28-Jul-2021 14:10:56 laguerre_integrands_test(): MATLAB/Octave version 9.9.0.1467703 (R2020b) Test laguerre_integrands(). laguerre_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 20 Problem Title 1 "1 / ( x * log(x)^2 )" 2 "1 / ( x * log(x)^(3/2) )" 3 "1 / ( x^1.01 )" 4 "Sine integral" 5 "Fresnel integral" 6 "Complementary error function" 7 "Bessel function" 8 "Debye function" 9 "Gamma(Z=5) function" 10 "1/(1+x*x)" 11 "1 / ( (1+x) * sqrt(x) )" 12 "exp ( - x ) * cos ( x )" 13 "sin(x) / x" 14 "sin ( exp(-x) + exp(-4x) )" 15 "log(x) / ( 1 + 100 x^2 )" 16 "cos(pi x / 2 ) / sqrt(x)" 17 "exp ( - x / 2^beta ) * cos ( x ) / sqrt ( x )" 18 "x^2 * exp ( - x / 2^beta )" 19 "x^(beta-1) / ( 1 + 10 x )^2" 20 "1 / ( 2^beta * ( ( x - 1 )^2 + (1/4)^beta ) * ( x - 2 ) )" laguerre_integrands_TEST02 P00_ALPHA returns the lower limit of integration. P00_EXACT returns the "exact" integral. Problem ALPHA EXACT 1 2.000000 0.1952475419827644 2 2.000000 0.3251084827899133 3 2.000000 13.6280000000000001 4 2.000000 -0.0046848541335081 5 2.000000 0.0015897286158592 6 2.000000 0.0005610371114839 7 2.000000 0.1626689100000000 8 0.000000 1.6449340668482264 9 0.000000 24.0000000000000000 10 0.000000 1.5707963267948966 11 0.000000 3.1415926535897931 12 0.000000 0.5000000000000000 13 0.000000 1.5707963267948966 14 0.000000 1.0634618101722400 15 0.000000 -0.3616892206207732 16 0.000000 1.0000000000000000 17 0.000000 1.3760433900907161 18 0.000000 16.0000000000000000 19 0.000000 0.4967294132898050 20 0.000000 -2.3936758682828221 laguerre_integrands_TEST03 P00_GAUSS_LAGUERRE applies a Gauss-Laguerre rule to estimate an integral on [ALPHA,+oo). Exact Problem Order Estimate Error 1 0.195248 1 0.101600 9.364716e-02 2 0.127915 6.733264e-02 4 0.145108 5.014002e-02 8 0.155432 3.981564e-02 16 0.162236 3.301119e-02 32 0.167086 2.816178e-02 64 0.170693 2.455474e-02 2 0.325108 1 0.106492 2.186163e-01 2 0.137350 1.877584e-01 4 0.161013 1.640951e-01 8 0.178329 1.467796e-01 16 0.191424 1.336844e-01 32 0.201636 1.234726e-01 64 0.209815 1.152935e-01 3 13.628000 1 0.121287 1.350671e+01 2 0.188887 1.343911e+01 4 0.270119 1.335788e+01 8 0.358719 1.326928e+01 16 0.449969 1.317803e+01 32 0.541454 1.308655e+01 64 0.632226 1.299577e+01 4 -0.004685 1 0.017305 2.198990e-02 2 -0.042621 3.793568e-02 4 -0.058719 5.403450e-02 8 -0.040797 3.611254e-02 16 -0.039259 3.457387e-02 32 -0.000240 4.444902e-03 64 -0.025383 2.069792e-02 5 0.001590 1 0.000000 1.589729e-03 2 -0.383132 3.847219e-01 4 -1.399239 1.400829e+00 8 -2.052914 2.054504e+00 16 -0.067543 6.913231e-02 32 1.125133 1.123543e+00 64 -4.590484 4.592073e+00 6 0.000561 1 0.000045 5.156372e-04 2 0.000259 3.020811e-04 4 0.000512 4.885287e-05 8 0.000564 2.831324e-06 16 0.000561 2.935408e-08 32 0.000561 2.106960e-12 64 0.000561 3.793623e-16 7 0.162669 1 0.193130 3.046158e-02 2 0.034668 1.280014e-01 4 0.036719 1.259501e-01 8 0.039504 1.231653e-01 16 0.097083 6.558583e-02 32 0.100708 6.196045e-02 64 0.107105 5.556368e-02 8 1.644934 1 1.581977 6.295736e-02 2 1.644834 1.005366e-04 4 1.644921 1.277597e-05 8 1.644934 9.479829e-08 16 1.644934 3.167400e-11 32 1.644934 1.842970e-14 64 1.644934 1.760814e-13 9 24.000000 1 1.000000 2.300000e+01 2 20.000000 4.000000e+00 4 24.000000 0.000000e+00 8 24.000000 3.552714e-15 16 24.000000 2.131628e-14 32 24.000000 3.552714e-14 64 24.000000 1.776357e-13 10 1.570796 1 1.359141 2.116554e-01 2 1.493257 7.753935e-02 4 1.501190 6.960681e-02 8 1.533760 3.703629e-02 16 1.553738 1.705859e-02 32 1.562483 8.313623e-03 64 1.566725 4.071501e-03 11 3.141593 1 1.359141 1.782452e+00 2 1.809041 1.332552e+00 4 2.184724 9.568689e-01 8 2.465064 6.765287e-01 16 2.665269 4.763241e-01 32 2.806387 3.352053e-01 64 2.905521 2.360718e-01 12 0.500000 1 0.540302 4.030231e-02 2 0.570209 7.020877e-02 4 0.502494 2.493705e-03 8 0.500001 1.206275e-06 16 0.500000 4.188150e-11 32 0.500000 1.626477e-14 64 0.500000 1.271760e-13 13 1.570796 1 2.287355 7.165590e-01 2 1.096108 4.746880e-01 4 1.206083 3.647132e-01 8 1.026964 5.438324e-01 16 1.439952 1.308439e-01 32 1.136135 4.346612e-01 64 1.349069 2.217275e-01 14 1.063462 1 1.023886 3.957620e-02 2 1.077657 1.419552e-02 4 1.097414 3.395216e-02 8 1.071809 8.347010e-03 16 1.063471 9.532385e-06 32 1.063453 9.142719e-06 64 1.063462 1.218448e-08 15 -0.361689 1 0.000000 3.616892e-01 2 -0.018536 3.431534e-01 4 -0.074581 2.871078e-01 8 -0.192106 1.695836e-01 16 -0.318721 4.296828e-02 32 -0.353294 8.395524e-03 64 -0.351930 9.759357e-03 16 1.000000 1 0.000000 1.000000e+00 2 2.672486 1.672486e+00 4 -0.242586 1.242586e+00 8 1.764991 7.649915e-01 16 2.825504 1.825504e+00 32 4.562707 3.562707e+00 64 -2.864909 3.864909e+00 17 1.376043 1 1.143820 2.322234e-01 2 0.453928 9.221158e-01 4 0.810611 5.654328e-01 8 1.065977 3.100663e-01 16 1.161538 2.145054e-01 32 1.223187 1.528561e-01 64 1.267582 1.084612e-01 18 16.000000 1 1.648721 1.435128e+01 2 9.803824 6.196176e+00 4 15.635946 3.640543e-01 8 15.999711 2.893818e-04 16 16.000000 3.654854e-11 32 16.000000 1.065814e-14 64 16.000000 1.474376e-13 19 0.496729 1 0.022465 4.742643e-01 2 0.044548 4.521811e-01 4 0.087700 4.090295e-01 8 0.159062 3.376678e-01 16 0.248273 2.484560e-01 32 0.328154 1.685758e-01 64 0.383161 1.135689e-01 20 -2.393676 1 -5.436564 3.042888e+00 2 -1.027037 1.366638e+00 4 -5.284233 2.890557e+00 8 -0.053775 2.339901e+00 16 0.850874 3.244550e+00 32 -2.954142 5.604659e-01 64 -2.493568 9.989189e-02 laguerre_integrands_TEST04 P00_EXP_TRANSFORM. applies an exponential tranform to estimate an integral on [ALPHA,+oo) as a transformed integral on (0,exp(-ALPHA)] and applying a Gauss-Legendre rule. Exact Problem Order Estimate Error 1 0.195248 1 0.102397 9.285049e-02 2 0.115146 8.010170e-02 4 0.122829 7.241870e-02 8 0.128835 6.641220e-02 16 0.133495 6.175243e-02 32 0.137146 5.810171e-02 64 0.140064 5.518335e-02 2 0.325108 1 0.101920 2.231882e-01 2 0.116144 2.089643e-01 4 0.126716 1.983927e-01 8 0.135293 1.898159e-01 16 0.142154 1.829544e-01 32 0.147686 1.774225e-01 64 0.152222 1.728863e-01 3 13.628000 1 0.099513 1.352849e+01 2 0.126993 1.350101e+01 4 0.153374 1.347463e+01 8 0.177288 1.345071e+01 16 0.198463 1.342954e+01 32 0.217138 1.341086e+01 64 0.233692 1.339431e+01 4 -0.004685 1 0.043575 4.825965e-02 2 -0.006000 1.315217e-03 4 -0.041913 3.722853e-02 8 -0.025927 2.124250e-02 16 0.011800 1.648456e-02 32 0.017410 2.209450e-02 64 -0.009113 4.428015e-03 5 0.001590 1 0.104775 1.031850e-01 2 0.173538 1.719480e-01 4 -0.473965 4.755547e-01 8 0.011009 9.419494e-03 16 0.216734 2.151441e-01 32 -0.144816 1.464056e-01 64 0.184938 1.833479e-01 6 0.000561 1 0.000192 3.693976e-04 2 0.000576 1.468050e-05 4 0.000561 1.395944e-07 8 0.000561 1.645604e-11 16 0.000561 1.810618e-16 32 0.000561 5.421011e-19 64 0.000561 1.084202e-19 7 0.162669 1 0.196625 3.395643e-02 2 0.186723 2.405397e-02 4 0.137365 2.530348e-02 8 0.115478 4.719102e-02 16 0.146863 1.580629e-02 32 0.183056 2.038677e-02 64 0.178131 1.546194e-02 8 1.644934 1 1.386294 2.586397e-01 2 1.547120 9.781386e-02 4 1.614078 3.085613e-02 8 1.636221 8.712950e-03 16 1.642617 2.317292e-03 32 1.644336 5.976363e-04 64 1.644782 1.517576e-04 9 24.000000 1 0.230835 2.376916e+01 2 2.920191 2.107981e+01 4 9.305913 1.469409e+01 8 15.782961 8.217039e+00 16 20.089634 3.910366e+00 32 22.339979 1.660021e+00 64 23.350631 6.493694e-01 10 1.570796 1 1.350938 2.198585e-01 2 1.292775 2.780216e-01 4 1.357305 2.134913e-01 8 1.399142 1.716545e-01 16 1.429304 1.414920e-01 32 1.451272 1.195241e-01 64 1.467677 1.031197e-01 11 3.141593 1 1.418804 1.722788e+00 2 1.794481 1.347112e+00 4 2.062349 1.079244e+00 8 2.242766 8.988266e-01 16 2.362904 7.786890e-01 32 2.445056 6.965371e-01 64 2.503935 6.376572e-01 12 0.500000 1 0.769239 2.692389e-01 2 0.494195 5.805282e-03 4 0.464401 3.559899e-02 8 0.494232 5.768370e-03 16 0.501907 1.906578e-03 32 0.500620 6.198009e-04 64 0.499939 6.051693e-05 13 1.570796 1 1.843653 2.728562e-01 2 2.150018 5.792219e-01 4 1.887081 3.162845e-01 8 1.418356 1.524400e-01 16 1.304519 2.662777e-01 32 1.567615 3.181751e-03 64 1.756936 1.861398e-01 14 1.063462 1 1.066605 3.143537e-03 2 1.086001 2.253877e-02 4 1.063331 1.312007e-04 8 1.063462 1.691666e-09 16 1.063462 2.220446e-16 32 1.063462 4.440892e-16 64 1.063462 2.220446e-16 15 -0.361689 1 -0.014946 3.467433e-01 2 -0.133081 2.286082e-01 4 -0.345925 1.576432e-02 8 -0.357263 4.426690e-03 16 -0.363602 1.912828e-03 32 -0.364854 3.165168e-03 64 -0.364930 3.240597e-03 16 1.000000 1 1.113573 1.135730e-01 2 -0.239747 1.239747e+00 4 0.389418 6.105815e-01 8 1.736133 7.361333e-01 16 0.623858 3.761420e-01 32 0.613312 3.866885e-01 64 1.599290 5.992897e-01 17 1.376043 1 1.553893 1.778492e-01 2 1.212941 1.631021e-01 4 0.939947 4.360959e-01 8 1.078436 2.976074e-01 16 1.353795 2.224801e-02 32 1.440716 6.467230e-02 64 1.374836 1.207029e-03 18 16.000000 1 0.679463 1.532054e+01 2 2.659561 1.334044e+01 4 5.458895 1.054110e+01 8 8.313838 7.686162e+00 16 10.728535 5.271465e+00 32 12.545793 3.454207e+00 64 13.812740 2.187260e+00 19 0.496729 1 0.038186 4.585429e-01 2 0.121232 3.754974e-01 4 0.274752 2.219774e-01 8 0.389742 1.069878e-01 16 0.443148 5.358157e-02 32 0.469610 2.711971e-02 64 0.483005 1.372401e-02 20 -2.393676 1 -2.223385 1.702911e-01 2 -4.979531 2.585855e+00 4 -1.790995 6.026808e-01 8 -1.358694 1.034982e+00 16 -0.704593 1.689083e+00 32 6.813467 9.207143e+00 64 -3.408214 1.014538e+00 laguerre_integrands_TEST05 P00_RAT_TRANSFORM. applies a rational tranform to estimate an integral on [ALPHA,+oo) as a transformed integral on (0,1/(1+ALPHA)] and applying a Gauss-Legendre rule. Exact Problem Order Estimate Error 1 0.195248 1 0.125393 6.985439e-02 2 0.161260 3.398742e-02 4 0.171148 2.409986e-02 8 0.175379 1.986845e-02 16 0.178502 1.674529e-02 32 0.180849 1.439828e-02 64 0.182650 1.259777e-02 2 0.325108 1 0.159078 1.660302e-01 2 0.194318 1.307906e-01 4 0.209901 1.152071e-01 8 0.220796 1.043126e-01 16 0.229533 9.557565e-02 32 0.236585 8.852344e-02 64 0.242364 8.274488e-02 3 13.628000 1 0.319619 1.330838e+01 2 0.450904 1.317710e+01 4 0.601571 1.302643e+01 8 0.763468 1.286453e+01 16 0.930418 1.269758e+01 32 1.098846 1.252915e+01 64 1.266864 1.236114e+01 4 -0.004685 1 -0.311463 3.067782e-01 2 0.241453 2.461381e-01 4 -0.192659 1.879742e-01 8 -0.339783 3.350978e-01 16 -0.146902 1.422170e-01 32 -0.115974 1.112893e-01 64 -0.237789 2.331039e-01 5 0.001590 1 -0.000000 1.589729e-03 2 -4.119966 4.121556e+00 4 -10.889254 1.089084e+01 8 41.522181 4.152059e+01 16 202.350180 2.023486e+02 32 -603.211877 6.032135e+02 64 -2630.294659 2.630296e+03 6 0.000561 1 0.000000 5.610371e-04 2 0.000126 4.352971e-04 4 0.000582 2.087117e-05 8 0.000561 3.154660e-08 16 0.000561 2.469774e-12 32 0.000561 4.336809e-19 64 0.000561 4.336809e-19 7 0.162669 1 -0.317343 4.800121e-01 2 0.076203 8.646597e-02 4 0.186351 2.368255e-02 8 -0.223540 3.862085e-01 16 -0.343201 5.058697e-01 32 -0.117109 2.797779e-01 64 -0.237837 4.005057e-01 8 1.644934 1 2.327907 6.829728e-01 2 1.725976 8.104223e-02 4 1.675123 3.018852e-02 8 1.643986 9.479904e-04 16 1.644932 1.745888e-06 32 1.644934 6.092769e-09 64 1.644934 2.242651e-14 9 24.000000 1 1.471518 2.252848e+01 2 52.008747 2.800875e+01 4 8.464497 1.553550e+01 8 22.679092 1.320908e+00 16 24.028670 2.866965e-02 32 24.000012 1.206287e-05 64 24.000000 3.802292e-10 10 1.570796 1 2.000000 4.292037e-01 2 1.500000 7.079633e-02 4 1.568627 2.168876e-03 8 1.570794 1.914249e-06 16 1.570796 1.452394e-12 32 1.570796 4.440892e-16 64 1.570796 4.440892e-16 11 3.141593 1 2.000000 1.141593e+00 2 2.449490 6.921029e-01 4 2.755404 3.861882e-01 8 2.936842 2.047506e-01 16 3.036071 1.055221e-01 32 3.088014 5.357882e-02 64 3.114595 2.699787e-02 12 0.500000 1 0.795064 2.950644e-01 2 0.370271 1.297295e-01 4 0.402708 9.729242e-02 8 0.480352 1.964804e-02 16 0.498803 1.197231e-03 32 0.499991 8.853198e-06 64 0.500000 2.386332e-09 13 1.570796 1 3.365884 1.795088e+00 2 -0.875962 2.446758e+00 4 4.217633 2.646837e+00 8 0.252453 1.318343e+00 16 0.204644 1.366152e+00 32 2.336418 7.656219e-01 64 -1.308716 2.879512e+00 14 1.063462 1 1.506666 4.432041e-01 2 0.987107 7.635497e-02 4 1.088213 2.475125e-02 8 1.064248 7.859805e-04 16 1.063457 4.385197e-06 32 1.063462 5.507590e-10 64 1.063462 2.220446e-16 15 -0.361689 1 0.000000 3.616892e-01 2 -0.118844 2.428456e-01 4 -0.344891 1.679799e-02 8 -0.352298 9.391018e-03 16 -0.359399 2.290214e-03 32 -0.361098 5.914095e-04 64 -0.361539 1.502252e-04 16 1.000000 1 0.000000 1.000000e+00 2 6.707126 5.707126e+00 4 -6.453605 7.453605e+00 8 -7.568561 8.568561e+00 16 31.368465 3.036847e+01 32 -79.713340 8.071334e+01 64 -1.624902 2.624902e+00 17 1.376043 1 1.683151 3.071080e-01 2 -0.493307 1.869351e+00 4 1.196720 1.793238e-01 8 0.646258 7.297857e-01 16 1.374959 1.084880e-03 32 1.395947 1.990387e-02 64 1.358315 1.772853e-02 18 16.000000 1 2.426123 1.357388e+01 2 24.180590 8.180590e+00 4 12.577617 3.422383e+00 8 17.206017 1.206017e+00 16 16.004154 4.153986e-03 32 16.000070 7.032069e-05 64 16.000000 9.463719e-10 19 0.496729 1 0.033058 4.636716e-01 2 0.118649 3.780805e-01 4 0.275815 2.209147e-01 8 0.390405 1.063247e-01 16 0.443513 5.321601e-02 32 0.469882 2.684698e-02 64 0.483223 1.350624e-02 20 -2.393676 1 -8.000000 5.606324e+00 2 0.123711 2.517387e+00 4 37.269722 3.966340e+01 8 -2.677678 2.840019e-01 16 -3.378910 9.852339e-01 32 -27.245485 2.485181e+01 64 -2.464336 7.066011e-02 laguerre_integrands_test(): Normal end of execution. 28-Jul-2021 14:10:56