#! /usr/bin/env python3 # def hammersley ( i, m, n ): #*****************************************************************************80 # ## hammersley() computes an element of a Hammersley sequence. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 19 August 2016 # # Author: # # John Burkardt # # Reference: # # John Hammersley, # Monte Carlo methods for solving multivariable problems, # Proceedings of the New York Academy of Science, # Volume 86, 1960, pages 844-874. # # Input: # # integer I, the index of the element of the sequence. # 0 <= I. # # integer M, the spatial dimension. # 1 <= M <= 100. # # integer N, the "base" for the first component. # 1 <= N. # # Output: # # real R(M), the element of the sequence with index I. # import numpy as np i = int ( i ) t = np.ones ( m - 1 ) t = i * t # # Carry out the computation. # prime_inv = np.zeros ( m - 1 ) for j in range ( 0, m - 1 ): prime_inv[j] = 1.0 / float ( prime ( j ) ) r = np.zeros ( m ) r[0] = float ( i % ( n + 1 ) ) / float ( n ) while ( 0 < np.sum ( t ) ): for j in range ( 0, m - 1 ): d = ( t[j] % prime ( j ) ) r[j+1] = r[j+1] + float ( d ) * prime_inv[j] prime_inv[j] = prime_inv[j] / prime ( j ) t[j] = ( t[j] // prime ( j ) ) return r def hammersley_test ( ): #*****************************************************************************80 # ## hammersley_test() tests hammersley(). # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 19 August 2016 # # Author: # # John Burkardt # import platform print ( '' ) print ( 'hammersley_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' hammersley() returns the I-th element of an M-dimensional' ) print ( ' Hammersley sequence.' ) print ( '' ) print ( ' I HAMMERSLEY(I)' ) n = 16 for m in range ( 1, 4 ): print ( '' ) print ( ' Use M = %d' % ( m ) ) print ( ' Use N = %d' % ( n ) ) print ( '' ) for i in range ( 0, 11 ): r = hammersley ( i, m, n ) print ( ' %3d' % ( i ), end = '' ) for j in range ( 0, m ): print ( ' %14.8f' % ( r[j] ), end = '' ) print ( '' ) # # Terminate. # print ( '' ) print ( 'hammersley_test' ) print ( ' Normal end of execution.' ) return def hammersley_inverse ( r, m, n ): #*****************************************************************************80 # ## hammersley_inverse() inverts an element of the Hammersley sequence. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 18 August 2016 # # Author: # # John Burkardt # # Input: # # real R(M), the I-th element of the Hammersley sequence. # 0 <= R < 1.0 # # integer M, the spatial dimension. # # integer N, the "base" for the first component. # 1 <= N. # # Output: # # integer I, the index of the element of the sequence. # import numpy as np for i in range ( 0, m ): if ( r[i] < 0.0 or 1.0 < r[i] ): print ( '' ) print ( 'hammersley_inverse - Fatal error!' ) print ( ' 0 <= R <= 1.0 is required.' ) print ( ' but R[%d] = %g' % ( i, r[i] ) ) raise Exception ( 'hammersley_inverse - Fatal error!' ) if ( m < 1 ): print ( '' ) print ( 'hammersley_inverse - Fatal error!' ) print ( ' 1 <= M is required.' ) raise Exception ( 'hammersley_inverse - Fatal error!' ) if ( n < 1 ): print ( '' ) print ( 'hammersley_inverse - Fatal error!' ) print ( ' 1 <= N is required.' ) raise Exception ( 'hammersley_inverse - Fatal error!' ) # # Invert using the second component, if possible. # if ( 2 <= m ): i = 0 t = r[1] p = 1 while ( t != 0.0 ): t = t * 2.0 d = np.floor ( t ) i = i + d * p p = p * 2 t = t - d # # Invert using component 1. # else: i = round ( n * r[0] ) return i def hammersley_inverse_test ( ): #*****************************************************************************80 # ## hammersley_inverse_test() tests hammersley_inverse(). # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 19 August 2016 # # Author: # # John Burkardt # import platform print ( '' ) print ( 'hammersley_inverse_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' hammersley_inverse inverts an element of a Hammersley sequence.' ) print ( '' ) print ( ' I R=HAMMERSLEY(I,3) hammersley_inverse(R,3)' ) print ( '' ) m = 3 n = 16 for i in range ( 0, 11 ): r = hammersley ( i, m, n ) i2 = hammersley_inverse ( r, m, n ) print ( ' %3d' % ( i ), end = '' ) for j in range ( 0, m ): print ( ' %14.8f' % ( r[j] ), end = '' ) print ( ' %3d' % ( i2 ) ) # # Terminate. # print ( '' ) print ( 'hammersley_inverse_test' ) print ( ' Normal end of execution.' ) return def hammersley_sequence ( i1, i2, m, n ): #*****************************************************************************80 # ## hammersley_sequence() computes elements I1 through I2 of a Hammersley sequence. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 19 August 2016 # # Author: # # John Burkardt # # Reference: # # John Hammersley, # On the efficiency of certain quasi-random sequences of points # in evaluating multi-dimensional integrals, # Numerische Mathematik, # Volume 2, pages 84-90, 1960. # # Input: # # integer I1, I2, the indices of the first and last elements # of the sequence. 0 <= I1, I2. # # integer M, the spatial dimension. # 1 <= M <= 100. # # integer N, the "base" for the first component. # 1 <= N. # # Output: # # real R(M,abs(I1-I2)+1), the elements of the sequence with # indices I1 through I2. # import numpy as np if ( i1 <= i2 ): i3 = +1 else: i3 = -1 l = abs ( i2 - i1 ) + 1 r = np.zeros ( [ m, l ] ) k = 0 for i in range ( i1, i2 + i3, i3 ): t = np.ones ( m - 1 ) t = t * i # # Carry out the computation. # prime_inv = np.zeros ( m - 1 ) for j in range ( 0, m - 1 ): prime_inv[j] = 1.0 / prime ( j ) r[0,k] = float ( i % ( n + 1 ) ) / float ( n ) while ( 0 < np.sum ( t ) ): for j in range ( 0, m - 1 ): d = ( t[j] % prime ( j ) ) r[j+1,k] = r[j+1,k] + float ( d ) * prime_inv[j] prime_inv[j] = prime_inv[j] / prime ( j ) t[j] = ( t[j] // prime ( j ) ) k = k + 1 return r def hammersley_sequence_test ( ): #*****************************************************************************80 # ## hammersley_sequence_test() tests hammersley_sequence(). # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 19 August 2016 # # Author: # # John Burkardt # import platform print ( '' ) print ( 'hammersley_sequence_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' hammersley_sequence returns the elements I1 through I2' ) print ( ' of an M-dimensional Hammersley sequence.' ) n = 16 for m in range ( 1, 4 ): print ( '' ) print ( ' hammersley_sequence(0,10,%d,%d):' % ( m, n ) ) r = hammersley_sequence ( 0, 10, m, n ) r8mat_print ( m, 11, r, ' R:' ) m = 3 print ( '' ) print ( ' hammersley_sequence(10,0,%d,%d):' % ( m, n ) ) r = hammersley_sequence ( 10, 0, m, n ) r8mat_print ( m, 11, r, ' R:' ) # # Terminate. # print ( '' ) print ( 'hammersley_sequence_test' ) print ( ' Normal end of execution.' ) return def prime ( n ): #*****************************************************************************80 # ## prime() returns returns any of the first prime_MAX prime numbers. # # Discussion: # # prime_MAX is 1600, and the largest prime stored is 13499. # # Thanks to Bart Vandewoestyne for pointing out a typo, 18 February 2005. # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 05 December 2014 # # Author: # # John Burkardt # # Reference: # # Milton Abramowitz and Irene Stegun, # Handbook of Mathematical Functions, # US Department of Commerce, 1964, pages 870-873. # # Daniel Zwillinger, # CRC Standard Mathematical Tables and Formulae, # 30th Edition, # CRC Press, 1996, pages 95-98. # # Input: # # integer N, the index of the desired prime number. # In general, is should be true that 0 <= N < prime_MAX. # # Output: # # integer P, the N-th prime. # import numpy as np prime_max = 1600 prime_vector = np.array ( [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \ 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, \ 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, \ 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, \ 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, \ 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, \ 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, \ 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, \ 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, \ 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, \ 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, \ 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, \ 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, \ 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, \ 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, \ 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, \ 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, \ 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, \ 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, \ 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, \ 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, \ 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, \ 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, \ 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, \ 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, \ 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, \ 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, \ 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, \ 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, \ 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, \ 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, \ 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, \ 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, \ 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, \ 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, \ 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, \ 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, \ 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, \ 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, \ 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, \ 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, \ 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, \ 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, \ 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, \ 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, \ 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, \ 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, \ 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, \ 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, \ 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, \ 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, \ 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, \ 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, \ 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, \ 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, \ 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, \ 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, \ 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, \ 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, \ 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, \ 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, \ 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, \ 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, \ 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, \ 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, \ 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, \ 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, \ 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, \ 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, \ 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, \ 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, \ 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, \ 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, \ 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, \ 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, \ 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, \ 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, \ 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, \ 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, \ 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, \ 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, \ 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, \ 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, \ 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, \ 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, \ 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, \ 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, \ 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, \ 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, \ 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, \ 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, \ 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, \ 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, \ 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, \ 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, \ 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, \ 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, \ 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, \ 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, \ 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, \ 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, \ 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, \ 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, \ 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, \ 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, \ 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, \ 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, \ 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, \ 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, \ 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, \ 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, \ 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, \ 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, \ 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, \ 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, \ 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, \ 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, \ 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, \ 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, \ 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, \ 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, \ 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, \ 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973,10007, \ 10009,10037,10039,10061,10067,10069,10079,10091,10093,10099, \ 10103,10111,10133,10139,10141,10151,10159,10163,10169,10177, \ 10181,10193,10211,10223,10243,10247,10253,10259,10267,10271, \ 10273,10289,10301,10303,10313,10321,10331,10333,10337,10343, \ 10357,10369,10391,10399,10427,10429,10433,10453,10457,10459, \ 10463,10477,10487,10499,10501,10513,10529,10531,10559,10567, \ 10589,10597,10601,10607,10613,10627,10631,10639,10651,10657, \ 10663,10667,10687,10691,10709,10711,10723,10729,10733,10739, \ 10753,10771,10781,10789,10799,10831,10837,10847,10853,10859, \ 10861,10867,10883,10889,10891,10903,10909,10937,10939,10949, \ 10957,10973,10979,10987,10993,11003,11027,11047,11057,11059, \ 11069,11071,11083,11087,11093,11113,11117,11119,11131,11149, \ 11159,11161,11171,11173,11177,11197,11213,11239,11243,11251, \ 11257,11261,11273,11279,11287,11299,11311,11317,11321,11329, \ 11351,11353,11369,11383,11393,11399,11411,11423,11437,11443, \ 11447,11467,11471,11483,11489,11491,11497,11503,11519,11527, \ 11549,11551,11579,11587,11593,11597,11617,11621,11633,11657, \ 11677,11681,11689,11699,11701,11717,11719,11731,11743,11777, \ 11779,11783,11789,11801,11807,11813,11821,11827,11831,11833, \ 11839,11863,11867,11887,11897,11903,11909,11923,11927,11933, \ 11939,11941,11953,11959,11969,11971,11981,11987,12007,12011, \ 12037,12041,12043,12049,12071,12073,12097,12101,12107,12109, \ 12113,12119,12143,12149,12157,12161,12163,12197,12203,12211, \ 12227,12239,12241,12251,12253,12263,12269,12277,12281,12289, \ 12301,12323,12329,12343,12347,12373,12377,12379,12391,12401, \ 12409,12413,12421,12433,12437,12451,12457,12473,12479,12487, \ 12491,12497,12503,12511,12517,12527,12539,12541,12547,12553, \ 12569,12577,12583,12589,12601,12611,12613,12619,12637,12641, \ 12647,12653,12659,12671,12689,12697,12703,12713,12721,12739, \ 12743,12757,12763,12781,12791,12799,12809,12821,12823,12829, \ 12841,12853,12889,12893,12899,12907,12911,12917,12919,12923, \ 12941,12953,12959,12967,12973,12979,12983,13001,13003,13007, \ 13009,13033,13037,13043,13049,13063,13093,13099,13103,13109, \ 13121,13127,13147,13151,13159,13163,13171,13177,13183,13187, \ 13217,13219,13229,13241,13249,13259,13267,13291,13297,13309, \ 13313,13327,13331,13337,13339,13367,13381,13397,13399,13411, \ 13417,13421,13441,13451,13457,13463,13469,13477,13487,13499 ] ) if ( n < 0 or prime_max <= n ): print ( '' ) print ( 'PRIME - Fatal error!' ) print ( ' 0 <= N < %d' % ( prime_max ) ) raise Exception ( 'PRIME - Fatal error!' ) return prime_vector[n] def prime_test ( ): #*****************************************************************************80 # ## prime_test() tests prime(). # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 05 December 2014 # # Author: # # John Burkardt # import platform print ( '' ) print ( 'prime_test' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' PRIME returns primes from a table.' ) print ( '' ) for i in range ( 0, 11 ): print ( ' %4d %6d' % ( i, prime(i) ) ) prime_max = 1600 print ( '' ) for i in range ( prime_max - 10, prime_max ): print ( ' %4d %6d' % ( i, prime(i) ) ) # # Terminate. # print ( '' ) print ( 'prime_test' ) print ( ' Normal end of execution.' ) return 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: ', 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_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 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 hammersley_tests ( ): #*****************************************************************************80 # ## hammersley_tests() tests hammersley(). # # Licensing: # # This code is distributed under the MIT license. # # Modified: # # 19 August 2016 # # Author: # # John Burkardt # import platform print ( '' ) print ( 'hammersley_tests()' ) print ( ' Python version: %s' % ( platform.python_version ( ) ) ) print ( ' Test hammersley()' ) hammersley_test ( ) hammersley_inverse_test ( ) hammersley_sequence_test ( ) # # Terminate. # print ( '' ) print ( 'hammersley_tests:' ) print ( ' Normal end of execution.' ) return if ( __name__ == '__main__' ): timestamp ( ) hammersley_tests ( ) timestamp ( )