07-Jan-2022 22:41:09 lattice_rule_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 Test lattice_rule(). lattice_rule_test01(): FIBONACCI_LATTICE_Q applies a Fibonacci lattice rule to integrate a function over the unit square. These Fibonacci rules are only available in 2D. The spatial dimension DIM_NUM = 2 K M EXACT ESTIMATE ERROR 3 2 1.000000 0.446909 5.530914e-01 4 3 1.000000 0.579554 4.204462e-01 5 5 1.000000 0.762059 2.379409e-01 6 8 1.000000 0.830249 1.697507e-01 7 13 1.000000 0.904672 9.532787e-02 8 21 1.000000 0.933352 6.664773e-02 9 34 1.000000 0.962800 3.719982e-02 10 55 1.000000 0.974212 2.578824e-02 11 89 1.000000 0.985652 1.434808e-02 12 144 1.000000 0.990092 9.908297e-03 13 233 1.000000 0.994496 5.504474e-03 14 377 1.000000 0.996205 3.794537e-03 15 610 1.000000 0.997893 2.106574e-03 16 987 1.000000 0.998549 1.451040e-03 17 1597 1.000000 0.999195 8.053118e-04 18 2584 1.000000 0.999445 5.545202e-04 lattice_rule_test02(): FIBONACCI_LATTICE_T applies a symmetric Fibonacci lattice rule --------- to integrate a function over the unit square. These Fibonacci rules are only available in 2D.= The spatial dimension DIM_NUM = 2 K M EXACT ESTIMATE ERROR 3 2 1.000000 0.446909 5.530914e-01 4 3 1.000000 0.579554 4.204462e-01 5 5 1.000000 0.762059 2.379409e-01 6 8 1.000000 0.830249 1.697507e-01 7 13 1.000000 0.904672 9.532787e-02 8 21 1.000000 0.933352 6.664773e-02 9 34 1.000000 0.962800 3.719982e-02 10 55 1.000000 0.974212 2.578824e-02 11 89 1.000000 0.985652 1.434808e-02 12 144 1.000000 0.990092 9.908297e-03 13 233 1.000000 0.994496 5.504474e-03 14 377 1.000000 0.996205 3.794537e-03 15 610 1.000000 0.997893 2.106574e-03 16 987 1.000000 0.998549 1.451040e-03 17 1597 1.000000 0.999195 8.053118e-04 18 2584 1.000000 0.999445 5.545202e-04 lattice_rule_test03(): FIBONACCI_LATTICE_B applies an optimal Fibonacci lattice rule ------- to integrate a function over the unit square. These Fibonacci rules are only available in 2D. The spatial dimension DIM_NUM = 2 K M EXACT ESTIMATE ERROR 3 2 1.000000 0.446909 5.530914e-01 4 3 1.000000 0.579554 4.204462e-01 5 5 1.000000 0.762059 2.379409e-01 6 8 1.000000 0.830249 1.697507e-01 7 13 1.000000 0.904672 9.532787e-02 8 21 1.000000 0.933352 6.664773e-02 9 34 1.000000 0.962800 3.719982e-02 10 55 1.000000 0.974212 2.578824e-02 11 89 1.000000 0.985652 1.434808e-02 12 144 1.000000 0.990092 9.908297e-03 13 233 1.000000 0.994496 5.504474e-03 14 377 1.000000 0.996205 3.794537e-03 15 610 1.000000 0.997893 2.106574e-03 16 987 1.000000 0.998549 1.451040e-03 17 1597 1.000000 0.999195 8.053118e-04 18 2584 1.000000 0.999445 5.545202e-04 lattice_rule_test04(): FIBONACCI_LATTICE_Q1 applies a Fibonacci lattice rule to integrate a function over the unit square. A nonlinear coordinate transformation is applied. These Fibonacci rules are only available in 2D. The spatial dimension DIM_NUM = 2 K M EXACT ESTIMATE ERROR 3 2 1.000000 1.005544 5.544437e-03 4 3 1.000000 0.999689 3.109315e-04 5 5 1.000000 1.110427 1.104273e-01 6 8 1.000000 1.062375 6.237528e-02 7 13 1.000000 1.033491 3.349101e-02 8 21 1.000000 1.015904 1.590427e-02 9 34 1.000000 1.007314 7.313868e-03 10 55 1.000000 1.003248 3.248174e-03 11 89 1.000000 1.001418 1.417980e-03 12 144 1.000000 1.000608 6.082856e-04 13 233 1.000000 1.000258 2.580455e-04 14 377 1.000000 1.000108 1.083183e-04 15 610 1.000000 1.000045 4.511364e-05 16 987 1.000000 1.000019 1.865663e-05 17 1597 1.000000 1.000008 7.671221e-06 18 2584 1.000000 1.000003 3.138114e-06 lattice_rule_test05(): FIBONACCI_LATTICE_Q2 applies a Fibonacci lattice rule to integrate a function over the unit square. A nonlinear coordinate transformation is applied. These Fibonacci rules are only available in 2D. The spatial dimension DIM_NUM = 2 K M EXACT ESTIMATE ERROR 3 2 1.000000 1.571163 5.711632e-01 4 3 1.000000 1.202250 2.022497e-01 5 5 1.000000 1.104584 1.045838e-01 6 8 1.000000 1.047741 4.774133e-02 7 13 1.000000 1.002023 2.023026e-03 8 21 1.000000 1.001297 1.297083e-03 9 34 1.000000 1.000048 4.835884e-05 10 55 1.000000 1.000061 6.147685e-05 11 89 1.000000 1.000000 3.056390e-07 12 144 1.000000 1.000003 3.025015e-06 13 233 1.000000 1.000000 4.990474e-08 14 377 1.000000 1.000000 1.577330e-07 15 610 1.000000 1.000000 4.528133e-09 16 987 1.000000 1.000000 8.511837e-09 17 1597 1.000000 1.000000 2.964128e-10 18 2584 1.000000 1.000000 4.674179e-10 LATTICE_RULE_TEST06 FIBONACCI_LATTICE_Q3 applies a Fibonacci lattice rule to integrate a function over the unit square. A nonlinear coordinate transformation is applied. These Fibonacci rules are only available in 2D. The spatial dimension DIM_NUM = 2 K M EXACT ESTIMATE ERROR 3 2 1.000000 1.787635 7.876346e-01 4 3 1.000000 1.222003 2.220034e-01 5 5 1.000000 1.050162 5.016247e-02 6 8 1.000000 1.031257 3.125671e-02 7 13 1.000000 0.997689 2.311129e-03 8 21 1.000000 1.000829 8.294076e-04 9 34 1.000000 0.999971 2.884616e-05 10 55 1.000000 1.000047 4.672840e-05 11 89 1.000000 0.999998 1.587487e-06 12 144 1.000000 1.000003 2.602494e-06 13 233 1.000000 1.000000 8.833584e-08 14 377 1.000000 1.000000 1.450382e-07 15 610 1.000000 1.000000 4.921388e-09 16 987 1.000000 1.000000 8.082873e-09 17 1597 1.000000 1.000000 2.742468e-10 18 2584 1.000000 1.000000 4.504459e-10 LATTICE_RULE_TEST07 LATTICE applies a lattice rule to integrate a function over the unit hypercube. The spatial dimension DIM_NUM = 2 The lattice rule order M will vary. The lattice generator vector: 1: 1 2: 2 I M EXACT ESTIMATE ERROR 1 5 1.000000 0.760192 2.398078e-01 2 13 1.000000 0.957115 4.288469e-02 3 23 1.000000 1.016074 1.607433e-02 4 37 1.000000 1.045954 4.595445e-02 5 47 1.000000 1.056536 5.653614e-02 6 61 1.000000 1.065578 6.557783e-02 7 73 1.000000 1.070590 7.058956e-02 8 89 1.000000 1.075183 7.518326e-02 9 103 1.000000 1.078039 7.803859e-02 10 113 1.000000 1.079647 7.964715e-02 LATTICE_RULE_TEST08 LATTICE applies a lattice rule to integrate a function over the unit hypercube. The spatial dimension DIM_NUM = 2 The lattice rule order M is fixed at 53 The lattice generator vector Z will vary. M Z(1) Z(2) EXACT ESTIMATE ERROR 53 1 1 1.000000 1.160699 1.606990e-01 53 1 2 1.000000 1.060990 6.098964e-02 53 1 3 1.000000 1.030924 3.092355e-02 53 1 4 1.000000 1.007034 7.034133e-03 53 1 5 1.000000 1.002994 2.993813e-03 53 1 6 1.000000 1.008962 8.962211e-03 53 1 7 1.000000 0.989136 1.086436e-02 53 1 8 1.000000 0.991666 8.334306e-03 53 1 9 1.000000 1.009447 9.447495e-03 53 1 10 1.000000 0.984278 1.572192e-02 53 1 11 1.000000 0.992941 7.059098e-03 53 1 12 1.000000 0.982027 1.797255e-02 53 1 13 1.000000 0.943609 5.639061e-02 53 1 14 1.000000 0.988858 1.114157e-02 53 1 15 1.000000 0.961547 3.845288e-02 53 1 16 1.000000 0.984685 1.531469e-02 53 1 17 1.000000 0.956489 4.351148e-02 53 1 18 1.000000 1.033181 3.318068e-02 53 1 19 1.000000 0.989046 1.095425e-02 53 1 20 1.000000 0.992422 7.578166e-03 53 1 21 1.000000 0.947847 5.215291e-02 53 1 22 1.000000 0.969229 3.077120e-02 53 1 23 1.000000 0.975920 2.408029e-02 53 1 24 1.000000 0.958501 4.149878e-02 53 1 25 1.000000 0.956891 4.310919e-02 53 1 26 1.000000 0.892870 1.071301e-01 53 1 27 1.000000 1.064199 6.419864e-02 53 1 28 1.000000 0.995337 4.662618e-03 53 1 29 1.000000 0.993437 6.563027e-03 53 1 30 1.000000 0.975887 2.411343e-02 53 1 31 1.000000 0.982584 1.741584e-02 53 1 32 1.000000 1.004630 4.629864e-03 53 1 33 1.000000 0.959544 4.045583e-02 53 1 34 1.000000 0.962793 3.720709e-02 53 1 35 1.000000 0.920861 7.913924e-02 53 1 36 1.000000 0.995744 4.256061e-03 53 1 37 1.000000 0.967209 3.279134e-02 53 1 38 1.000000 0.990532 9.467639e-03 53 1 39 1.000000 0.962977 3.702263e-02 53 1 40 1.000000 1.009396 9.396158e-03 53 1 41 1.000000 0.969779 3.022093e-02 53 1 42 1.000000 0.958991 4.100916e-02 53 1 43 1.000000 0.967607 3.239340e-02 53 1 44 1.000000 0.942910 5.709018e-02 53 1 45 1.000000 0.960280 3.972012e-02 53 1 46 1.000000 0.962918 3.708191e-02 53 1 47 1.000000 0.943370 5.662998e-02 53 1 48 1.000000 0.949423 5.057660e-02 53 1 49 1.000000 0.945882 5.411779e-02 53 1 50 1.000000 0.922956 7.704364e-02 53 1 51 1.000000 0.895767 1.042335e-01 53 1 52 1.000000 0.811426 1.885735e-01 LATTICE_RULE_TEST085 LATTICE is a lattice rule for periodic functions. However, we apply it to a nonperiodic function just to see how it does. The spatial dimension DIM_NUM = 2 The lattice generator vector: 1: 1 2: 2 K M EXACT ESTIMATE ERROR 3 2 1.000000 0.000000 1.000000e+00 4 3 1.000000 0.579554 4.204462e-01 5 5 1.000000 0.760192 2.398078e-01 6 8 1.000000 0.745448 2.545515e-01 7 13 1.000000 0.957115 4.288469e-02 8 21 1.000000 1.008645 8.645009e-03 9 34 1.000000 1.008523 8.523065e-03 10 55 1.000000 1.062260 6.226049e-02 11 89 1.000000 1.075183 7.518326e-02 12 144 1.000000 1.075288 7.528812e-02 13 233 1.000000 1.088207 8.820672e-02 14 377 1.000000 1.091296 9.129640e-02 15 610 1.000000 1.091330 9.132955e-02 16 987 1.000000 1.094392 9.439191e-02 17 1597 1.000000 1.095124 9.512351e-02 18 2584 1.000000 1.095132 9.513179e-02 LATTICE_RULE_TEST09 LATTICE_NP0 applies a lattice rule to a nonperiodic function by reflecting the function about the midpoint and averaging. The spatial dimension DIM_NUM = 2 The lattice generator vector: 1: 1 2: 2 M EXACT ESTIMATE ERROR 2 1.000000 1.220921 2.209212e-01 3 1.000000 1.048856 4.885598e-02 5 1.000000 1.019957 1.995732e-02 8 1.000000 1.017478 1.747847e-02 13 1.000000 1.005253 5.253032e-03 21 1.000000 1.003643 3.642565e-03 34 1.000000 1.003465 3.465166e-03 55 1.000000 1.002785 2.784894e-03 89 1.000000 1.002694 2.694300e-03 144 1.000000 1.002684 2.684427e-03 233 1.000000 1.002646 2.646484e-03 377 1.000000 1.002641 2.641435e-03 610 1.000000 1.002641 2.640884e-03 987 1.000000 1.002639 2.638770e-03 1597 1.000000 1.002638 2.638489e-03 2584 1.000000 1.002638 2.638458e-03 LATTICE_RULE_TEST10 LATTICE_NP1 applies a lattice rule to a nonperiodic function using a nonlinear transformation, to integrate a function over the unit square. The spatial dimension DIM_NUM = 2 The lattice generator vector: 1: 1 2: 2 M EXACT ESTIMATE ERROR 2 1.000000 0.000000 1.000000e+00 3 1.000000 0.999689 3.109315e-04 5 1.000000 1.101893 1.018931e-01 8 1.000000 0.997195 2.804576e-03 13 1.000000 1.064038 6.403783e-02 21 1.000000 1.057889 5.788897e-02 34 1.000000 1.051019 5.101900e-02 55 1.000000 1.054574 5.457401e-02 89 1.000000 1.054223 5.422259e-02 144 1.000000 1.053839 5.383924e-02 233 1.000000 1.054037 5.403702e-02 377 1.000000 1.054017 5.401742e-02 610 1.000000 1.053996 5.399605e-02 987 1.000000 1.054007 5.400707e-02 1597 1.000000 1.054006 5.400598e-02 2584 1.000000 1.054005 5.400479e-02 LATTICE_RULE_TEST11 MONTE_CARLO applies a Monte Carlo scheme to estimate the integral of a function over the unit hypercube. The spatial dimension DIM_NUM = 2 M EXACT ESTIMATE ERROR 100 1.000000 0.994672 5.328346e-03 1000 1.000000 1.012975 1.297514e-02 10000 1.000000 1.003861 3.861270e-03 100000 1.000000 0.999300 6.995806e-04 LATTICE_RULE_TEST12 LATTICE_PRINT prints out the lattice generated by a single generator vector. The spatial dimension DIM_NUM = 2 The generator vector: 1: 1 2: 3 The total lattice: 1 0 0 2 1 3 3 2 6 4 3 1 5 4 4 6 5 7 7 6 2 8 7 5 LATTICE_RULE_TEST13 FIND_Z20 finds the optimal lattice generator Z with Fourier coefficient smoothness ALPHA = 2, and copy exponent 0, for a rank 1 "method of good lattice points" rule. The spatial dimension DIM_NUM = 2 M Z(1) Z(2) (M = Fibonacci) 2 1 1 3 1 1 5 1 2 8 1 3 13 1 5 21 1 8 34 1 10 55 1 21 (M = 2**K) 4 1 2 8 1 3 16 1 6 32 1 14 64 1 27 128 1 54 256 1 94 512 1 198 1024 1 298 (M = 3*2**K) 6 1 2 12 1 5 24 1 10 48 1 14 96 1 22 192 1 73 384 1 146 768 1 225 1536 1 674 3072 1 849 (M = Prime) 113 1 35 173 1 64 229 1 94 281 1 109 349 1 135 409 1 169 463 1 179 541 1 200 LATTICE_RULE_TEST14 FIBONACCI_LATTICE_Q_NODES... The spatial dimension DIM_NUM = 2 The Fibonacci index K = 12 The Fibonacci value M = 144 The Fibonacci lattice nodes: Row: 1 2 Col 1: 0 0 2: 0.00694444 0.618056 3: 0.0138889 0.236111 4: 0.0208333 0.854167 5: 0.0277778 0.472222 6: 0.0347222 0.0902778 7: 0.0416667 0.708333 8: 0.0486111 0.326389 9: 0.0555556 0.944444 10: 0.0625 0.5625 11: 0.0694444 0.180556 12: 0.0763889 0.798611 13: 0.0833333 0.416667 14: 0.0902778 0.0347222 15: 0.0972222 0.652778 16: 0.104167 0.270833 17: 0.111111 0.888889 18: 0.118056 0.506944 19: 0.125 0.125 20: 0.131944 0.743056 21: 0.138889 0.361111 22: 0.145833 0.979167 23: 0.152778 0.597222 24: 0.159722 0.215278 25: 0.166667 0.833333 26: 0.173611 0.451389 27: 0.180556 0.0694444 28: 0.1875 0.6875 29: 0.194444 0.305556 30: 0.201389 0.923611 31: 0.208333 0.541667 32: 0.215278 0.159722 33: 0.222222 0.777778 34: 0.229167 0.395833 35: 0.236111 0.0138889 36: 0.243056 0.631944 37: 0.25 0.25 38: 0.256944 0.868056 39: 0.263889 0.486111 40: 0.270833 0.104167 41: 0.277778 0.722222 42: 0.284722 0.340278 43: 0.291667 0.958333 44: 0.298611 0.576389 45: 0.305556 0.194444 46: 0.3125 0.8125 47: 0.319444 0.430556 48: 0.326389 0.0486111 49: 0.333333 0.666667 50: 0.340278 0.284722 51: 0.347222 0.902778 52: 0.354167 0.520833 53: 0.361111 0.138889 54: 0.368056 0.756944 55: 0.375 0.375 56: 0.381944 0.993056 57: 0.388889 0.611111 58: 0.395833 0.229167 59: 0.402778 0.847222 60: 0.409722 0.465278 61: 0.416667 0.0833333 62: 0.423611 0.701389 63: 0.430556 0.319444 64: 0.4375 0.9375 65: 0.444444 0.555556 66: 0.451389 0.173611 67: 0.458333 0.791667 68: 0.465278 0.409722 69: 0.472222 0.0277778 70: 0.479167 0.645833 71: 0.486111 0.263889 72: 0.493056 0.881944 73: 0.5 0.5 74: 0.506944 0.118056 75: 0.513889 0.736111 76: 0.520833 0.354167 77: 0.527778 0.972222 78: 0.534722 0.590278 79: 0.541667 0.208333 80: 0.548611 0.826389 81: 0.555556 0.444444 82: 0.5625 0.0625 83: 0.569444 0.680556 84: 0.576389 0.298611 85: 0.583333 0.916667 86: 0.590278 0.534722 87: 0.597222 0.152778 88: 0.604167 0.770833 89: 0.611111 0.388889 90: 0.618056 0.00694444 91: 0.625 0.625 92: 0.631944 0.243056 93: 0.638889 0.861111 94: 0.645833 0.479167 95: 0.652778 0.0972222 96: 0.659722 0.715278 97: 0.666667 0.333333 98: 0.673611 0.951389 99: 0.680556 0.569444 100: 0.6875 0.1875 101: 0.694444 0.805556 102: 0.701389 0.423611 103: 0.708333 0.0416667 104: 0.715278 0.659722 105: 0.722222 0.277778 106: 0.729167 0.895833 107: 0.736111 0.513889 108: 0.743056 0.131944 109: 0.75 0.75 110: 0.756944 0.368056 111: 0.763889 0.986111 112: 0.770833 0.604167 113: 0.777778 0.222222 114: 0.784722 0.840278 115: 0.791667 0.458333 116: 0.798611 0.0763889 117: 0.805556 0.694444 118: 0.8125 0.3125 119: 0.819444 0.930556 120: 0.826389 0.548611 121: 0.833333 0.166667 122: 0.840278 0.784722 123: 0.847222 0.402778 124: 0.854167 0.0208333 125: 0.861111 0.638889 126: 0.868056 0.256944 127: 0.875 0.875 128: 0.881944 0.493056 129: 0.888889 0.111111 130: 0.895833 0.729167 131: 0.902778 0.347222 132: 0.909722 0.965278 133: 0.916667 0.583333 134: 0.923611 0.201389 135: 0.930556 0.819444 136: 0.9375 0.4375 137: 0.944444 0.0555556 138: 0.951389 0.673611 139: 0.958333 0.291667 140: 0.965278 0.909722 141: 0.972222 0.527778 142: 0.979167 0.145833 143: 0.986111 0.763889 144: 0.993056 0.381944 lattice_rule_test(): Normal end of execution. 07-Jan-2022 22:41:20