Fri Dec 26 15:23:38 2025 triangle_integrands_test(): python version: 3.10.12 numpy version: 1.26.4 Test triangle_integrands(). triangle_integrands_test01(): get_prob_num() reports the number of problems. p00_title() returns a title for each problem. The number of problems available is 22 The problem titles: 1 f(x,y) = 2 2 f(x,y) = 6 * x 3 f(x,y) = 6 * y 4 f(x,y) = 12 * x^2 5 f(x,y) = 24 * x*y 6 f(x,y) = 12 * y^2 7 f(x,y) = 20 * x^3 8 f(x,y) = 30 * x^4 9 f(x,y) = 42 * x^5 10 f(x,y) = x^(-0.2) on ((1,0),(5,0),(5,1)) 11 f(x,y) = (x+y)^(-0.2) 12 f(x,y) = (1-x-y)^(-0.2) on ((-1,-3),(3,-2),(-1,2)) 13 f(x,y) = (x*y)^(-0.2) on ((0,0),(-7,0),(0,-3)) 14 f(x,y) = 1/sqrt(x) + 1/sqrt(y) + 1/sqrt(x+y) 15 f(x,y) = 1/sqrt(1-x-y) 16 f(x,y) = log(x*y) 17 f(x,y) = 1/sqrt(|x-1/4|) + 1/sqrt(|y-1/2|) 18 f(x,y) = log ( x + y ) 19 f(x,y) = sin ( x ) cos ( 5 y ) 20 f(x,y) = sin ( 11 x ) cos ( y ) 21 f(x,y) = 1 / r = 1 / sqrt ( x^2 + y^2 ) 22 f(x,y) = log ( r ) / r triangle_integrands_test02(): p00_monte_carlo() applies a Monte Carlo rule. Problem Exact Pts Approx Error f(x,y) = 2 1 1.000000 1 1.000000 0.000000 2 1.000000 0.000000 4 1.000000 0.000000 8 1.000000 0.000000 16 1.000000 0.000000 32 1.000000 0.000000 64 1.000000 0.000000 128 1.000000 0.000000 256 1.000000 0.000000 512 1.000000 0.000000 1024 1.000000 0.000000 2048 1.000000 0.000000 4096 1.000000 0.000000 8192 1.000000 0.000000 16384 1.000000 0.000000 32768 1.000000 0.000000 f(x,y) = 6 * x 2 1.000000 1 1.496874 0.496874 2 0.590810 0.409190 4 1.102271 0.102271 8 1.383224 0.383224 16 1.296024 0.296024 32 1.147984 0.147984 64 0.869016 0.130984 128 0.993158 0.006842 256 0.991931 0.008069 512 1.053714 0.053714 1024 0.960203 0.039797 2048 1.000657 0.000657 4096 1.002793 0.002793 8192 0.990217 0.009783 16384 1.004170 0.004170 32768 0.994433 0.005567 f(x,y) = 6 * y 3 1.000000 1 1.766858 0.766858 2 0.672287 0.327713 4 0.836631 0.163369 8 1.036122 0.036122 16 0.891900 0.108100 32 1.106275 0.106275 64 0.981961 0.018039 128 1.064728 0.064728 256 0.923396 0.076604 512 0.979067 0.020933 1024 0.990462 0.009538 2048 0.975521 0.024479 4096 1.012814 0.012814 8192 0.991760 0.008240 16384 1.008410 0.008410 32768 1.001648 0.001648 f(x,y) = 12 * x^2 4 1.000000 1 0.517724 0.482276 2 2.478818 1.478818 4 0.528718 0.471282 8 0.392481 0.607519 16 1.976740 0.976740 32 1.005486 0.005486 64 1.164211 0.164211 128 1.029867 0.029867 256 1.055453 0.055453 512 0.957492 0.042508 1024 0.995850 0.004150 2048 1.034568 0.034568 4096 1.015895 0.015895 8192 1.009978 0.009978 16384 0.997131 0.002869 32768 1.004831 0.004831 f(x,y) = 24 * x*y 5 1.000000 1 0.249096 0.750904 2 1.097017 0.097017 4 0.894496 0.105504 8 1.488198 0.488198 16 0.858586 0.141414 32 1.126441 0.126441 64 0.972167 0.027833 128 1.152860 0.152860 256 0.997562 0.002438 512 0.985474 0.014526 1024 1.032581 0.032581 2048 1.024481 0.024481 4096 0.978202 0.021798 8192 0.996282 0.003718 16384 0.997889 0.002111 32768 0.992709 0.007291 f(x,y) = 12 * y^2 6 1.000000 1 0.001856 0.998144 2 0.668455 0.331545 4 1.238185 0.238185 8 0.871272 0.128728 16 0.580525 0.419475 32 0.867382 0.132618 64 0.800147 0.199853 128 1.032745 0.032745 256 1.055580 0.055580 512 0.969209 0.030791 1024 0.994048 0.005952 2048 1.014900 0.014900 4096 0.966990 0.033010 8192 1.021547 0.021547 16384 0.999945 0.000055 32768 0.997132 0.002868 f(x,y) = 20 * x^3 7 1.000000 1 0.006157 0.993843 2 1.213970 0.213970 4 0.538822 0.461178 8 0.983177 0.016823 16 0.826093 0.173907 32 0.978822 0.021178 64 0.858152 0.141848 128 0.859853 0.140147 256 1.070234 0.070234 512 1.064102 0.064102 1024 1.041310 0.041310 2048 1.049329 0.049329 4096 1.018353 0.018353 8192 0.981459 0.018541 16384 1.004713 0.004713 32768 0.995940 0.004060 f(x,y) = 30 * x^4 8 1.000000 1 0.627917 0.372083 2 1.015687 0.015687 4 3.254842 2.254842 8 2.045845 1.045845 16 0.883546 0.116454 32 0.390312 0.609688 64 1.133083 0.133083 128 1.086783 0.086783 256 1.069037 0.069037 512 0.865460 0.134540 1024 0.852581 0.147419 2048 1.067120 0.067120 4096 0.999242 0.000758 8192 1.028492 0.028492 16384 1.008552 0.008552 32768 1.004558 0.004558 f(x,y) = 42 * x^5 9 1.000000 1 0.108978 0.891022 2 0.001496 0.998504 4 0.802042 0.197958 8 0.395361 0.604639 16 0.559435 0.440565 32 1.216462 0.216462 64 0.715701 0.284299 128 1.074064 0.074064 256 1.347816 0.347816 512 1.055378 0.055378 1024 0.926219 0.073781 2048 0.985559 0.014441 4096 1.070130 0.070130 8192 0.975951 0.024049 16384 0.994722 0.005278 32768 1.013947 0.013947 f(x,y) = x^(-0.2) on ((1,0),(5,0),(5,1)) 10 1.000000 1 2.242642 1.242642 2 1.844253 0.844253 4 0.301892 0.698108 8 1.044276 0.044276 16 0.636564 0.363436 32 0.809343 0.190657 64 0.774779 0.225221 128 0.914949 0.085051 256 1.144607 0.144607 512 0.831435 0.168565 1024 0.968620 0.031380 2048 0.965668 0.034332 4096 0.972865 0.027135 8192 1.063180 0.063180 16384 1.079163 0.079163 32768 0.955161 0.044839 f(x,y) = (x+y)^(-0.2) 11 1.000000 1 0.959331 0.040669 2 1.129933 0.129933 4 0.968253 0.031747 8 1.018719 0.018719 16 1.013200 0.013200 32 0.971331 0.028669 64 1.006570 0.006570 128 0.987638 0.012362 256 0.991709 0.008291 512 1.007806 0.007806 1024 0.997928 0.002072 2048 0.996949 0.003051 4096 0.998306 0.001694 8192 0.999156 0.000844 16384 0.998122 0.001878 32768 0.999204 0.000796 f(x,y) = (1-x-y)^(-0.2) on ((-1,-3),(3,-2),(-1,2)) 12 1.000000 1 1.064037 0.064037 2 0.815053 0.184947 4 1.148211 0.148211 8 0.982066 0.017934 16 1.039013 0.039013 32 0.975167 0.024833 64 1.062958 0.062958 128 0.991534 0.008466 256 1.054216 0.054216 512 1.020369 0.020369 1024 0.987490 0.012510 2048 0.993333 0.006667 4096 1.001884 0.001884 8192 0.995764 0.004236 16384 0.998124 0.001876 32768 1.000316 0.000316 f(x,y) = (x*y)^(-0.2) on ((0,0),(-7,0),(0,-3)) 13 1.000000 1 0.815041 0.184959 2 1.118033 0.118033 4 0.826792 0.173208 8 0.878287 0.121713 16 1.073182 0.073182 32 0.974850 0.025150 64 0.954397 0.045603 128 0.998549 0.001451 256 0.985337 0.014663 512 0.976935 0.023065 1024 1.003166 0.003166 2048 0.998867 0.001133 4096 1.000519 0.000519 8192 0.999593 0.000407 16384 1.002712 0.002712 32768 0.998986 0.001014 f(x,y) = 1/sqrt(x) + 1/sqrt(y) + 1/sqrt(x+y) 14 1.000000 1 0.985538 0.014462 2 0.862944 0.137056 4 0.763490 0.236510 8 0.843100 0.156900 16 0.854139 0.145861 32 0.952388 0.047612 64 0.974978 0.025022 128 1.170324 0.170324 256 0.969385 0.030615 512 0.991432 0.008568 1024 0.989705 0.010295 2048 1.018221 0.018221 4096 0.994996 0.005004 8192 0.986778 0.013222 16384 1.015572 0.015572 32768 0.997806 0.002194 f(x,y) = 1/sqrt(1-x-y) 15 1.000000 1 0.594004 0.405996 2 0.690434 0.309566 4 0.745518 0.254482 8 0.793118 0.206882 16 1.087583 0.087583 32 1.126612 0.126612 64 0.872431 0.127569 128 0.976511 0.023489 256 0.982202 0.017798 512 0.974368 0.025632 1024 1.053099 0.053099 2048 1.017196 0.017196 4096 1.028328 0.028328 8192 0.997537 0.002463 16384 1.029337 0.029337 32768 0.989777 0.010223 f(x,y) = log(x*y) 16 1.000000 1 0.614309 0.385691 2 1.232866 0.232866 4 1.290170 0.290170 8 1.204348 0.204348 16 1.330858 0.330858 32 0.908565 0.091435 64 1.031903 0.031903 128 1.018685 0.018685 256 0.961352 0.038648 512 1.002172 0.002172 1024 1.018168 0.018168 2048 1.004524 0.004524 4096 1.009508 0.009508 8192 0.995927 0.004073 16384 1.001701 0.001701 32768 0.998872 0.001128 f(x,y) = 1/sqrt(|x-1/4|) + 1/sqrt(|y-1/2|) 17 1.000000 1 0.803706 0.196294 2 0.691492 0.308508 4 0.737281 0.262719 8 0.976511 0.023489 16 0.811922 0.188078 32 1.346136 0.346136 64 1.403964 0.403964 128 1.199760 0.199760 256 0.921568 0.078432 512 0.955709 0.044291 1024 1.007821 0.007821 2048 1.037846 0.037846 4096 0.996219 0.003781 8192 0.984599 0.015401 16384 0.990910 0.009090 32768 1.003696 0.003696 f(x,y) = log ( x + y ) 18 1.000000 1 0.339752 0.660248 2 0.490002 0.509998 4 1.609707 0.609707 8 2.081011 1.081011 16 1.022128 0.022128 32 0.931095 0.068905 64 0.953323 0.046677 128 0.916633 0.083367 256 1.154158 0.154158 512 0.998598 0.001402 1024 0.916663 0.083337 2048 0.994809 0.005191 4096 0.993116 0.006884 8192 1.002257 0.002257 16384 0.989143 0.010857 32768 1.001918 0.001918 f(x,y) = sin ( x ) cos ( 5 y ) 19 1.000000 1 -0.494695 1.494695 2 0.585888 0.414112 4 -0.298814 1.298814 8 2.602463 1.602463 16 0.176463 0.823537 32 1.264041 0.264041 64 1.350093 0.350093 128 0.826936 0.173064 256 1.110030 0.110030 512 1.188900 0.188900 1024 1.000145 0.000145 2048 0.959857 0.040143 4096 0.995434 0.004566 8192 1.008556 0.008556 16384 1.006702 0.006702 32768 1.006873 0.006873 f(x,y) = sin ( 11 x ) cos ( y ) 20 1.000000 1 3.246602 2.246602 2 2.052579 1.052579 4 -1.941444 2.941444 8 0.753488 0.246512 16 1.449960 0.449960 32 1.810901 0.810901 64 0.583220 0.416780 128 0.620629 0.379371 256 0.956706 0.043294 512 1.012610 0.012610 1024 0.886346 0.113654 2048 0.991605 0.008395 4096 0.913210 0.086790 8192 1.018976 0.018976 16384 1.003590 0.003590 32768 0.994377 0.005623 f(x,y) = 1 / r = 1 / sqrt ( x^2 + y^2 ) 21 1.000000 1 0.811596 0.188404 2 0.543881 0.456119 4 1.203546 0.203546 8 0.723463 0.276537 16 1.747790 0.747790 32 1.209984 0.209984 64 1.000060 0.000060 128 0.964694 0.035306 256 0.972855 0.027145 512 0.963537 0.036463 1024 0.988294 0.011706 2048 0.961689 0.038311 4096 1.012519 0.012519 8192 1.021002 0.021002 16384 1.001368 0.001368 32768 1.001653 0.001653 f(x,y) = log ( r ) / r 22 1.000000 1 0.254883 0.745117 2 0.459222 0.540778 4 0.648298 0.351702 8 0.961119 0.038881 16 2.952289 1.952289 32 0.579882 0.420118 64 1.437650 0.437650 128 1.480918 0.480918 256 1.067483 0.067483 512 0.872304 0.127696 1024 1.007991 0.007991 2048 0.897013 0.102987 4096 0.876114 0.123886 8192 1.012129 0.012129 16384 1.001653 0.001653 32768 0.956995 0.043005 triangle_integrands_test03(): p00_vertex_sub() applies a vertex rule with subdivision. Problem Exact Pts Approx Error f(x,y) = 2 1 1.000000 3 1.000000 0.000000 6 1.000000 0.000000 24 1.000000 0.000000 84 1.000000 0.000000 300 1.000000 0.000000 f(x,y) = 6 * x 2 1.000000 3 1.000000 0.000000 6 1.000000 0.000000 24 0.625000 0.375000 84 0.531250 0.468750 300 0.507812 0.492188 f(x,y) = 6 * y 3 1.000000 3 1.000000 0.000000 6 1.000000 0.000000 24 0.625000 0.375000 84 0.531250 0.468750 300 0.507812 0.492188 f(x,y) = 12 * x^2 4 1.000000 3 2.000000 1.000000 6 1.250000 0.250000 24 0.718750 0.281250 84 0.578125 0.421875 300 0.533203 0.466797 f(x,y) = 24 * x*y 5 1.000000 3 0.000000 1.000000 6 0.750000 0.250000 24 0.531250 0.468750 84 0.484375 0.515625 300 0.482422 0.517578 f(x,y) = 12 * y^2 6 1.000000 3 2.000000 1.000000 6 1.250000 0.250000 24 0.718750 0.281250 84 0.578125 0.421875 300 0.533203 0.466797 f(x,y) = 20 * x^3 7 1.000000 3 3.333333 2.333333 6 1.458333 0.458333 24 0.794271 0.205729 84 0.619466 0.380534 300 0.557088 0.442912 f(x,y) = 30 * x^4 8 1.000000 3 5.000000 4.000000 6 1.718750 0.718750 24 0.874023 0.125977 84 0.660339 0.339661 300 0.580639 0.419361 f(x,y) = 42 * x^5 9 1.000000 3 7.000000 6.000000 6 2.078125 1.078125 24 0.965576 0.034424 84 0.702019 0.297981 300 0.604101 0.395899 f(x,y) = x^(-0.2) on ((1,0),(5,0),(5,1)) 10 1.000000 Skip this problem, it has edge singularities. f(x,y) = (x+y)^(-0.2) 11 1.000000 Skip this problem, it has vertex singularities. f(x,y) = (1-x-y)^(-0.2) on ((-1,-3),(3,-2),(-1,2)) 12 1.000000 Skip this problem, it has edge singularities. f(x,y) = (x*y)^(-0.2) on ((0,0),(-7,0),(0,-3)) 13 1.000000 Skip this problem, it has edge singularities. f(x,y) = 1/sqrt(x) + 1/sqrt(y) + 1/sqrt(x+y) 14 1.000000 Skip this problem, it has edge singularities. f(x,y) = 1/sqrt(1-x-y) 15 1.000000 Skip this problem, it has edge singularities. f(x,y) = log(x*y) 16 1.000000 Skip this problem, it has edge singularities. f(x,y) = 1/sqrt(|x-1/4|) + 1/sqrt(|y-1/2|) 17 1.000000 Skip this problem, it has internal singularities. f(x,y) = log ( x + y ) 18 1.000000 Skip this problem, it has vertex singularities. f(x,y) = sin ( x ) cos ( 5 y ) 19 1.000000 3 3.257551 2.257551 6 1.091193 0.091193 24 0.743131 0.256869 84 0.622029 0.377971 300 0.564706 0.435294 f(x,y) = sin ( 11 x ) cos ( y ) 20 1.000000 3 -1.950026 2.950026 6 -2.424940 3.424940 24 0.002806 0.997194 84 0.333582 0.666418 300 0.429274 0.570726 f(x,y) = 1 / r = 1 / sqrt ( x^2 + y^2 ) 21 1.000000 Skip this problem, it has vertex singularities. f(x,y) = log ( r ) / r 22 1.000000 Skip this problem, it has vertex singularities. triangle_integrands_test04(): p00_wandzura05_sub() applies a Wandzura rule with subdivision. Problem Exact Pts Approx Error f(x,y) = 2 1 1.000000 7 1.000000 0.000000 Accuracy acceptable f(x,y) = 6 * x 2 1.000000 7 1.000000 0.000000 Accuracy acceptable f(x,y) = 6 * y 3 1.000000 7 1.000000 0.000000 Accuracy acceptable f(x,y) = 12 * x^2 4 1.000000 7 1.000000 0.000000 Accuracy acceptable f(x,y) = 24 * x*y 5 1.000000 7 1.000000 0.000000 Accuracy acceptable f(x,y) = 12 * y^2 6 1.000000 7 1.000000 0.000000 Accuracy acceptable f(x,y) = 20 * x^3 7 1.000000 7 1.000000 0.000000 Accuracy acceptable f(x,y) = 30 * x^4 8 1.000000 7 1.000000 0.000000 Accuracy acceptable f(x,y) = 42 * x^5 9 1.000000 7 1.000000 0.000000 Accuracy acceptable f(x,y) = x^(-0.2) on ((1,0),(5,0),(5,1)) 10 1.000000 7 0.798157 0.201843 28 0.883684 0.116316 112 0.933087 0.066913 448 0.961538 0.038462 1792 0.977901 0.022099 7168 0.987305 0.012695 f(x,y) = (x+y)^(-0.2) 11 1.000000 7 0.998396 0.001604 28 0.999539 0.000461 112 0.999868 0.000132 448 0.999962 0.000038 1792 0.999989 0.000011 7168 0.999997 0.000003 f(x,y) = (1-x-y)^(-0.2) on ((-1,-3),(3,-2),(-1,2)) 12 1.000000 7 0.972572 0.027428 28 0.983878 0.016122 112 0.990635 0.009365 448 0.994591 0.005409 1792 0.996884 0.003116 7168 0.998208 0.001792 f(x,y) = (x*y)^(-0.2) on ((0,0),(-7,0),(0,-3)) 13 1.000000 7 0.949590 0.050410 28 0.970474 0.029526 112 0.982868 0.017132 448 0.990106 0.009894 1792 0.994300 0.005700 7168 0.996721 0.003279 f(x,y) = 1/sqrt(x) + 1/sqrt(y) + 1/sqrt(x+y) 14 1.000000 7 0.873246 0.126754 28 0.909736 0.090264 112 0.935949 0.064051 448 0.954630 0.045370 1792 0.967890 0.032110 7168 0.977285 0.022715 f(x,y) = 1/sqrt(1-x-y) 15 1.000000 7 0.843800 0.156200 28 0.887964 0.112036 112 0.920217 0.079783 448 0.943387 0.056613 1792 0.959898 0.040102 7168 0.971619 0.028381 f(x,y) = log(x*y) 16 1.000000 7 0.950264 0.049736 28 0.974388 0.025612 112 0.987008 0.012992 448 0.993457 0.006543 1792 0.996717 0.003283 7168 0.998356 0.001644 f(x,y) = 1/sqrt(|x-1/4|) + 1/sqrt(|y-1/2|) 17 1.000000 7 0.864525 0.135475 28 0.965722 0.034278 112 0.913986 0.086014 448 0.939178 0.060822 1792 0.956993 0.043007 7168 0.969589 0.030411 f(x,y) = log ( x + y ) 18 1.000000 7 0.991081 0.008919 28 0.997766 0.002234 112 0.999442 0.000558 448 0.999860 0.000140 1792 0.999965 0.000035 7168 0.999991 0.000009 f(x,y) = sin ( x ) cos ( 5 y ) 19 1.000000 7 0.975991 0.024009 28 0.999840 0.000160 112 0.999998 0.000002 448 1.000000 0.000000 Accuracy acceptable f(x,y) = sin ( 11 x ) cos ( y ) 20 1.000000 7 0.051313 0.948687 28 1.019286 0.019286 112 1.000129 0.000129 448 1.000002 0.000002 1792 1.000000 0.000000 Accuracy acceptable f(x,y) = 1 / r = 1 / sqrt ( x^2 + y^2 ) 21 1.000000 7 0.973846 0.026154 28 0.987025 0.012975 112 0.993511 0.006489 448 0.996756 0.003244 1792 0.998378 0.001622 7168 0.999189 0.000811 f(x,y) = log ( r ) / r 22 1.000000 7 0.862016 0.137984 28 0.923815 0.076185 112 0.958237 0.041763 448 0.977284 0.022716 1792 0.987725 0.012275 7168 0.993404 0.006596 triangle_integrands_test(): Normal end of execution. Fri Dec 26 15:23:38 2025