08-Jan-2022 07:59:21 quadrule_fast_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 Test quadrule_fast(). quadrule_fast_test01 CLENSHAW_CURTIS_RULE_COMPUTE computes a Clenshaw-Curtis quadrature rule. CLENSHAW_CURTIS_RULE_SET sets a Clenshaw-Curtis quadrature rule. Compare: (X1,W1) from CLENSHAW_CURTIS_RULE_SET (X2,W2) from CLENSHAW_CURTIS_RULE_COMPUTE Order W1 W2 X1 X2 1 2.000000 2.000000 0.000000 0.000000 4 0.111111 0.111111 -1.000000 -1.000000 0.888889 0.888889 -0.500000 -0.500000 0.888889 0.888889 0.500000 0.500000 0.111111 0.111111 1.000000 1.000000 7 0.028571 0.028571 -1.000000 -1.000000 0.253968 0.253968 -0.866025 -0.866025 0.457143 0.457143 -0.500000 -0.500000 0.520635 0.520635 0.000000 0.000000 0.457143 0.457143 0.500000 0.500000 0.253968 0.253968 0.866025 0.866025 0.028571 0.028571 1.000000 1.000000 10 0.012346 0.012346 -1.000000 -1.000000 0.116567 0.116567 -0.939693 -0.939693 0.225284 0.225284 -0.766044 -0.766044 0.301940 0.301940 -0.500000 -0.500000 0.343863 0.343863 -0.173648 -0.173648 0.343863 0.343863 0.173648 0.173648 0.301940 0.301940 0.500000 0.500000 0.225284 0.225284 0.766044 0.766044 0.116567 0.116567 0.939693 0.939693 0.012346 0.012346 1.000000 1.000000 quadrule_fast_test02 CLENSHAW_CURTIS_RULE_COMPUTE computes a Clenshaw-Curtis rule; The integration interval is [-1,1]. Quadrature order will vary. Integrand will vary. Order F1 F2 F3 1 -0.160000 2.222222 1.414214 2 1.758664 0.689655 1.931852 3 0.479555 1.711367 1.586760 4 0.479555 1.542837 1.103539 5 0.479377 1.581331 1.523155 6 0.479422 1.584456 1.491432 7 0.479428 1.581814 1.330965 8 0.479428 1.582296 1.483961 9 0.479428 1.582221 1.476494 10 0.479428 1.582228 1.391306 11 0.479428 1.582235 1.473921 12 0.479428 1.582233 1.470589 13 0.479428 1.582233 1.415409 14 0.479428 1.582233 1.469335 15 0.479428 1.582233 1.467614 16 0.479428 1.582233 1.428316 Exact 0.479428 1.582233 1.460447 quadrule_fast_test03 CLENSHAW_CURTIS_INTEGRATE applies a Clenshaw-Curtis rule; CLENSHAW_CURTIS_INTEGRATE_FAST also applies a Clenshaw-Curtis rule, using an efficient MATLAB scheme; The integration interval is [-1,1]. Quadrature order will vary. Integrand will vary. CLENSHAW_CURTIS_INTEGRATE Order F1 F2 F3 1 -0.160000 2.222222 1.414214 2 1.758664 0.689655 1.931852 3 0.479555 1.711367 1.586760 4 0.479555 1.542837 1.103539 5 0.479377 1.581331 1.523155 6 0.479422 1.584456 1.491432 7 0.479428 1.581814 1.330965 8 0.479428 1.582296 1.483961 9 0.479428 1.582221 1.476494 10 0.479428 1.582228 1.391306 11 0.479428 1.582235 1.473921 12 0.479428 1.582233 1.470589 13 0.479428 1.582233 1.415409 14 0.479428 1.582233 1.469335 15 0.479428 1.582233 1.467614 16 0.479428 1.582233 1.428316 Exact 0.479428 1.582233 1.460447 CLENSHAW_CURTIS_INTEGRATE_FAST Order F1 F2 F3 1 -0.160000 2.222222 1.414214 2 1.758664 0.689655 1.931852 3 0.479555 1.711367 1.586760 4 0.479555 1.542837 1.103539 5 0.479377 1.581331 1.523155 6 0.479422 1.584456 1.491432 7 0.479428 1.581814 1.330965 8 0.479428 1.582296 1.483961 9 0.479428 1.582221 1.476494 10 0.479428 1.582228 1.391306 11 0.479428 1.582235 1.473921 12 0.479428 1.582233 1.470589 13 0.479428 1.582233 1.415409 14 0.479428 1.582233 1.469335 15 0.479428 1.582233 1.467614 16 0.479428 1.582233 1.428316 Exact 0.479428 1.582233 1.460447 quadrule_fast_test04 GAUSS_LEGENDRE_RULE_COMPUTE computes a Clenshaw-Curtis quadrature rule. GAUSS_LEGENDRE_RULE_SET sets a Clenshaw-Curtis quadrature rule. Compare: (X1,W1) from GAUSS_LEGENDRE_RULE_SET (X2,W2) from GAUSS_LEGENDRE_RULE_COMPUTE Order W1 W2 X1 X2 1 2.000000 2.000000 0.000000 0.000000 4 0.347855 0.347855 -0.861136 -0.861136 0.652145 0.652145 -0.339981 -0.339981 0.652145 0.652145 0.339981 0.339981 0.347855 0.347855 0.861136 0.861136 7 0.129485 0.129485 -0.949108 -0.949108 0.279705 0.279705 -0.741531 -0.741531 0.381830 0.381830 -0.405845 -0.405845 0.417959 0.417959 0.000000 -0.000000 0.381830 0.381830 0.405845 0.405845 0.279705 0.279705 0.741531 0.741531 0.129485 0.129485 0.949108 0.949108 10 0.066671 0.066671 -0.973907 -0.973907 0.149451 0.149451 -0.865063 -0.865063 0.219086 0.219086 -0.679410 -0.679410 0.269267 0.269267 -0.433395 -0.433395 0.295524 0.295524 -0.148874 -0.148874 0.295524 0.295524 0.148874 0.148874 0.269267 0.269267 0.433395 0.433395 0.219086 0.219086 0.679410 0.679410 0.149451 0.149451 0.865063 0.865063 0.066671 0.066671 0.973907 0.973907 quadrule_fast_test05 GAUSS_LEGENDRE_RULE_COMPUTE computes a Gauss-Legendre rule; The integration interval is [-1,1]. Quadrature order will vary. Integrand will vary. Order F1 F2 F3 1 -0.160000 2.222222 1.414214 2 0.479457 1.487603 1.316074 3 0.479306 1.585026 1.546873 4 0.479428 1.585060 1.473444 5 0.479428 1.581279 1.415842 6 0.479428 1.582407 1.492684 7 0.479428 1.582218 1.467885 8 0.479428 1.582231 1.437063 9 0.479428 1.582234 1.478380 10 0.479428 1.582233 1.465284 11 0.479428 1.582233 1.445525 12 0.479428 1.582233 1.472232 13 0.479428 1.582233 1.463895 14 0.479428 1.582233 1.449879 15 0.479428 1.582233 1.468941 16 0.479428 1.582233 1.463059 Exact 0.479428 1.582233 1.460447 quadrule_fast_test06 GAUSS_LEGENDRE_INTEGRATE_FAST applies a Gauss-Legendre rule; The integration interval is [-1,1]. Quadrature order will vary. Integrand will vary. Order F1 F2 F3 1 0.479457 1.487603 1.316074 2 0.479306 1.585026 1.546873 3 0.479428 1.585060 1.473444 4 0.479428 1.581279 1.415842 5 0.479428 1.582407 1.492684 6 0.479428 1.582218 1.467885 7 0.479428 1.582231 1.437063 8 0.479428 1.582234 1.478380 9 0.479428 1.582233 1.465284 10 0.479428 1.582233 1.445525 11 0.479428 1.582233 1.472232 12 0.479428 1.582233 1.463895 13 0.479428 1.582233 1.449879 14 0.479428 1.582233 1.468941 15 0.479428 1.582233 1.463059 16 0.479428 1.582233 1.452461 Exact 0.479428 1.582233 1.460447 quadrule_fast_test07 FEJER1_RULE_COMPUTE computes a Fejer type 1 quadrature rule. FEJER1_RULE_SET sets a Fejer type 1 quadrature rule. Compare: (X1,W1) from FEJER1_RULE_SET (X2,W2) from FEJER1_RULE_COMPUTE Order W1 W2 X1 X2 1 2.000000 2.000000 0.000000 0.000000 4 0.264298 0.264298 -0.923880 -0.923880 0.735702 0.735702 -0.382683 -0.382683 0.735702 0.735702 0.382683 0.382683 0.264298 0.264298 0.923880 0.923880 7 0.086716 0.086716 -0.974928 -0.974928 0.287831 0.287831 -0.781831 -0.781831 0.398242 0.398242 -0.433884 -0.433884 0.454422 0.454422 0.000000 0.000000 0.398242 0.398242 0.433884 0.433884 0.287831 0.287831 0.781831 0.781831 0.086716 0.086716 0.974928 0.974928 10 0.042939 0.042939 -0.987688 -0.987688 0.145875 0.145875 -0.891007 -0.891007 0.220317 0.220317 -0.707107 -0.707107 0.280879 0.280879 -0.453990 -0.453990 0.309989 0.309989 -0.156434 -0.156434 0.309989 0.309989 0.156434 0.156434 0.280879 0.280879 0.453990 0.453990 0.220317 0.220317 0.707107 0.707107 0.145875 0.145875 0.891007 0.891007 0.042939 0.042939 0.987688 0.987688 quadrule_fast_test08 FEJER1_RULE_COMPUTE computes a Fejer Type 1 rule; The integration interval is [-1,1]. Quadrature order will vary. Integrand will vary. Order F1 F2 F3 1 -0.160000 2.222222 1.414214 2 0.799000 1.212121 1.553774 3 0.479333 1.636326 1.574017 4 0.479500 1.590820 1.430641 5 0.479444 1.579551 1.452333 6 0.479433 1.582389 1.497600 7 0.479428 1.582105 1.452331 8 0.479428 1.582221 1.457881 9 0.479428 1.582256 1.480872 10 0.479428 1.582239 1.455389 11 0.479428 1.582230 1.457906 12 0.479428 1.582232 1.473613 13 0.479428 1.582233 1.457493 14 0.479428 1.582233 1.458820 15 0.479428 1.582233 1.469883 16 0.479428 1.582233 1.458228 Exact 0.479428 1.582233 1.460447 quadrule_fast_test09 FEJER1_INTEGRATE_FAST applies a Fejer Type 1 rule; The integration interval is [-1,1]. Quadrature order will vary. Integrand will vary. Order F1 F2 F3 1 -0.160000 2.222222 1.414214 2 0.799000 1.212121 1.553774 3 0.479333 1.636326 1.574017 4 0.479500 1.590820 1.430641 5 0.479444 1.579551 1.452333 6 0.479433 1.582389 1.497600 7 0.479428 1.582105 1.452331 8 0.479428 1.582221 1.457881 9 0.479428 1.582256 1.480872 10 0.479428 1.582239 1.455389 11 0.479428 1.582230 1.457906 12 0.479428 1.582232 1.473613 13 0.479428 1.582233 1.457493 14 0.479428 1.582233 1.458820 15 0.479428 1.582233 1.469883 16 0.479428 1.582233 1.458228 Exact 0.479428 1.582233 1.460447 quadrule_fast_test10 FEJER2_RULE_COMPUTE computes a Fejer type 2 quadrature rule. FEJER2_RULE_SET sets a Fejer type 2 quadrature rule. Compare: (X1,W1) from FEJER2_RULE_SET (X2,W2) from FEJER2_RULE_COMPUTE Order W1 W2 X1 X2 1 2.000000 2.000000 0.000000 0.000000 4 0.425464 0.425464 -0.809017 -0.809017 0.574536 0.574536 -0.309017 -0.309017 0.574536 0.574536 0.309017 0.309017 0.425464 0.425464 0.809017 0.809017 7 0.177965 0.177965 -0.923880 -0.923880 0.247619 0.247619 -0.707107 -0.707107 0.393464 0.393464 -0.382683 -0.382683 0.361905 0.361905 0.000000 0.000000 0.393464 0.393464 0.382683 0.382683 0.247619 0.247619 0.707107 0.707107 0.177965 0.177965 0.923880 0.923880 10 0.094420 0.094420 -0.959493 -0.959493 0.141135 0.141135 -0.841254 -0.841254 0.226387 0.226387 -0.654861 -0.654861 0.253051 0.253051 -0.415415 -0.415415 0.285007 0.285007 -0.142315 -0.142315 0.285007 0.285007 0.142315 0.142315 0.253051 0.253051 0.415415 0.415415 0.226387 0.226387 0.654861 0.654861 0.141135 0.141135 0.841254 0.841254 0.094420 0.094420 0.959493 0.959493 quadrule_fast_test11 FEJER2_RULE_COMPUTE computes a Fejer Type 2 rule; The integration interval is [-1,1]. Quadrature order will vary. Integrand will vary. Order F1 F2 F3 1 -0.160000 2.222222 1.414214 2 0.319667 1.649485 1.000000 3 0.479333 1.548822 1.507254 4 0.479389 1.572934 1.491146 5 0.479400 1.583000 1.360391 6 0.479416 1.582978 1.487936 7 0.479428 1.582556 1.473305 8 0.479428 1.582317 1.386020 9 0.479428 1.582129 1.472308 10 0.479428 1.582195 1.470924 11 0.479428 1.582250 1.420479 12 0.479428 1.582240 1.469975 13 0.479428 1.582231 1.466859 14 0.479428 1.582232 1.426865 15 0.479428 1.582233 1.466397 16 0.479428 1.582233 1.465994 Exact 0.479428 1.582233 1.460447 quadrule_fast_test12 FEJER2_INTEGRATE_FAST applies a Fejer Type 2 rule; The integration interval is [-1,1]. Quadrature order will vary. Integrand will vary. Order F1 F2 F3 1 -0.160000 2.222222 1.414214 2 -0.160000 2.222222 1.414214 3 0.319667 1.649485 1.000000 4 0.479333 1.548822 1.507254 5 0.479389 1.572934 1.491146 6 0.479400 1.583000 1.360391 7 0.479416 1.582978 1.487936 8 0.479428 1.582556 1.473305 9 0.479428 1.582317 1.386020 10 0.479428 1.582129 1.472308 11 0.479428 1.582195 1.470924 12 0.479428 1.582250 1.420479 13 0.479428 1.582240 1.469975 14 0.479428 1.582231 1.466859 15 0.479428 1.582232 1.426865 16 0.479428 1.582233 1.466397 Exact 0.479428 1.582233 1.460447 quadrule_fast_test(): Normal end of execution. 08-Jan-2022 07:59:21