22:
1272:
1776:
3100:
The description given above is a simplified procedures used in the stepsize control for explicit RK solvers. A more detailed treatment can be found in Hairer's textbook. The ODE solver in many programming languages uses this procedure as the default strategy for adaptive stepsize control, which adds
2677:
824:. Take the new step size to be one half of the original step size, and apply two steps of Euler's method. This second solution is presumably more accurate. Since we have to apply Euler's method twice, the local error is (in the worst case) twice the original error.
1071:
1065:
2825:
1638:
158:. There, scenarios emerge where one can take large time steps when the spacecraft is far from the Earth and Moon, but if the spacecraft gets close to colliding with one of the planetary bodies, then small time steps are needed.
2902:
923:
146:. Using an adaptive stepsize is of particular importance when there is a large variation in the size of the derivative. For example, when modeling the motion of a satellite about the earth as a standard
1996:
1578:
2250:
665:
2481:
3097:, which has lower order. The above prediction formula is plausible in a sense that it enlarges the step if the estimated local error is smaller than the tolerance and it shrinks the step otherwise.
1368:
279:
1889:
557:
2091:
Similar methods can be developed for higher order methods, such as the 4th-order Runge–Kutta method. Also, a global error tolerance can be achieved by scaling the local error to global scope.
3059:
2421:
1801:
is a safety factor to ensure success on the next try. The minimum and maximum are to prevent extreme changes from the previous stepsize. This should, in principle give an error of about
1267:{\displaystyle \tau _{n+1}^{(1)}=c\left({\frac {h}{2}}\right)^{2}+c\left({\frac {h}{2}}\right)^{2}=2c\left({\frac {h}{2}}\right)^{2}={\frac {1}{2}}ch^{2}={\frac {1}{2}}\tau _{n+1}^{(0)}}
2005:
accurate in the local scope (second order in the global scope), but since there is no error estimate for it, this doesn't help in reducing the number of steps. This technique is called
739:
393:
1827:
929:
2710:
2310:
131:
3095:
2473:
1626:
413:
1471:
2718:
2084:, using an optimal number of steps given a local error tolerance. A drawback is that the step size may become prohibitively small, especially when using the low-order
2959:
2932:
2444:
2340:
2143:
2042:
1799:
822:
775:
154:
may be sufficient. However things are more difficult if one wishes to model the motion of a spacecraft taking into account both the Earth and the Moon as in the
1771:{\displaystyle h\rightarrow 0.9\times h\times \min \left(\max \left(\left({\frac {\text{tol}}{2\left|\tau _{n+1}^{(1)}\right|}}\right)^{1/2},0.3\right),2\right)}
2082:
2062:
1604:
1391:
1429:
2712:
is the unnormalized error. To normalize it, we compare it against a user-defined tolerance, which consists of the absolute tolerance and relative tolerance:
3185:
2831:
830:
2104:
163:
39:
3115:
3170:
3156:
105:
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1479:
2672:{\displaystyle {\textrm {err}}_{n}(h)={\tilde {y}}_{n+1}-y_{n+1}=h({\tilde {\psi }}(t_{n},y_{n},h_{n})-\psi (t_{n},y_{n},h_{n}))}
2151:
86:
568:
58:
2115:
methods. These methods are considered to be more computationally efficient, but have lower accuracy in their error estimates.
43:
1278:
784:
is not known to us. Let us now apply Euler's method again with a different step size to generate a second approximation to
65:
1832:
467:
191:
3190:
2967:
72:
2345:
2112:
32:
2006:
54:
684:
1060:{\displaystyle y_{n+1}^{(1)}=y_{n+{\frac {1}{2}}}+{\frac {h}{2}}f(t_{n+{\frac {1}{2}}},y_{n+{\frac {1}{2}}})}
350:
3134:
E. Hairer, S. P. Norsett G. Wanner, “Solving
Ordinary Differential Equations I: Nonstiff Problems”, Sec. II.
2100:
1804:
292:
may denote vectors (in which case this equation represents a system of coupled ODEs in several variables).
2684:
2256:
135:
2108:
3071:
2449:
3110:
678:
is sufficiently smooth) the local truncation error is proportional to the square of the step size:
671:
159:
139:
2820:{\displaystyle {\textrm {tol}}_{n}={\textrm {Atol}}+{\textrm {Rtol}}\cdot \max(|y_{n}|,|y_{n-1}|)}
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398:
155:
123:
1434:
79:
3166:
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2015:
1891:, we consider the step successful, and the error estimate is used to improve the solution:
179:
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1606:
should be modified to achieve the desired accuracy. For example, if a local tolerance of
801:
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2118:
To illustrate the ideas of embedded method, consider the following scheme which update
2067:
2047:
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3179:
2099:
Adaptive stepsize methods that use a so-called 'embedded' error estimate include the
2085:
175:
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182:
methods are preferred due to their superior convergence and stability properties.
174:
For simplicity, the following example uses the simplest integration method, the
166:
are examples of a numerical integration methods which use an adaptive stepsize.
143:
119:
21:
3147:
William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery,
2897:{\displaystyle E_{n}={\textrm {norm}}({\textrm {err}}_{n}/{\textrm {tol}}_{n})}
2044:, this theory facilitates our controllable integration of the ODE from point
918:{\displaystyle y_{n+{\frac {1}{2}}}=y_{n}+{\frac {h}{2}}f(t_{n},y_{n})}
448:, the Euler method gives approximations to the corresponding values of
562:
The local truncation error of this approximation is defined by
15:
3101:
other engineering parameters to make the system more stable.
1991:{\displaystyle y_{n+1}^{(2)}=y_{n+1}^{(1)}+\tau _{n+1}^{(1)}}
1573:{\displaystyle y_{n+1}^{(1)}-y_{n+1}^{(0)}=\tau _{n+1}^{(1)}}
138:) in order to control the errors of the method and to ensure
1586:
The local error estimate can be used to decide how stepsize
2245:{\displaystyle y_{n+1}=y_{n}+h_{n}\psi (t_{n},y_{n},h_{n})}
660:{\displaystyle \tau _{n+1}^{(0)}=y(t_{n+1})-y_{n+1}^{(0)}}
1363:{\displaystyle y_{n+1}^{(1)}+\tau _{n+1}^{(1)}=y(t+h)}
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numerical solution of ordinary differential equations
3151:, Second Edition, CAMBRIDGE UNIVERSITY PRESS, 1992.
320:), and we are interested in finding the solution at
1884:{\displaystyle |\tau _{n+1}^{(1)}|<{\text{tol}}}
1583:This local error estimate is third order accurate.
1431:. In reality its rate of change is proportional to
798:). We get a second solution, which we label with a
552:{\displaystyle y_{n+1}^{(0)}=y_{n}+hf(t_{n},y_{n})}
46:. Unsourced material may be challenged and removed.
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3053:
2953:
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1990:
1883:
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1473:. Subtracting solutions gives the error estimate:
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751:We have marked this solution and its error with a
733:
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387:
273:
274:{\displaystyle y'(t)=f(t,y(t)),\qquad y(a)=y_{a}}
3054:{\displaystyle h_{n}=h_{n-1}(1/E_{n})^{1/(q+1)}}
2759:
1668:
1660:
3165:, Second Edition, John Wiley & Sons, 1989.
347:denote the solution that we compute. We write
8:
3068:is the order corresponding to the RK method
2416:{\displaystyle h_{n}=g(t_{n},y_{n},h_{n-1})}
178:; in practice, higher-order methods such as
2342:is predicted from the previous information
150:, a fixed time-stepping method such as the
2475:. The error then can be simply written as
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2191:
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2017:
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486:
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469:
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358:
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265:
193:
106:Learn how and when to remove this message
734:{\displaystyle \tau _{n+1}^{(0)}=ch^{2}}
415:is the error in the numerical solution.
3127:
2426:For embedded RK method, computation of
388:{\displaystyle y_{b}+\varepsilon =y(b)}
2012:Beginning with an initial stepsize of
1822:{\displaystyle 0.9\times {\text{tol}}}
2907:Then we compare the normalized error
748:is some constant of proportionality.
7:
44:adding citations to reliable sources
2705:{\displaystyle {\textrm {err}}_{n}}
2305:{\displaystyle t_{n+1}=t_{n}+h_{n}}
185:Consider the initial value problem
3116:Adaptive numerical differentiation
14:
2446:includes a lower order RK method
674:, it can be shown that (provided
3186:Numerical differential equations
3090:{\displaystyle {\tilde {\psi }}}
2468:{\displaystyle {\tilde {\psi }}}
130:is used in some methods for the
20:
2934:against 1 to get the predicted
336:) denote the exact solution at
245:
134:(including the special case of
31:needs additional citations for
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3046:
3034:
3022:
3000:
2891:
2855:
2814:
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2762:
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2663:
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2615:
2576:
2570:
2561:
2518:
2505:
2499:
2459:
2410:
2365:
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1983:
1977:
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1947:
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1393:is constant over the interval
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307:) and the initial conditions (
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1:
1373:Here, we assume error factor
1621:{\displaystyle {\text{tol}}}
408:{\displaystyle \varepsilon }
3207:
2001:This solution is actually
1466:{\displaystyle y^{(3)}(t)}
295:We are given the function
1628:is allowed, we could let
2095:Embedded error estimates
2007:Richardson extrapolation
3149:Numerical Recipes in C
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771:
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553:
409:
389:
275:
3161:Kendall E. Atkinson,
3092:
3056:
2956:
2954:{\displaystyle h_{n}}
2929:
2927:{\displaystyle E_{n}}
2899:
2822:
2707:
2674:
2470:
2441:
2439:{\displaystyle \psi }
2418:
2337:
2335:{\displaystyle h_{n}}
2307:
2247:
2140:
2138:{\displaystyle y_{n}}
2079:
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2039:
2037:{\displaystyle h=b-a}
1993:
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136:numerical integration
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2319:
2257:
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2122:
2105:Runge–Kutta–Fehlberg
2068:
2048:
2016:
1898:
1833:
1829:in the next try. If
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1072:
930:
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468:
399:
351:
192:
164:Runge–Kutta–Fehlberg
140:stability properties
55:"Adaptive step size"
40:improve this article
3111:Adaptive quadrature
1987:
1957:
1927:
1867:
1794:{\displaystyle 0.9}
1720:
1569:
1539:
1509:
1338:
1308:
1263:
1101:
959:
817:{\displaystyle (1)}
770:{\displaystyle (0)}
714:
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497:
3191:Numerical analysis
3163:Numerical Analysis
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767:
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688:
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156:Three-body problem
128:adaptive step size
124:numerical analysis
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2753:
2743:
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2077:{\displaystyle b}
2057:{\displaystyle a}
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1817:
1726:
1685:
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1599:{\displaystyle h}
1386:{\displaystyle c}
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2614:
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2600:
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2587:
2575:
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2498:
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2238:
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2225:
2224:
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2211:
2196:
2195:
2183:
2182:
2170:
2169:
2144:
2142:
2141:
2136:
2134:
2133:
2101:Bogacki–Shampine
2083:
2081:
2080:
2075:
2063:
2061:
2060:
2055:
2043:
2041:
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2035:
1997:
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1989:
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1956:
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1708:
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1602:
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1579:
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1571:
1568:
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1538:
1527:
1508:
1497:
1472:
1470:
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1464:
1453:
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1430:
1428:
1427:
1424:{\displaystyle }
1422:
1392:
1390:
1389:
1384:
1369:
1367:
1366:
1361:
1337:
1326:
1307:
1296:
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1228:
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1210:
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1066:
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1063:
1058:
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1052:
1051:
1043:
1027:
1026:
1025:
1017:
998:
990:
985:
984:
983:
975:
958:
947:
924:
922:
921:
916:
911:
910:
898:
897:
882:
874:
869:
868:
856:
855:
854:
846:
823:
821:
820:
815:
776:
774:
773:
768:
740:
738:
737:
732:
730:
729:
713:
702:
672:Taylor's theorem
666:
664:
663:
658:
655:
644:
623:
622:
597:
586:
558:
556:
555:
550:
545:
544:
532:
531:
510:
509:
496:
485:
418:For a sequence (
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406:
394:
392:
391:
386:
363:
362:
280:
278:
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272:
270:
269:
202:
160:Romberg's method
111:
104:
100:
97:
91:
89:
48:
24:
16:
3206:
3205:
3201:
3200:
3199:
3197:
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3176:
3175:
3144:
3142:Further reading
3139:
3138:
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3129:
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3107:
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3011:
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2592:
2579:
2539:
2511:
2485:
2480:
2479:
2448:
2447:
2428:
2427:
2394:
2381:
2368:
2349:
2344:
2343:
2322:
2317:
2316:
2292:
2279:
2260:
2255:
2254:
2229:
2216:
2203:
2187:
2174:
2155:
2150:
2149:
2125:
2120:
2119:
2097:
2066:
2065:
2046:
2045:
2014:
2013:
1896:
1895:
1831:
1830:
1803:
1802:
1783:
1782:
1690:
1686:
1677:
1676:
1675:
1671:
1667:
1663:
1637:
1636:
1608:
1607:
1588:
1587:
1478:
1477:
1438:
1433:
1432:
1395:
1394:
1375:
1374:
1277:
1276:
1214:
1174:
1173:
1140:
1139:
1109:
1108:
1070:
1069:
1031:
1005:
963:
928:
927:
902:
889:
860:
834:
829:
828:
800:
799:
797:
753:
752:
721:
683:
682:
608:
567:
566:
536:
523:
501:
466:
465:
460:
439:
427:) of values of
426:
397:
396:
354:
349:
348:
345:
319:
261:
195:
190:
189:
172:
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
3204:
3202:
3194:
3193:
3188:
3178:
3177:
3174:
3173:
3159:
3143:
3140:
3137:
3136:
3126:
3125:
3123:
3120:
3119:
3118:
3113:
3106:
3103:
3083:
3080:
3064:The parameter
3062:
3061:
3048:
3045:
3042:
3039:
3036:
3032:
3028:
3024:
3018:
3014:
3009:
3005:
3002:
2997:
2994:
2991:
2987:
2983:
2978:
2974:
2948:
2944:
2921:
2917:
2905:
2904:
2893:
2888:
2875:
2869:
2857:
2847:
2842:
2838:
2827:
2816:
2812:
2806:
2803:
2800:
2796:
2791:
2787:
2783:
2777:
2773:
2768:
2764:
2761:
2758:
2748:
2738:
2733:
2699:
2680:
2679:
2668:
2665:
2660:
2656:
2652:
2647:
2643:
2639:
2634:
2630:
2626:
2623:
2620:
2617:
2612:
2608:
2604:
2599:
2595:
2591:
2586:
2582:
2578:
2572:
2569:
2563:
2560:
2557:
2552:
2549:
2546:
2542:
2538:
2533:
2530:
2527:
2520:
2517:
2510:
2507:
2504:
2501:
2496:
2461:
2458:
2435:
2412:
2407:
2404:
2401:
2397:
2393:
2388:
2384:
2380:
2375:
2371:
2367:
2364:
2361:
2356:
2352:
2329:
2325:
2315:The next step
2313:
2312:
2299:
2295:
2291:
2286:
2282:
2278:
2273:
2270:
2267:
2263:
2252:
2241:
2236:
2232:
2228:
2223:
2219:
2215:
2210:
2206:
2202:
2199:
2194:
2190:
2186:
2181:
2177:
2173:
2168:
2165:
2162:
2158:
2132:
2128:
2113:Dormand–Prince
2096:
2093:
2073:
2053:
2033:
2030:
2027:
2024:
2021:
1999:
1998:
1985:
1982:
1979:
1974:
1971:
1968:
1964:
1960:
1955:
1952:
1949:
1944:
1941:
1938:
1934:
1930:
1925:
1922:
1919:
1914:
1911:
1908:
1904:
1875:
1871:
1865:
1862:
1859:
1854:
1851:
1848:
1844:
1839:
1813:
1810:
1790:
1779:
1778:
1766:
1762:
1759:
1755:
1751:
1748:
1743:
1739:
1735:
1730:
1723:
1718:
1715:
1712:
1707:
1704:
1701:
1697:
1693:
1689:
1680:
1674:
1670:
1666:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1595:
1581:
1580:
1567:
1564:
1561:
1556:
1553:
1550:
1546:
1542:
1537:
1534:
1531:
1526:
1523:
1520:
1516:
1512:
1507:
1504:
1501:
1496:
1493:
1490:
1486:
1462:
1459:
1456:
1451:
1448:
1445:
1441:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1382:
1371:
1370:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1336:
1333:
1330:
1325:
1322:
1319:
1315:
1311:
1306:
1303:
1300:
1295:
1292:
1289:
1285:
1274:
1261:
1258:
1255:
1250:
1247:
1244:
1240:
1234:
1231:
1226:
1221:
1217:
1213:
1208:
1205:
1200:
1195:
1190:
1185:
1182:
1177:
1172:
1169:
1166:
1161:
1156:
1151:
1148:
1143:
1138:
1135:
1130:
1125:
1120:
1117:
1112:
1107:
1104:
1099:
1096:
1093:
1088:
1085:
1082:
1078:
1067:
1056:
1049:
1046:
1041:
1038:
1034:
1030:
1023:
1020:
1015:
1012:
1008:
1004:
1001:
996:
993:
988:
981:
978:
973:
970:
966:
962:
957:
954:
951:
946:
943:
940:
936:
925:
914:
909:
905:
901:
896:
892:
888:
885:
880:
877:
872:
867:
863:
859:
852:
849:
844:
841:
837:
813:
810:
807:
792:
766:
763:
760:
742:
741:
728:
724:
720:
717:
712:
709:
706:
701:
698:
695:
691:
668:
667:
654:
651:
648:
643:
640:
637:
633:
629:
626:
621:
618:
615:
611:
607:
604:
601:
596:
593:
590:
585:
582:
579:
575:
560:
559:
548:
543:
539:
535:
530:
526:
522:
519:
516:
513:
508:
504:
500:
495:
492:
489:
484:
481:
478:
474:
456:
435:
422:
404:
384:
381:
378:
375:
372:
369:
366:
361:
357:
343:
315:
282:
281:
268:
264:
260:
257:
254:
251:
248:
244:
241:
238:
235:
232:
229:
226:
223:
220:
217:
214:
211:
208:
205:
201:
198:
171:
168:
114:
113:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
3203:
3192:
3189:
3187:
3184:
3183:
3181:
3172:
3171:0-471-62489-6
3168:
3164:
3160:
3158:
3157:0-521-43108-5
3154:
3150:
3146:
3145:
3141:
3131:
3128:
3121:
3117:
3114:
3112:
3109:
3108:
3104:
3102:
3098:
3078:
3067:
3043:
3040:
3037:
3030:
3026:
3016:
3012:
3007:
3003:
2995:
2992:
2989:
2985:
2981:
2976:
2972:
2964:
2963:
2962:
2946:
2942:
2919:
2915:
2886:
2873:
2867:
2845:
2840:
2836:
2828:
2804:
2801:
2798:
2794:
2785:
2775:
2771:
2756:
2746:
2736:
2731:
2715:
2714:
2713:
2697:
2658:
2654:
2650:
2645:
2641:
2637:
2632:
2628:
2621:
2618:
2610:
2606:
2602:
2597:
2593:
2589:
2584:
2580:
2567:
2558:
2555:
2550:
2547:
2544:
2540:
2536:
2531:
2528:
2525:
2515:
2508:
2502:
2494:
2478:
2477:
2476:
2456:
2433:
2424:
2405:
2402:
2399:
2395:
2391:
2386:
2382:
2378:
2373:
2369:
2362:
2359:
2354:
2350:
2327:
2323:
2297:
2293:
2289:
2284:
2280:
2276:
2271:
2268:
2265:
2261:
2253:
2234:
2230:
2226:
2221:
2217:
2213:
2208:
2204:
2197:
2192:
2188:
2184:
2179:
2175:
2171:
2166:
2163:
2160:
2156:
2148:
2147:
2146:
2130:
2126:
2116:
2114:
2110:
2106:
2102:
2094:
2092:
2089:
2087:
2071:
2051:
2031:
2028:
2025:
2022:
2019:
2010:
2008:
2004:
1980:
1972:
1969:
1966:
1962:
1958:
1950:
1942:
1939:
1936:
1932:
1928:
1920:
1912:
1909:
1906:
1902:
1894:
1893:
1892:
1873:
1860:
1852:
1849:
1846:
1842:
1811:
1808:
1788:
1764:
1760:
1757:
1753:
1749:
1746:
1741:
1737:
1733:
1728:
1721:
1713:
1705:
1702:
1699:
1695:
1691:
1687:
1678:
1672:
1664:
1657:
1654:
1651:
1648:
1642:
1635:
1634:
1633:
1632:evolve like:
1631:
1593:
1584:
1562:
1554:
1551:
1548:
1544:
1540:
1532:
1524:
1521:
1518:
1514:
1510:
1502:
1494:
1491:
1488:
1484:
1476:
1475:
1474:
1457:
1446:
1439:
1415:
1412:
1409:
1406:
1403:
1380:
1354:
1351:
1348:
1342:
1339:
1331:
1323:
1320:
1317:
1313:
1309:
1301:
1293:
1290:
1287:
1283:
1275:
1256:
1248:
1245:
1242:
1238:
1232:
1229:
1224:
1219:
1215:
1211:
1206:
1203:
1198:
1193:
1188:
1183:
1180:
1175:
1170:
1167:
1164:
1159:
1154:
1149:
1146:
1141:
1136:
1133:
1128:
1123:
1118:
1115:
1110:
1105:
1102:
1094:
1086:
1083:
1080:
1076:
1068:
1047:
1044:
1039:
1036:
1032:
1028:
1021:
1018:
1013:
1010:
1006:
999:
994:
991:
986:
979:
976:
971:
968:
964:
960:
952:
944:
941:
938:
934:
926:
907:
903:
899:
894:
890:
883:
878:
875:
870:
865:
861:
857:
850:
847:
842:
839:
835:
827:
826:
825:
808:
795:
791:
787:
783:
780:The value of
778:
761:
749:
747:
726:
722:
718:
715:
707:
699:
696:
693:
689:
681:
680:
679:
677:
673:
649:
641:
638:
635:
631:
627:
619:
616:
613:
609:
602:
599:
591:
583:
580:
577:
573:
565:
564:
563:
541:
537:
533:
528:
524:
517:
514:
511:
506:
502:
498:
490:
482:
479:
476:
472:
464:
463:
462:
459:
455:
451:
447:
443:
438:
434:
430:
425:
421:
416:
402:
379:
373:
370:
367:
364:
359:
355:
346:
339:
335:
331:
327:
324: =
323:
318:
314:
310:
306:
302:
298:
293:
291:
287:
266:
262:
258:
252:
246:
242:
233:
227:
224:
221:
215:
212:
206:
199:
196:
188:
187:
186:
183:
181:
177:
169:
167:
165:
161:
157:
153:
149:
145:
141:
137:
133:
129:
125:
121:
110:
107:
99:
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
3162:
3148:
3130:
3099:
3065:
3063:
2906:
2681:
2425:
2314:
2117:
2098:
2090:
2086:Euler method
2011:
2002:
2000:
1780:
1629:
1585:
1582:
1372:
793:
789:
785:
781:
779:
750:
745:
743:
675:
669:
561:
457:
453:
449:
445:
441:
436:
432:
428:
423:
419:
417:
341:
337:
333:
329:
325:
321:
316:
312:
308:
304:
300:
296:
294:
289:
285:
283:
184:
176:Euler method
173:
152:Euler method
148:Kepler orbit
127:
117:
102:
96:October 2012
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
2003:third order
180:Runge–Kutta
144:A-stability
120:mathematics
3180:Categories
3122:References
340:, and let
66:newspapers
3082:~
3079:ψ
2993:−
2802:−
2757:⋅
2622:ψ
2619:−
2571:~
2568:ψ
2537:−
2519:~
2460:~
2457:ψ
2434:ψ
2403:−
2198:ψ
2109:Cash–Karp
2029:−
1963:τ
1843:τ
1812:×
1696:τ
1658:×
1652:×
1646:→
1545:τ
1511:−
1314:τ
1239:τ
1077:τ
690:τ
628:−
574:τ
403:ε
368:ε
3105:See also
395:, where
200:′
142:such as
670:and by
431:, with
170:Example
80:scholar
3169:
3155:
744:where
328:. Let
284:where
126:, an
82:
75:
68:
61:
53:
461:) as
87:JSTOR
73:books
3167:ISBN
3153:ISBN
2851:norm
2752:Rtol
2742:Atol
2111:and
1874:<
1781:The
288:and
162:and
122:and
59:news
2881:tol
2862:err
2760:max
2726:tol
2692:err
2489:err
2064:to
1878:tol
1816:tol
1809:0.9
1789:0.9
1750:0.3
1684:tol
1669:max
1661:min
1649:0.9
1615:tol
118:In
42:by
3182::
2961::
2423:.
2145::
2107:,
2103:,
2088:.
2009:.
796:+1
777:.
446:nh
444:+
440:=
311:,
3066:q
3047:)
3044:1
3041:+
3038:q
3035:(
3031:/
3027:1
3023:)
3017:n
3013:E
3008:/
3004:1
3001:(
2996:1
2990:n
2986:h
2982:=
2977:n
2973:h
2947:n
2943:h
2920:n
2916:E
2892:)
2887:n
2874:/
2868:n
2856:(
2846:=
2841:n
2837:E
2815:)
2811:|
2805:1
2799:n
2795:y
2790:|
2786:,
2782:|
2776:n
2772:y
2767:|
2763:(
2747:+
2737:=
2732:n
2698:n
2667:)
2664:)
2659:n
2655:h
2651:,
2646:n
2642:y
2638:,
2633:n
2629:t
2625:(
2616:)
2611:n
2607:h
2603:,
2598:n
2594:y
2590:,
2585:n
2581:t
2577:(
2562:(
2559:h
2556:=
2551:1
2548:+
2545:n
2541:y
2532:1
2529:+
2526:n
2516:y
2509:=
2506:)
2503:h
2500:(
2495:n
2411:)
2406:1
2400:n
2396:h
2392:,
2387:n
2383:y
2379:,
2374:n
2370:t
2366:(
2363:g
2360:=
2355:n
2351:h
2328:n
2324:h
2298:n
2294:h
2290:+
2285:n
2281:t
2277:=
2272:1
2269:+
2266:n
2262:t
2240:)
2235:n
2231:h
2227:,
2222:n
2218:y
2214:,
2209:n
2205:t
2201:(
2193:n
2189:h
2185:+
2180:n
2176:y
2172:=
2167:1
2164:+
2161:n
2157:y
2131:n
2127:y
2072:b
2052:a
2032:a
2026:b
2023:=
2020:h
1984:)
1981:1
1978:(
1973:1
1970:+
1967:n
1959:+
1954:)
1951:1
1948:(
1943:1
1940:+
1937:n
1933:y
1929:=
1924:)
1921:2
1918:(
1913:1
1910:+
1907:n
1903:y
1870:|
1864:)
1861:1
1858:(
1853:1
1850:+
1847:n
1838:|
1765:)
1761:2
1758:,
1754:)
1747:,
1742:2
1738:/
1734:1
1729:)
1722:|
1717:)
1714:1
1711:(
1706:1
1703:+
1700:n
1692:|
1688:2
1679:(
1673:(
1665:(
1655:h
1643:h
1630:h
1594:h
1566:)
1563:1
1560:(
1555:1
1552:+
1549:n
1541:=
1536:)
1533:0
1530:(
1525:1
1522:+
1519:n
1515:y
1506:)
1503:1
1500:(
1495:1
1492:+
1489:n
1485:y
1461:)
1458:t
1455:(
1450:)
1447:3
1444:(
1440:y
1419:]
1416:h
1413:+
1410:t
1407:,
1404:t
1401:[
1381:c
1358:)
1355:h
1352:+
1349:t
1346:(
1343:y
1340:=
1335:)
1332:1
1329:(
1324:1
1321:+
1318:n
1310:+
1305:)
1302:1
1299:(
1294:1
1291:+
1288:n
1284:y
1260:)
1257:0
1254:(
1249:1
1246:+
1243:n
1233:2
1230:1
1225:=
1220:2
1216:h
1212:c
1207:2
1204:1
1199:=
1194:2
1189:)
1184:2
1181:h
1176:(
1171:c
1168:2
1165:=
1160:2
1155:)
1150:2
1147:h
1142:(
1137:c
1134:+
1129:2
1124:)
1119:2
1116:h
1111:(
1106:c
1103:=
1098:)
1095:1
1092:(
1087:1
1084:+
1081:n
1055:)
1048:2
1045:1
1040:+
1037:n
1033:y
1029:,
1022:2
1019:1
1014:+
1011:n
1007:t
1003:(
1000:f
995:2
992:h
987:+
980:2
977:1
972:+
969:n
965:y
961:=
956:)
953:1
950:(
945:1
942:+
939:n
935:y
913:)
908:n
904:y
900:,
895:n
891:t
887:(
884:f
879:2
876:h
871:+
866:n
862:y
858:=
851:2
848:1
843:+
840:n
836:y
812:)
809:1
806:(
794:n
790:t
788:(
786:y
782:c
765:)
762:0
759:(
746:c
727:2
723:h
719:c
716:=
711:)
708:0
705:(
700:1
697:+
694:n
676:f
653:)
650:0
647:(
642:1
639:+
636:n
632:y
625:)
620:1
617:+
614:n
610:t
606:(
603:y
600:=
595:)
592:0
589:(
584:1
581:+
578:n
547:)
542:n
538:y
534:,
529:n
525:t
521:(
518:f
515:h
512:+
507:n
503:y
499:=
494:)
491:0
488:(
483:1
480:+
477:n
473:y
458:n
454:t
452:(
450:y
442:a
437:n
433:t
429:t
424:n
420:t
383:)
380:b
377:(
374:y
371:=
365:+
360:b
356:y
344:b
342:y
338:b
334:b
332:(
330:y
326:b
322:t
317:a
313:y
309:a
305:y
303:,
301:t
299:(
297:f
290:f
286:y
267:a
263:y
259:=
256:)
253:a
250:(
247:y
243:,
240:)
237:)
234:t
231:(
228:y
225:,
222:t
219:(
216:f
213:=
210:)
207:t
204:(
197:y
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
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