#! /usr/bin/env python3 # def hypersphere01_area ( m ): #*****************************************************************************80 # ## hypersphere01_area() returns the surface area of the unit hypersphere. # # Discussion: # # M Area # # 2 2 * PI # 3 4 * PI # 4 ( 2 / 1) * PI^2 # 5 ( 8 / 3) * PI^2 # 6 ( 1 / 1) * PI^3 # 7 (16 / 15) * PI^3 # 8 ( 1 / 3) * PI^4 # 9 (32 / 105) * PI^4 # 10 ( 1 / 12) * PI^5 # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 04 January 2014 # # Author: # # John Burkardt # # Input: # # integer M, the spatial dimension. # # Output: # # real VALUE, the area of the unit hypersphere. # import numpy as np if ( ( m % 2 ) == 0 ): m_half = ( m // 2 ) value = 2.0 * np.pi ** m_half for i in range ( 1, m_half ): value = value / float ( i ) else: m_half = ( ( m - 1 ) // 2 ) value = np.pi ** m_half * 2.0 ** m for i in range ( m_half + 1, 2 * m_half + 1 ): value = value / float ( i ) return value def hypersphere01_area_test ( ) : #*****************************************************************************80 # ## hypersphere01_area_test() tests hypersphere01_area(). # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 21 June 2015 # # Author: # # John Burkardt # import platform print ( '' ) print ( 'hypersphere01_area_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' hypersphere01_area returns the volume of the unit hypersphere.' ) print ( '' ) print ( ' M Area' ) print ( '' ) for m in range ( 1, 11 ): value = hypersphere01_area ( m ) print ( ' %2d %g' % ( m, value ) ) # # Terminate. # print ( '' ) print ( 'hypersphere01_area_test' ) print ( ' Normal end of execution.' ) return def hypersphere01_monomial_integral ( m, e ): #*****************************************************************************80 # ## hypersphere01_monomial_integral(): monomial integrals on the unit hypersphere. # # Discussion: # # The integration region is # # sum ( 1 <= I <= M ) X(I)^2 = 1. # # The monomial is F(X) = product ( 1 <= I <= M ) X(I)^E(I) # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 22 June 2015 # # Author: # # John Burkardt # # Input: # # integer M, the spatial dimension. # # integer E(M), the exponents of X(1) through X(M). # Each exponent must be nonnegative. # # Output: # # real INTEGRAL, the integral. # from scipy.special import gamma for i in range ( 0, m ): if ( e[i] < 0 ): print ( '' ) print ( 'hypersphere01_monomial_integral - Fatal error!' ) print ( ' All exponents must be nonnegative.' ) raise Exception ( 'hypersphere01_monomial_integral - Fatal error!' ) for i in range ( 0, m ): if ( ( e[i] % 2 ) == 1 ): integral = 0.0 return integral integral = 2.0 for i in range ( 0, m ): arg = 0.5 * float ( e[i] + 1 ) integral = integral * gamma ( arg ) s = 0 for i in range ( 0, m ): s = s + float ( e[i] + 1 ) arg = 0.5 * float ( s ) integral = integral / gamma ( arg ) return integral def hypersphere01_monomial_integral_test ( ): #*****************************************************************************80 # ## hypersphere01_monomial_integral_test() tests hypersphere01_monomial_integral(). # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 07 January 2014 # # Author: # # John Burkardt # import numpy as np import platform m = 3 n = 4192 test_num = 20 print ( '' ) print ( 'hypersphere01_monomial_integral' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' hypersphere01_monomial_integral returns the integral of' ) print ( ' a monomial over the surface of the unit hypersphere in 3D.' ) print ( ' Compare with a Monte Carlo estimate.' ) # # Get sample points. # x = hypersphere01_sample ( m, n ) print ( '' ) print ( ' Number of sample points used is %d' % ( n ) ) # # Randomly choose X,Y,Z exponents between (0,0,0) and (8,8,8). # 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 ( '' ) 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 = hypersphere01_area ( m ) * np.sum ( value ) / float ( n ) exact = hypersphere01_monomial_integral ( m, e ) error = abs ( result - exact ) for i in range ( 0, m ): print ( ' %2d' % ( e[i] ), end = '' ) print ( ' %14.6g %14.6g %10.2g' % ( result, exact, error ) ) # # Terminate. # print ( '' ) print ( 'hypersphere_monomial_integral_test' ) print ( ' Normal end of execution.' ) return def hypersphere01_sample ( m, n ): #*****************************************************************************80 # ## hypersphere01_sample() uniformly samples the surface of the unit hypersphere. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 22 June 2015 # # Author: # # John Burkardt # # Input: # # integer M, the spatial dimension. # # integer N, the number of points. # # Output: # # real X(M,N), the points. # import numpy as np 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 hypersphere01_sample_test ( ): #*****************************************************************************80 # ## hypersphere01_sample_test() tests hypersphere01_sample(). # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 22 June 2015 # # Author: # # John Burkardt # import platform print ( '' ) print ( 'hypersphere01_sample_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' hypersphere01_sample samples the unit hypersphere' ) print ( ' in M dimensions.' ) m = 3 n = 10 x = hypersphere01_sample ( m, n ) r8mat_transpose_print ( m, n, x, ' Sample points on the unit hypersphere.' ) # # Terminate. # print ( '' ) print ( 'hypersphere01_sample_test' ) print ( ' Normal end of execution.' ) return def hypersphere_integrals_test ( ): #*****************************************************************************80 # ## hypersphere_integrals_test() tests hypersphere_integrals(). # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 22 June 2015 # # Author: # # John Burkardt # import platform print ( '' ) print ( 'hypersphere_integrals_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' Test hypersphere_integrals().' ) hypersphere01_area_test ( ) hypersphere01_monomial_integral_test ( ) hypersphere01_sample_test ( ) # # Terminate. # print ( '' ) print ( 'hypersphere_integrals_test:' ) print ( ' Normal end of execution.' ) return 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_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: # # 08 September 2018 # # 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 ( title, end = '' ) if ( 0 < n ): for i in range ( 0, n ): print ( ' %d' % ( a[i] ), end = '' ) if ( ( i + 1 ) % 20 == 0 or i == n - 1 ): print ( '' ) else: print ( '(empty vector)' ) 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_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: ', end = '' ) for j in range ( j2lo, j2hi + 1 ): print ( '%7d ' % ( j ), end = '' ) print ( '' ) print ( ' Row' ) i2lo = max ( ilo, 0 ) i2hi = min ( ihi, m ) for i in range ( i2lo, i2hi + 1 ): print ( '%7d :' % ( i ), end = '' ) for j in range ( j2lo, j2hi + 1 ): print ( '%12g ' % ( a[i,j] ), end = '' ) print ( '' ) 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_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: ', end = '' ) for i in range ( i2lo, i2hi + 1 ): print ( '%7d ' % ( i ), end = '' ) print ( '' ) print ( ' Col' ) j2lo = max ( jlo, 0 ) j2hi = min ( jhi, n - 1 ) for j in range ( j2lo, j2hi + 1 ): print ( '%7d :' % ( j ), end = '' ) for i in range ( i2lo, i2hi + 1 ): print ( '%12g ' % ( a[i,j] ), end = '' ) print ( '' ) 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 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 if ( __name__ == '__main__' ): timestamp ( ) hypersphere_integrals_test ( ) timestamp ( )