23 March 2020 10:07:06 AM laguerre_integrands_test C++ version Test 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 ) )". TEST02 P00_ALPHA returns the lower limit of integration. P00_EXACT returns the "exact" integral. Problem ALPHA EXACT 1 2 0.1952475419827644 2 2 0.3251084827899133 3 2 13.628 4 2 -0.004684854133508064 5 2 0.001589728615859233 6 2 0.0005610371114838712 7 2 0.16266891 8 0 1.644934066848226 9 0 24 10 0 1.570796326794897 11 0 3.141592653589793 12 0 0.5 13 0 1.570796326794897 14 0 1.06346181017224 15 0 -0.3616892206207732 16 0 1 17 0 1.376043390090716 18 0 16 19 0 0.496729413289805 20 0 -2.393675868282822 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 0.1016 0.0936472 2 0.127915 0.0673326 4 0.145108 0.05014 8 0.155432 0.0398156 16 0.162236 0.0330112 32 0.167086 0.0281618 64 0.170693 0.0245547 2 0.325108 1 0.106492 0.218616 2 0.13735 0.187758 4 0.161013 0.164095 8 0.178329 0.14678 16 0.191424 0.133684 32 0.201636 0.123473 64 0.209815 0.115293 3 13.628 1 0.121287 13.5067 2 0.188887 13.4391 4 0.270119 13.3579 8 0.358719 13.2693 16 0.449969 13.178 32 0.541454 13.0865 64 0.632226 12.9958 4 -0.00468485 1 0.017305 0.0219899 2 -0.0426205 0.0379357 4 -0.0587194 0.0540345 8 -0.0407974 0.0361125 16 -0.0392587 0.0345739 32 -0.000239952 0.0044449 64 -0.0253828 0.0206979 5 0.00158973 1 2.02735e-16 0.00158973 2 -0.383132 0.384722 4 -1.39924 1.40083 8 -2.05291 2.0545 16 -0.0675426 0.0691323 32 1.12513 1.12354 64 -4.59048 4.59207 6 0.000561037 1 4.53999e-05 0.000515637 2 0.000258956 0.000302081 4 0.000512184 4.88529e-05 8 0.000563868 2.83132e-06 16 0.000561008 2.93541e-08 32 0.000561037 2.10696e-12 64 0.000561037 3.79362e-16 7 0.162669 1 0.19313 0.0304616 2 0.0346675 0.128001 4 0.0367188 0.12595 8 0.0395037 0.123165 16 0.0970831 0.0655858 32 0.100708 0.0619605 64 0.107105 0.0555637 8 1.64493 1 1.58198 0.0629574 2 1.64483 0.000100537 4 1.64492 1.2776e-05 8 1.64493 9.47983e-08 16 1.64493 3.1674e-11 32 1.64493 1.77636e-14 64 1.64493 1.76081e-13 9 24 1 1 23 2 20 4 4 24 0 8 24 3.55271e-15 16 24 2.4869e-14 32 24 3.90799e-14 64 24 1.77636e-13 10 1.5708 1 1.35914 0.211655 2 1.49326 0.0775394 4 1.50119 0.0696068 8 1.53376 0.0370363 16 1.55374 0.0170586 32 1.56248 0.00831362 64 1.56672 0.0040715 11 3.14159 1 1.35914 1.78245 2 1.80904 1.33255 4 2.18472 0.956869 8 2.46506 0.676529 16 2.66527 0.476324 32 2.80639 0.335205 64 2.90552 0.236072 12 0.5 1 0.540302 0.0403023 2 0.570209 0.0702088 4 0.502494 0.00249371 8 0.500001 1.20627e-06 16 0.5 4.18814e-11 32 0.5 1.62093e-14 64 0.5 1.27176e-13 13 1.5708 1 2.28736 0.716559 2 1.09611 0.474688 4 1.20608 0.364713 8 1.02696 0.543832 16 1.43995 0.130844 32 1.13614 0.434661 64 1.34907 0.221727 14 1.06346 1 1.02389 0.0395762 2 1.07766 0.0141955 4 1.09741 0.0339522 8 1.07181 0.00834701 16 1.06347 9.53238e-06 32 1.06345 9.14272e-06 64 1.06346 1.21845e-08 15 -0.361689 1 0 0.361689 2 -0.0185358 0.343153 4 -0.0745814 0.287108 8 -0.192106 0.169584 16 -0.318721 0.0429683 32 -0.353294 0.00839552 64 -0.35193 0.00975936 16 1 1 1.66447e-16 1 2 2.67249 1.67249 4 -0.242586 1.24259 8 1.76499 0.764991 16 2.8255 1.8255 32 4.56271 3.56271 64 -2.86491 3.86491 17 1.37604 1 1.14382 0.232223 2 0.453928 0.922116 4 0.810611 0.565433 8 1.06598 0.310066 16 1.16154 0.214505 32 1.22319 0.152856 64 1.26758 0.108461 18 16 1 1.64872 14.3513 2 9.80382 6.19618 4 15.6359 0.364054 8 15.9997 0.000289382 16 16 3.65485e-11 32 16 1.42109e-14 64 16 1.47438e-13 19 0.496729 1 0.0224651 0.474264 2 0.0445483 0.452181 4 0.0876999 0.409029 8 0.159062 0.337668 16 0.248273 0.248456 32 0.328154 0.168576 64 0.383161 0.113569 20 -2.39368 1 -5.43656 3.04289 2 -1.02704 1.36664 4 -5.28423 2.89056 8 -0.0537746 2.3399 16 0.850874 3.24455 32 -2.95414 0.560466 64 -2.49357 0.0998919 TEST04 P00_EXP_TRANSFORM applies an exponential transform 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 0.0928505 2 0.115146 0.0801017 4 0.122829 0.0724187 8 0.128835 0.0664122 16 0.133495 0.0617524 32 0.137146 0.0581017 64 0.140064 0.0551833 2 0.325108 1 0.10192 0.223188 2 0.116144 0.208964 4 0.126716 0.198393 8 0.135293 0.189816 16 0.142154 0.182954 32 0.147686 0.177423 64 0.152222 0.172886 3 13.628 1 0.0995127 13.5285 2 0.126993 13.501 4 0.153374 13.4746 8 0.177288 13.4507 16 0.198463 13.4295 32 0.217138 13.4109 64 0.233692 13.3943 4 -0.00468485 1 0.0435748 0.0482597 2 -0.00600007 0.00131522 4 -0.0419134 0.0372285 8 -0.0259274 0.0212425 16 0.0117997 0.0164846 32 0.0174096 0.0220945 64 -0.00911287 0.00442802 5 0.00158973 1 0.104775 0.103185 2 0.173538 0.171948 4 -0.473965 0.475555 8 0.0110092 0.00941949 16 0.216734 0.215144 32 -0.144816 0.146406 64 0.184938 0.183348 6 0.000561037 1 0.000191639 0.000369398 2 0.000575718 1.46805e-05 4 0.000561177 1.39594e-07 8 0.000561037 1.6456e-11 16 0.000561037 1.81062e-16 32 0.000561037 4.33681e-19 64 0.000561037 2.1684e-19 7 0.162669 1 0.196625 0.0339564 2 0.186723 0.024054 4 0.137365 0.0253035 8 0.115478 0.047191 16 0.146863 0.0158063 32 0.183056 0.0203868 64 0.178131 0.0154619 8 1.64493 1 1.38629 0.25864 2 1.54712 0.0978139 4 1.61408 0.0308561 8 1.63622 0.00871295 16 1.64262 0.00231729 32 1.64434 0.000597636 64 1.64478 0.000151758 9 24 1 0.230835 23.7692 2 2.92019 21.0798 4 9.30591 14.6941 8 15.783 8.21704 16 20.0896 3.91037 32 22.34 1.66002 64 23.3506 0.649369 10 1.5708 1 1.35094 0.219858 2 1.29277 0.278022 4 1.35731 0.213491 8 1.39914 0.171655 16 1.4293 0.141492 32 1.45127 0.119524 64 1.46768 0.10312 11 3.14159 1 1.4188 1.72279 2 1.79448 1.34711 4 2.06235 1.07924 8 2.24277 0.898827 16 2.3629 0.778689 32 2.44506 0.696537 64 2.50394 0.637657 12 0.5 1 0.769239 0.269239 2 0.494195 0.00580528 4 0.464401 0.035599 8 0.494232 0.00576837 16 0.501907 0.00190658 32 0.50062 0.000619801 64 0.499939 6.05169e-05 13 1.5708 1 1.84365 0.272856 2 2.15002 0.579222 4 1.88708 0.316285 8 1.41836 0.15244 16 1.30452 0.266278 32 1.56761 0.00318175 64 1.75694 0.18614 14 1.06346 1 1.06661 0.00314354 2 1.086 0.0225388 4 1.06333 0.000131201 8 1.06346 1.69167e-09 16 1.06346 2.22045e-16 32 1.06346 4.44089e-16 64 1.06346 2.22045e-16 15 -0.361689 1 -0.0149459 0.346743 2 -0.133081 0.228608 4 -0.345925 0.0157643 8 -0.357263 0.00442669 16 -0.363602 0.00191283 32 -0.364854 0.00316517 64 -0.36493 0.0032406 16 1 1 1.11357 0.113573 2 -0.239747 1.23975 4 0.389418 0.610582 8 1.73613 0.736133 16 0.623858 0.376142 32 0.613312 0.386688 64 1.59929 0.59929 17 1.37604 1 1.55389 0.177849 2 1.21294 0.163102 4 0.939947 0.436096 8 1.07844 0.297607 16 1.3538 0.022248 32 1.44072 0.0646723 64 1.37484 0.00120703 18 16 1 0.679463 15.3205 2 2.65956 13.3404 4 5.4589 10.5411 8 8.31384 7.68616 16 10.7285 5.27147 32 12.5458 3.45421 64 13.8127 2.18726 19 0.496729 1 0.0381865 0.458543 2 0.121232 0.375497 4 0.274752 0.221977 8 0.389742 0.106988 16 0.443148 0.0535816 32 0.46961 0.0271197 64 0.483005 0.013724 20 -2.39368 1 -2.22338 0.170291 2 -4.97953 2.58585 4 -1.791 0.602681 8 -1.35869 1.03498 16 -0.704593 1.68908 32 6.81347 9.20714 64 -3.40821 1.01454 TEST05 P00_RAT_TRANSFORM applies a rational transform 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 0.0698544 2 0.16126 0.0339874 4 0.171148 0.0240999 8 0.175379 0.0198685 16 0.178502 0.0167453 32 0.180849 0.0143983 64 0.18265 0.0125978 2 0.325108 1 0.159078 0.16603 2 0.194318 0.130791 4 0.209901 0.115207 8 0.220796 0.104313 16 0.229533 0.0955756 32 0.236585 0.0885234 64 0.242364 0.0827449 3 13.628 1 0.319619 13.3084 2 0.450904 13.1771 4 0.601571 13.0264 8 0.763468 12.8645 16 0.930418 12.6976 32 1.09885 12.5292 64 1.26686 12.3611 4 -0.00468485 1 -0.311463 0.306778 2 0.241453 0.246138 4 -0.192659 0.187974 8 -0.339783 0.335098 16 -0.146902 0.142217 32 -0.115974 0.111289 64 -0.237789 0.233104 5 0.00158973 1 -2.34775e-14 0.00158973 2 -4.11997 4.12156 4 -10.8893 10.8908 8 41.5222 41.5206 16 202.35 202.349 32 -603.212 603.213 64 -2630.29 2630.3 6 0.000561037 1 2.25543e-11 0.000561037 2 0.00012574 0.000435297 4 0.000581908 2.08712e-05 8 0.000561006 3.15466e-08 16 0.000561037 2.46977e-12 32 0.000561037 3.25261e-19 64 0.000561037 4.33681e-19 7 0.162669 1 -0.317343 0.480012 2 0.0762029 0.086466 4 0.186351 0.0236826 8 -0.22354 0.386209 16 -0.343201 0.50587 32 -0.117109 0.279778 64 -0.237837 0.400506 8 1.64493 1 2.32791 0.682973 2 1.72598 0.0810422 4 1.67512 0.0301885 8 1.64399 0.00094799 16 1.64493 1.74589e-06 32 1.64493 6.09277e-09 64 1.64493 2.22045e-14 9 24 1 1.47152 22.5285 2 52.0087 28.0087 4 8.4645 15.5355 8 22.6791 1.32091 16 24.0287 0.0286696 32 24 1.20629e-05 64 24 3.8024e-10 10 1.5708 1 2 0.429204 2 1.5 0.0707963 4 1.56863 0.00216888 8 1.57079 1.91425e-06 16 1.5708 1.45262e-12 32 1.5708 2.22045e-16 64 1.5708 1.33227e-15 11 3.14159 1 2 1.14159 2 2.44949 0.692103 4 2.7554 0.386188 8 2.93684 0.204751 16 3.03607 0.105522 32 3.08801 0.0535788 64 3.11459 0.0269979 12 0.5 1 0.795064 0.295064 2 0.370271 0.129729 4 0.402708 0.0972924 8 0.480352 0.019648 16 0.498803 0.00119723 32 0.499991 8.8532e-06 64 0.5 2.38633e-09 13 1.5708 1 3.36588 1.79509 2 -0.875962 2.44676 4 4.21763 2.64684 8 0.252453 1.31834 16 0.204644 1.36615 32 2.33642 0.765622 64 -1.30872 2.87951 14 1.06346 1 1.50667 0.443204 2 0.987107 0.076355 4 1.08821 0.0247513 8 1.06425 0.00078598 16 1.06346 4.3852e-06 32 1.06346 5.50759e-10 64 1.06346 0 15 -0.361689 1 0 0.361689 2 -0.118844 0.242846 4 -0.344891 0.016798 8 -0.352298 0.00939102 16 -0.359399 0.00229021 32 -0.361098 0.00059141 64 -0.361539 0.000150225 16 1 1 2.44929e-16 1 2 6.70713 5.70713 4 -6.45361 7.45361 8 -7.56856 8.56856 16 31.3685 30.3685 32 -79.7133 80.7133 64 -1.6249 2.6249 17 1.37604 1 1.68315 0.307108 2 -0.493307 1.86935 4 1.19672 0.179324 8 0.646258 0.729786 16 1.37496 0.00108488 32 1.39595 0.0199039 64 1.35831 0.0177285 18 16 1 2.42612 13.5739 2 24.1806 8.18059 4 12.5776 3.42238 8 17.206 1.20602 16 16.0042 0.00415399 32 16.0001 7.03207e-05 64 16 9.46375e-10 19 0.496729 1 0.0330579 0.463672 2 0.118649 0.378081 4 0.275815 0.220915 8 0.390405 0.106325 16 0.443513 0.053216 32 0.469882 0.026847 64 0.483223 0.0135062 20 -2.39368 1 -8 5.60632 2 0.123711 2.51739 4 37.2697 39.6634 8 -2.67768 0.284002 16 -3.37891 0.985234 32 -27.2455 24.8518 64 -2.46434 0.0706601 laguerre_integrands_test Normal end of execution. 23 March 2020 10:07:06 AM