21 October 2023 1:15:38.893 PM hermite_integrands_test(): FORTRAN77 version Test 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 "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)" TEST02 P00_EXACT returns the "exact" integral. Problem EXACT 1 0.6520493321732922 2 1.772453850905516 3 1.209199576156145 4 1.253314137315500 5 1.047197551196598 6 1.329340388179137 7 0.3450971117607857 8 3.008823566113644 TEST03 P00_GAUSS_HERMITE applies a Gauss-Hermite rule to estimate an integral on (-oo,+oo). Problem Order Estimate Exact Error 1 1 1.77245 0.652049 1.12040 1 2 0.276403 0.652049 0.375646 1 4 0.641433 0.652049 0.106160E-01 1 8 0.652049 0.652049 0.542791E-06 1 16 0.652049 0.652049 0.793587E-12 1 32 0.652049 0.652049 0.205269E-11 1 64 0.652049 0.652049 0.119527E-11 2 1 1.77245 1.77245 0.551581E-11 2 2 1.77245 1.77245 0.551625E-11 2 4 1.77245 1.77245 0.551603E-11 2 8 1.77245 1.77245 0.551603E-11 2 16 1.77245 1.77245 0.693356E-11 2 32 1.77245 1.77245 0.554645E-11 2 64 1.77245 1.77245 0.678746E-11 3 1 0.886227 1.20920 0.322973 3 2 0.960529 1.20920 0.248670 3 4 1.10876 1.20920 0.100440 3 8 1.18010 1.20920 0.291040E-01 3 16 1.20387 1.20920 0.532531E-02 3 32 1.20870 1.20920 0.497008E-03 3 64 1.20918 1.20920 0.175953E-04 4 1 0.00000 1.25331 1.25331 4 2 1.40102 1.25331 0.147703 4 4 1.58047 1.25331 0.327152 4 8 2.12994 1.25331 0.876630 4 16 1.55094 1.25331 0.297630 4 32 2.29436 1.25331 1.04105 4 64 -0.498965 1.25331 1.75228 5 1 0.886227 1.04720 0.160971 5 2 0.830710 1.04720 0.216487 5 4 0.947590 1.04720 0.996079E-01 5 8 1.00288 1.04720 0.443171E-01 5 16 1.02654 1.04720 0.206547E-01 5 32 1.03729 1.04720 0.990572E-02 5 64 1.04239 1.04720 0.480589E-02 6 1 0.00000 1.32934 1.32934 6 2 0.443113 1.32934 0.886227 6 4 1.32934 1.32934 0.413714E-11 6 8 1.32934 1.32934 0.413780E-11 6 16 1.32934 1.32934 0.118106E-10 6 32 1.32934 1.32934 0.431810E-11 6 64 1.32934 1.32934 0.667910E-11 7 1 0.00000 0.345097 0.345097 7 2 0.673749 0.345097 0.328652 7 4 0.348155 0.345097 0.305786E-02 7 8 0.345097 0.345097 0.274859E-08 7 16 0.345097 0.345097 0.121564E-11 7 32 0.345097 0.345097 0.105588E-11 7 64 0.345097 0.345097 0.170219E-11 8 1 1.77245 3.00882 1.23637 8 2 2.54451 3.00882 0.464317 8 4 2.94657 3.00882 0.622519E-01 8 8 3.00785 3.00882 0.972964E-03 8 16 3.00882 3.00882 0.170381E-06 8 32 3.00882 3.00882 0.126961E-10 8 64 3.00882 3.00882 0.127107E-10 TEST04 P00_TURING applies a Turing procedure to estimate an integral on (-oo,+oo). Using a tolerance of TOL = 0.100000E-03 Problem H N Estimate Exact Error 1 1.000000 9 0.670109 0.652049 0.180601E-01 1 0.500000 15 0.652049 0.652049 0.178494E-07 1 0.250000 27 0.652047 0.652049 0.192583E-05 1 0.125000 53 0.652046 0.652049 0.376749E-05 1 0.062500 103 0.652041 0.652049 0.816511E-05 1 0.031250 203 0.652038 0.652049 0.116531E-04 1 0.015625 303 0.651546 0.652049 0.503673E-03 2 1.000000 9 1.77264 1.77245 0.183354E-03 2 0.500000 15 1.77245 1.77245 0.114154E-06 2 0.250000 27 1.77245 1.77245 0.284744E-05 2 0.125000 53 1.77245 1.77245 0.482675E-05 2 0.062500 103 1.77244 1.77245 0.935244E-05 2 0.031250 203 1.77244 1.77245 0.128587E-04 2 0.015625 405 1.77244 1.77245 0.135593E-04 3 1.000000 21 1.21422 1.20920 0.501647E-02 3 0.500000 39 1.20915 1.20920 0.507025E-04 3 0.250000 75 1.20911 1.20920 0.846013E-04 3 0.125000 149 1.20911 1.20920 0.902339E-04 3 0.062500 297 1.20911 1.20920 0.931438E-04 3 0.031250 591 1.20910 1.20920 0.976260E-04 3 0.015625 1181 1.20910 1.20920 0.983947E-04 4 1.000000 67199 34.7886 1.25331 33.5353 4 0.500000 72231 192.753 1.25331 191.500 4 0.250000 144461 71.5169 1.25331 70.2635 4 0.125000 101939 32.4151 1.25331 31.1618 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.062500 200001 7.77694 1.25331 6.52362 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.031250 200001 2.38535 1.25331 1.13204 4 0.015625 29071 3.99397 1.25331 2.74066 5 1.000000 47 1.05625 1.04720 0.904800E-02 5 0.500000 93 1.04615 1.04720 0.104531E-02 5 0.250000 183 1.04610 1.04720 0.110157E-02 5 0.125000 363 1.04608 1.04720 0.111985E-02 5 0.062500 725 1.04607 1.04720 0.112294E-02 5 0.031250 1447 1.04607 1.04720 0.112760E-02 5 0.015625 2891 1.04607 1.04720 0.112994E-02 6 1.000000 9 1.34191 1.32934 0.125689E-01 6 0.500000 17 1.32934 1.32934 0.666991E-06 6 0.250000 33 1.32934 1.32934 0.270826E-05 6 0.125000 65 1.32934 1.32934 0.481629E-05 6 0.062500 3 0.189991E-05 1.32934 1.32934 6 0.031250 3 0.595465E-07 1.32934 1.32934 6 0.015625 3 0.186219E-08 1.32934 1.32934 7 1.000000 9 0.334355 0.345097 0.107424E-01 7 0.500000 17 0.345097 0.345097 0.675187E-08 7 0.250000 31 0.345098 0.345097 0.649404E-06 7 0.125000 59 0.345101 0.345097 0.352199E-05 7 0.062500 115 0.345105 0.345097 0.748815E-05 7 0.031250 227 0.345108 0.345097 0.107085E-04 7 0.015625 451 0.345110 0.345097 0.127481E-04 8 1.000000 11 3.00865 3.00882 0.171790E-03 8 0.500000 21 3.00882 3.00882 0.115982E-05 8 0.250000 39 3.00881 3.00882 0.955875E-05 8 0.125000 77 3.00881 3.00882 0.135440E-04 8 0.062500 153 3.00881 3.00882 0.159144E-04 8 0.031250 305 3.00881 3.00882 0.171969E-04 8 0.015625 607 3.00880 3.00882 0.192433E-04 Using a tolerance of TOL = 0.100000E-06 Problem H N Estimate Exact Error 1 1.000000 9 0.670109 0.652049 0.180601E-01 1 0.500000 17 0.652049 0.652049 0.147556E-08 1 0.250000 33 0.652049 0.652049 0.512318E-08 1 0.125000 63 0.652049 0.652049 0.116741E-07 1 0.062500 125 0.652049 0.652049 0.112942E-07 1 0.031250 249 0.652049 0.652049 0.107237E-07 1 0.015625 495 0.652049 0.652049 0.936090E-08 2 1.000000 11 1.77264 1.77245 0.183354E-03 2 0.500000 19 1.77245 1.77245 0.139608E-10 2 0.250000 35 1.77245 1.77245 0.889602E-09 2 0.125000 67 1.77245 1.77245 0.537981E-08 2 0.062500 133 1.77245 1.77245 0.728341E-08 2 0.031250 265 1.77245 1.77245 0.839373E-08 2 0.015625 527 1.77245 1.77245 0.102632E-07 3 1.000000 35 1.21424 1.20920 0.504286E-02 3 0.500000 67 1.20921 1.20920 0.693830E-05 3 0.250000 131 1.20920 1.20920 0.771297E-07 3 0.125000 259 1.20920 1.20920 0.932342E-07 3 0.062500 517 1.20920 1.20920 0.962408E-07 3 0.031250 1033 1.20920 1.20920 0.977683E-07 3 0.015625 2065 1.20920 1.20920 0.985381E-07 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 1.000000 200001 301.392 1.25331 300.139 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.500000 200001 120.152 1.25331 118.899 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.250000 200001 27.1654 1.25331 25.9121 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.125000 200001 23.0552 1.25331 21.8018 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.062500 200001 7.77694 1.25331 6.52362 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.031250 200001 2.38535 1.25331 1.13204 P00_TURING - Warning! Number of steps exceeded N_TOO_MANY = 100000 4 0.015625 200001 7.81267 1.25331 6.55936 5 1.000000 455 1.05728 1.04720 0.100802E-01 5 0.500000 907 1.04721 1.04720 0.974568E-05 5 0.250000 1811 1.04719 1.04720 0.112661E-04 5 0.125000 3619 1.04719 1.04720 0.112848E-04 5 0.062500 7235 1.04719 1.04720 0.112942E-04 5 0.031250 14467 1.04719 1.04720 0.112989E-04 5 0.015625 28933 1.04719 1.04720 0.112997E-04 6 1.000000 11 1.34191 1.32934 0.125690E-01 6 0.500000 21 1.32934 1.32934 0.669520E-10 6 0.250000 41 1.32934 1.32934 0.442681E-09 6 0.125000 79 1.32934 1.32934 0.313102E-08 6 0.062500 155 1.32934 1.32934 0.772569E-08 6 0.031250 309 1.32934 1.32934 0.899104E-08 6 0.015625 617 1.32934 1.32934 0.967355E-08 7 1.000000 11 0.334355 0.345097 0.107424E-01 7 0.500000 19 0.345097 0.345097 0.100232E-09 7 0.250000 37 0.345097 0.345097 0.125071E-09 7 0.125000 71 0.345097 0.345097 0.183235E-08 7 0.062500 141 0.345097 0.345097 0.310895E-08 7 0.031250 279 0.345097 0.345097 0.583390E-08 7 0.015625 557 0.345097 0.345097 0.645572E-08 8 1.000000 15 3.00865 3.00882 0.171657E-03 8 0.500000 27 3.00882 3.00882 0.624212E-08 8 0.250000 51 3.00882 3.00882 0.195499E-08 8 0.125000 99 3.00882 3.00882 0.687079E-08 8 0.062500 195 3.00882 3.00882 0.124539E-07 8 0.031250 389 3.00882 3.00882 0.137587E-07 8 0.015625 777 3.00882 3.00882 0.144441E-07 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 1.77245 1.77245 0.551581E-11 0 2 1.77245 1.77245 0.551625E-11 0 3 1.77245 1.77245 0.551537E-11 1 1 0.00000 0.00000 0.00000 1 2 0.00000 0.00000 0.00000 1 3 0.00000 0.00000 0.00000 2 1 0.00000 0.886227 0.886227 2 2 0.886227 0.886227 0.275835E-11 2 3 0.886227 0.886227 0.275768E-11 2 4 0.886227 0.886227 0.275813E-11 3 1 0.00000 0.00000 0.00000 3 2 0.00000 0.00000 0.00000 3 3 0.00000 0.00000 0.00000 3 4 -0.555112E-16 0.00000 0.555112E-16 4 1 0.00000 1.32934 1.32934 4 2 0.443113 1.32934 0.886227 4 3 1.32934 1.32934 0.413669E-11 4 4 1.32934 1.32934 0.413714E-11 4 5 1.32934 1.32934 0.413669E-11 5 1 0.00000 0.00000 0.00000 5 2 0.00000 0.00000 0.00000 5 3 0.00000 0.00000 0.00000 5 4 -0.111022E-15 0.00000 0.111022E-15 5 5 0.111022E-15 0.00000 0.111022E-15 6 1 0.00000 3.32335 3.32335 6 2 0.221557 3.32335 3.10179 6 3 1.99401 3.32335 1.32934 6 4 3.32335 3.32335 0.103428E-10 6 5 3.32335 3.32335 0.103419E-10 6 6 3.32335 3.32335 0.297553E-10 TEST06 P00_MONTE_CARLO applies a Monte Carlo rule to estimate Hermite-weighted integrals on (-oo,+oo). Problem Order Estimate Exact Error 1 128 0.710875 0.652049 0.588260E-01 1 512 0.671919 0.652049 0.198693E-01 1 2048 0.658156 0.652049 0.610633E-02 1 8192 0.672415 0.652049 0.203659E-01 1 32768 0.653297 0.652049 0.124760E-02 1 131072 0.648537 0.652049 0.351272E-02 1 524288 0.651170 0.652049 0.879272E-03 2 128 1.75646 1.77245 0.159953E-01 2 512 1.77985 1.77245 0.739278E-02 2 2048 1.77717 1.77245 0.471585E-02 2 8192 1.77541 1.77245 0.295467E-02 2 32768 1.76978 1.77245 0.267609E-02 2 131072 1.76995 1.77245 0.250111E-02 2 524288 1.77225 1.77245 0.204005E-03 3 128 1.20100 1.20920 0.820257E-02 3 512 1.19308 1.20920 0.161190E-01 3 2048 1.19718 1.20920 0.120156E-01 3 8192 1.19618 1.20920 0.130233E-01 3 32768 1.23844 1.20920 0.292399E-01 3 131072 1.21142 1.20920 0.221715E-02 3 524288 1.20744 1.20920 0.175582E-02 4 128 1.31349 1.25331 0.601719E-01 4 512 0.899154 1.25331 0.354160 4 2048 1.51914 1.25331 0.265823 4 8192 1.40893 1.25331 0.155618 4 32768 3.59332 1.25331 2.34000 4 131072 2.55785 1.25331 1.30454 4 524288 1.12211 1.25331 0.131203 5 128 1.01431 1.04720 0.328851E-01 5 512 1.02373 1.04720 0.234704E-01 5 2048 1.00400 1.04720 0.431960E-01 5 8192 1.01413 1.04720 0.330681E-01 5 32768 1.03937 1.04720 0.782650E-02 5 131072 1.03111 1.04720 0.160852E-01 5 524288 1.02676 1.04720 0.204419E-01 6 128 1.38202 1.32934 0.526798E-01 6 512 1.27475 1.32934 0.545863E-01 6 2048 1.30954 1.32934 0.198033E-01 6 8192 1.32287 1.32934 0.647487E-02 6 32768 1.33616 1.32934 0.681863E-02 6 131072 1.33575 1.32934 0.640874E-02 6 524288 1.32996 1.32934 0.617572E-03 7 128 0.301840 0.345097 0.432569E-01 7 512 0.349257 0.345097 0.415954E-02 7 2048 0.348001 0.345097 0.290397E-02 7 8192 0.337856 0.345097 0.724081E-02 7 32768 0.342165 0.345097 0.293211E-02 7 131072 0.344584 0.345097 0.513491E-03 7 524288 0.345320 0.345097 0.222933E-03 8 128 3.03968 3.00882 0.308557E-01 8 512 3.01436 3.00882 0.553363E-02 8 2048 3.00635 3.00882 0.247459E-02 8 8192 3.01035 3.00882 0.152182E-02 8 32768 3.01233 3.00882 0.351009E-02 8 131072 3.01079 3.00882 0.196609E-02 8 524288 3.00877 3.00882 0.512057E-04 hermite_integrands_test(): Normal end of execution. 21 October 2023 1:15:39.369 PM