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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
938:
1873:
2075:
331:
1472:
2319:
783:
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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633:
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512:
1116:
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197:
2443:
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833:
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978:
520:. 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
368:
2843:
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.
2068:
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
1864:
2675:{\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}}}}
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865:
705:
679:
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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.
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2912:
238:
2921:
1382:
1042:
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
1371:
is a tool that can be used to generate derivative approximation methods for any stencil with any derivative order (provided a solution exists).
3241:
3457:
3440:
3013:
2988:
2059:{\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 .}
551:
this is a more accurate approximation to the tangent line than the one-sided estimation. However, although the slope is being computed at
2930:
988:
3279:
Lantoine, G.; Russell, R. P.; Dargent, Th. (2012). "Using multicomplex variables for automatic computation of high-order derivatives".
2236:{\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}}}}
3070:
1063:
facilities may fail to attend to the details of actual computer arithmetic and instead apply the axioms of mathematics to deduce that
565:
2958:
2500:
extracts the last, “most imaginary” component. The method can be applied to mixed derivatives, e.g. for a second-order derivative
714:
The symmetric difference quotient is employed as the method of approximating the derivative in a number of calculators, including
444:
140:
3579:
2873:
214:
The slope of this secant line differs from the slope of the tangent line by an amount that is approximately proportional to
2281:
1101:
517:
2688:
235:
is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line:
3574:
3470:
Ahnert, Karsten; Abel, Markus (2007). "Numerical differentiation of experimental data: local versus global methods".
3031:
Numerical
Differentiation of Analytic Functions, B Fornberg – ACM Transactions on Mathematical Software (TOMS), 1981.
2448:
751:
3353:
3118:
3545:
2869:
3435:
Differential
Quadrature and Its Application in Engineering: Engineering Applications, Chang Shu, Springer, 2000,
2885:
764:
1015:
will be changed (by rounding or truncation) to a nearby machine-representable number, with the consequence that
1074:
2406:
2324:
1655:
The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if
3214:
1056:
h := sqrt(eps) * x; xph := x + h; dx := xph - x; slope := (F(xph) - F(x)) / dx;
811:
3550:
3199:
222:
approaches zero, the slope of the secant line approaches the slope of the tangent line. Therefore, the true
3043:
Using
Complex Variables to Estimate Derivatives of Real Functions, W. Squire, G. Trapp – SIAM REVIEW, 1998.
3487:
3164:
2822:{\displaystyle f^{(n)}(a)={\frac {n!}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{(z-a)^{n+1}}}\,\mathrm {d} z,}
1641:{\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)}
1106:
Higher-order methods for approximating the derivative, as well as methods for higher derivatives, exist.
2906:
2897:
2857:
2830:
2246:
1060:
57:
3401:
Abate, J; Dubner, H (March 1968). "A New Method for
Generating Power Series Expansions of Functions".
3155:
Martins, J. R. R. A.; Sturdza, P.; Alonso, J. J. (2003). "The
Complex-Step Derivative Approximation".
3540:
3479:
3410:
3375:
2069:
1703:
methods. For example, the first derivative can be calculated by the complex-step derivative formula:
1676:
336:
3546:
Boost. Math numerical differentiation, including finite differencing and the complex step derivative
3492:
3169:
1301:{\displaystyle f'(x)={\frac {-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)}{12h}}+{\frac {h^{4}}{30}}f^{(5)}(c),}
346:
2861:
1700:
204:
17:
3513:
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3182:
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1833:
1110:
746:
370:
208:
73:
42:
1821:{\displaystyle f'(x)={\frac {\Im (f(x+\mathrm {i} h))}{h}}+O(h^{2}),\quad \mathrm {i^{2}} :=-1.}
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3453:
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3259:
3100:
3066:
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2984:
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3534:
393:
Another two-point formula is to compute the slope of a nearby secant line through the points
3497:
3418:
3383:
3288:
3174:
2972:
2865:
855:
1679:, real-valued on the real line, which can be evaluated at points in the complex plane near
1311:
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840:
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684:
658:
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3379:
3352:
Ablowitz, M. J., Fokas, A. S.,(2003). Complex variables: introduction and applications.
711:
due to numbers being represented and calculations being performed in limited precision.
3366:
Lyness, J. N.; Moler, C. B. (1967). "Numerical differentiation of analytic functions".
2386:
2366:
1682:
1658:
772:
708:
638:
2868:. There are various methods for determining the weight coefficients, for example, the
1830:
The recommended step size to obtain accurate derivatives for a range of conditions is
1005:
almost certainly will not be exactly representable in that precision. This means that
3568:
2973:
1867:
1368:
3517:
3300:
3186:
3004:
Tamara
Lefcourt Ruby; James Sellers; Lisa Korf; Jeremy Van Horn; Mike Munn (2014).
2915: – Methods used to find numerical solutions of ordinary differential equations
200:
862:
that balances the rounding error against the secant error for optimum accuracy is
3222:
82:
3057:
3501:
61:
53:
3509:
3316:"Computation of higher-order derivatives using the multi-complex step method"
763:
An important consideration in practice when the function is calculated using
64:
using values of the function and perhaps other knowledge about the function.
3292:
3218:
3106:
933:{\displaystyle h=2{\sqrt {\varepsilon \left|{\frac {f(x)}{f''(x)}}\right|}}}
49:
3452:
Advanced
Differential Quadrature Methods, Yingyan Zhang, CRC Press, 2009,
3178:
983:
For computer calculations the problems are exacerbated because, although
137:, and it can be either positive or negative. The slope of this line is
2876:. There are further methods for computing derivatives from noisy data.
30:
3338:
3422:
3387:
786:
For basic central differences, the optimal step is the cube-root of
3333:
2975:
Windows on
Teaching Math: Cases of Middle and Secondary Classrooms
727:
723:
719:
715:
376:
Equivalently, the slope could be estimated by employing positions
81:
A simple two-point estimation is to compute the slope of a nearby
29:
3558:
1049:
will not be a round number in binary; it is 0.000110011001100...
78:
The simplest method is to use finite difference approximations.
2684:
A C++ implementation of multicomplex arithmetics is available.
1109:
Given below is the five-point method for the first derivative (
27:
Use of numerical analysis to estimate derivatives of functions
2687:
In general, derivatives of any order can be calculated using
326:{\displaystyle f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}.}
2565:
2455:
2413:
2331:
2113:
1467:{\displaystyle f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}}
1367:
For other stencil configurations and derivative orders, the
980:), and to employ it will require knowledge of the function.
808:
that is small without producing a large rounding error is
771:. If chosen too small, the subtraction will yield a large
373:, calculating the derivative directly can be unintuitive.
2888: – Numerical calculations carrying along derivatives
779:
and due to cancellation will produce a value of zero if
2926:
Pages displaying short descriptions of redirect targets
2917:
Pages displaying short descriptions of redirect targets
2902:
Pages displaying short descriptions of redirect targets
3056:
Martins, Joaquim R. R. A.; Ning, Andrew (2021-10-01).
1994:
1947:
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2314:{\displaystyle \mathrm {i} ^{(1)}\equiv \mathrm {i} }
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790:. For the numerical derivative formula evaluated at
2839:A method based on numerical inversion of a complex
1038:; the two function evaluations will not be exactly
2860:, where weighted sums are used in methods such as
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775:. In fact, all the finite-difference formulae are
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362:
325:
191:
2909: – Methods of calculating definite integrals
1611:
1598:
1517:
1407:
767:of finite precision is the choice of step size,
628:{\displaystyle R={\frac {-f^{(3)}(c)}{6}}h^{2},}
263:
2383:th component of a multicomplex number of level
3541:NAG Library numerical differentiation routines
3039:
3037:
2894: – A point and its four nearest neighbors
2493:{\displaystyle {\mathcal {C}}_{n^{2}-1}^{(n)}}
2949:Richard L. Burden, J. Douglas Faires (2000),
8:
759:due to both rounding error and formula error
507:{\displaystyle {\frac {f(x+h)-f(x-h)}{2h}}.}
3027:
3025:
2872:. Differential quadrature is used to solve
755:Example showing the difficulty of choosing
3491:
3168:
3157:ACM Transactions on Mathematical Software
3051:
3049:
2913:Numerical ordinary differential equations
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2278:denote the multicomplex imaginary units;
2256:
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2199:
2194:
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2136:
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2072:, resulting in multicomplex derivatives.
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2016:
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1369:Finite Difference Coefficients Calculator
1313:
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192:{\displaystyle {\frac {f(x+h)-f(x)}{h}}.}
144:
142:
2438:{\displaystyle {\mathcal {C}}_{0}^{(n)}}
2356:{\displaystyle {\mathcal {C}}_{k}^{(n)}}
1077:and similar languages, a directive that
750:
2942:
2924: – Algorithm to smooth data points
2922:Numerical smoothing and differentiation
2900: – Algorithm to smooth data points
828:{\displaystyle {\sqrt {\varepsilon }}x}
3551:Differentiation With(out) a Difference
3264:: CS1 maint: archived copy as title (
3257:
3200:Differentiation With(out) a Difference
7:
3334:"mcx (multicomplex algebra library)"
3142:Elementary Real and Complex Analysis
3130:Abramowitz & Stegun, Table 25.2.
1379:Using Newton's difference quotient,
730:, all of which use this method with
3006:Kaplan AP Calculus AB & BC 2015
2971:Katherine Klippert Merseth (2003).
2931:List of numerical-analysis software
1053:A possible approach is as follows:
989:representable floating-point number
3008:. Kaplan Publishing. p. 299.
2979:. Teachers College Press. p.
2809:
2634:
2604:
2547:
2540:
2511:
2307:
2287:
2271:{\displaystyle \mathrm {i} ^{(k)}}
2252:
2195:
2165:
1997:
1919:
1890:
1866:. This formula can be obtained by
1799:
1752:
1733:
1602:
707:. This error does not include the
25:
18:Adaptive numerical differentiation
991:in some precision (32 or 64-bit,
562:The estimation error is given by
2445:extracts the real component and
1474:the following can be shown (for
3472:Computer Physics Communications
3059:Engineering Design Optimization
1796:
555:, the value of the function at
3065:. Cambridge University Press.
2874:partial differential equations
2829:where the integration is done
2789:
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363:{\displaystyle {\frac {0}{0}}}
311:
305:
296:
284:
270:
256:
250:
177:
171:
162:
150:
1:
1102:Finite difference coefficient
846:is typically of the order of
518:symmetric difference quotient
516:This formula is known as the
207:(also known as a first-order
133:represents a small change in
547:. Hence for small values of
441:. The slope of this line is
3314:Verheyleweghen, A. (2014).
1859:{\displaystyle h=10^{-200}}
3596:
3354:Cambridge University Press
2953:, (7th Ed), Brooks/Cole.
1099:
744:
125:. Choosing a small number
71:
3535:Numerical Differentiation
3502:10.1016/j.cpc.2007.03.009
2886:Automatic differentiation
2689:Cauchy's integral formula
1059:However, with computers,
765:floating-point arithmetic
47:numerical differentiation
1651:Complex-variable methods
973:{\displaystyle f''(x)=0}
37:estimation of derivative
3293:10.1145/2168773.2168774
3085:Sauer, Timothy (2012).
2847:Differential quadrature
3281:ACM Trans. Math. Softw
2823:
2676:
2494:
2439:
2397:
2377:
2363:operator extracts the
2357:
2315:
2272:
2237:
2060:
1860:
1822:
1693:
1669:
1642:
1569:
1468:
1358:
1302:
974:
934:
839: = 0), where the
829:
760:
701:
675:
655:is some point between
649:
629:
541:
508:
364:
327:
193:
38:
3580:Differential calculus
3356:. Check theorem 2.6.2
3179:10.1145/838250.838251
2907:Numerical integration
2898:Savitzky-Golay filter
2870:Savitzky–Golay filter
2858:numerical integration
2824:
2677:
2495:
2440:
2398:
2378:
2358:
2316:
2273:
2238:
2061:
1861:
1823:
1694:
1670:
1643:
1549:
1469:
1359:
1357:{\displaystyle c\in }
1303:
1100:Further information:
1061:compiler optimization
975:
935:
830:
754:
702:
676:
650:
630:
542:
540:{\displaystyle h^{2}}
509:
365:
328:
194:
72:Further information:
58:mathematical function
33:
3332:Bell, I. H. (2019).
2695:
2504:
2449:
2407:
2387:
2367:
2325:
2282:
2247:
2076:
2070:multicomplex numbers
1874:
1834:
1707:
1683:
1677:holomorphic function
1659:
1485:
1383:
1312:
1117:
1096:Higher-order methods
987:necessarily holds a
944:
866:
812:
685:
659:
639:
566:
524:
445:
347:
239:
141:
3484:2007CoPhC.177..764A
3415:1968SJNA....5..102A
3403:SIAM J. Numer. Anal
3380:1967SJNA....4..202L
3368:SIAM J. Numer. Anal
2586:
2489:
2434:
2352:
2147:
1113:in one dimension):
1087:will prevent this.
1073:are the same. With
700:{\displaystyle x+h}
674:{\displaystyle x-h}
205:difference quotient
199:This expression is
85:through the points
3575:Numerical analysis
3087:Numerical Analysis
2951:Numerical Analysis
2892:Five-point stencil
2819:
2672:
2562:
2490:
2452:
2435:
2410:
2393:
2373:
2353:
2328:
2311:
2268:
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2110:
2056:
2010:
1961:
1856:
1818:
1689:
1665:
1638:
1531:
1464:
1421:
1375:Higher derivatives
1354:
1298:
1111:five-point stencil
970:
930:
825:
761:
747:Adaptive step size
697:
671:
645:
625:
537:
504:
371:indeterminate form
360:
335:Since immediately
323:
277:
209:divided difference
189:
74:Finite differences
68:Finite differences
43:numerical analysis
39:
3458:978-1-4200-8248-7
3441:978-1-85233-209-9
3101:Numerical Recipes
3089:. Pearson. p.248.
3015:978-1-61865-686-5
2990:978-0-8077-4279-2
2841:Laplace transform
2805:
2747:
2670:
2554:
2396:{\displaystyle n}
2376:{\displaystyle k}
2231:
2009:
1960:
1769:
1699:, then there are
1692:{\displaystyle x}
1668:{\displaystyle f}
1609:
1547:
1516:
1462:
1406:
1268:
1248:
1085:volatile variable
940:(though not when
928:
922:
835:(though not when
820:
648:{\displaystyle c}
610:
559:is not involved.
499:
358:
318:
262:
184:
35:Finite difference
16:(Redirected from
3587:
3537:from wolfram.com
3522:
3521:
3495:
3467:
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3450:
3444:
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3320:
3311:
3305:
3304:
3276:
3270:
3269:
3263:
3255:
3253:
3252:
3246:
3240:. Archived from
3239:
3231:
3225:
3221:blog, posted by
3212:
3206:
3197:
3191:
3190:
3172:
3152:
3146:
3145:
3140:Shilov, George.
3137:
3131:
3128:
3122:
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3110:
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3083:
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3064:
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3019:
3001:
2995:
2994:
2978:
2968:
2962:
2947:
2927:
2918:
2903:
2866:Trapezoidal rule
2862:Simpson's method
2828:
2826:
2825:
2820:
2812:
2806:
2804:
2803:
2802:
2774:
2760:
2758:
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2312:
2310:
2302:
2301:
2290:
2277:
2275:
2274:
2269:
2267:
2266:
2255:
2242:
2240:
2239:
2234:
2232:
2230:
2229:
2220:
2210:
2209:
2198:
2180:
2179:
2168:
2146:
2135:
2128:
2127:
2117:
2116:
2108:
2094:
2093:
2065:
2063:
2062:
2057:
2037:
2036:
2021:
2020:
2011:
2008:
2000:
1995:
1980:
1972:
1971:
1962:
1959:
1948:
1933:
1922:
1893:
1865:
1863:
1862:
1857:
1855:
1854:
1827:
1825:
1824:
1819:
1808:
1807:
1806:
1789:
1788:
1770:
1765:
1755:
1731:
1717:
1698:
1696:
1695:
1690:
1674:
1672:
1671:
1666:
1647:
1645:
1644:
1639:
1616:
1615:
1614:
1601:
1594:
1593:
1568:
1563:
1548:
1546:
1545:
1533:
1530:
1503:
1502:
1480:
1473:
1471:
1470:
1465:
1463:
1458:
1423:
1420:
1393:
1363:
1361:
1360:
1355:
1307:
1305:
1304:
1299:
1285:
1284:
1269:
1264:
1263:
1254:
1249:
1247:
1239:
1141:
1127:
1082:
1072:
1068:
1048:
1041:
1037:
1029:
1014:
1004:
986:
979:
977:
976:
971:
954:
939:
937:
936:
931:
929:
927:
923:
921:
911:
902:
888:
879:
861:
858:. A formula for
856:double precision
853:
851:
845:
834:
832:
831:
826:
821:
816:
807:
803:
793:
782:
770:
758:
736:
706:
704:
703:
698:
680:
678:
677:
672:
654:
652:
651:
646:
634:
632:
631:
626:
621:
620:
611:
606:
596:
595:
576:
558:
554:
550:
546:
544:
543:
538:
536:
535:
513:
511:
510:
505:
500:
498:
490:
449:
440:
416:
389:
385:
369:
367:
366:
361:
359:
351:
342:
332:
330:
329:
324:
319:
314:
279:
276:
249:
233:
229:
221:
217:
198:
196:
195:
190:
185:
180:
145:
136:
132:
128:
124:
100:
21:
3595:
3594:
3590:
3589:
3588:
3586:
3585:
3584:
3565:
3564:
3555:Nicholas Higham
3531:
3526:
3525:
3493:10.1.1.752.3843
3478:(10): 764–774.
3469:
3468:
3464:
3451:
3447:
3434:
3430:
3423:10.1137/0705008
3400:
3399:
3395:
3388:10.1137/0704019
3365:
3364:
3360:
3351:
3347:
3331:
3330:
3326:
3318:
3313:
3312:
3308:
3278:
3277:
3273:
3256:
3250:
3248:
3244:
3237:
3235:"Archived copy"
3233:
3232:
3228:
3213:
3209:
3204:Nicholas Higham
3198:
3194:
3170:10.1.1.141.8002
3154:
3153:
3149:
3139:
3138:
3134:
3129:
3125:
3117:
3113:
3097:
3093:
3084:
3080:
3073:
3062:
3055:
3054:
3047:
3042:
3035:
3030:
3023:
3016:
3003:
3002:
2998:
2991:
2970:
2969:
2965:
2948:
2944:
2939:
2925:
2916:
2901:
2882:
2849:
2788:
2775:
2761:
2749:
2736:
2728:
2698:
2693:
2692:
2660:
2632:
2602:
2561:
2539:
2510:
2509:
2502:
2501:
2461:
2447:
2446:
2405:
2404:
2385:
2384:
2365:
2364:
2323:
2322:
2285:
2280:
2279:
2250:
2245:
2244:
2221:
2193:
2163:
2119:
2109:
2079:
2074:
2073:
2022:
2012:
2001:
1973:
1963:
1952:
1926:
1872:
1871:
1843:
1832:
1831:
1798:
1780:
1732:
1710:
1705:
1704:
1681:
1680:
1657:
1656:
1653:
1596:
1579:
1537:
1488:
1483:
1482:
1475:
1424:
1386:
1381:
1380:
1377:
1310:
1309:
1270:
1255:
1240:
1142:
1120:
1115:
1114:
1104:
1098:
1093:
1078:
1070:
1064:
1057:
1052:
1043:
1039:
1035:
1016:
1006:
996:
984:
947:
942:
941:
904:
903:
889:
883:
864:
863:
859:
849:
847:
843:
841:machine epsilon
810:
809:
805:
804:, a choice for
795:
791:
788:machine epsilon
780:
777:ill-conditioned
768:
756:
749:
743:
731:
683:
682:
657:
656:
637:
636:
612:
581:
577:
564:
563:
556:
552:
548:
527:
522:
521:
491:
450:
443:
442:
418:
394:
387:
377:
345:
344:
340:
280:
242:
237:
236:
231:
225:
219:
215:
146:
139:
138:
134:
130:
126:
102:
86:
76:
70:
28:
23:
22:
15:
12:
11:
5:
3593:
3591:
3583:
3582:
3577:
3567:
3566:
3563:
3562:
3548:
3543:
3538:
3530:
3529:External links
3527:
3524:
3523:
3462:
3445:
3428:
3409:(1): 102–112.
3393:
3374:(2): 202–210.
3358:
3345:
3324:
3306:
3271:
3226:
3207:
3192:
3163:(3): 245–262.
3147:
3132:
3123:
3111:
3091:
3078:
3072:978-1108833417
3071:
3045:
3033:
3021:
3014:
2996:
2989:
2963:
2941:
2940:
2938:
2935:
2934:
2933:
2928:
2919:
2910:
2904:
2895:
2889:
2881:
2878:
2848:
2845:
2818:
2815:
2811:
2801:
2798:
2795:
2791:
2787:
2784:
2781:
2778:
2773:
2770:
2767:
2764:
2756:
2752:
2745:
2742:
2739:
2734:
2731:
2725:
2722:
2719:
2716:
2711:
2708:
2705:
2701:
2667:
2663:
2658:
2655:
2652:
2647:
2644:
2641:
2636:
2631:
2628:
2625:
2622:
2617:
2614:
2611:
2606:
2601:
2598:
2595:
2592:
2589:
2584:
2581:
2578:
2573:
2567:
2558:
2552:
2549:
2545:
2542:
2537:
2534:
2531:
2528:
2525:
2522:
2517:
2513:
2487:
2484:
2481:
2476:
2473:
2468:
2464:
2457:
2432:
2429:
2426:
2421:
2415:
2392:
2372:
2350:
2347:
2344:
2339:
2333:
2309:
2305:
2300:
2297:
2294:
2289:
2265:
2262:
2259:
2254:
2228:
2224:
2219:
2216:
2213:
2208:
2205:
2202:
2197:
2192:
2189:
2186:
2183:
2178:
2175:
2172:
2167:
2162:
2159:
2156:
2153:
2150:
2145:
2142:
2139:
2134:
2131:
2126:
2122:
2115:
2106:
2103:
2100:
2097:
2092:
2089:
2086:
2082:
2055:
2052:
2049:
2046:
2043:
2040:
2035:
2032:
2029:
2025:
2019:
2015:
2007:
2004:
1999:
1992:
1989:
1986:
1983:
1979:
1976:
1970:
1966:
1958:
1955:
1951:
1945:
1942:
1939:
1936:
1932:
1929:
1925:
1921:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1892:
1888:
1885:
1882:
1879:
1853:
1850:
1846:
1842:
1839:
1817:
1814:
1811:
1805:
1801:
1795:
1792:
1787:
1783:
1779:
1776:
1773:
1768:
1764:
1761:
1758:
1754:
1750:
1747:
1744:
1741:
1738:
1735:
1729:
1726:
1723:
1720:
1716:
1713:
1688:
1664:
1652:
1649:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1613:
1608:
1605:
1600:
1592:
1589:
1586:
1582:
1578:
1575:
1572:
1567:
1562:
1559:
1556:
1552:
1544:
1540:
1536:
1529:
1526:
1523:
1519:
1515:
1512:
1509:
1506:
1501:
1498:
1495:
1491:
1461:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1419:
1416:
1413:
1409:
1405:
1402:
1399:
1396:
1392:
1389:
1376:
1373:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1297:
1294:
1291:
1288:
1283:
1280:
1277:
1273:
1267:
1262:
1258:
1252:
1246:
1243:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1139:
1136:
1133:
1130:
1126:
1123:
1097:
1094:
1092:
1089:
1055:
1050:
1025:) −
969:
966:
963:
960:
957:
953:
950:
926:
920:
917:
914:
910:
907:
901:
898:
895:
892:
886:
882:
877:
874:
871:
824:
819:
773:rounding error
742:
739:
709:rounding error
696:
693:
690:
670:
667:
664:
644:
624:
619:
615:
609:
605:
602:
599:
594:
591:
588:
584:
580:
574:
571:
534:
530:
503:
497:
494:
489:
486:
483:
480:
477:
474:
471:
468:
465:
462:
459:
456:
453:
357:
354:
322:
317:
313:
310:
307:
304:
301:
298:
295:
292:
289:
286:
283:
275:
272:
269:
265:
261:
258:
255:
252:
248:
245:
224:derivative of
188:
183:
179:
176:
173:
170:
167:
164:
161:
158:
155:
152:
149:
69:
66:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3592:
3581:
3578:
3576:
3573:
3572:
3570:
3560:
3556:
3552:
3549:
3547:
3544:
3542:
3539:
3536:
3533:
3532:
3528:
3519:
3515:
3511:
3507:
3503:
3499:
3494:
3489:
3485:
3481:
3477:
3473:
3466:
3463:
3459:
3455:
3449:
3446:
3442:
3438:
3432:
3429:
3424:
3420:
3416:
3412:
3408:
3404:
3397:
3394:
3389:
3385:
3381:
3377:
3373:
3369:
3362:
3359:
3355:
3349:
3346:
3341:
3340:
3335:
3328:
3325:
3317:
3310:
3307:
3302:
3298:
3294:
3290:
3286:
3282:
3275:
3272:
3267:
3261:
3247:on 2014-01-09
3243:
3236:
3230:
3227:
3224:
3220:
3216:
3211:
3208:
3205:
3201:
3196:
3193:
3188:
3184:
3180:
3176:
3171:
3166:
3162:
3158:
3151:
3148:
3143:
3136:
3133:
3127:
3124:
3120:
3115:
3112:
3108:
3104:
3102:
3095:
3092:
3088:
3082:
3079:
3074:
3068:
3061:
3060:
3052:
3050:
3046:
3040:
3038:
3034:
3028:
3026:
3022:
3017:
3011:
3007:
3000:
2997:
2992:
2986:
2982:
2977:
2976:
2967:
2964:
2960:
2959:0-534-38216-9
2956:
2952:
2946:
2943:
2936:
2932:
2929:
2923:
2920:
2914:
2911:
2908:
2905:
2899:
2896:
2893:
2890:
2887:
2884:
2883:
2879:
2877:
2875:
2871:
2867:
2863:
2859:
2855:
2846:
2844:
2842:
2837:
2834:
2832:
2816:
2813:
2799:
2796:
2793:
2785:
2782:
2779:
2768:
2762:
2754:
2750:
2743:
2740:
2737:
2732:
2729:
2723:
2717:
2706:
2699:
2690:
2685:
2682:
2665:
2661:
2650:
2642:
2629:
2626:
2623:
2620:
2612:
2599:
2596:
2590:
2579:
2571:
2556:
2550:
2543:
2532:
2529:
2526:
2520:
2515:
2482:
2474:
2471:
2466:
2462:
2427:
2419:
2390:
2370:
2345:
2337:
2303:
2295:
2260:
2226:
2222:
2211:
2203:
2190:
2187:
2184:
2181:
2173:
2160:
2157:
2151:
2140:
2132:
2129:
2124:
2120:
2104:
2098:
2087:
2080:
2071:
2066:
2053:
2050:
2047:
2041:
2030:
2023:
2017:
2013:
2005:
2002:
1990:
1984:
1977:
1974:
1968:
1964:
1956:
1953:
1949:
1943:
1937:
1930:
1927:
1923:
1915:
1909:
1903:
1900:
1894:
1886:
1883:
1877:
1869:
1868:Taylor series
1851:
1848:
1844:
1840:
1837:
1828:
1815:
1812:
1809:
1803:
1793:
1785:
1781:
1774:
1771:
1766:
1756:
1748:
1745:
1739:
1727:
1721:
1714:
1711:
1702:
1686:
1678:
1662:
1650:
1648:
1632:
1629:
1626:
1623:
1617:
1606:
1603:
1590:
1587:
1584:
1576:
1573:
1565:
1560:
1557:
1554:
1550:
1542:
1538:
1534:
1527:
1521:
1513:
1507:
1496:
1489:
1478:
1459:
1452:
1446:
1443:
1437:
1434:
1431:
1425:
1417:
1411:
1403:
1397:
1390:
1387:
1374:
1372:
1370:
1365:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1318:
1315:
1295:
1289:
1278:
1271:
1265:
1260:
1256:
1250:
1244:
1241:
1233:
1230:
1227:
1224:
1218:
1215:
1209:
1206:
1203:
1197:
1194:
1191:
1185:
1182:
1179:
1173:
1170:
1167:
1161:
1158:
1155:
1152:
1146:
1143:
1137:
1131:
1124:
1121:
1112:
1107:
1103:
1095:
1091:Other methods
1090:
1088:
1086:
1081:
1076:
1067:
1062:
1054:
1046:
1033:
1028:
1024:
1021: +
1020:
1013:
1009:
1003:
999:
994:
990:
981:
967:
964:
958:
951:
948:
924:
915:
908:
905:
896:
890:
884:
880:
875:
872:
869:
857:
842:
838:
822:
817:
802:
798:
789:
784:
778:
774:
766:
753:
748:
740:
738:
735: = 0.001
734:
729:
725:
721:
717:
712:
710:
694:
691:
688:
668:
665:
662:
642:
622:
617:
613:
607:
600:
589:
582:
578:
572:
569:
560:
532:
528:
519:
514:
501:
495:
492:
484:
481:
478:
472:
469:
463:
460:
457:
451:
438:
435: +
434:
430:
426:
423: +
422:
414:
411: −
410:
406:
402:
399: −
398:
391:
384:
381: −
380:
374:
372:
355:
352:
338:
333:
320:
315:
308:
302:
299:
293:
290:
287:
281:
273:
267:
259:
253:
246:
243:
234:
228:
212:
210:
206:
202:
186:
181:
174:
168:
165:
159:
156:
153:
147:
122:
119: +
118:
114:
110:
107: +
106:
98:
94:
90:
84:
79:
75:
67:
65:
63:
59:
55:
52:estimate the
51:
48:
44:
36:
32:
19:
3475:
3471:
3465:
3448:
3431:
3406:
3402:
3396:
3371:
3367:
3361:
3348:
3337:
3327:
3309:
3284:
3280:
3274:
3249:. Retrieved
3242:the original
3229:
3210:
3195:
3160:
3156:
3150:
3141:
3135:
3126:
3114:
3099:
3094:
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3107:Chapter 5.7
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3569:Categories
3251:2012-11-24
3098:Following
2937:References
2856:, meaning
2854:quadrature
2243:where the
745:See also:
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3510:0010-4655
3488:CiteSeerX
3219:MathWorks
3165:CiteSeerX
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