| number of | ||||||
| subintervals | Left sum | Rite sum | Trap sum | Mid sum | Simpson | |
| n | ||||||
| 1 | 1.000000000000000 | 0.367879441171442 | 0.683939720585721 | 0.778800783071405 | 0.747180428909510 | |
| 2 | 0.889400391535702 | 0.573340112121424 | 0.731370251828563 | 0.754597943772199 | 0.746855379790987 | |
| 4 | 0.821999167653951 | 0.663969027946812 | 0.742984097800381 | 0.748747131891009 | 0.746826120527467 | |
| 8 | 0.785373149772480 | 0.706358079918910 | 0.745865614845695 | 0.747303578730748 | 0.746824257435730 | |
| 16 | 0.766338364251614 | 0.726830829324829 | 0.746584596788222 | 0.746943912516367 | 0.746824140606985 | |
| 32 | 0.756641138383991 | 0.736887370920598 | 0.746764254652294 | 0.746854072623361 | 0.746824133299672 | |
| 64 | 0.751747605503676 | 0.741870721771980 | 0.746809163637828 | 0.746831617445408 | 0.746824132842881 | |
| 128 | 0.749289611474542 | 0.744351169608694 | 0.746820390541618 | 0.746826003950687 | 0.746824132814330 | |
| 256 | 0.748057807712614 | 0.745588586779690 | 0.746823197246152 | 0.746824600595743 | 0.746824132812546 | |
| 512 | 0.747441204154178 | 0.746206593687716 | 0.746823898920947 | 0.746824249758177 | 0.746824132812434 | |
| 1,024 | 0.747132726956178 | 0.746515421722947 | 0.746824074339562 | 0.746824162048860 | 0.746824132812427 | |
| 2,048 | 0.746978444502519 | 0.746669791885903 | 0.746824118194211 | 0.746824140121534 | 0.746824132812426 | |
| 4,096 | 0.746901292312026 | 0.746746966003718 | 0.746824129157872 | 0.746824134639701 | 0.746824132812425 | |
| 8,192 | 0.746862713475864 | 0.746785550321710 | 0.746824131898787 | 0.746824133269247 | 0.746824132812427 | |
| 16,384 | 0.746843423372555 | 0.746804841795478 | 0.746824132584017 | 0.746824132926631 | 0.746824132812426 | |
| 32,768 | 0.746833778149593 | 0.746814487361054 | 0.746824132755324 | 0.746824132840976 | 0.746824132812425 | |
| 65,536 | 0.746828955495285 | 0.746819310101015 | 0.746824132798150 | 0.746824132819563 | 0.746824132812425 | |
| 131,072 | 0.746826544157424 | 0.746821721460289 | 0.746824132808856 | 0.746824132814204 | 0.746824132812421 | |
| 262,144 | 0.746825338485814 | 0.746822927137246 | 0.746824132811530 | 0.746824132812877 | 0.746824132812428 | |
| 524,288 | 0.746824735649345 | 0.746823529975062 | 0.746824132812203 | 0.746824132812558 | 0.746824132812439 | |
| 1,048,576 | 0.746824434230951 | 0.746823831393810 | 0.746824132812381 | 0.746824132812476 | 0.746824132812444 | |
| Data generated with MATLAB 5.2 by Matt Kawski, int(exp(-x^2),x=0..1) | ||||||
| Best estimate of the errors -- using as comparison the values of Simpson's for n between 1000 and 65000 | ||||||
| 1 | 0.253175867187575 | -0.378944691640983 | -0.062884412226704 | 0.031976650258980 | 0.000356296097085 | |
| 2 | 0.142576258723277 | -0.173484020691001 | -0.015453880983862 | 0.007773810959774 | 0.000031246978562 | |
| 4 | 0.075175034841526 | -0.082855104865613 | -0.003840035012044 | 0.001922999078584 | 0.000001987715042 | |
| 8 | 0.038549016960055 | -0.040466052893515 | -0.000958517966730 | 0.000479445918323 | 0.000000124623305 | |
| 16 | 0.019514231439189 | -0.019993303487596 | -0.000239536024203 | 0.000119779703942 | 0.000000007794560 | |
| 32 | 0.009817005571566 | -0.009936761891827 | -0.000059878160131 | 0.000029939810936 | 0.000000000487247 | |
| 64 | 0.004923472691251 | -0.004953411040445 | -0.000014969174597 | 0.000007484632983 | 0.000000000030456 | |
| 128 | 0.002465478662117 | -0.002472963203731 | -0.000003742270807 | 0.000001871138262 | 0.000000000001905 | |
| 256 | 0.001233674900189 | -0.001235546032735 | -0.000000935566273 | 0.000000467783318 | 0.000000000000121 | |
| 512 | 0.000617071341753 | -0.000617539124709 | -0.000000233891478 | 0.000000116945752 | 0.000000000000009 | |
| 1,024 | 0.000308594143753 | -0.000308711089478 | -0.000000058472863 | 0.000000029236435 | 0.000000000000002 | |
| 2,048 | 0.000154311690094 | -0.000154340926522 | -0.000000014618214 | 0.000000007309109 | 0.000000000000001 | |
| 4,096 | 0.000077159499601 | -0.000077166808707 | -0.000000003654553 | 0.000000001827276 | 0.000000000000000 | |
| 8,192 | 0.000038580663439 | -0.000038582490715 | -0.000000000913638 | 0.000000000456822 | 0.000000000000002 | |
| 16,384 | 0.000019290560130 | -0.000019291016947 | -0.000000000228408 | 0.000000000114206 | 0.000000000000001 | |
| 32,768 | 0.000009645337168 | -0.000009645451371 | -0.000000000057101 | 0.000000000028551 | 0.000000000000000 | |
| 65,536 | 0.000004822682860 | -0.000004822711410 | -0.000000000014275 | 0.000000000007138 | 0.000000000000000 | |
| 131,072 | 0.000002411344999 | -0.000002411352136 | -0.000000000003569 | 0.000000000001779 | -0.000000000000004 | |
| 262,144 | 0.000001205673389 | -0.000001205675179 | -0.000000000000895 | 0.000000000000452 | 0.000000000000003 | |
| 524,288 | 0.000000602836920 | -0.000000602837363 | -0.000000000000222 | 0.000000000000133 | 0.000000000000014 | |
| 1,048,576 | 0.000000301418526 | -0.000000301418615 | -0.000000000000044 | 0.000000000000051 | 0.000000000000019 | |
| Ratios of the subsequent errors | ||||||
| from n= | to n= | |||||
| 1 | 2 | 1.775722476 | 2.184320436 | 4.069166334 | 4.113381509 | 11.402577576 |
| 2 | 4 | 1.896590524 | 2.093824164 | 4.024411479 | 4.042545338 | 15.720049354 |
| 4 | 8 | 1.950115483 | 2.047521291 | 4.006221214 | 4.010877985 | 15.949785972 |
| 8 | 16 | 1.975430961 | 2.023980325 | 4.001560809 | 4.002730868 | 15.988497754 |
| 16 | 32 | 1.987798754 | 2.012054199 | 4.000390521 | 4.000683378 | 15.997142454 |
| 32 | 64 | 1.993918965 | 2.006044282 | 4.000097650 | 4.000170884 | 15.998406987 |
| 64 | 128 | 1.996964227 | 2.003026585 | 4.000024415 | 4.000042720 | 15.987120462 |
| 128 | 256 | 1.998483281 | 2.001514422 | 4.000006109 | 4.000010668 | 15.742201835 |
| 256 | 512 | 1.999241930 | 2.000757496 | 4.000001544 | 4.000002652 | 13.456790123 |
| 512 | 1,024 | 1.999621037 | 2.000378819 | 4.000000444 | 4.000000403 | 4.500000000 |
| 1,024 | 2,048 | 1.999810536 | 2.000189428 | 4.000000486 | 3.999999878 | 2.000000000 |
| 2,048 | 4,096 | 1.999905273 | 2.000094718 | 4.000000547 | 4.000002734 | #DIV/0! |
| 4,096 | 8,192 | 1.999952637 | 2.000047360 | 4.000001337 | 3.999973510 | 0.000000000 |
| 8,192 | 16,384 | 1.999976319 | 2.000023681 | 4.000026734 | 3.999983474 | 2.000000000 |
| 16,384 | 32,768 | 1.999988159 | 2.000011840 | 4.000069995 | 4.000062217 | #DIV/0! |
| 32,768 | 65,536 | 1.999994080 | 2.000005920 | 4.000062219 | 3.999891123 | #DIV/0! |
| 65,536 | 131,072 | 1.999997040 | 2.000002960 | 3.999688929 | 4.012294059 | 0.000000000 |
| 131,072 | 262,144 | 1.999998524 | 2.000001475 | 3.987966754 | 3.936133628 | -1.333333333 |
| 262,144 | 524,288 | 1.999999252 | 2.000000751 | 4.030500000 | 3.398163606 | 0.214285714 |
| 524,288 | 1,048,576 | 1.999999562 | 2.000000442 | 5.050505051 | 2.610021786 | 0.736842105 |