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20:
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Differential quadrature is the approximation of derivatives by using weighted sums of function values. Differential quadrature is of practical interest because its allows one to compute derivatives from noisy data. The name is in analogy with
927:
1862:
2064:
320:
1461:
2308:
772:
is small enough. If too large, the calculation of the slope of the secant line will be more accurately calculated, but the estimate of the slope of the tangent by using the secant could be worse.
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501:
1105:
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967:
509:. In this case the first-order errors cancel, so the slope of these secant lines differ from the slope of the tangent line by an amount that is approximately proportional to
357:
2832:
was developed by Abate and Dubner. An algorithm that can be used without requiring knowledge about the method or the character of the function was developed by
Fornberg.
2057:
The complex-step derivative formula is only valid for calculating first-order derivatives. A generalization of the above for calculating derivatives of any order employs
1853:
2664:{\displaystyle {\frac {\partial ^{2}f(x,y)}{\partial x\,\partial y}}\approx {\frac {{\mathcal {C}}_{3}^{(2)}(f(x+\mathrm {i} ^{(1)}h,y+\mathrm {i} ^{(2)}h))}{h^{2}}}}
1351:
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854:
694:
668:
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Using complex variables for numerical differentiation was started by Lyness and Moler in 1967. Their algorithm is applicable to higher-order derivatives.
3254:
2901:
227:
2910:
1371:
1031:
apart. In this regard, since most decimal fractions are recurring sequences in binary (just as 1/3 is in decimal) a seemingly round step such as
1360:
is a tool that can be used to generate derivative approximation methods for any stencil with any derivative order (provided a solution exists).
3230:
3446:
3429:
3002:
2977:
2048:{\displaystyle f(x+\mathrm {i} h)=f(x)+\mathrm {i} hf'(x)-{\tfrac {1}{2!}}h^{2}f''(x)-{\tfrac {\mathrm {i} }{3!}}h^{3}f^{(3)}(x)+\cdots .}
540:
this is a more accurate approximation to the tangent line than the one-sided estimation. However, although the slope is being computed at
2919:
977:
3268:
Lantoine, G.; Russell, R. P.; Dargent, Th. (2012). "Using multicomplex variables for automatic computation of high-order derivatives".
2225:{\displaystyle f^{(n)}(x)\approx {\frac {{\mathcal {C}}_{n^{2}-1}^{(n)}(f(x+\mathrm {i} ^{(1)}h+\cdots +\mathrm {i} ^{(n)}h))}{h^{n}}}}
3059:
1052:
facilities may fail to attend to the details of actual computer arithmetic and instead apply the axioms of mathematics to deduce that
554:
2947:
2489:
extracts the last, “most imaginary” component. The method can be applied to mixed derivatives, e.g. for a second-order derivative
703:
The symmetric difference quotient is employed as the method of approximating the derivative in a number of calculators, including
433:
129:
3568:
2862:
203:
The slope of this secant line differs from the slope of the tangent line by an amount that is approximately proportional to
2270:
1090:
506:
2677:
224:
is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line:
3563:
3459:
Ahnert, Karsten; Abel, Markus (2007). "Numerical differentiation of experimental data: local versus global methods".
3020:
Numerical
Differentiation of Analytic Functions, B Fornberg – ACM Transactions on Mathematical Software (TOMS), 1981.
2437:
740:
3342:
3107:
3534:
2858:
3424:
Differential
Quadrature and Its Application in Engineering: Engineering Applications, Chang Shu, Springer, 2000,
2874:
753:
1004:
will be changed (by rounding or truncation) to a nearby machine-representable number, with the consequence that
1063:
2395:
2313:
1644:
The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if
3203:
1045:
h := sqrt(eps) * x; xph := x + h; dx := xph - x; slope := (F(xph) - F(x)) / dx;
800:
3539:
3188:
211:
approaches zero, the slope of the secant line approaches the slope of the tangent line. Therefore, the true
3032:
Using
Complex Variables to Estimate Derivatives of Real Functions, W. Squire, G. Trapp – SIAM REVIEW, 1998.
3476:
3153:
2811:{\displaystyle f^{(n)}(a)={\frac {n!}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{(z-a)^{n+1}}}\,\mathrm {d} z,}
1630:{\displaystyle f^{(n)}(x)=\lim _{h\to 0}{\frac {1}{h^{n}}}\sum _{k=0}^{n}(-1)^{k+n}{\binom {n}{k}}f(x+kh)}
1095:
Higher-order methods for approximating the derivative, as well as methods for higher derivatives, exist.
2895:
2886:
2846:
2819:
2235:
1049:
46:
3390:
Abate, J; Dubner, H (March 1968). "A New Method for
Generating Power Series Expansions of Functions".
3144:
Martins, J. R. R. A.; Sturdza, P.; Alonso, J. J. (2003). "The
Complex-Step Derivative Approximation".
3529:
3468:
3399:
3364:
2058:
1692:
methods. For example, the first derivative can be calculated by the complex-step derivative formula:
1665:
325:
3535:
Boost. Math numerical differentiation, including finite differencing and the complex step derivative
3481:
3158:
1290:{\displaystyle f'(x)={\frac {-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)}{12h}}+{\frac {h^{4}}{30}}f^{(5)}(c),}
335:
2850:
1689:
193:
3502:
3285:
3171:
2880:
1822:
1099:
735:
359:
197:
62:
31:
1810:{\displaystyle f'(x)={\frac {\Im (f(x+\mathrm {i} h))}{h}}+O(h^{2}),\quad \mathrm {i^{2}} :=-1.}
3494:
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23:
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382:
Another two-point formula is to compute the slope of a nearby secant line through the points
3486:
3407:
3372:
3277:
3163:
2961:
2854:
844:
1668:, real-valued on the real line, which can be evaluated at points in the complex plane near
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512:
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829:
776:
765:
673:
647:
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3368:
3341:
Ablowitz, M. J., Fokas, A. S.,(2003). Complex variables: introduction and applications.
700:
due to numbers being represented and calculations being performed in limited precision.
3355:
Lyness, J. N.; Moler, C. B. (1967). "Numerical differentiation of analytic functions".
2375:
2355:
1671:
1647:
761:
697:
627:
2857:. There are various methods for determining the weight coefficients, for example, the
1819:
The recommended step size to obtain accurate derivatives for a range of conditions is
994:
almost certainly will not be exactly representable in that precision. This means that
3557:
2962:
1856:
1357:
3506:
3289:
3175:
2993:
Tamara
Lefcourt Ruby; James Sellers; Lisa Korf; Jeremy Van Horn; Mike Munn (2014).
2904: – Methods used to find numerical solutions of ordinary differential equations
189:
851:
that balances the rounding error against the secant error for optimum accuracy is
3211:
71:
3046:
3490:
50:
42:
3498:
3305:"Computation of higher-order derivatives using the multi-complex step method"
752:
An important consideration in practice when the function is calculated using
53:
using values of the function and perhaps other knowledge about the function.
3281:
3207:
3095:
922:{\displaystyle h=2{\sqrt {\varepsilon \left|{\frac {f(x)}{f''(x)}}\right|}}}
38:
3441:
Advanced
Differential Quadrature Methods, Yingyan Zhang, CRC Press, 2009,
3167:
972:
For computer calculations the problems are exacerbated because, although
126:, and it can be either positive or negative. The slope of this line is
2865:. There are further methods for computing derivatives from noisy data.
19:
3327:
3411:
3376:
775:
For basic central differences, the optimal step is the cube-root of
3322:
2964:
Windows on
Teaching Math: Cases of Middle and Secondary Classrooms
716:
712:
708:
704:
365:
Equivalently, the slope could be estimated by employing positions
70:
A simple two-point estimation is to compute the slope of a nearby
18:
3547:
1038:
will not be a round number in binary; it is 0.000110011001100...
67:
The simplest method is to use finite difference approximations.
2673:
A C++ implementation of multicomplex arithmetics is available.
1098:
Given below is the five-point method for the first derivative (
16:
Use of numerical analysis to estimate derivatives of functions
2676:
In general, derivatives of any order can be calculated using
315:{\displaystyle f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}.}
2554:
2444:
2402:
2320:
2102:
1456:{\displaystyle f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}}
1356:
For other stencil configurations and derivative orders, the
969:), and to employ it will require knowledge of the function.
797:
that is small without producing a large rounding error is
760:. If chosen too small, the subtraction will yield a large
362:, calculating the derivative directly can be unintuitive.
2877: – Numerical calculations carrying along derivatives
768:
and due to cancellation will produce a value of zero if
2915:
Pages displaying short descriptions of redirect targets
2906:
Pages displaying short descriptions of redirect targets
2891:
Pages displaying short descriptions of redirect targets
3045:
Martins, Joaquim R. R. A.; Ning, Andrew (2021-10-01).
1983:
1936:
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2495:
2440:
2398:
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2358:
2316:
2303:{\displaystyle \mathrm {i} ^{(1)}\equiv \mathrm {i} }
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132:
779:. For the numerical derivative formula evaluated at
2828:A method based on numerical inversion of a complex
1027:; the two function evaluations will not be exactly
2849:, where weighted sums are used in methods such as
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816:
764:. In fact, all the finite-difference formulae are
688:
662:
636:
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528:
495:
351:
314:
180:
2898: – Methods of calculating definite integrals
1600:
1587:
1506:
1396:
756:of finite precision is the choice of step size,
617:{\displaystyle R={\frac {-f^{(3)}(c)}{6}}h^{2},}
252:
2372:th component of a multicomplex number of level
3530:NAG Library numerical differentiation routines
3028:
3026:
2883: – A point and its four nearest neighbors
2482:{\displaystyle {\mathcal {C}}_{n^{2}-1}^{(n)}}
2938:Richard L. Burden, J. Douglas Faires (2000),
8:
748:due to both rounding error and formula error
496:{\displaystyle {\frac {f(x+h)-f(x-h)}{2h}}.}
3016:
3014:
2861:. Differential quadrature is used to solve
744:Example showing the difficulty of choosing
3480:
3157:
3146:ACM Transactions on Mathematical Software
3040:
3038:
2902:Numerical ordinary differential equations
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2267:denote the multicomplex imaginary units;
2245:
2240:
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2183:
2158:
2153:
2125:
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2107:
2101:
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2061:, resulting in multicomplex derivatives.
2015:
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1956:
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1358:Finite Difference Coefficients Calculator
1302:
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339:
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181:{\displaystyle {\frac {f(x+h)-f(x)}{h}}.}
133:
131:
2427:{\displaystyle {\mathcal {C}}_{0}^{(n)}}
2345:{\displaystyle {\mathcal {C}}_{k}^{(n)}}
1066:and similar languages, a directive that
739:
2931:
2913: – Algorithm to smooth data points
2911:Numerical smoothing and differentiation
2889: – Algorithm to smooth data points
817:{\displaystyle {\sqrt {\varepsilon }}x}
3540:Differentiation With(out) a Difference
3253:: CS1 maint: archived copy as title (
3246:
3189:Differentiation With(out) a Difference
7:
3323:"mcx (multicomplex algebra library)"
3131:Elementary Real and Complex Analysis
3119:Abramowitz & Stegun, Table 25.2.
1368:Using Newton's difference quotient,
719:, all of which use this method with
2995:Kaplan AP Calculus AB & BC 2015
2960:Katherine Klippert Merseth (2003).
2920:List of numerical-analysis software
1042:A possible approach is as follows:
978:representable floating-point number
2997:. Kaplan Publishing. p. 299.
2968:. Teachers College Press. p.
2798:
2623:
2593:
2536:
2529:
2500:
2296:
2276:
2260:{\displaystyle \mathrm {i} ^{(k)}}
2241:
2184:
2154:
1986:
1908:
1879:
1855:. This formula can be obtained by
1788:
1741:
1722:
1591:
696:. This error does not include the
14:
980:in some precision (32 or 64-bit,
551:The estimation error is given by
2434:extracts the real component and
1463:the following can be shown (for
3461:Computer Physics Communications
3048:Engineering Design Optimization
1785:
544:, the value of the function at
3054:. Cambridge University Press.
2863:partial differential equations
2818:where the integration is done
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2016:
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950:
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352:{\displaystyle {\frac {0}{0}}}
300:
294:
285:
273:
259:
245:
239:
166:
160:
151:
139:
1:
1091:Finite difference coefficient
835:is typically of the order of
507:symmetric difference quotient
505:This formula is known as the
196:(also known as a first-order
122:represents a small change in
536:. Hence for small values of
430:. The slope of this line is
3303:Verheyleweghen, A. (2014).
1848:{\displaystyle h=10^{-200}}
3585:
3343:Cambridge University Press
2942:, (7th Ed), Brooks/Cole.
1088:
733:
114:. Choosing a small number
60:
3524:Numerical Differentiation
3491:10.1016/j.cpc.2007.03.009
2875:Automatic differentiation
2678:Cauchy's integral formula
1048:However, with computers,
754:floating-point arithmetic
36:numerical differentiation
1640:Complex-variable methods
962:{\displaystyle f''(x)=0}
26:estimation of derivative
3282:10.1145/2168773.2168774
3074:Sauer, Timothy (2012).
2836:Differential quadrature
3270:ACM Trans. Math. Softw
2812:
2665:
2483:
2428:
2386:
2366:
2352:operator extracts the
2346:
2304:
2261:
2226:
2049:
1849:
1811:
1682:
1658:
1631:
1558:
1457:
1347:
1291:
963:
923:
828: = 0), where the
818:
749:
690:
664:
644:is some point between
638:
618:
530:
497:
353:
316:
182:
27:
3569:Differential calculus
3345:. Check theorem 2.6.2
3168:10.1145/838250.838251
2896:Numerical integration
2887:Savitzky-Golay filter
2859:Savitzky–Golay filter
2847:numerical integration
2813:
2666:
2484:
2429:
2387:
2367:
2347:
2305:
2262:
2227:
2050:
1850:
1812:
1683:
1659:
1632:
1538:
1458:
1348:
1346:{\displaystyle c\in }
1292:
1089:Further information:
1050:compiler optimization
964:
924:
819:
743:
691:
665:
639:
619:
531:
529:{\displaystyle h^{2}}
498:
354:
317:
183:
61:Further information:
47:mathematical function
22:
3321:Bell, I. H. (2019).
2684:
2493:
2438:
2396:
2376:
2356:
2314:
2271:
2236:
2065:
2059:multicomplex numbers
1863:
1823:
1696:
1672:
1666:holomorphic function
1648:
1474:
1372:
1301:
1106:
1085:Higher-order methods
976:necessarily holds a
933:
855:
801:
674:
648:
628:
555:
513:
434:
336:
228:
130:
3473:2007CoPhC.177..764A
3404:1968SJNA....5..102A
3392:SIAM J. Numer. Anal
3369:1967SJNA....4..202L
3357:SIAM J. Numer. Anal
2575:
2478:
2423:
2341:
2136:
1102:in one dimension):
1076:will prevent this.
1062:are the same. With
689:{\displaystyle x+h}
663:{\displaystyle x-h}
194:difference quotient
188:This expression is
74:through the points
3564:Numerical analysis
3076:Numerical Analysis
2940:Numerical Analysis
2881:Five-point stencil
2808:
2661:
2551:
2479:
2441:
2424:
2399:
2382:
2362:
2342:
2317:
2300:
2257:
2222:
2099:
2045:
1999:
1950:
1845:
1807:
1678:
1654:
1627:
1520:
1453:
1410:
1364:Higher derivatives
1343:
1287:
1100:five-point stencil
959:
919:
814:
750:
736:Adaptive step size
686:
660:
634:
614:
526:
493:
360:indeterminate form
349:
324:Since immediately
312:
266:
198:divided difference
178:
63:Finite differences
57:Finite differences
32:numerical analysis
28:
3447:978-1-4200-8248-7
3430:978-1-85233-209-9
3090:Numerical Recipes
3078:. Pearson. p.248.
3004:978-1-61865-686-5
2979:978-0-8077-4279-2
2830:Laplace transform
2794:
2736:
2659:
2543:
2385:{\displaystyle n}
2365:{\displaystyle k}
2220:
1998:
1949:
1758:
1688:, then there are
1681:{\displaystyle x}
1657:{\displaystyle f}
1598:
1536:
1505:
1451:
1395:
1257:
1237:
1074:volatile variable
929:(though not when
917:
911:
824:(though not when
809:
637:{\displaystyle c}
599:
548:is not involved.
488:
347:
307:
251:
173:
24:Finite difference
3576:
3526:from wolfram.com
3511:
3510:
3484:
3456:
3450:
3439:
3433:
3422:
3416:
3415:
3387:
3381:
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3333:
3332:
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3311:
3309:
3300:
3294:
3293:
3265:
3259:
3258:
3252:
3244:
3242:
3241:
3235:
3229:. Archived from
3228:
3220:
3214:
3210:blog, posted by
3201:
3195:
3186:
3180:
3179:
3161:
3141:
3135:
3134:
3129:Shilov, George.
3126:
3120:
3117:
3111:
3105:
3099:
3085:
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3065:
3053:
3042:
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3018:
3009:
3008:
2990:
2984:
2983:
2967:
2957:
2951:
2936:
2916:
2907:
2892:
2855:Trapezoidal rule
2851:Simpson's method
2817:
2815:
2814:
2809:
2801:
2795:
2793:
2792:
2791:
2763:
2749:
2747:
2746:
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2701:
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2290:
2279:
2266:
2264:
2263:
2258:
2256:
2255:
2244:
2231:
2229:
2228:
2223:
2221:
2219:
2218:
2209:
2199:
2198:
2187:
2169:
2168:
2157:
2135:
2124:
2117:
2116:
2106:
2105:
2097:
2083:
2082:
2054:
2052:
2051:
2046:
2026:
2025:
2010:
2009:
2000:
1997:
1989:
1984:
1969:
1961:
1960:
1951:
1948:
1937:
1922:
1911:
1882:
1854:
1852:
1851:
1846:
1844:
1843:
1816:
1814:
1813:
1808:
1797:
1796:
1795:
1778:
1777:
1759:
1754:
1744:
1720:
1706:
1687:
1685:
1684:
1679:
1663:
1661:
1660:
1655:
1636:
1634:
1633:
1628:
1605:
1604:
1603:
1590:
1583:
1582:
1557:
1552:
1537:
1535:
1534:
1522:
1519:
1492:
1491:
1469:
1462:
1460:
1459:
1454:
1452:
1447:
1412:
1409:
1382:
1352:
1350:
1349:
1344:
1296:
1294:
1293:
1288:
1274:
1273:
1258:
1253:
1252:
1243:
1238:
1236:
1228:
1130:
1116:
1071:
1061:
1057:
1037:
1030:
1026:
1018:
1003:
993:
975:
968:
966:
965:
960:
943:
928:
926:
925:
920:
918:
916:
912:
910:
900:
891:
877:
868:
850:
847:. A formula for
845:double precision
842:
840:
834:
823:
821:
820:
815:
810:
805:
796:
792:
782:
771:
759:
747:
725:
695:
693:
692:
687:
669:
667:
666:
661:
643:
641:
640:
635:
623:
621:
620:
615:
610:
609:
600:
595:
585:
584:
565:
547:
543:
539:
535:
533:
532:
527:
525:
524:
502:
500:
499:
494:
489:
487:
479:
438:
429:
405:
378:
374:
358:
356:
355:
350:
348:
340:
331:
321:
319:
318:
313:
308:
303:
268:
265:
238:
222:
218:
210:
206:
187:
185:
184:
179:
174:
169:
134:
125:
121:
117:
113:
89:
3584:
3583:
3579:
3578:
3577:
3575:
3574:
3573:
3554:
3553:
3544:Nicholas Higham
3520:
3515:
3514:
3482:10.1.1.752.3843
3467:(10): 764–774.
3458:
3457:
3453:
3440:
3436:
3423:
3419:
3412:10.1137/0705008
3389:
3388:
3384:
3377:10.1137/0704019
3354:
3353:
3349:
3340:
3336:
3320:
3319:
3315:
3307:
3302:
3301:
3297:
3267:
3266:
3262:
3245:
3239:
3237:
3233:
3226:
3224:"Archived copy"
3222:
3221:
3217:
3202:
3198:
3193:Nicholas Higham
3187:
3183:
3159:10.1.1.141.8002
3143:
3142:
3138:
3128:
3127:
3123:
3118:
3114:
3106:
3102:
3086:
3082:
3073:
3069:
3062:
3051:
3044:
3043:
3036:
3031:
3024:
3019:
3012:
3005:
2992:
2991:
2987:
2980:
2959:
2958:
2954:
2937:
2933:
2928:
2914:
2905:
2890:
2871:
2838:
2777:
2764:
2750:
2738:
2725:
2717:
2687:
2682:
2681:
2649:
2621:
2591:
2550:
2528:
2499:
2498:
2491:
2490:
2450:
2436:
2435:
2394:
2393:
2374:
2373:
2354:
2353:
2312:
2311:
2274:
2269:
2268:
2239:
2234:
2233:
2210:
2182:
2152:
2108:
2098:
2068:
2063:
2062:
2011:
2001:
1990:
1962:
1952:
1941:
1915:
1861:
1860:
1832:
1821:
1820:
1787:
1769:
1721:
1699:
1694:
1693:
1670:
1669:
1646:
1645:
1642:
1585:
1568:
1526:
1477:
1472:
1471:
1464:
1413:
1375:
1370:
1369:
1366:
1299:
1298:
1259:
1244:
1229:
1131:
1109:
1104:
1103:
1093:
1087:
1082:
1067:
1059:
1053:
1046:
1041:
1032:
1028:
1024:
1005:
995:
985:
973:
936:
931:
930:
893:
892:
878:
872:
853:
852:
848:
838:
836:
832:
830:machine epsilon
799:
798:
794:
793:, a choice for
784:
780:
777:machine epsilon
769:
766:ill-conditioned
757:
745:
738:
732:
720:
672:
671:
646:
645:
626:
625:
601:
570:
566:
553:
552:
545:
541:
537:
516:
511:
510:
480:
439:
432:
431:
407:
383:
376:
366:
334:
333:
329:
269:
231:
226:
225:
220:
214:
208:
204:
135:
128:
127:
123:
119:
115:
91:
75:
65:
59:
17:
12:
11:
5:
3582:
3580:
3572:
3571:
3566:
3556:
3555:
3552:
3551:
3537:
3532:
3527:
3519:
3518:External links
3516:
3513:
3512:
3451:
3434:
3417:
3398:(1): 102–112.
3382:
3363:(2): 202–210.
3347:
3334:
3313:
3295:
3260:
3215:
3196:
3181:
3152:(3): 245–262.
3136:
3121:
3112:
3100:
3080:
3067:
3061:978-1108833417
3060:
3034:
3022:
3010:
3003:
2985:
2978:
2952:
2930:
2929:
2927:
2924:
2923:
2922:
2917:
2908:
2899:
2893:
2884:
2878:
2870:
2867:
2837:
2834:
2807:
2804:
2800:
2790:
2787:
2784:
2780:
2776:
2773:
2770:
2767:
2762:
2759:
2756:
2753:
2745:
2741:
2734:
2731:
2728:
2723:
2720:
2714:
2711:
2708:
2705:
2700:
2697:
2694:
2690:
2656:
2652:
2647:
2644:
2641:
2636:
2633:
2630:
2625:
2620:
2617:
2614:
2611:
2606:
2603:
2600:
2595:
2590:
2587:
2584:
2581:
2578:
2573:
2570:
2567:
2562:
2556:
2547:
2541:
2538:
2534:
2531:
2526:
2523:
2520:
2517:
2514:
2511:
2506:
2502:
2476:
2473:
2470:
2465:
2462:
2457:
2453:
2446:
2421:
2418:
2415:
2410:
2404:
2381:
2361:
2339:
2336:
2333:
2328:
2322:
2298:
2294:
2289:
2286:
2283:
2278:
2254:
2251:
2248:
2243:
2217:
2213:
2208:
2205:
2202:
2197:
2194:
2191:
2186:
2181:
2178:
2175:
2172:
2167:
2164:
2161:
2156:
2151:
2148:
2145:
2142:
2139:
2134:
2131:
2128:
2123:
2120:
2115:
2111:
2104:
2095:
2092:
2089:
2086:
2081:
2078:
2075:
2071:
2044:
2041:
2038:
2035:
2032:
2029:
2024:
2021:
2018:
2014:
2008:
2004:
1996:
1993:
1988:
1981:
1978:
1975:
1972:
1968:
1965:
1959:
1955:
1947:
1944:
1940:
1934:
1931:
1928:
1925:
1921:
1918:
1914:
1910:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1881:
1877:
1874:
1871:
1868:
1842:
1839:
1835:
1831:
1828:
1806:
1803:
1800:
1794:
1790:
1784:
1781:
1776:
1772:
1768:
1765:
1762:
1757:
1753:
1750:
1747:
1743:
1739:
1736:
1733:
1730:
1727:
1724:
1718:
1715:
1712:
1709:
1705:
1702:
1677:
1653:
1641:
1638:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1602:
1597:
1594:
1589:
1581:
1578:
1575:
1571:
1567:
1564:
1561:
1556:
1551:
1548:
1545:
1541:
1533:
1529:
1525:
1518:
1515:
1512:
1508:
1504:
1501:
1498:
1495:
1490:
1487:
1484:
1480:
1450:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1408:
1405:
1402:
1398:
1394:
1391:
1388:
1385:
1381:
1378:
1365:
1362:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1286:
1283:
1280:
1277:
1272:
1269:
1266:
1262:
1256:
1251:
1247:
1241:
1235:
1232:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1128:
1125:
1122:
1119:
1115:
1112:
1086:
1083:
1081:
1078:
1044:
1039:
1014:) −
958:
955:
952:
949:
946:
942:
939:
915:
909:
906:
903:
899:
896:
890:
887:
884:
881:
875:
871:
866:
863:
860:
813:
808:
762:rounding error
731:
728:
698:rounding error
685:
682:
679:
659:
656:
653:
633:
613:
608:
604:
598:
594:
591:
588:
583:
580:
577:
573:
569:
563:
560:
523:
519:
492:
486:
483:
478:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
346:
343:
311:
306:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
264:
261:
258:
254:
250:
247:
244:
241:
237:
234:
213:derivative of
177:
172:
168:
165:
162:
159:
156:
153:
150:
147:
144:
141:
138:
58:
55:
15:
13:
10:
9:
6:
4:
3:
2:
3581:
3570:
3567:
3565:
3562:
3561:
3559:
3549:
3545:
3541:
3538:
3536:
3533:
3531:
3528:
3525:
3522:
3521:
3517:
3508:
3504:
3500:
3496:
3492:
3488:
3483:
3478:
3474:
3470:
3466:
3462:
3455:
3452:
3448:
3444:
3438:
3435:
3431:
3427:
3421:
3418:
3413:
3409:
3405:
3401:
3397:
3393:
3386:
3383:
3378:
3374:
3370:
3366:
3362:
3358:
3351:
3348:
3344:
3338:
3335:
3330:
3329:
3324:
3317:
3314:
3306:
3299:
3296:
3291:
3287:
3283:
3279:
3275:
3271:
3264:
3261:
3256:
3250:
3236:on 2014-01-09
3232:
3225:
3219:
3216:
3213:
3209:
3205:
3200:
3197:
3194:
3190:
3185:
3182:
3177:
3173:
3169:
3165:
3160:
3155:
3151:
3147:
3140:
3137:
3132:
3125:
3122:
3116:
3113:
3109:
3104:
3101:
3097:
3093:
3091:
3084:
3081:
3077:
3071:
3068:
3063:
3057:
3050:
3049:
3041:
3039:
3035:
3029:
3027:
3023:
3017:
3015:
3011:
3006:
3000:
2996:
2989:
2986:
2981:
2975:
2971:
2966:
2965:
2956:
2953:
2949:
2948:0-534-38216-9
2945:
2941:
2935:
2932:
2925:
2921:
2918:
2912:
2909:
2903:
2900:
2897:
2894:
2888:
2885:
2882:
2879:
2876:
2873:
2872:
2868:
2866:
2864:
2860:
2856:
2852:
2848:
2844:
2835:
2833:
2831:
2826:
2823:
2821:
2805:
2802:
2788:
2785:
2782:
2774:
2771:
2768:
2757:
2751:
2743:
2739:
2732:
2729:
2726:
2721:
2718:
2712:
2706:
2695:
2688:
2679:
2674:
2671:
2654:
2650:
2639:
2631:
2618:
2615:
2612:
2609:
2601:
2588:
2585:
2579:
2568:
2560:
2545:
2539:
2532:
2521:
2518:
2515:
2509:
2504:
2471:
2463:
2460:
2455:
2451:
2416:
2408:
2379:
2359:
2334:
2326:
2292:
2284:
2249:
2215:
2211:
2200:
2192:
2179:
2176:
2173:
2170:
2162:
2149:
2146:
2140:
2129:
2121:
2118:
2113:
2109:
2093:
2087:
2076:
2069:
2060:
2055:
2042:
2039:
2036:
2030:
2019:
2012:
2006:
2002:
1994:
1991:
1979:
1973:
1966:
1963:
1957:
1953:
1945:
1942:
1938:
1932:
1926:
1919:
1916:
1912:
1904:
1898:
1892:
1889:
1883:
1875:
1872:
1866:
1858:
1857:Taylor series
1840:
1837:
1833:
1829:
1826:
1817:
1804:
1801:
1798:
1792:
1782:
1774:
1770:
1763:
1760:
1755:
1745:
1737:
1734:
1728:
1716:
1710:
1703:
1700:
1691:
1675:
1667:
1651:
1639:
1637:
1621:
1618:
1615:
1612:
1606:
1595:
1592:
1579:
1576:
1573:
1565:
1562:
1554:
1549:
1546:
1543:
1539:
1531:
1527:
1523:
1516:
1510:
1502:
1496:
1485:
1478:
1467:
1448:
1441:
1435:
1432:
1426:
1423:
1420:
1414:
1406:
1400:
1392:
1386:
1379:
1376:
1363:
1361:
1359:
1354:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1307:
1304:
1284:
1278:
1267:
1260:
1254:
1249:
1245:
1239:
1233:
1230:
1222:
1219:
1216:
1213:
1207:
1204:
1198:
1195:
1192:
1186:
1183:
1180:
1174:
1171:
1168:
1162:
1159:
1156:
1150:
1147:
1144:
1141:
1135:
1132:
1126:
1120:
1113:
1110:
1101:
1096:
1092:
1084:
1080:Other methods
1079:
1077:
1075:
1070:
1065:
1056:
1051:
1043:
1035:
1022:
1017:
1013:
1010: +
1009:
1002:
998:
992:
988:
983:
979:
970:
956:
953:
947:
940:
937:
913:
904:
897:
894:
885:
879:
873:
869:
864:
861:
858:
846:
831:
827:
811:
806:
791:
787:
778:
773:
767:
763:
755:
742:
737:
729:
727:
724: = 0.001
723:
718:
714:
710:
706:
701:
699:
683:
680:
677:
657:
654:
651:
631:
611:
606:
602:
596:
589:
578:
571:
567:
561:
558:
549:
521:
517:
508:
503:
490:
484:
481:
473:
470:
467:
461:
458:
452:
449:
446:
440:
427:
424: +
423:
419:
415:
412: +
411:
403:
400: −
399:
395:
391:
388: −
387:
380:
373:
370: −
369:
363:
361:
344:
341:
327:
322:
309:
304:
297:
291:
288:
282:
279:
276:
270:
262:
256:
248:
242:
235:
232:
223:
217:
201:
199:
195:
191:
175:
170:
163:
157:
154:
148:
145:
142:
136:
111:
108: +
107:
103:
99:
96: +
95:
87:
83:
79:
73:
68:
64:
56:
54:
52:
48:
44:
41:estimate the
40:
37:
33:
25:
21:
3464:
3460:
3454:
3437:
3420:
3395:
3391:
3385:
3360:
3356:
3350:
3337:
3326:
3316:
3298:
3273:
3269:
3263:
3238:. Retrieved
3231:the original
3218:
3199:
3184:
3149:
3145:
3139:
3130:
3124:
3115:
3103:
3088:
3083:
3075:
3070:
3047:
2994:
2988:
2963:
2955:
2939:
2934:
2842:
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3212:Cleve Moler
3096:Chapter 5.7
2820:numerically
1859:expansion:
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332:results in
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3558:Categories
3240:2012-11-24
3087:Following
2926:References
2845:, meaning
2843:quadrature
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734:See also:
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43:derivative
39:algorithms
3499:0010-4655
3477:CiteSeerX
3208:MathWorks
3154:CiteSeerX
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730:Step size
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