#! /usr/bin/env python3 # def i4vec_print ( n, a, title ): #*****************************************************************************80 # ## i4vec_print() prints an I4VEC. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 31 August 2014 # # Author: # # John Burkardt # # Input: # # integer N, the dimension of the vector. # # integer A(N), the vector to be printed. # # string TITLE, a title. # print ( '' ) print ( title ) print ( '' ) for i in range ( 0, n ): print ( '%6d %6d' % ( i, a[i] ) ) return def i4vec_print_test ( ): #*****************************************************************************80 # ## i4vec_print_test() tests i4vec_print. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 25 September 2016 # # Author: # # John Burkardt # import numpy as np import platform print ( '' ) print ( 'i4vec_print_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' i4vec_print prints an I4VEC.' ) n = 4 v = np.array ( [ 91, 92, 93, 94 ], dtype = np.int32 ) i4vec_print ( n, v, ' Here is an I4VEC:' ) # # Terminate. # print ( '' ) print ( 'i4vec_print_test:' ) print ( ' Normal end of execution.' ) return def i4vec_transpose_print ( n, a, title ): #*****************************************************************************80 # ## i4vec_transpose_print prints an I4VEC "transposed". # # Example: # # A = (/ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 /) # TITLE = 'My vector: ' # # My vector: # # 1 2 3 4 5 # 6 7 8 9 10 # 11 # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 02 June 2015 # # Author: # # John Burkardt # # Input: # # integer N, the number of components of the vector. # # integer A(N), the vector to be printed. # # string TITLE, a title. # if ( 0 < len ( title ) ): print ( '' ) print ( title ) if ( 0 < n ): for i in range ( 0, n ): print ( '%8d' % ( a[i] ) ), if ( ( i + 1 ) % 10 == 0 or i == n - 1 ): print ( '' ) else: print ( ' (empty vector)' ) return def i4vec_transpose_print_test ( ): #*****************************************************************************80 # ## i4vec_transpose_print_test() tests i4vec_transpose_print. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 07 April 2015 # # Author: # # John Burkardt # import numpy as np import platform print ( '' ) print ( 'i4vec_transpose_print_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' i4vec_transpose_print prints an I4VEC' ) print ( ' with 5 entries to a row, and an optional title.' ) n = 12 a = np.zeros ( n, dtype = np.int32 ) for i in range ( 0, n ): a[i] = i + 1 i4vec_transpose_print ( n, a, ' My array: ' ) # # Terminate. # print ( '' ) print ( 'i4vec_transpose_print_test:' ) print ( ' Normal end of execution.' ) return def monomial_value ( m, n, 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: # # 07 April 2015 # # Author: # # John Burkardt # # Input: # # integer M, the spatial dimension. # # integer N, the number of evaluation points. # # integer E(M), the exponents. # # real X(M,N), the point coordinates. # # Output: # # real V(N), the monomial values. # import numpy as np v = np.ones ( n ) for i in range ( 0, m ): if ( 0 != e[i] ): for j in range ( 0, n ): v[j] = v[j] * x[i,j] ** e[i] return v def r8mat_print ( m, n, a, title ): #*****************************************************************************80 # ## r8mat_print prints an R8MAT. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 31 August 2014 # # Author: # # John Burkardt # # Input: # # integer M, the number of rows in A. # # integer N, the number of columns in A. # # real A(M,N), the matrix. # # string TITLE, a title. # r8mat_print_some ( m, n, a, 0, 0, m - 1, n - 1, title ) return def r8mat_print_test ( ): #*****************************************************************************80 # ## r8mat_print_test() tests r8mat_print. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 10 February 2015 # # Author: # # John Burkardt # import numpy as np import platform print ( '' ) print ( 'r8mat_print_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' r8mat_print prints an R8MAT.' ) m = 4 n = 6 v = np.array ( [ \ [ 11.0, 12.0, 13.0, 14.0, 15.0, 16.0 ], [ 21.0, 22.0, 23.0, 24.0, 25.0, 26.0 ], [ 31.0, 32.0, 33.0, 34.0, 35.0, 36.0 ], [ 41.0, 42.0, 43.0, 44.0, 45.0, 46.0 ] ], dtype = np.float64 ) r8mat_print ( m, n, v, ' Here is an R8MAT:' ) # # Terminate. # print ( '' ) print ( 'r8mat_print_test:' ) print ( ' Normal end of execution.' ) return def r8mat_print_some ( m, n, a, ilo, jlo, ihi, jhi, title ): #*****************************************************************************80 # ## r8mat_print_some prints out a portion of an R8MAT. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 10 February 2015 # # Author: # # John Burkardt # # Input: # # integer M, N, the number of rows and columns of the matrix. # # real A(M,N), an M by N matrix to be printed. # # integer ILO, JLO, the first row and column to print. # # integer IHI, JHI, the last row and column to print. # # string TITLE, a title. # incx = 5 print ( '' ) print ( title ) if ( m <= 0 or n <= 0 ): print ( '' ) print ( ' (None)' ) return for j2lo in range ( max ( jlo, 0 ), min ( jhi + 1, n ), incx ): j2hi = j2lo + incx - 1 j2hi = min ( j2hi, n ) j2hi = min ( j2hi, jhi ) print ( '' ) print ( ' Col: ' ), for j in range ( j2lo, j2hi + 1 ): print ( '%7d ' % ( j ) ), print ( '' ) print ( ' Row' ) i2lo = max ( ilo, 0 ) i2hi = min ( ihi, m ) for i in range ( i2lo, i2hi + 1 ): print ( '%7d :' % ( i ) ), for j in range ( j2lo, j2hi + 1 ): print ( '%12g ' % ( a[i,j] ) ), print ( '' ) return def r8mat_print_some_test ( ): #*****************************************************************************80 # ## r8mat_print_some_test() tests r8mat_print_some. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 31 October 2014 # # Author: # # John Burkardt # import numpy as np import platform print ( '' ) print ( 'r8mat_print_some_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' r8mat_print_some prints some of an R8MAT.' ) m = 4 n = 6 v = np.array ( [ \ [ 11.0, 12.0, 13.0, 14.0, 15.0, 16.0 ], [ 21.0, 22.0, 23.0, 24.0, 25.0, 26.0 ], [ 31.0, 32.0, 33.0, 34.0, 35.0, 36.0 ], [ 41.0, 42.0, 43.0, 44.0, 45.0, 46.0 ] ], dtype = np.float64 ) r8mat_print_some ( m, n, v, 0, 3, 2, 5, ' Here is an R8MAT:' ) # # Terminate. # print ( '' ) print ( 'r8mat_print_some_test:' ) print ( ' Normal end of execution.' ) return def r8mat_transpose_print ( m, n, a, title ): #*****************************************************************************80 # ## r8mat_transpose_print prints an R8MAT, transposed. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 31 August 2014 # # Author: # # John Burkardt # # Input: # # integer M, the number of rows in A. # # integer N, the number of columns in A. # # real A(M,N), the matrix. # # string TITLE, a title. # r8mat_transpose_print_some ( m, n, a, 0, 0, m - 1, n - 1, title ) return def r8mat_transpose_print_test ( ): #*****************************************************************************80 # ## r8mat_transpose_print_test() tests r8mat_transpose_print. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 31 October 2014 # # Author: # # John Burkardt # import numpy as np import platform print ( '' ) print ( 'r8mat_transpose_print_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' r8mat_transpose_print prints an R8MAT.' ) m = 4 n = 3 v = np.array ( [ \ [ 11.0, 12.0, 13.0 ], [ 21.0, 22.0, 23.0 ], [ 31.0, 32.0, 33.0 ], [ 41.0, 42.0, 43.0 ] ], dtype = np.float64 ) r8mat_transpose_print ( m, n, v, ' Here is an R8MAT, transposed:' ) # # Terminate. # print ( '' ) print ( 'r8mat_transpose_print_test:' ) print ( ' Normal end of execution.' ) return def r8mat_transpose_print_some ( m, n, a, ilo, jlo, ihi, jhi, title ): #*****************************************************************************80 # ## r8mat_transpose_print_some prints a portion of an R8MAT, transposed. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 13 November 2014 # # Author: # # John Burkardt # # Input: # # integer M, N, the number of rows and columns of the matrix. # # real A(M,N), an M by N matrix to be printed. # # integer ILO, JLO, the first row and column to print. # # integer IHI, JHI, the last row and column to print. # # string TITLE, a title. # incx = 5 print ( '' ) print ( title ) if ( m <= 0 or n <= 0 ): print ( '' ) print ( ' (None)' ) return for i2lo in range ( max ( ilo, 0 ), min ( ihi, m - 1 ), incx ): i2hi = i2lo + incx - 1 i2hi = min ( i2hi, m - 1 ) i2hi = min ( i2hi, ihi ) print ( '' ) print ( ' Row: ' ), for i in range ( i2lo, i2hi + 1 ): print ( '%7d ' % ( i ) ), print ( '' ) print ( ' Col' ) j2lo = max ( jlo, 0 ) j2hi = min ( jhi, n - 1 ) for j in range ( j2lo, j2hi + 1 ): print ( '%7d :' % ( j ) ), for i in range ( i2lo, i2hi + 1 ): print ( '%12g ' % ( a[i,j] ) ), print ( '' ) return def r8mat_transpose_print_some_test ( ): #*****************************************************************************80 # ## r8mat_transpose_print_some_test() tests r8mat_transpose_print_some. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 31 October 2014 # # Author: # # John Burkardt # import numpy as np import platform print ( '' ) print ( 'r8mat_transpose_print_some_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' r8mat_transpose_print_some prints some of an R8MAT, transposed.' ) m = 4 n = 6 v = np.array ( [ \ [ 11.0, 12.0, 13.0, 14.0, 15.0, 16.0 ], [ 21.0, 22.0, 23.0, 24.0, 25.0, 26.0 ], [ 31.0, 32.0, 33.0, 34.0, 35.0, 36.0 ], [ 41.0, 42.0, 43.0, 44.0, 45.0, 46.0 ] ], dtype = np.float64 ) r8mat_transpose_print_some ( m, n, v, 0, 3, 2, 5, ' R8MAT, rows 0:2, cols 3:5:' ) # # Terminate. # print ( '' ) print ( 'r8mat_transpose_print_some_test:' ) print ( ' Normal end of execution.' ) return def r8vec_print ( n, a, title ): #*****************************************************************************80 # ## r8vec_print prints an R8VEC. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 31 August 2014 # # Author: # # John Burkardt # # Input: # # integer N, the dimension of the vector. # # real A(N), the vector to be printed. # # string TITLE, a title. # print ( '' ) print ( title ) print ( '' ) for i in range ( 0, n ): print ( '%6d: %12g' % ( i, a[i] ) ) def r8vec_print_test ( ): #*****************************************************************************80 # ## r8vec_print_test() tests r8vec_print. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 29 October 2014 # # Author: # # John Burkardt # import numpy as np import platform print ( '' ) print ( 'r8vec_print_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' r8vec_print prints an R8VEC.' ) n = 4 v = np.array ( [ 123.456, 0.000005, -1.0E+06, 3.14159265 ], dtype = np.float64 ) r8vec_print ( n, v, ' Here is an R8VEC:' ) # # Terminate. # print ( '' ) print ( 'r8vec_print_test:' ) print ( ' Normal end of execution.' ) return def sphere01_area ( ): #*****************************************************************************80 # ## sphere01_area returns the area of the surface of the unit sphere in 3D. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 24 June 2015 # # Author: # # John Burkardt # # Output: # # real VALUE, the area. # import numpy as np r = 1.0 value = 4.0 * np.pi * r * r return value def sphere01_area_test ( ) : #*****************************************************************************80 # ## sphere01_area_test() tests sphere01_area. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 23 June 2015 # # Author: # # John Burkardt # import platform print ( '' ) print ( 'sphere01_area_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' sphere01_area returns the volume of the unit sphere.' ) print ( '' ) value = sphere01_area ( ) print ( ' sphere01_area() = %g' % ( value ) ) # # Terminate. # print ( '' ) print ( 'sphere01_area_test' ) print ( ' Normal end of execution.' ) return def sphere01_monomial_integral ( e ): #*****************************************************************************80 # ## sphere01_monomial_integral returns monomial integrals on the unit sphere. # # Discussion: # # The integration region is # # X^2 + Y^2 + Z^2 = 1. # # The monomial is F(X,Y,Z) = X^E(1) * Y^E(2) * Z^E(3). # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 23 June 2015 # # Author: # # John Burkardt # # Reference: # # Philip Davis, Philip Rabinowitz, # Methods of Numerical Integration, # Second Edition, # Academic Press, 1984, page 263. # # Input: # # integer E(3), the exponents of X, Y and Z in the # monomial. Each exponent must be nonnegative. # # Output: # # real INTEGRAL, the integral. # from scipy.special import gamma import numpy as np for i in range ( 0, 3 ): if ( e[i] < 0 ): print ( '' ) print ( 'sphere01_monomial_integral - Fatal error!' ) print ( ' All exponents must be nonnegative.' ) raise Exception ( 'sphere01_monomial_integral - Fatal error!' ) if ( e[0] == 0 and e[1] == 0 and e[2] == 0 ): integral = 2.0 * np.sqrt ( ( np.pi ) ** 3 ) / gamma ( 1.5 ) return integral for i in range ( 0, 3 ): if ( ( e[i] % 2 ) == 1 ): integral = 0.0 return integral integral = 2.0 e_sum = 0 for i in range ( 0, 3 ): integral = integral * gamma ( 0.5 * float ( e[i] + 1 ) ) e_sum = e_sum + e[i] + 1 integral = integral / gamma ( 0.5 * float ( e_sum ) ) return integral def sphere01_monomial_integral_test ( ): #*****************************************************************************80 # ## sphere01_monomial_integral_test() tests sphere01_monomial_integral. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 22 June 2015 # # Author: # # John Burkardt # import numpy as np import platform m = 3 print ( '' ) print ( 'sphere01_monomial_integral_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' sphere01_monomial_integral returns the integral of a monomial' ) print ( ' over the surface of the unit sphere in 3D.' ) print ( ' Compare with a Monte Carlo estimate.' ) # # Get sample points. # n = 8192 x = sphere01_sample ( n ) # # Randomly choose X,Y,Z exponents between (0,0,0) and (9,9,9). # print ( '' ) print ( ' If any exponent is odd, the integral is zero.' ) print ( ' We will restrict this test to randomly chosen even exponents.' ) print ( '' ) print ( ' Ex Ey Ez MC-Estimate Exact Error' ) print ( '' ) test_num = 20 for test in range ( 0, test_num ): e = np.random.random_integers ( 0, 4, size = m ) for i in range ( 0, m ): e[i] = e[i] * 2 value = monomial_value ( m, n, e, x ) result = sphere01_area ( ) * np.sum ( value ) / float ( n ) exact = sphere01_monomial_integral ( e ) error = abs ( result - exact ) for i in range ( 0, m ): print ( ' %2d' % ( e[i] ) ), print ( ' %14.6g %14.6g %10.2g' % ( result, exact, error ) ) # # Terminate. # print ( '' ) print ( 'sphere01_monomial_integral_test' ) print ( ' Normal end of execution.' ) return def sphere01_monte_carlo_test ( ): #*****************************************************************************80 # ## sphere01_monte_carlo_test uses the Monte Carlo method on the unit sphere. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 14 November 2016 # # Author: # # John Burkardt # import numpy as np e_test = np.array ( [ \ [ 0, 0, 0 ], \ [ 2, 0, 0 ], \ [ 0, 2, 0 ], \ [ 0, 0, 2 ], \ [ 4, 0, 0 ], \ [ 2, 2, 0 ], \ [ 0, 0, 4 ] ] ) print ( '' ) print ( 'sphere01_monte_carlo_test' ) print ( ' Use sphere01_sample to estimate integrals over ' ) print ( ' the surface of the unit sphere.' ) print ( '' ) print ( ' N 1 X^2 Y^2' ), print ( ' Z^2 X^4 X^2Y^2 Z^4' ) print ( '' ) n = 1 e = np.zeros ( 3, dtype = np.int32 ) while ( n <= 65536 ): x = sphere01_sample ( n ) print ( ' %8d' % ( n ) ), for j in range ( 0, 7 ): e[0:3] = e_test[j,0:3] value = monomial_value ( 3, n, e, x ) result = sphere01_area ( ) * np.sum ( value[0:n] ) / float ( n ) print ( ' %14f' % ( result ) ), print ( '' ) n = 2 * n print ( '' ) print ( ' Exact' ), for j in range ( 0, 7 ): e[0:3] = e_test[j,0:3] result = sphere01_monomial_integral ( e ) print ( ' %14f' % ( result ) ), print ( '' ) return def sphere01_sample ( n ): #*****************************************************************************80 # ## sphere01_sample samples points from the surface of the unit sphere in 3D. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 23 June 2015 # # Author: # # John Burkardt # # Reference: # # Russell Cheng, # Random Variate Generation, # in Handbook of Simulation, # edited by Jerry Banks, # Wiley, 1998, pages 168. # # Reuven Rubinstein, # Monte Carlo Optimization, Simulation, and Sensitivity # of Queueing Networks, # Krieger, 1992, # ISBN: 0894647644, # LC: QA298.R79. # # Input: # # integer N, the number of points. # # Output: # # real X(3,N), the points. # import numpy as np m = 3 x = np.zeros ( [ m, n ] ) for j in range ( 0, n ): # # Fill a vector with normally distributed values. # v = np.random.normal ( 0.0, 1.0, size = m ) # # Compute the length of the vector. # norm = np.linalg.norm ( v ) # # Normalize the vector. # for i in range ( 0, m ): x[i,j] = v[i] / norm return x def sphere01_sample_test ( ): #*****************************************************************************80 # ## sphere01_sample_test() tests sphere01_sample. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 23 June 2015 # # Author: # # John Burkardt # import platform print ( '' ) print ( 'sphere01_sample_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' sphere01_sample samples the unit sphere.' ) m = 3 n = 10 x = sphere01_sample ( n ) r8mat_transpose_print ( m, n, x, ' Sample points on the unit sphere.' ) # # Terminate. # print ( '' ) print ( 'sphere01_sample_test' ) print ( ' Normal end of execution.' ) return def timestamp ( ): #*****************************************************************************80 # ## timestamp() prints the date as a timestamp. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 06 April 2013 # # Author: # # John Burkardt # import time t = time.time ( ) print ( time.ctime ( t ) ) return None def sphere_monte_carlo_tests ( ): #*****************************************************************************80 # ## sphere_monte_carlo_tests() tests sphere_monte_carlo(). # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 13 November 2016 # # Author: # # John Burkardt # import platform print ( '' ) print ( 'sphere_monte_carlo_tests():' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' Test sphere_monte_carlo().' ) i4vec_print_test ( ) i4vec_transpose_print_test ( ) r8mat_print_test ( ) r8mat_print_some_test ( ) r8mat_transpose_print_test ( ) r8mat_transpose_print_some_test ( ) r8vec_print_test ( ) sphere01_area_test ( ) sphere01_monomial_integral_test ( ) sphere01_monte_carlo_test ( ) sphere01_sample_test ( ) # # Terminate. # print ( '' ) print ( 'sphere_monte_carlo_tests():' ) print ( ' Normal end of execution.' ) return if ( __name__ == '__main__' ): timestamp ( ) sphere_monte_carlo_tests ( ) timestamp ( )