28-Jul-2021 10:51:04 hermite_integrands_test() MATLAB/Octave version 9.9.0.1467703 (R2020b) Test hermite_integrands(). HERMITE_INTEGRANDS_TEST01 P00_PROBLEM_NUM returns the number of problems. P00_TITLE returns the title of a problem. P00_PROBLEM_NUM: number of problems is 8 Problem Title 1 "exp(-x*x) * cos(2*omega*x)" 2 "exp(-x*x)" 3 "exp(-px) / ( 1 + exp(-qx) )" 4 "sin(x^2)" 5 "1/( (1+x^2) sqrt(4+3x^2) )" 6 "x^m exp(-x*x)" 7 "x^2 cos ( x ) exp(-x*x)" 8 "sqrt(1+x*x/2) * exp(-x*x/2)" HERMITE_INTEGRANDS_TEST02 P00_EXACT returns the "exact" integral. Problem EXACT 1 0.6520493321732922 2 1.7724538509055159 3 1.2091995761561452 4 1.2533141373155001 5 1.0471975511965976 6 1.3293403881791370 7 0.3450971117607858 8 3.0088235661136435 HERMITE_INTEGRANDS_TEST03 P00_GAUSS_HERMITE applies a Gauss-Hermite rule to estimate an integral on (-oo,+oo). Problem Order Estimate Exact Error 1 1 1.772454 0.652049 1.120405e+00 1 2 0.276403 0.652049 3.756463e-01 1 4 0.641433 0.652049 1.061603e-02 1 8 0.652049 0.652049 5.427909e-07 1 16 0.652049 0.652049 7.949197e-13 1 32 0.652049 0.652049 2.052580e-12 1 64 0.652049 0.652049 1.213807e-12 2 1 1.772454 1.772454 5.515810e-12 2 2 1.772454 1.772454 5.516254e-12 2 4 1.772454 1.772454 5.516032e-12 2 8 1.772454 1.772454 5.516032e-12 2 16 1.772454 1.772454 6.937562e-12 2 32 1.772454 1.772454 5.545564e-12 2 64 1.772454 1.772454 6.838308e-12 3 1 0.886227 1.209200 3.229727e-01 3 2 0.960529 1.209200 2.486705e-01 3 4 1.108759 1.209200 1.004405e-01 3 8 1.180096 1.209200 2.910395e-02 3 16 1.203874 1.209200 5.325315e-03 3 32 1.208703 1.209200 4.970083e-04 3 64 1.209182 1.209200 1.759527e-05 4 1 0.000000 1.253314 1.253314e+00 4 2 1.401017 1.253314 1.477027e-01 4 4 1.580466 1.253314 3.271523e-01 4 8 2.129944 1.253314 8.766300e-01 4 16 1.550944 1.253314 2.976301e-01 4 32 2.294360 1.253314 1.041046e+00 4 64 -0.498965 1.253314 1.752279e+00 5 1 0.886227 1.047198 1.609706e-01 5 2 0.830710 1.047198 2.164873e-01 5 4 0.947590 1.047198 9.960791e-02 5 8 1.002880 1.047198 4.431715e-02 5 16 1.026543 1.047198 2.065470e-02 5 32 1.037292 1.047198 9.905724e-03 5 64 1.042392 1.047198 4.805893e-03 6 1 0.000000 1.329340 1.329340e+00 6 2 0.443113 1.329340 8.862269e-01 6 4 1.329340 1.329340 4.137357e-12 6 8 1.329340 1.329340 4.138245e-12 6 16 1.329340 1.329340 1.181366e-11 6 32 1.329340 1.329340 4.317435e-12 6 64 1.329340 1.329340 6.716849e-12 7 1 0.000000 0.345097 3.450971e-01 7 2 0.673749 0.345097 3.286521e-01 7 4 0.348155 0.345097 3.057863e-03 7 8 0.345097 0.345097 2.748593e-09 7 16 0.345097 0.345097 1.216527e-12 7 32 0.345097 0.345097 1.056044e-12 7 64 0.345097 0.345097 1.712019e-12 8 1 1.772454 3.008824 1.236370e+00 8 2 2.544506 3.008824 4.643171e-01 8 4 2.946572 3.008824 6.225193e-02 8 8 3.007851 3.008824 9.729635e-04 8 16 3.008823 3.008824 1.703806e-07 8 32 3.008824 3.008824 1.269695e-11 8 64 3.008824 3.008824 1.279599e-11 HERMITE_INTEGRANDS_TEST04 P00_TURING applies a Turing procedure to estimate an integral on (-oo,+oo). Using a tolerance of TOL = 0.000100 Problem H N Estimate Exact Error 1 1.000000 9 0.670109 0.652049 1.806009e-02 1 0.500000 15 0.652049 0.652049 1.784943e-08 1 0.250000 27 0.652047 0.652049 1.925832e-06 1 0.125000 53 0.652046 0.652049 3.767487e-06 1 0.062500 103 0.652041 0.652049 8.165106e-06 1 0.031250 203 0.652038 0.652049 1.165309e-05 1 0.015625 303 0.651546 0.652049 5.036730e-04 2 1.000000 9 1.772637 1.772454 1.833539e-04 2 0.500000 15 1.772454 1.772454 1.141544e-07 2 0.250000 27 1.772451 1.772454 2.847444e-06 2 0.125000 53 1.772449 1.772454 4.826746e-06 2 0.062500 103 1.772444 1.772454 9.352444e-06 2 0.031250 203 1.772441 1.772454 1.285874e-05 2 0.015625 405 1.772440 1.772454 1.355934e-05 3 1.000000 21 1.214216 1.209200 5.016467e-03 3 0.500000 39 1.209149 1.209200 5.070255e-05 3 0.250000 75 1.209115 1.209200 8.460130e-05 3 0.125000 149 1.209109 1.209200 9.023385e-05 3 0.062500 297 1.209106 1.209200 9.314378e-05 3 0.031250 591 1.209102 1.209200 9.762598e-05 3 0.015625 1181 1.209101 1.209200 9.839471e-05 4 1.000000 67199 34.788617 1.253314 3.353530e+01 4 0.500000 72231 192.753398 1.253314 1.915001e+02 4 0.250000 144461 71.516851 1.253314 7.026354e+01 4 0.125000 101939 32.415085 1.253314 3.116177e+01 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.062500 200001 7.776937 1.253314 6.523623e+00 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.031250 200001 2.385352 1.253314 1.132037e+00 4 0.015625 29071 3.993971 1.253314 2.740657e+00 5 1.000000 47 1.056246 1.047198 9.047995e-03 5 0.500000 93 1.046152 1.047198 1.045314e-03 5 0.250000 183 1.046096 1.047198 1.101573e-03 5 0.125000 363 1.046078 1.047198 1.119851e-03 5 0.062500 725 1.046075 1.047198 1.122944e-03 5 0.031250 1447 1.046070 1.047198 1.127599e-03 5 0.015625 2891 1.046068 1.047198 1.129937e-03 6 1.000000 9 1.341909 1.329340 1.256894e-02 6 0.500000 17 1.329340 1.329340 6.669907e-07 6 0.250000 33 1.329338 1.329340 2.708257e-06 6 0.125000 65 1.329336 1.329340 4.816289e-06 6 0.062500 3 0.000002 1.329340 1.329338e+00 6 0.031250 3 0.000000 1.329340 1.329340e+00 6 0.015625 3 0.000000 1.329340 1.329340e+00 7 1.000000 9 0.334355 0.345097 1.074235e-02 7 0.500000 17 0.345097 0.345097 6.751869e-09 7 0.250000 31 0.345098 0.345097 6.494042e-07 7 0.125000 59 0.345101 0.345097 3.521986e-06 7 0.062500 115 0.345105 0.345097 7.488147e-06 7 0.031250 227 0.345108 0.345097 1.070852e-05 7 0.015625 451 0.345110 0.345097 1.274814e-05 8 1.000000 11 3.008652 3.008824 1.717898e-04 8 0.500000 21 3.008822 3.008824 1.159817e-06 8 0.250000 39 3.008814 3.008824 9.558749e-06 8 0.125000 77 3.008810 3.008824 1.354403e-05 8 0.062500 153 3.008808 3.008824 1.591441e-05 8 0.031250 305 3.008806 3.008824 1.719691e-05 8 0.015625 607 3.008804 3.008824 1.924333e-05 Using a tolerance of TOL = 0.000000 Problem H N Estimate Exact Error 1 1.000000 9 0.670109 0.652049 1.806009e-02 1 0.500000 17 0.652049 0.652049 1.475562e-09 1 0.250000 33 0.652049 0.652049 5.123175e-09 1 0.125000 63 0.652049 0.652049 1.167414e-08 1 0.062500 125 0.652049 0.652049 1.129419e-08 1 0.031250 249 0.652049 0.652049 1.072373e-08 1 0.015625 495 0.652049 0.652049 9.360897e-09 2 1.000000 11 1.772637 1.772454 1.833539e-04 2 0.500000 19 1.772454 1.772454 1.396083e-11 2 0.250000 35 1.772454 1.772454 8.896017e-10 2 0.125000 67 1.772454 1.772454 5.379811e-09 2 0.062500 133 1.772454 1.772454 7.283415e-09 2 0.031250 265 1.772454 1.772454 8.393727e-09 2 0.015625 527 1.772454 1.772454 1.026324e-08 3 1.000000 35 1.214242 1.209200 5.042865e-03 3 0.500000 67 1.209207 1.209200 6.938303e-06 3 0.250000 131 1.209199 1.209200 7.712970e-08 3 0.125000 259 1.209199 1.209200 9.323423e-08 3 0.062500 517 1.209199 1.209200 9.624077e-08 3 0.031250 1033 1.209199 1.209200 9.776828e-08 3 0.015625 2065 1.209199 1.209200 9.853810e-08 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 1.000000 200001 301.392135 1.253314 3.001388e+02 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.500000 200001 120.152472 1.253314 1.188992e+02 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.250000 200001 27.165435 1.253314 2.591212e+01 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.125000 200001 23.055157 1.253314 2.180184e+01 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.062500 200001 7.776937 1.253314 6.523623e+00 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.031250 200001 2.385352 1.253314 1.132037e+00 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.015625 200001 7.812673 1.253314 6.559359e+00 5 1.000000 455 1.057278 1.047198 1.008025e-02 5 0.500000 907 1.047207 1.047198 9.745685e-06 5 0.250000 1811 1.047186 1.047198 1.126607e-05 5 0.125000 3619 1.047186 1.047198 1.128483e-05 5 0.062500 7235 1.047186 1.047198 1.129419e-05 5 0.031250 14467 1.047186 1.047198 1.129888e-05 5 0.015625 28933 1.047186 1.047198 1.129966e-05 6 1.000000 11 1.341909 1.329340 1.256896e-02 6 0.500000 21 1.329340 1.329340 6.695200e-11 6 0.250000 41 1.329340 1.329340 4.426806e-10 6 0.125000 79 1.329340 1.329340 3.131023e-09 6 0.062500 155 1.329340 1.329340 7.725692e-09 6 0.031250 309 1.329340 1.329340 8.991035e-09 6 0.015625 617 1.329340 1.329340 9.673546e-09 7 1.000000 11 0.334355 0.345097 1.074235e-02 7 0.500000 19 0.345097 0.345097 1.002318e-10 7 0.250000 37 0.345097 0.345097 1.250707e-10 7 0.125000 71 0.345097 0.345097 1.832351e-09 7 0.062500 141 0.345097 0.345097 3.108951e-09 7 0.031250 279 0.345097 0.345097 5.833900e-09 7 0.015625 557 0.345097 0.345097 6.455718e-09 8 1.000000 15 3.008652 3.008824 1.716568e-04 8 0.500000 27 3.008824 3.008824 6.242118e-09 8 0.250000 51 3.008824 3.008824 1.954992e-09 8 0.125000 99 3.008824 3.008824 6.870790e-09 8 0.062500 195 3.008824 3.008824 1.245387e-08 8 0.031250 389 3.008824 3.008824 1.375869e-08 8 0.015625 777 3.008824 3.008824 1.444410e-08 HERMITE_INTEGRANDS_TEST05 P00_GAUSS_HERMITE applies a Gauss-Hermite rule to estimate the integral x^m exp(-x*x) over (-oo,+oo). M Order Estimate Exact Error 0 1 1.772454 1.772454 0.000000 0 2 1.772454 1.772454 0.000000 0 3 1.772454 1.772454 0.000000 1 1 -0.000000 0.000000 0.000000 1 2 0.000000 0.000000 0.000000 1 3 0.000000 0.000000 0.000000 2 1 0.000000 0.886227 0.886227 2 2 0.886227 0.886227 0.000000 2 3 0.886227 0.886227 0.000000 2 4 0.886227 0.886227 0.000000 3 1 -0.000000 0.000000 0.000000 3 2 0.000000 0.000000 0.000000 3 3 0.000000 0.000000 0.000000 3 4 0.000000 0.000000 0.000000 4 1 0.000000 1.329340 1.329340 4 2 0.443113 1.329340 0.886227 4 3 1.329340 1.329340 0.000000 4 4 1.329340 1.329340 0.000000 4 5 1.329340 1.329340 0.000000 5 1 -0.000000 0.000000 0.000000 5 2 0.000000 0.000000 0.000000 5 3 0.000000 0.000000 0.000000 5 4 0.000000 0.000000 0.000000 5 5 0.000000 0.000000 0.000000 6 1 0.000000 3.323351 3.323351 6 2 0.221557 3.323351 3.101794 6 3 1.994011 3.323351 1.329340 6 4 3.323351 3.323351 0.000000 6 5 3.323351 3.323351 0.000000 6 6 3.323351 3.323351 0.000000 HERMITE_INTEGRANDS_TEST06 P00_MONTE_CARLO applies a weighted Monte Carlo rule to estimate an integral on (-oo,+oo). Problem Order Estimate Exact Error 1 128 0.490381 0.652049 1.616680e-01 1 512 0.762021 0.652049 1.099716e-01 1 2048 0.674161 0.652049 2.211175e-02 1 8192 0.670697 0.652049 1.864756e-02 1 32768 0.646809 0.652049 5.240040e-03 1 131072 0.652048 0.652049 1.347331e-06 1 524288 0.653214 0.652049 1.164371e-03 2 128 1.834263 1.772454 6.180957e-02 2 512 1.777657 1.772454 5.203193e-03 2 2048 1.747348 1.772454 2.510561e-02 2 8192 1.785707 1.772454 1.325320e-02 2 32768 1.769381 1.772454 3.072392e-03 2 131072 1.771953 1.772454 5.008155e-04 2 524288 1.770930 1.772454 1.524178e-03 3 128 1.147541 1.209200 6.165843e-02 3 512 1.183112 1.209200 2.608729e-02 3 2048 1.205920 1.209200 3.279453e-03 3 8192 1.201655 1.209200 7.544393e-03 3 32768 1.188914 1.209200 2.028560e-02 3 131072 1.199025 1.209200 1.017489e-02 3 524288 1.200335 1.209200 8.864154e-03 4 128 -1.265116 1.253314 2.518430e+00 4 512 2.420975 1.253314 1.167661e+00 4 2048 2.263289 1.253314 1.009974e+00 4 8192 1.750069 1.253314 4.967551e-01 4 32768 0.140227 1.253314 1.113087e+00 4 131072 1.603488 1.253314 3.501743e-01 4 524288 0.748236 1.253314 5.050779e-01 5 128 0.975091 1.047198 7.210658e-02 5 512 1.084107 1.047198 3.690986e-02 5 2048 0.998472 1.047198 4.872590e-02 5 8192 1.018355 1.047198 2.884217e-02 5 32768 1.014410 1.047198 3.278765e-02 5 131072 1.020457 1.047198 2.674087e-02 5 524288 1.022942 1.047198 2.425512e-02 6 128 1.174619 1.329340 1.547217e-01 6 512 1.319957 1.329340 9.383522e-03 6 2048 1.295262 1.329340 3.407828e-02 6 8192 1.347523 1.329340 1.818219e-02 6 32768 1.328381 1.329340 9.592199e-04 6 131072 1.332245 1.329340 2.904962e-03 6 524288 1.328577 1.329340 7.633968e-04 7 128 0.309146 0.345097 3.595065e-02 7 512 0.354887 0.345097 9.789678e-03 7 2048 0.329768 0.345097 1.532940e-02 7 8192 0.351470 0.345097 6.373041e-03 7 32768 0.345679 0.345097 5.820766e-04 7 131072 0.344980 0.345097 1.169378e-04 7 524288 0.346310 0.345097 1.213018e-03 8 128 3.024201 3.008824 1.537694e-02 8 512 3.016907 3.008824 8.083120e-03 8 2048 3.007582 3.008824 1.241313e-03 8 8192 3.005636 3.008824 3.187189e-03 8 32768 3.006880 3.008824 1.943953e-03 8 131072 3.008123 3.008824 7.007180e-04 8 524288 3.009369 3.008824 5.449362e-04 hermite_integrands_test(): Normal end of execution. 28-Jul-2021 10:51:07