 0.2271654000920070, \ 0.0682727115900655, \ 0.0682727115900655, \ 0.7045618883179275, \ 0.2271654000920070, \ 0.7045618883179275, \ 0.2271654000920070, \ 0.0682727115900655, \ 0.0823227847933820, \ 0.5336092397180819, \ 0.3840679754885361, \ 0.5336092397180819, \ 0.3840679754885361, \ 0.0823227847933820, \ 0.0823227847933820, \ 0.5336092397180819, \ 0.3840679754885361, \ 0.5336092397180819, \ 0.3840679754885361, \ 0.0823227847933820, \ 0.2958566157742775, \ 0.1659343236267790, \ 0.5382090605989435, \ 0.1659343236267790, \ 0.5382090605989435, \ 0.2958566157742775, \ 0.2958566157742775, \ 0.1659343236267790, \ 0.5382090605989435, \ 0.1659343236267790, \ 0.5382090605989435, \ 0.2958566157742775, \ 0.0091538552254334, \ 0.5873912217089957, \ 0.4034549230655710, \ 0.5873912217089957, \ 0.4034549230655710, \ 0.0091538552254334, \ 0.0091538552254334, \ 0.5873912217089957, \ 0.4034549230655710, \ 0.5873912217089957, \ 0.4034549230655710, \ 0.0091538552254334, \ 0.0492455683577680, \ 0.1216404492275597, \ 0.8291139824146723, \ 0.1216404492275597, \ 0.8291139824146723, \ 0.0492455683577680, \ 0.0492455683577680, \ 0.1216404492275597, \ 0.8291139824146723, \ 0.1216404492275597, \ 0.8291139824146723, \ 0.0492455683577680, \ 0.0077537345196640, \ 0.0420809533226036, \ 0.9501653121577324, \ 0.0420809533226036, \ 0.9501653121577324, \ 0.0077537345196640, \ 0.0077537345196640, \ 0.0420809533226036, \ 0.9501653121577324, \ 0.0420809533226036, \ 0.9501653121577324, \ 0.0077537345196640, \ 0.0228219169717125, \ 0.4516351625773616, \ 0.5255429204509259, \ 0.4516351625773616, \ 0.5255429204509259, \ 0.0228219169717125, \ 0.0228219169717125, \ 0.4516351625773616, \ 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0.0122678034366560, \ 0.0122678034366560, \ 0.0644058552263157, \ 0.9233263413370284, \ 0.0644058552263157, \ 0.9233263413370284, \ 0.0122678034366560, \ 0.0044793265613672, \ 0.2681069831840066, \ 0.7274136902546262, \ 0.2681069831840066, \ 0.7274136902546262, \ 0.0044793265613672, \ 0.0044793265613672, \ 0.2681069831840066, \ 0.7274136902546262, \ 0.2681069831840066, \ 0.7274136902546262, \ 0.0044793265613672, \ 0.0117334535244670, \ 0.1727205252244992, \ 0.8155460212510338, \ 0.1727205252244992, \ 0.8155460212510338, \ 0.0117334535244670, \ 0.0117334535244670, \ 0.1727205252244992, \ 0.8155460212510338, \ 0.1727205252244992, \ 0.8155460212510338, \ 0.0117334535244670, \ 0.5446775610503505, \ 0.1118329953359960, \ 0.3434894436136536, \ 0.1118329953359960, \ 0.3434894436136536, \ 0.5446775610503505, \ 0.5446775610503505, \ 0.1118329953359960, \ 0.3434894436136536, \ 0.1118329953359960, \ 0.3434894436136536, \ 0.5446775610503505, \ 0.0080175874177025, \ 0.5899536365475773, \ 0.4020287760347202, \ 0.5899536365475773, \ 0.4020287760347202, \ 0.0080175874177025, \ 0.0080175874177025, \ 0.5899536365475773, \ 0.4020287760347202, \ 0.5899536365475773, \ 0.4020287760347202, \ 0.0080175874177025, \ 0.2067498834662211, \ 0.0961540597861764, \ 0.6970960567476024, \ 0.0961540597861764, \ 0.6970960567476024, \ 0.2067498834662211, \ 0.2067498834662211, \ 0.0961540597861764, \ 0.6970960567476024, \ 0.0961540597861764, \ 0.6970960567476024, \ 0.2067498834662211, \ 0.0117746553239110, \ 0.3068968355307726, \ 0.6813285091453164, \ 0.3068968355307726, \ 0.6813285091453164, \ 0.0117746553239110, \ 0.0117746553239110, \ 0.3068968355307726, \ 0.6813285091453164, \ 0.3068968355307726, \ 0.6813285091453164, \ 0.0117746553239110, \ 0.0174063356195718, \ 0.0798486447852760, \ 0.9027450195951523, \ 0.0798486447852760, \ 0.9027450195951523, \ 0.0174063356195718, \ 0.0174063356195718, \ 0.0798486447852760, \ 0.9027450195951523, \ 0.0798486447852760, \ 0.9027450195951523, \ 0.0174063356195718, \ 0.1618562004775664, \ 0.0360777790627768, \ 0.8020660204596567, \ 0.0360777790627768, \ 0.8020660204596567, \ 0.1618562004775664, \ 0.1618562004775664, \ 0.0360777790627768, \ 0.8020660204596567, \ 0.0360777790627768, \ 0.8020660204596567, \ 0.1618562004775664 ] ) z = np.array ( [ \ 0.9534387216382612, \ 0.9534387216382612, \ 0.9534387216382612, \ 0.0465612783617388, \ 0.0465612783617388, \ 0.0465612783617388, \ 0.7595655027404185, \ 0.7595655027404185, \ 0.7595655027404185, \ 0.2404344972595815, \ 0.2404344972595815, \ 0.2404344972595815, \ 0.8487633516520783, \ 0.8487633516520783, \ 0.8487633516520783, \ 0.1512366483479217, \ 0.1512366483479217, \ 0.1512366483479217, \ 0.7352481798898231, \ 0.7352481798898231, \ 0.7352481798898231, \ 0.2647518201101769, \ 0.2647518201101769, \ 0.2647518201101769, \ 0.3114476852954629, \ 0.3114476852954629, \ 0.3114476852954629, \ 0.6885523147045371, \ 0.6885523147045371, \ 0.6885523147045371, \ 0.3645354584203186, \ 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0.8283291277845266, \ 0.8283291277845266, \ 0.8283291277845266, \ 0.8283291277845266, \ 0.8283291277845266, \ 0.1716708722154733, \ 0.1716708722154733, \ 0.1716708722154733, \ 0.1716708722154733, \ 0.1716708722154733, \ 0.1716708722154733, \ 0.9370173839969559, \ 0.9370173839969559, \ 0.9370173839969559, \ 0.9370173839969559, \ 0.9370173839969559, \ 0.9370173839969559, \ 0.0629826160030441, \ 0.0629826160030441, \ 0.0629826160030441, \ 0.0629826160030441, \ 0.0629826160030441, \ 0.0629826160030441, \ 0.9939758307965130, \ 0.9939758307965130, \ 0.9939758307965130, \ 0.9939758307965130, \ 0.9939758307965130, \ 0.9939758307965130, \ 0.0060241692034870, \ 0.0060241692034870, \ 0.0060241692034870, \ 0.0060241692034870, \ 0.0060241692034870, \ 0.0060241692034870, \ 0.5786001946994722, \ 0.5786001946994722, \ 0.5786001946994722, \ 0.5786001946994722, \ 0.5786001946994722, \ 0.5786001946994722, \ 0.4213998053005278, \ 0.4213998053005278, \ 0.4213998053005278, \ 0.4213998053005278, \ 0.4213998053005278, \ 0.4213998053005278, \ 0.8755215159278712, \ 0.8755215159278712, \ 0.8755215159278712, \ 0.8755215159278712, \ 0.8755215159278712, \ 0.8755215159278712, \ 0.1244784840721288, \ 0.1244784840721288, \ 0.1244784840721288, \ 0.1244784840721288, \ 0.1244784840721288, \ 0.1244784840721288, \ 0.9893261490512200, \ 0.9893261490512200, \ 0.9893261490512200, \ 0.9893261490512200, \ 0.9893261490512200, \ 0.9893261490512200, \ 0.0106738509487800, \ 0.0106738509487800, \ 0.0106738509487800, \ 0.0106738509487800, \ 0.0106738509487800, \ 0.0106738509487800, \ 0.9893821585030006, \ 0.9893821585030006, \ 0.9893821585030006, \ 0.9893821585030006, \ 0.9893821585030006, \ 0.9893821585030006, \ 0.0106178414969994, \ 0.0106178414969994, \ 0.0106178414969994, \ 0.0106178414969994, \ 0.0106178414969994, \ 0.0106178414969994, \ 0.7966282776345323, \ 0.7966282776345323, \ 0.7966282776345323, \ 0.7966282776345323, \ 0.7966282776345323, \ 0.7966282776345323, \ 0.2033717223654677, \ 0.2033717223654677, \ 0.2033717223654677, \ 0.2033717223654677, \ 0.2033717223654677, \ 0.2033717223654677 ] ) w = np.array ( [ \ 0.0022576928480960, \ 0.0022576928480960, \ 0.0022576928480960, \ 0.0022576928480960, \ 0.0022576928480960, \ 0.0022576928480960, \ 0.0035220009936119, \ 0.0035220009936119, \ 0.0035220009936119, \ 0.0035220009936119, \ 0.0035220009936119, \ 0.0035220009936119, \ 0.0044617234977870, \ 0.0044617234977870, \ 0.0044617234977870, \ 0.0044617234977870, \ 0.0044617234977870, \ 0.0044617234977870, \ 0.0003707999359392, \ 0.0003707999359392, \ 0.0003707999359392, \ 0.0003707999359392, \ 0.0003707999359392, \ 0.0003707999359392, \ 0.0024236611427218, \ 0.0024236611427218, \ 0.0024236611427218, \ 0.0024236611427218, \ 0.0024236611427218, \ 0.0024236611427218, \ 0.0028149967000162, \ 0.0028149967000162, \ 0.0028149967000162, \ 0.0028149967000162, \ 0.0028149967000162, \ 0.0028149967000162, \ 0.0053376421669576, \ 0.0053376421669576, \ 0.0053376421669576, \ 0.0053376421669576, \ 0.0053376421669576, \ 0.0053376421669576, \ 0.0085742666644647, \ 0.0085742666644647, \ 0.0085742666644647, \ 0.0085742666644647, \ 0.0085742666644647, \ 0.0085742666644647, \ 0.0022657258714867, \ 0.0022657258714867, \ 0.0022657258714867, \ 0.0022657258714867, \ 0.0022657258714867, \ 0.0022657258714867, \ 0.0029551579597074, \ 0.0029551579597074, \ 0.0029551579597074, \ 0.0029551579597074, \ 0.0029551579597074, \ 0.0029551579597074, \ 0.0039194664968460, \ 0.0039194664968460, \ 0.0039194664968460, \ 0.0039194664968460, \ 0.0039194664968460, \ 0.0039194664968460, \ 0.0006030674708196, \ 0.0006030674708196, \ 0.0006030674708196, \ 0.0006030674708196, \ 0.0006030674708196, \ 0.0006030674708196, \ 0.0045779809945426, \ 0.0045779809945426, \ 0.0045779809945426, \ 0.0045779809945426, \ 0.0045779809945426, \ 0.0045779809945426, \ 0.0024306636423047, \ 0.0024306636423047, \ 0.0024306636423047, \ 0.0024306636423047, \ 0.0024306636423047, \ 0.0024306636423047, \ 0.0021599699258395, \ 0.0021599699258395, \ 0.0021599699258395, \ 0.0021599699258395, \ 0.0021599699258395, \ 0.0021599699258395, \ 0.0079010510892335, \ 0.0079010510892335, \ 0.0079010510892335, \ 0.0079010510892335, \ 0.0079010510892335, \ 0.0079010510892335, \ 0.0021738453915315, \ 0.0021738453915315, \ 0.0021738453915315, \ 0.0021738453915315, \ 0.0021738453915315, \ 0.0021738453915315, \ 0.0022486463151748, \ 0.0022486463151748, \ 0.0022486463151748, \ 0.0022486463151748, \ 0.0022486463151748, \ 0.0022486463151748, \ 0.0003442548195872, \ 0.0003442548195872, \ 0.0003442548195872, \ 0.0003442548195872, \ 0.0003442548195872, \ 0.0003442548195872, \ 0.0080699355816128, \ 0.0080699355816128, \ 0.0080699355816128, \ 0.0080699355816128, \ 0.0080699355816128, \ 0.0080699355816128, \ 0.0023877231635901, \ 0.0023877231635901, \ 0.0023877231635901, \ 0.0023877231635901, \ 0.0023877231635901, \ 0.0023877231635901, \ 0.0023877231635901, \ 0.0023877231635901, \ 0.0023877231635901, \ 0.0023877231635901, \ 0.0023877231635901, \ 0.0023877231635901, \ 0.0034305282210409, \ 0.0034305282210409, \ 0.0034305282210409, \ 0.0034305282210409, \ 0.0034305282210409, \ 0.0034305282210409, \ 0.0034305282210409, \ 0.0034305282210409, \ 0.0034305282210409, \ 0.0034305282210409, \ 0.0034305282210409, \ 0.0034305282210409, \ 0.0060295453672165, \ 0.0060295453672165, \ 0.0060295453672165, \ 0.0060295453672165, \ 0.0060295453672165, \ 0.0060295453672165, \ 0.0060295453672165, \ 0.0060295453672165, \ 0.0060295453672165, \ 0.0060295453672165, \ 0.0060295453672165, \ 0.0060295453672165, \ 0.0032768238791581, \ 0.0032768238791581, \ 0.0032768238791581, \ 0.0032768238791581, \ 0.0032768238791581, \ 0.0032768238791581, \ 0.0032768238791581, \ 0.0032768238791581, \ 0.0032768238791581, \ 0.0032768238791581, \ 0.0032768238791581, \ 0.0032768238791581, \ 0.0036635492105407, \ 0.0036635492105407, \ 0.0036635492105407, \ 0.0036635492105407, \ 0.0036635492105407, \ 0.0036635492105407, \ 0.0036635492105407, \ 0.0036635492105407, \ 0.0036635492105407, \ 0.0036635492105407, \ 0.0036635492105407, \ 0.0036635492105407, \ 0.0044244882628077, \ 0.0044244882628077, \ 0.0044244882628077, \ 0.0044244882628077, \ 0.0044244882628077, \ 0.0044244882628077, \ 0.0044244882628077, \ 0.0044244882628077, \ 0.0044244882628077, \ 0.0044244882628077, \ 0.0044244882628077, \ 0.0044244882628077, \ 0.0027600635232146, \ 0.0027600635232146, \ 0.0027600635232146, \ 0.0027600635232146, \ 0.0027600635232146, \ 0.0027600635232146, \ 0.0027600635232146, \ 0.0027600635232146, \ 0.0027600635232146, \ 0.0027600635232146, \ 0.0027600635232146, \ 0.0027600635232146, \ 0.0013921098894439, \ 0.0013921098894439, \ 0.0013921098894439, \ 0.0013921098894439, \ 0.0013921098894439, \ 0.0013921098894439, \ 0.0013921098894439, \ 0.0013921098894439, \ 0.0013921098894439, \ 0.0013921098894439, \ 0.0013921098894439, \ 0.0013921098894439, \ 0.0021384358726425, \ 0.0021384358726425, \ 0.0021384358726425, \ 0.0021384358726425, \ 0.0021384358726425, \ 0.0021384358726425, \ 0.0021384358726425, \ 0.0021384358726425, \ 0.0021384358726425, \ 0.0021384358726425, \ 0.0021384358726425, \ 0.0021384358726425, \ 0.0005166067073489, \ 0.0005166067073489, \ 0.0005166067073489, \ 0.0005166067073489, \ 0.0005166067073489, \ 0.0005166067073489, \ 0.0005166067073489, \ 0.0005166067073489, \ 0.0005166067073489, \ 0.0005166067073489, \ 0.0005166067073489, \ 0.0005166067073489, \ 0.0019539531411333, \ 0.0019539531411333, \ 0.0019539531411333, \ 0.0019539531411333, \ 0.0019539531411333, \ 0.0019539531411333, \ 0.0019539531411333, \ 0.0019539531411333, \ 0.0019539531411333, \ 0.0019539531411333, \ 0.0019539531411333, \ 0.0019539531411333, \ 0.0012550085983638, \ 0.0012550085983638, \ 0.0012550085983638, \ 0.0012550085983638, \ 0.0012550085983638, \ 0.0012550085983638, \ 0.0012550085983638, \ 0.0012550085983638, \ 0.0012550085983638, \ 0.0012550085983638, \ 0.0012550085983638, \ 0.0012550085983638, \ 0.0010423341141818, \ 0.0010423341141818, \ 0.0010423341141818, \ 0.0010423341141818, \ 0.0010423341141818, \ 0.0010423341141818, \ 0.0010423341141818, \ 0.0010423341141818, \ 0.0010423341141818, \ 0.0010423341141818, \ 0.0010423341141818, \ 0.0010423341141818, \ 0.0012793420418862, \ 0.0012793420418862, \ 0.0012793420418862, \ 0.0012793420418862, \ 0.0012793420418862, \ 0.0012793420418862, \ 0.0012793420418862, \ 0.0012793420418862, \ 0.0012793420418862, \ 0.0012793420418862, \ 0.0012793420418862, \ 0.0012793420418862, \ 0.0010877722341374, \ 0.0010877722341374, \ 0.0010877722341374, \ 0.0010877722341374, \ 0.0010877722341374, \ 0.0010877722341374, \ 0.0010877722341374, \ 0.0010877722341374, \ 0.0010877722341374, \ 0.0010877722341374, \ 0.0010877722341374, \ 0.0010877722341374, \ 0.0008506443220554, \ 0.0008506443220554, \ 0.0008506443220554, \ 0.0008506443220554, \ 0.0008506443220554, \ 0.0008506443220554, \ 0.0008506443220554, \ 0.0008506443220554, \ 0.0008506443220554, \ 0.0008506443220554, \ 0.0008506443220554, \ 0.0008506443220554, \ 0.0010717384362760, \ 0.0010717384362760, \ 0.0010717384362760, \ 0.0010717384362760, \ 0.0010717384362760, \ 0.0010717384362760, \ 0.0010717384362760, \ 0.0010717384362760, \ 0.0010717384362760, \ 0.0010717384362760, \ 0.0010717384362760, \ 0.0010717384362760, \ 0.0012223898383723, \ 0.0012223898383723, \ 0.0012223898383723, \ 0.0012223898383723, \ 0.0012223898383723, \ 0.0012223898383723, \ 0.0012223898383723, \ 0.0012223898383723, \ 0.0012223898383723, \ 0.0012223898383723, \ 0.0012223898383723, \ 0.0012223898383723, \ 0.0014264821649004, \ 0.0014264821649004, \ 0.0014264821649004, \ 0.0014264821649004, \ 0.0014264821649004, \ 0.0014264821649004, \ 0.0014264821649004, \ 0.0014264821649004, \ 0.0014264821649004, \ 0.0014264821649004, \ 0.0014264821649004, \ 0.0014264821649004, \ 0.0038341778997734, \ 0.0038341778997734, \ 0.0038341778997734, \ 0.0038341778997734, \ 0.0038341778997734, \ 0.0038341778997734, \ 0.0038341778997734, \ 0.0038341778997734, \ 0.0038341778997734, \ 0.0038341778997734, \ 0.0038341778997734, \ 0.0038341778997734, \ 0.0004758021594541, \ 0.0004758021594541, \ 0.0004758021594541, \ 0.0004758021594541, \ 0.0004758021594541, \ 0.0004758021594541, \ 0.0004758021594541, \ 0.0004758021594541, \ 0.0004758021594541, \ 0.0004758021594541, \ 0.0004758021594541, \ 0.0004758021594541, \ 0.0003711298716255, \ 0.0003711298716255, \ 0.0003711298716255, \ 0.0003711298716255, \ 0.0003711298716255, \ 0.0003711298716255, \ 0.0003711298716255, \ 0.0003711298716255, \ 0.0003711298716255, \ 0.0003711298716255, \ 0.0003711298716255, \ 0.0003711298716255, \ 0.0027364096600290, \ 0.0027364096600290, \ 0.0027364096600290, \ 0.0027364096600290, \ 0.0027364096600290, \ 0.0027364096600290, \ 0.0027364096600290, \ 0.0027364096600290, \ 0.0027364096600290, \ 0.0027364096600290, \ 0.0027364096600290, \ 0.0027364096600290 ] ) return x, y, z, w def rule19 ( ): #*****************************************************************************80 # ## rule19() returns the prism rule of precision 19. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4935398626809867, \ 0.4935398626809867, \ 0.0129202746380266, \ 0.2767659367008514, \ 0.2767659367008514, \ 0.4464681265982973, \ 0.0545777948502843, \ 0.0545777948502843, \ 0.8908444102994315, \ 0.0139969735218886, \ 0.0139969735218886, \ 0.9720060529562228, \ 0.1418740786751734, \ 0.1418740786751734, \ 0.7162518426496531, \ 0.1418740786751734, \ 0.1418740786751734, \ 0.7162518426496531, \ 0.0083955990101041, \ 0.0083955990101041, \ 0.9832088019797917, \ 0.0083955990101041, \ 0.0083955990101041, \ 0.9832088019797917, \ 0.1035973359771996, \ 0.1035973359771996, \ 0.7928053280456009, \ 0.1035973359771996, \ 0.1035973359771996, \ 0.7928053280456009, \ 0.3956080996828434, \ 0.3956080996828434, \ 0.2087838006343133, \ 0.3956080996828434, \ 0.3956080996828434, \ 0.2087838006343133, \ 0.0140019311977436, \ 0.0140019311977436, \ 0.9719961376045129, \ 0.0140019311977436, \ 0.0140019311977436, \ 0.9719961376045129, \ 0.4801897764603656, \ 0.4801897764603656, \ 0.0396204470792688, \ 0.4801897764603656, \ 0.4801897764603656, \ 0.0396204470792688, \ 0.1623484102426076, \ 0.1623484102426076, \ 0.6753031795147848, \ 0.1623484102426076, \ 0.1623484102426076, \ 0.6753031795147848, \ 0.4972236736522475, \ 0.4972236736522475, \ 0.0055526526955051, \ 0.4972236736522475, \ 0.4972236736522475, \ 0.0055526526955051, \ 0.1990216655966096, \ 0.1990216655966096, \ 0.6019566688067807, \ 0.1990216655966096, \ 0.1990216655966096, \ 0.6019566688067807, \ 0.1202889101503504, \ 0.1202889101503504, \ 0.7594221796992991, \ 0.1202889101503504, \ 0.1202889101503504, \ 0.7594221796992991, \ 0.0532191526098831, \ 0.0532191526098831, \ 0.8935616947802337, \ 0.0532191526098831, \ 0.0532191526098831, \ 0.8935616947802337, \ 0.4012796612843944, \ 0.4012796612843944, \ 0.1974406774312114, \ 0.4012796612843944, \ 0.4012796612843944, \ 0.1974406774312114, \ 0.1089883669604236, \ 0.1089883669604236, \ 0.7820232660791528, \ 0.1089883669604236, \ 0.1089883669604236, \ 0.7820232660791528, \ 0.4472748897677901, \ 0.4472748897677901, \ 0.1054502204644198, \ 0.4472748897677901, \ 0.4472748897677901, \ 0.1054502204644198, \ 0.0692335448571578, \ 0.0692335448571578, \ 0.8615329102856844, \ 0.0692335448571578, \ 0.0692335448571578, \ 0.8615329102856844, \ 0.2475145234973103, \ 0.2475145234973103, \ 0.5049709530053793, \ 0.2475145234973103, \ 0.2475145234973103, \ 0.5049709530053793, \ 0.4533377451558121, \ 0.4533377451558121, \ 0.0933245096883758, \ 0.4533377451558121, \ 0.4533377451558121, \ 0.0933245096883758, \ 0.0180837411268107, \ 0.0180837411268107, \ 0.9638325177463786, \ 0.0180837411268107, \ 0.0180837411268107, \ 0.9638325177463786, \ 0.1297897131077726, \ 0.0069174965695053, \ 0.1297897131077726, \ 0.8632927903227221, \ 0.0069174965695053, \ 0.8632927903227221, \ 0.3779418721990778, \ 0.0310061742359910, \ 0.3779418721990778, \ 0.5910519535649312, \ 0.0310061742359910, \ 0.5910519535649312, \ 0.0644631311955231, \ 0.2203251202566757, \ 0.0644631311955231, \ 0.7152117485478011, \ 0.2203251202566757, \ 0.7152117485478011, \ 0.1625899734623090, \ 0.2824754920851798, \ 0.1625899734623090, \ 0.5549345344525113, \ 0.2824754920851798, \ 0.5549345344525113, \ 0.1625899734623090, \ 0.2824754920851798, \ 0.1625899734623090, \ 0.5549345344525113, \ 0.2824754920851798, \ 0.5549345344525113, \ 0.0081184148094398, \ 0.0547883511283449, \ 0.0081184148094398, \ 0.9370932340622153, \ 0.0547883511283449, \ 0.9370932340622153, \ 0.0081184148094398, \ 0.0547883511283449, \ 0.0081184148094398, \ 0.9370932340622153, \ 0.0547883511283449, \ 0.9370932340622153, \ 0.6582273496659641, \ 0.1178005364736766, \ 0.6582273496659641, \ 0.2239721138603593, \ 0.1178005364736766, \ 0.2239721138603593, \ 0.6582273496659641, \ 0.1178005364736766, \ 0.6582273496659641, \ 0.2239721138603593, \ 0.1178005364736766, \ 0.2239721138603593, \ 0.2876874815978630, \ 0.0355268541454468, \ 0.2876874815978630, \ 0.6767856642566902, \ 0.0355268541454468, \ 0.6767856642566902, \ 0.2876874815978630, \ 0.0355268541454468, \ 0.2876874815978630, \ 0.6767856642566902, \ 0.0355268541454468, \ 0.6767856642566902, \ 0.2839484400308330, \ 0.2073917255967352, \ 0.2839484400308330, \ 0.5086598343724318, \ 0.2073917255967352, \ 0.5086598343724318, \ 0.2839484400308330, \ 0.2073917255967352, \ 0.2839484400308330, \ 0.5086598343724318, \ 0.2073917255967352, \ 0.5086598343724318, \ 0.0282457003834953, \ 0.0862230026956853, \ 0.0282457003834953, \ 0.8855312969208194, \ 0.0862230026956853, \ 0.8855312969208194, \ 0.0282457003834953, \ 0.0862230026956853, \ 0.0282457003834953, \ 0.8855312969208194, \ 0.0862230026956853, \ 0.8855312969208194, \ 0.3915043117238656, \ 0.0056196516036861, \ 0.3915043117238656, \ 0.6028760366724483, \ 0.0056196516036861, \ 0.6028760366724483, \ 0.3915043117238656, \ 0.0056196516036861, \ 0.3915043117238656, \ 0.6028760366724483, \ 0.0056196516036861, \ 0.6028760366724483, \ 0.0737667613724345, \ 0.3389265718024346, \ 0.0737667613724345, \ 0.5873066668251310, \ 0.3389265718024346, \ 0.5873066668251310, \ 0.0737667613724345, \ 0.3389265718024346, \ 0.0737667613724345, \ 0.5873066668251310, \ 0.3389265718024346, \ 0.5873066668251310, \ 0.1582419524500696, \ 0.0080205188978798, \ 0.1582419524500696, \ 0.8337375286520506, \ 0.0080205188978798, \ 0.8337375286520506, \ 0.1582419524500696, \ 0.0080205188978798, \ 0.1582419524500696, \ 0.8337375286520506, \ 0.0080205188978798, \ 0.8337375286520506, \ 0.2270465393157224, \ 0.0761183063986999, \ 0.2270465393157224, \ 0.6968351542855776, \ 0.0761183063986999, \ 0.6968351542855776, \ 0.2270465393157224, \ 0.0761183063986999, \ 0.2270465393157224, \ 0.6968351542855776, \ 0.0761183063986999, \ 0.6968351542855776, \ 0.0107402320272877, \ 0.0674860196923538, \ 0.0107402320272877, \ 0.9217737482803585, \ 0.0674860196923538, \ 0.9217737482803585, \ 0.0107402320272877, \ 0.0674860196923538, \ 0.0107402320272877, \ 0.9217737482803585, \ 0.0674860196923538, \ 0.9217737482803585, \ 0.0450629754431290, \ 0.1495289831435912, \ 0.0450629754431290, \ 0.8054080414132798, \ 0.1495289831435912, \ 0.8054080414132798, \ 0.0450629754431290, \ 0.1495289831435912, \ 0.0450629754431290, \ 0.8054080414132798, \ 0.1495289831435912, \ 0.8054080414132798, \ 0.3899263834124326, \ 0.0536599901206231, \ 0.3899263834124326, \ 0.5564136264669443, \ 0.0536599901206231, \ 0.5564136264669443, \ 0.3899263834124326, \ 0.0536599901206231, \ 0.3899263834124326, \ 0.5564136264669443, \ 0.0536599901206231, \ 0.5564136264669443, \ 0.0142616821078597, \ 0.4039638109964547, \ 0.0142616821078597, \ 0.5817745068956857, \ 0.4039638109964547, \ 0.5817745068956857, \ 0.0142616821078597, \ 0.4039638109964547, \ 0.0142616821078597, \ 0.5817745068956857, \ 0.4039638109964547, \ 0.5817745068956857, \ 0.3314062155351160, \ 0.0088330831166114, \ 0.3314062155351160, \ 0.6597607013482726, \ 0.0088330831166114, \ 0.6597607013482726, \ 0.3314062155351160, \ 0.0088330831166114, \ 0.3314062155351160, \ 0.6597607013482726, \ 0.0088330831166114, \ 0.6597607013482726, \ 0.1223644156202090, \ 0.0103923793593637, \ 0.1223644156202090, \ 0.8672432050204273, \ 0.0103923793593637, \ 0.8672432050204273, \ 0.1223644156202090, \ 0.0103923793593637, \ 0.1223644156202090, \ 0.8672432050204273, \ 0.0103923793593637, \ 0.8672432050204273, \ 0.2929652829983945, \ 0.1324448044285512, \ 0.2929652829983945, \ 0.5745899125730543, \ 0.1324448044285512, \ 0.5745899125730543, \ 0.2929652829983945, \ 0.1324448044285512, \ 0.2929652829983945, \ 0.5745899125730543, \ 0.1324448044285512, \ 0.5745899125730543, \ 0.2534748953911949, \ 0.0535147091599154, \ 0.2534748953911949, \ 0.6930103954488898, \ 0.0535147091599154, \ 0.6930103954488898, \ 0.2534748953911949, \ 0.0535147091599154, \ 0.2534748953911949, \ 0.6930103954488898, \ 0.0535147091599154, \ 0.6930103954488898, \ 0.0440785872085001, \ 0.1461188761758480, \ 0.0440785872085001, \ 0.8098025366156519, \ 0.1461188761758480, \ 0.8098025366156519, \ 0.0440785872085001, \ 0.1461188761758480, \ 0.0440785872085001, \ 0.8098025366156519, \ 0.1461188761758480, \ 0.8098025366156519, \ 0.5031457633505478, \ 0.1264604222462178, \ 0.5031457633505478, \ 0.3703938144032344, \ 0.1264604222462178, \ 0.3703938144032344, \ 0.5031457633505478, \ 0.1264604222462178, \ 0.5031457633505478, \ 0.3703938144032344, \ 0.1264604222462178, \ 0.3703938144032344, \ 0.0087499979841172, \ 0.2433623618774898, \ 0.0087499979841172, \ 0.7478876401383930, \ 0.2433623618774898, \ 0.7478876401383930, \ 0.0087499979841172, \ 0.2433623618774898, \ 0.0087499979841172, \ 0.7478876401383930, \ 0.2433623618774898, \ 0.7478876401383930, \ 0.4433279475253553, \ 0.2569581578007861, \ 0.4433279475253553, \ 0.2997138946738586, \ 0.2569581578007861, \ 0.2997138946738586, \ 0.4433279475253553, \ 0.2569581578007861, \ 0.4433279475253553, \ 0.2997138946738586, \ 0.2569581578007861, \ 0.2997138946738586, \ 0.2485412372956145, \ 0.0112771540337380, \ 0.2485412372956145, \ 0.7401816086706475, \ 0.0112771540337380, \ 0.7401816086706475, \ 0.2485412372956145, \ 0.0112771540337380, \ 0.2485412372956145, \ 0.7401816086706475, \ 0.0112771540337380, \ 0.7401816086706475 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.3333333333333333, \ 0.4935398626809867, \ 0.0129202746380266, \ 0.4935398626809867, \ 0.2767659367008514, \ 0.4464681265982973, \ 0.2767659367008514, \ 0.0545777948502843, \ 0.8908444102994315, \ 0.0545777948502843, \ 0.0139969735218886, \ 0.9720060529562228, \ 0.0139969735218886, \ 0.1418740786751734, \ 0.7162518426496531, \ 0.1418740786751734, \ 0.1418740786751734, \ 0.7162518426496531, \ 0.1418740786751734, \ 0.0083955990101041, \ 0.9832088019797917, \ 0.0083955990101041, \ 0.0083955990101041, \ 0.9832088019797917, \ 0.0083955990101041, \ 0.1035973359771996, \ 0.7928053280456009, \ 0.1035973359771996, \ 0.1035973359771996, \ 0.7928053280456009, \ 0.1035973359771996, \ 0.3956080996828434, \ 0.2087838006343133, \ 0.3956080996828434, \ 0.3956080996828434, \ 0.2087838006343133, \ 0.3956080996828434, \ 0.0140019311977436, \ 0.9719961376045129, \ 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0.1178005364736766, \ 0.0355268541454468, \ 0.2876874815978630, \ 0.6767856642566902, \ 0.2876874815978630, \ 0.6767856642566902, \ 0.0355268541454468, \ 0.0355268541454468, \ 0.2876874815978630, \ 0.6767856642566902, \ 0.2876874815978630, \ 0.6767856642566902, \ 0.0355268541454468, \ 0.2073917255967352, \ 0.2839484400308330, \ 0.5086598343724318, \ 0.2839484400308330, \ 0.5086598343724318, \ 0.2073917255967352, \ 0.2073917255967352, \ 0.2839484400308330, \ 0.5086598343724318, \ 0.2839484400308330, \ 0.5086598343724318, \ 0.2073917255967352, \ 0.0862230026956853, \ 0.0282457003834953, \ 0.8855312969208194, \ 0.0282457003834953, \ 0.8855312969208194, \ 0.0862230026956853, \ 0.0862230026956853, \ 0.0282457003834953, \ 0.8855312969208194, \ 0.0282457003834953, \ 0.8855312969208194, \ 0.0862230026956853, \ 0.0056196516036861, \ 0.3915043117238656, \ 0.6028760366724483, \ 0.3915043117238656, \ 0.6028760366724483, \ 0.0056196516036861, \ 0.0056196516036861, \ 0.3915043117238656, \ 0.6028760366724483, \ 0.3915043117238656, \ 0.6028760366724483, \ 0.0056196516036861, \ 0.3389265718024346, \ 0.0737667613724345, \ 0.5873066668251310, \ 0.0737667613724345, \ 0.5873066668251310, \ 0.3389265718024346, \ 0.3389265718024346, \ 0.0737667613724345, \ 0.5873066668251310, \ 0.0737667613724345, \ 0.5873066668251310, \ 0.3389265718024346, \ 0.0080205188978798, \ 0.1582419524500696, \ 0.8337375286520506, \ 0.1582419524500696, \ 0.8337375286520506, \ 0.0080205188978798, \ 0.0080205188978798, \ 0.1582419524500696, \ 0.8337375286520506, \ 0.1582419524500696, \ 0.8337375286520506, \ 0.0080205188978798, \ 0.0761183063986999, \ 0.2270465393157224, \ 0.6968351542855776, \ 0.2270465393157224, \ 0.6968351542855776, \ 0.0761183063986999, \ 0.0761183063986999, \ 0.2270465393157224, \ 0.6968351542855776, \ 0.2270465393157224, \ 0.6968351542855776, \ 0.0761183063986999, \ 0.0674860196923538, \ 0.0107402320272877, \ 0.9217737482803585, \ 0.0107402320272877, \ 0.9217737482803585, \ 0.0674860196923538, \ 0.0674860196923538, \ 0.0107402320272877, \ 0.9217737482803585, \ 0.0107402320272877, \ 0.9217737482803585, \ 0.0674860196923538, \ 0.1495289831435912, \ 0.0450629754431290, \ 0.8054080414132798, \ 0.0450629754431290, \ 0.8054080414132798, \ 0.1495289831435912, \ 0.1495289831435912, \ 0.0450629754431290, \ 0.8054080414132798, \ 0.0450629754431290, \ 0.8054080414132798, \ 0.1495289831435912, \ 0.0536599901206231, \ 0.3899263834124326, \ 0.5564136264669443, \ 0.3899263834124326, \ 0.5564136264669443, \ 0.0536599901206231, \ 0.0536599901206231, \ 0.3899263834124326, \ 0.5564136264669443, \ 0.3899263834124326, \ 0.5564136264669443, \ 0.0536599901206231, \ 0.4039638109964547, \ 0.0142616821078597, \ 0.5817745068956857, \ 0.0142616821078597, \ 0.5817745068956857, \ 0.4039638109964547, \ 0.4039638109964547, \ 0.0142616821078597, \ 0.5817745068956857, \ 0.0142616821078597, \ 0.5817745068956857, \ 0.4039638109964547, \ 0.0088330831166114, \ 0.3314062155351160, \ 0.6597607013482726, \ 0.3314062155351160, \ 0.6597607013482726, \ 0.0088330831166114, \ 0.0088330831166114, \ 0.3314062155351160, \ 0.6597607013482726, \ 0.3314062155351160, \ 0.6597607013482726, \ 0.0088330831166114, \ 0.0103923793593637, \ 0.1223644156202090, \ 0.8672432050204273, \ 0.1223644156202090, \ 0.8672432050204273, \ 0.0103923793593637, \ 0.0103923793593637, \ 0.1223644156202090, \ 0.8672432050204273, \ 0.1223644156202090, \ 0.8672432050204273, \ 0.0103923793593637, \ 0.1324448044285512, \ 0.2929652829983945, \ 0.5745899125730543, \ 0.2929652829983945, \ 0.5745899125730543, \ 0.1324448044285512, \ 0.1324448044285512, \ 0.2929652829983945, \ 0.5745899125730543, \ 0.2929652829983945, \ 0.5745899125730543, \ 0.1324448044285512, \ 0.0535147091599154, \ 0.2534748953911949, \ 0.6930103954488898, \ 0.2534748953911949, \ 0.6930103954488898, \ 0.0535147091599154, \ 0.0535147091599154, \ 0.2534748953911949, \ 0.6930103954488898, \ 0.2534748953911949, \ 0.6930103954488898, \ 0.0535147091599154, \ 0.1461188761758480, \ 0.0440785872085001, \ 0.8098025366156519, \ 0.0440785872085001, \ 0.8098025366156519, \ 0.1461188761758480, \ 0.1461188761758480, \ 0.0440785872085001, \ 0.8098025366156519, \ 0.0440785872085001, \ 0.8098025366156519, \ 0.1461188761758480, \ 0.1264604222462178, \ 0.5031457633505478, \ 0.3703938144032344, \ 0.5031457633505478, \ 0.3703938144032344, \ 0.1264604222462178, \ 0.1264604222462178, \ 0.5031457633505478, \ 0.3703938144032344, \ 0.5031457633505478, \ 0.3703938144032344, \ 0.1264604222462178, \ 0.2433623618774898, \ 0.0087499979841172, \ 0.7478876401383930, \ 0.0087499979841172, \ 0.7478876401383930, \ 0.2433623618774898, \ 0.2433623618774898, \ 0.0087499979841172, \ 0.7478876401383930, \ 0.0087499979841172, \ 0.7478876401383930, \ 0.2433623618774898, \ 0.2569581578007861, \ 0.4433279475253553, \ 0.2997138946738586, \ 0.4433279475253553, \ 0.2997138946738586, \ 0.2569581578007861, \ 0.2569581578007861, \ 0.4433279475253553, \ 0.2997138946738586, \ 0.4433279475253553, \ 0.2997138946738586, \ 0.2569581578007861, \ 0.0112771540337380, \ 0.2485412372956145, \ 0.7401816086706475, \ 0.2485412372956145, \ 0.7401816086706475, \ 0.0112771540337380, \ 0.0112771540337380, \ 0.2485412372956145, \ 0.7401816086706475, \ 0.2485412372956145, \ 0.7401816086706475, \ 0.0112771540337380 ] ) z = np.array ( [ \ 0.7622415410729753, \ 0.2377584589270247, \ 0.5372666694077471, \ 0.4627333305922529, \ 0.9783048646074941, \ 0.0216951353925058, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.9776297496930976, \ 0.9776297496930976, \ 0.9776297496930976, \ 0.0223702503069024, \ 0.0223702503069024, \ 0.0223702503069024, \ 0.8365993771201422, \ 0.8365993771201422, \ 0.8365993771201422, \ 0.1634006228798578, \ 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0.0052380601608220, \ 0.0052380601608220, \ 0.0052380601608220, \ 0.0052380601608220, \ 0.0052380601608220, \ 0.0052380601608220, \ 0.0052380601608220, \ 0.0052380601608220, \ 0.0052380601608220, \ 0.0052380601608220, \ 0.0052380601608220, \ 0.0052380601608220, \ 0.0007780035531901, \ 0.0007780035531901, \ 0.0007780035531901, \ 0.0007780035531901, \ 0.0007780035531901, \ 0.0007780035531901, \ 0.0007780035531901, \ 0.0007780035531901, \ 0.0007780035531901, \ 0.0007780035531901, \ 0.0007780035531901, \ 0.0007780035531901, \ 0.0027997199341340, \ 0.0027997199341340, \ 0.0027997199341340, \ 0.0027997199341340, \ 0.0027997199341340, \ 0.0027997199341340, \ 0.0027997199341340, \ 0.0027997199341340, \ 0.0027997199341340, \ 0.0027997199341340, \ 0.0027997199341340, \ 0.0027997199341340, \ 0.0018871269258399, \ 0.0018871269258399, \ 0.0018871269258399, \ 0.0018871269258399, \ 0.0018871269258399, \ 0.0018871269258399, \ 0.0018871269258399, \ 0.0018871269258399, \ 0.0018871269258399, \ 0.0018871269258399, \ 0.0018871269258399, \ 0.0018871269258399 ] ) return x, y, z, w def rule20 ( ): #*****************************************************************************80 # ## rule20() returns the prism rule of precision 20. # # Discussion: # # We suppose we are given a triangular prism P with vertices # (1,0,0), (0,1,0), (0,0,0), # (1,0,1), (0,1,1), (0,0,1), # We call a rule with n points, returning coordinates # x, y, z, and weights w. Then the integral I of f(x,y,z) over P is # approximated by Q as follows: # # Q = volume(P) * sum ( 1 <= i <= n ) w(i) * f(x(i),y(i),z(i)) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 May 2023 # # Author: # # John Burkardt # # Reference: # # Jan Jaskowiec, Natarajan Sukumar, # High order symmetric cubature rules for tetrahedra and prisms, # International Journal of Numerical Methods in Engineering, # Volume 122, Number 1, pages 148-171, 24 August 2020. # # Output: # # real x(n), y(n), z(n): the coordinates of quadrature points. # # real w(n): the quadrature weights, which sum to 1. # import numpy as np x = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.0144110873698983, \ 0.0144110873698983, \ 0.9711778252602034, \ 0.0594974134225245, \ 0.0594974134225245, \ 0.8810051731549510, \ 0.4716249531173324, \ 0.4716249531173324, \ 0.0567500937653351, \ 0.4716249531173324, \ 0.4716249531173324, \ 0.0567500937653351, \ 0.4084460043805324, \ 0.4084460043805324, \ 0.1831079912389352, \ 0.4084460043805324, \ 0.4084460043805324, \ 0.1831079912389352, \ 0.2544280410411358, \ 0.2544280410411358, \ 0.4911439179177284, \ 0.2544280410411358, \ 0.2544280410411358, \ 0.4911439179177284, \ 0.4774877095488475, \ 0.4774877095488475, \ 0.0450245809023050, \ 0.4774877095488475, \ 0.4774877095488475, \ 0.0450245809023050, \ 0.0080292636682187, \ 0.0080292636682187, \ 0.9839414726635627, \ 0.0080292636682187, \ 0.0080292636682187, \ 0.9839414726635627, \ 0.4968613511179721, \ 0.4968613511179721, \ 0.0062772977640557, \ 0.4968613511179721, \ 0.4968613511179721, \ 0.0062772977640557, \ 0.4952615769053836, \ 0.4952615769053836, \ 0.0094768461892327, \ 0.4952615769053836, \ 0.4952615769053836, \ 0.0094768461892327, \ 0.0353578097171312, \ 0.0353578097171312, \ 0.9292843805657376, \ 0.0353578097171312, \ 0.0353578097171312, \ 0.9292843805657376, \ 0.2045970282680664, \ 0.2045970282680664, \ 0.5908059434638671, \ 0.2045970282680664, \ 0.2045970282680664, \ 0.5908059434638671, \ 0.1406131665045265, \ 0.1406131665045265, \ 0.7187736669909469, \ 0.1406131665045265, \ 0.1406131665045265, \ 0.7187736669909469, \ 0.0982561053377771, \ 0.0982561053377771, \ 0.8034877893244458, \ 0.0982561053377771, \ 0.0982561053377771, \ 0.8034877893244458, \ 0.0505951664588140, \ 0.0505951664588140, \ 0.8988096670823720, \ 0.0505951664588140, \ 0.0505951664588140, \ 0.8988096670823720, \ 0.4494558038329708, \ 0.4494558038329708, \ 0.1010883923340583, \ 0.4494558038329708, \ 0.4494558038329708, \ 0.1010883923340583, \ 0.2188296334498393, \ 0.2188296334498393, \ 0.5623407331003214, \ 0.2188296334498393, \ 0.2188296334498393, \ 0.5623407331003214, \ 0.3839280551265914, \ 0.3839280551265914, \ 0.2321438897468171, \ 0.3839280551265914, \ 0.3839280551265914, \ 0.2321438897468171, \ 0.0599706591707595, \ 0.0599706591707595, \ 0.8800586816584810, \ 0.0599706591707595, \ 0.0599706591707595, \ 0.8800586816584810, \ 0.2735494152095392, \ 0.2735494152095392, \ 0.4529011695809215, \ 0.2735494152095392, \ 0.2735494152095392, \ 0.4529011695809215, \ 0.4907393281824849, \ 0.4907393281824849, \ 0.0185213436350301, \ 0.4907393281824849, \ 0.4907393281824849, \ 0.0185213436350301, \ 0.0114930108302544, \ 0.0114930108302544, \ 0.9770139783394911, \ 0.0114930108302544, \ 0.0114930108302544, \ 0.9770139783394911, \ 0.4048678509173179, \ 0.4048678509173179, \ 0.1902642981653643, \ 0.4048678509173179, \ 0.4048678509173179, \ 0.1902642981653643, \ 0.2917111538437177, \ 0.2917111538437177, \ 0.4165776923125646, \ 0.2917111538437177, \ 0.2917111538437177, \ 0.4165776923125646, \ 0.4459340787080981, \ 0.4459340787080981, \ 0.1081318425838039, \ 0.4459340787080981, \ 0.4459340787080981, \ 0.1081318425838039, \ 0.2107382124681030, \ 0.0313712403793080, \ 0.2107382124681030, \ 0.7578905471525890, \ 0.0313712403793080, \ 0.7578905471525890, \ 0.2031438036613264, \ 0.1114400380412596, \ 0.2031438036613264, \ 0.6854161582974140, \ 0.1114400380412596, \ 0.6854161582974140, \ 0.3734168270486775, \ 0.0362901207122587, \ 0.3734168270486775, \ 0.5902930522390638, \ 0.0362901207122587, \ 0.5902930522390638, \ 0.5331515253263366, \ 0.1335223252058013, \ 0.5331515253263366, \ 0.3333261494678621, \ 0.1335223252058013, \ 0.3333261494678621, \ 0.5331515253263366, \ 0.1335223252058013, \ 0.5331515253263366, \ 0.3333261494678621, \ 0.1335223252058013, \ 0.3333261494678621, \ 0.0031052202964266, \ 0.0650661770452139, \ 0.0031052202964266, \ 0.9318286026583595, \ 0.0650661770452139, \ 0.9318286026583595, \ 0.0031052202964266, \ 0.0650661770452139, \ 0.0031052202964266, \ 0.9318286026583595, \ 0.0650661770452139, \ 0.9318286026583595, \ 0.7387825606686489, \ 0.0932063677657537, \ 0.7387825606686489, \ 0.1680110715655975, \ 0.0932063677657537, \ 0.1680110715655975, \ 0.7387825606686489, \ 0.0932063677657537, \ 0.7387825606686489, \ 0.1680110715655975, \ 0.0932063677657537, \ 0.1680110715655975, \ 0.2402209181076414, \ 0.0342816904321169, \ 0.2402209181076414, \ 0.7254973914602417, \ 0.0342816904321169, \ 0.7254973914602417, \ 0.2402209181076414, \ 0.0342816904321169, \ 0.2402209181076414, \ 0.7254973914602417, \ 0.0342816904321169, \ 0.7254973914602417, \ 0.0816129838618181, \ 0.3087274419444517, \ 0.0816129838618181, \ 0.6096595741937303, \ 0.3087274419444517, \ 0.6096595741937303, \ 0.0816129838618181, \ 0.3087274419444517, \ 0.0816129838618181, \ 0.6096595741937303, \ 0.3087274419444517, \ 0.6096595741937303, \ 0.1728159430622389, \ 0.3228866908185781, \ 0.1728159430622389, \ 0.5042973661191831, \ 0.3228866908185781, \ 0.5042973661191831, \ 0.1728159430622389, \ 0.3228866908185781, \ 0.1728159430622389, \ 0.5042973661191831, \ 0.3228866908185781, \ 0.5042973661191831, \ 0.2247127800382396, \ 0.1327427633967232, \ 0.2247127800382396, \ 0.6425444565650372, \ 0.1327427633967232, \ 0.6425444565650372, \ 0.2247127800382396, \ 0.1327427633967232, \ 0.2247127800382396, \ 0.6425444565650372, \ 0.1327427633967232, \ 0.6425444565650372, \ 0.0285010351208627, \ 0.1058138309132070, \ 0.0285010351208627, \ 0.8656851339659303, \ 0.1058138309132070, \ 0.8656851339659303, \ 0.0285010351208627, \ 0.1058138309132070, \ 0.0285010351208627, \ 0.8656851339659303, \ 0.1058138309132070, \ 0.8656851339659303, \ 0.3896100790636586, \ 0.0109712296494740, \ 0.3896100790636586, \ 0.5994186912868675, \ 0.0109712296494740, \ 0.5994186912868675, \ 0.3896100790636586, \ 0.0109712296494740, \ 0.3896100790636586, \ 0.5994186912868675, \ 0.0109712296494740, \ 0.5994186912868675, \ 0.2493335646225524, \ 0.3397868885786173, \ 0.2493335646225524, \ 0.4108795467988303, \ 0.3397868885786173, \ 0.4108795467988303, \ 0.2493335646225524, \ 0.3397868885786173, \ 0.2493335646225524, \ 0.4108795467988303, \ 0.3397868885786173, \ 0.4108795467988303, \ 0.0070420236000393, \ 0.3177207979690921, \ 0.0070420236000393, \ 0.6752371784308686, \ 0.3177207979690921, \ 0.6752371784308686, \ 0.0070420236000393, \ 0.3177207979690921, \ 0.0070420236000393, \ 0.6752371784308686, \ 0.3177207979690921, \ 0.6752371784308686, \ 0.1229469656661441, \ 0.0156675786078851, \ 0.1229469656661441, \ 0.8613854557259707, \ 0.0156675786078851, \ 0.8613854557259707, \ 0.1229469656661441, \ 0.0156675786078851, \ 0.1229469656661441, \ 0.8613854557259707, \ 0.0156675786078851, \ 0.8613854557259707, \ 0.2259160262165568, \ 0.0521518846441873, \ 0.2259160262165568, \ 0.7219320891392559, \ 0.0521518846441873, \ 0.7219320891392559, \ 0.2259160262165568, \ 0.0521518846441873, \ 0.2259160262165568, \ 0.7219320891392559, \ 0.0521518846441873, \ 0.7219320891392559, \ 0.0094714933266370, \ 0.0496441415264223, \ 0.0094714933266370, \ 0.9408843651469406, \ 0.0496441415264223, \ 0.9408843651469406, \ 0.0094714933266370, \ 0.0496441415264223, \ 0.0094714933266370, \ 0.9408843651469406, \ 0.0496441415264223, \ 0.9408843651469406, \ 0.0524311424127234, \ 0.1452721772236918, \ 0.0524311424127234, \ 0.8022966803635848, \ 0.1452721772236918, \ 0.8022966803635848, \ 0.0524311424127234, \ 0.1452721772236918, \ 0.0524311424127234, \ 0.8022966803635848, \ 0.1452721772236918, \ 0.8022966803635848, \ 0.0088722025555106, \ 0.3816302433543118, \ 0.0088722025555106, \ 0.6094975540901776, \ 0.3816302433543118, \ 0.6094975540901776, \ 0.0088722025555106, \ 0.3816302433543118, \ 0.0088722025555106, \ 0.6094975540901776, \ 0.3816302433543118, \ 0.6094975540901776, \ 0.2592518129005248, \ 0.5551397100853696, \ 0.2592518129005248, \ 0.1856084770141057, \ 0.5551397100853696, \ 0.1856084770141057, \ 0.2592518129005248, \ 0.5551397100853696, \ 0.2592518129005248, \ 0.1856084770141057, \ 0.5551397100853696, \ 0.1856084770141057, \ 0.3527097683072426, \ 0.0454609714340920, \ 0.3527097683072426, \ 0.6018292602586655, \ 0.0454609714340920, \ 0.6018292602586655, \ 0.3527097683072426, \ 0.0454609714340920, \ 0.3527097683072426, \ 0.6018292602586655, \ 0.0454609714340920, \ 0.6018292602586655, \ 0.1450709676542589, \ 0.0118170254385161, \ 0.1450709676542589, \ 0.8431120069072251, \ 0.0118170254385161, \ 0.8431120069072251, \ 0.1450709676542589, \ 0.0118170254385161, \ 0.1450709676542589, \ 0.8431120069072251, \ 0.0118170254385161, \ 0.8431120069072251, \ 0.1874143309040665, \ 0.1371981166173774, \ 0.1874143309040665, \ 0.6753875524785560, \ 0.1371981166173774, \ 0.6753875524785560, \ 0.1874143309040665, \ 0.1371981166173774, \ 0.1874143309040665, \ 0.6753875524785560, \ 0.1371981166173774, \ 0.6753875524785560, \ 0.0681401608659605, \ 0.1498311852045099, \ 0.0681401608659605, \ 0.7820286539295297, \ 0.1498311852045099, \ 0.7820286539295297, \ 0.0681401608659605, \ 0.1498311852045099, \ 0.0681401608659605, \ 0.7820286539295297, \ 0.1498311852045099, \ 0.7820286539295297, \ 0.3340585216441213, \ 0.0549696286134690, \ 0.3340585216441213, \ 0.6109718497424097, \ 0.0549696286134690, \ 0.6109718497424097, \ 0.3340585216441213, \ 0.0549696286134690, \ 0.3340585216441213, \ 0.6109718497424097, \ 0.0549696286134690, \ 0.6109718497424097, \ 0.0678515797670052, \ 0.1205147270278029, \ 0.0678515797670052, \ 0.8116336932051920, \ 0.1205147270278029, \ 0.8116336932051920, \ 0.0678515797670052, \ 0.1205147270278029, \ 0.0678515797670052, \ 0.8116336932051920, \ 0.1205147270278029, \ 0.8116336932051920, \ 0.5023136055537912, \ 0.1076311573233394, \ 0.5023136055537912, \ 0.3900552371228693, \ 0.1076311573233394, \ 0.3900552371228693, \ 0.5023136055537912, \ 0.1076311573233394, \ 0.5023136055537912, \ 0.3900552371228693, \ 0.1076311573233394, \ 0.3900552371228693, \ 0.0078193997434242, \ 0.2556542053755758, \ 0.0078193997434242, \ 0.7365263948810000, \ 0.2556542053755758, \ 0.7365263948810000, \ 0.0078193997434242, \ 0.2556542053755758, \ 0.0078193997434242, \ 0.7365263948810000, \ 0.2556542053755758, \ 0.7365263948810000, \ 0.6609230621965876, \ 0.0983956958507364, \ 0.6609230621965876, \ 0.2406812419526760, \ 0.0983956958507364, \ 0.2406812419526760, \ 0.6609230621965876, \ 0.0983956958507364, \ 0.6609230621965876, \ 0.2406812419526760, \ 0.0983956958507364, \ 0.2406812419526760, \ 0.1784736485903538, \ 0.0017965399943124, \ 0.1784736485903538, \ 0.8197298114153339, \ 0.0017965399943124, \ 0.8197298114153339, \ 0.1784736485903538, \ 0.0017965399943124, \ 0.1784736485903538, \ 0.8197298114153339, \ 0.0017965399943124, \ 0.8197298114153339, \ 0.2532227359651285, \ 0.0131217590858908, \ 0.2532227359651285, \ 0.7336555049489807, \ 0.0131217590858908, \ 0.7336555049489807, \ 0.2532227359651285, \ 0.0131217590858908, \ 0.2532227359651285, \ 0.7336555049489807, \ 0.0131217590858908, \ 0.7336555049489807, \ 0.0144350536713311, \ 0.0686738759058901, \ 0.0144350536713311, \ 0.9168910704227788, \ 0.0686738759058901, \ 0.9168910704227788, \ 0.0144350536713311, \ 0.0686738759058901, \ 0.0144350536713311, \ 0.9168910704227788, \ 0.0686738759058901, \ 0.9168910704227788, \ 0.2794904874495157, \ 0.1274003485132245, \ 0.2794904874495157, \ 0.5931091640372598, \ 0.1274003485132245, \ 0.5931091640372598, \ 0.2794904874495157, \ 0.1274003485132245, \ 0.2794904874495157, \ 0.5931091640372598, \ 0.1274003485132245, \ 0.5931091640372598 ] ) y = np.array ( [ \ 0.3333333333333333, \ 0.3333333333333333, \ 0.0144110873698983, \ 0.9711778252602034, \ 0.0144110873698983, \ 0.0594974134225245, \ 0.8810051731549510, \ 0.0594974134225245, \ 0.4716249531173324, \ 0.0567500937653351, \ 0.4716249531173324, \ 0.4716249531173324, \ 0.0567500937653351, \ 0.4716249531173324, \ 0.4084460043805324, \ 0.1831079912389352, \ 0.4084460043805324, \ 0.4084460043805324, \ 0.1831079912389352, \ 0.4084460043805324, \ 0.2544280410411358, \ 0.4911439179177284, \ 0.2544280410411358, \ 0.2544280410411358, \ 0.4911439179177284, \ 0.2544280410411358, \ 0.4774877095488475, \ 0.0450245809023050, \ 0.4774877095488475, \ 0.4774877095488475, \ 0.0450245809023050, \ 0.4774877095488475, \ 0.0080292636682187, \ 0.9839414726635627, \ 0.0080292636682187, \ 0.0080292636682187, \ 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0.6753875524785560, \ 0.1371981166173774, \ 0.1498311852045099, \ 0.0681401608659605, \ 0.7820286539295297, \ 0.0681401608659605, \ 0.7820286539295297, \ 0.1498311852045099, \ 0.1498311852045099, \ 0.0681401608659605, \ 0.7820286539295297, \ 0.0681401608659605, \ 0.7820286539295297, \ 0.1498311852045099, \ 0.0549696286134690, \ 0.3340585216441213, \ 0.6109718497424097, \ 0.3340585216441213, \ 0.6109718497424097, \ 0.0549696286134690, \ 0.0549696286134690, \ 0.3340585216441213, \ 0.6109718497424097, \ 0.3340585216441213, \ 0.6109718497424097, \ 0.0549696286134690, \ 0.1205147270278029, \ 0.0678515797670052, \ 0.8116336932051920, \ 0.0678515797670052, \ 0.8116336932051920, \ 0.1205147270278029, \ 0.1205147270278029, \ 0.0678515797670052, \ 0.8116336932051920, \ 0.0678515797670052, \ 0.8116336932051920, \ 0.1205147270278029, \ 0.1076311573233394, \ 0.5023136055537912, \ 0.3900552371228693, \ 0.5023136055537912, \ 0.3900552371228693, \ 0.1076311573233394, \ 0.1076311573233394, \ 0.5023136055537912, \ 0.3900552371228693, \ 0.5023136055537912, \ 0.3900552371228693, \ 0.1076311573233394, \ 0.2556542053755758, \ 0.0078193997434242, \ 0.7365263948810000, \ 0.0078193997434242, \ 0.7365263948810000, \ 0.2556542053755758, \ 0.2556542053755758, \ 0.0078193997434242, \ 0.7365263948810000, \ 0.0078193997434242, \ 0.7365263948810000, \ 0.2556542053755758, \ 0.0983956958507364, \ 0.6609230621965876, \ 0.2406812419526760, \ 0.6609230621965876, \ 0.2406812419526760, \ 0.0983956958507364, \ 0.0983956958507364, \ 0.6609230621965876, \ 0.2406812419526760, \ 0.6609230621965876, \ 0.2406812419526760, \ 0.0983956958507364, \ 0.0017965399943124, \ 0.1784736485903538, \ 0.8197298114153339, \ 0.1784736485903538, \ 0.8197298114153339, \ 0.0017965399943124, \ 0.0017965399943124, \ 0.1784736485903538, \ 0.8197298114153339, \ 0.1784736485903538, \ 0.8197298114153339, \ 0.0017965399943124, \ 0.0131217590858908, \ 0.2532227359651285, \ 0.7336555049489807, \ 0.2532227359651285, \ 0.7336555049489807, \ 0.0131217590858908, \ 0.0131217590858908, \ 0.2532227359651285, \ 0.7336555049489807, \ 0.2532227359651285, \ 0.7336555049489807, \ 0.0131217590858908, \ 0.0686738759058901, \ 0.0144350536713311, \ 0.9168910704227788, \ 0.0144350536713311, \ 0.9168910704227788, \ 0.0686738759058901, \ 0.0686738759058901, \ 0.0144350536713311, \ 0.9168910704227788, \ 0.0144350536713311, \ 0.9168910704227788, \ 0.0686738759058901, \ 0.1274003485132245, \ 0.2794904874495157, \ 0.5931091640372598, \ 0.2794904874495157, \ 0.5931091640372598, \ 0.1274003485132245, \ 0.1274003485132245, \ 0.2794904874495157, \ 0.5931091640372598, \ 0.2794904874495157, \ 0.5931091640372598, \ 0.1274003485132245 ] ) z = np.array ( [ \ 0.9210083004731364, \ 0.0789916995268636, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.5000000000000000, \ 0.9076441430436997, \ 0.9076441430436997, \ 0.9076441430436997, \ 0.0923558569563003, \ 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0.0008460253945738, \ 0.0008460253945738, \ 0.0015975131973438, \ 0.0015975131973438, \ 0.0015975131973438, \ 0.0015975131973438, \ 0.0015975131973438, \ 0.0015975131973438, \ 0.0015975131973438, \ 0.0015975131973438, \ 0.0015975131973438, \ 0.0015975131973438, \ 0.0015975131973438, \ 0.0015975131973438, \ 0.0026781578703448, \ 0.0026781578703448, \ 0.0026781578703448, \ 0.0026781578703448, \ 0.0026781578703448, \ 0.0026781578703448, \ 0.0026781578703448, \ 0.0026781578703448, \ 0.0026781578703448, \ 0.0026781578703448, \ 0.0026781578703448, \ 0.0026781578703448, \ 0.0007373212126546, \ 0.0007373212126546, \ 0.0007373212126546, \ 0.0007373212126546, \ 0.0007373212126546, \ 0.0007373212126546, \ 0.0007373212126546, \ 0.0007373212126546, \ 0.0007373212126546, \ 0.0007373212126546, \ 0.0007373212126546, \ 0.0007373212126546, \ 0.0013900594133789, \ 0.0013900594133789, \ 0.0013900594133789, \ 0.0013900594133789, \ 0.0013900594133789, \ 0.0013900594133789, \ 0.0013900594133789, \ 0.0013900594133789, \ 0.0013900594133789, \ 0.0013900594133789, \ 0.0013900594133789, \ 0.0013900594133789, \ 0.0007964285504550, \ 0.0007964285504550, \ 0.0007964285504550, \ 0.0007964285504550, \ 0.0007964285504550, \ 0.0007964285504550, \ 0.0007964285504550, \ 0.0007964285504550, \ 0.0007964285504550, \ 0.0007964285504550, \ 0.0007964285504550, \ 0.0007964285504550, \ 0.0014601164425740, \ 0.0014601164425740, \ 0.0014601164425740, \ 0.0014601164425740, \ 0.0014601164425740, \ 0.0014601164425740, \ 0.0014601164425740, \ 0.0014601164425740, \ 0.0014601164425740, \ 0.0014601164425740, \ 0.0014601164425740, \ 0.0014601164425740, \ 0.0013483736840244, \ 0.0013483736840244, \ 0.0013483736840244, \ 0.0013483736840244, \ 0.0013483736840244, \ 0.0013483736840244, \ 0.0013483736840244, \ 0.0013483736840244, \ 0.0013483736840244, \ 0.0013483736840244, \ 0.0013483736840244, \ 0.0013483736840244, \ 0.0032860508781285, \ 0.0032860508781285, \ 0.0032860508781285, \ 0.0032860508781285, \ 0.0032860508781285, \ 0.0032860508781285, \ 0.0032860508781285, \ 0.0032860508781285, \ 0.0032860508781285, \ 0.0032860508781285, \ 0.0032860508781285, \ 0.0032860508781285, \ 0.0011247201187777, \ 0.0011247201187777, \ 0.0011247201187777, \ 0.0011247201187777, \ 0.0011247201187777, \ 0.0011247201187777, \ 0.0011247201187777, \ 0.0011247201187777, \ 0.0011247201187777, \ 0.0011247201187777, \ 0.0011247201187777, \ 0.0011247201187777, \ 0.0021271072748908, \ 0.0021271072748908, \ 0.0021271072748908, \ 0.0021271072748908, \ 0.0021271072748908, \ 0.0021271072748908, \ 0.0021271072748908, \ 0.0021271072748908, \ 0.0021271072748908, \ 0.0021271072748908, \ 0.0021271072748908, \ 0.0021271072748908, \ 0.0007135219223491, \ 0.0007135219223491, \ 0.0007135219223491, \ 0.0007135219223491, \ 0.0007135219223491, \ 0.0007135219223491, \ 0.0007135219223491, \ 0.0007135219223491, \ 0.0007135219223491, \ 0.0007135219223491, \ 0.0007135219223491, \ 0.0007135219223491, \ 0.0003719588400836, \ 0.0003719588400836, \ 0.0003719588400836, \ 0.0003719588400836, \ 0.0003719588400836, \ 0.0003719588400836, \ 0.0003719588400836, \ 0.0003719588400836, \ 0.0003719588400836, \ 0.0003719588400836, \ 0.0003719588400836, \ 0.0003719588400836, \ 0.0002151829924432, \ 0.0002151829924432, \ 0.0002151829924432, \ 0.0002151829924432, \ 0.0002151829924432, \ 0.0002151829924432, \ 0.0002151829924432, \ 0.0002151829924432, \ 0.0002151829924432, \ 0.0002151829924432, \ 0.0002151829924432, \ 0.0002151829924432, \ 0.0008163226512583, \ 0.0008163226512583, \ 0.0008163226512583, \ 0.0008163226512583, \ 0.0008163226512583, \ 0.0008163226512583, \ 0.0008163226512583, \ 0.0008163226512583, \ 0.0008163226512583, \ 0.0008163226512583, \ 0.0008163226512583, \ 0.0008163226512583 ] ) return x, y, z, w def timestamp ( ): #*****************************************************************************80 # ## timestamp() prints the date as a timestamp. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 21 August 2019 # # Author: # # John Burkardt # import time t = time.time ( ) print ( time.ctime ( t ) ) return if ( __name__ == '__main__' ): timestamp ( ) prism_jaskowiec_rule_test ( ) timestamp ( )