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1560:{\displaystyle {\begin{aligned}{\tilde {y}}_{i+1}&=y_{i}+hf(t_{i},y_{i}),\\{\hat {y}}_{i+1}&=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{i+1}){\bigr )},\\y_{i+1}&=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},y_{i})+f(t_{i+1},{\hat {y}}_{i+1}){\bigr )}.\end{aligned}}}
841:
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1117:{\displaystyle {\begin{aligned}{\tilde {y}}_{i+1}&=y_{i}+hf(t_{i},{\tilde {y}}_{i}),\\y_{i+1}&=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},{\tilde {y}}_{i})+f(t_{i+1},{\tilde {y}}_{i+1}){\bigr )}.\end{aligned}}}
556:
There are different variants of a predictor–corrector method, depending on how often the corrector method is applied. The
Predict–Evaluate–Correct–Evaluate (PECE) mode refers to the variant in the above example:
32:
The initial, "prediction" step, starts from a function fitted to the function-values and derivative-values at a preceding set of points to extrapolate ("anticipate") this function's value at a subsequent, new
368:
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Additionally, the corrector step can be repeated in the hope that this achieves an even better approximation to the true solution. If the corrector method is run twice, this yields the PECECE mode:
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821:{\displaystyle {\begin{aligned}{\tilde {y}}_{i+1}&=y_{i}+hf(t_{i},y_{i}),\\y_{i+1}&=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{i+1}){\bigr )}.\end{aligned}}}
28:
designed to integrate ordinary differential equations – to find an unknown function that satisfies a given differential equation. All such algorithms proceed in two steps:
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538:{\displaystyle y_{i+1}=y_{i}+{\tfrac {1}{2}}h{\bigl (}f(t_{i},y_{i})+f(t_{i+1},{\tilde {y}}_{i+1}){\bigr )}.}
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only once per step by using the method in
Predict–Evaluate–Correct (PEC) mode:
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Next, the corrector step: improve the initial guess using trapezoidal rule,
25:
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The next, "corrector" step refines the initial approximation by using the
1670:"Section 17.6. Multistep, Multivalue, and Predictor-Corrector Methods"
65:
for the predictor step and an implicit method for the corrector step.
1570:
The PECEC mode has one fewer function evaluation than PECECE mode.
1726:
1668:
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007).
59:
numerical solution of ordinary differential equations (ODEs)
202:
First, the predictor step: starting from the current value
363:{\displaystyle {\tilde {y}}_{i+1}=y_{i}+hf(t_{i},y_{i}).}
1676:(3rd ed.). New York: Cambridge University Press.
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Numerical methods for ordinary differential equations
1647:
Numerical
Methods for Ordinary Differential Equations
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to interpolate that unknown function's value at the
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1674:Numerical Recipes: The Art of Scientific Computing
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168:
61:, a predictor–corrector method typically uses an
169:{\displaystyle y'=f(t,y),\quad y(t_{0})=y_{0},}
69:Example: Euler method with the trapezoidal rule
73:A simple predictor–corrector method (known as
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1602:Mehrotra predictor–corrector method
229:, calculate an initial guess value
88:Consider the differential equation
264:{\displaystyle {\tilde {y}}_{i+1}}
14:
1881:Backward differentiation formula
1587:Backward differentiation formula
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1703:"Predictor-Corrector Methods"
81:(an explicit method) and the
179:and denote the step size by
1866:List of Runge–Kutta methods
1717:Predictor–corrector methods
22:predictor–corrector methods
1949:
1719:for differential equations
40:value of the function and
1871:Linear multistep method
1876:General linear methods
1856:Exponential integrator
1607:Numerical continuation
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552:PEC mode and PECE mode
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271:via the Euler method,
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85:(an implicit method).
1907:Symplectic integrator
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222:{\displaystyle y_{i}}
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57:When considering the
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1848:Higher-order methods
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1795:Second-order methods
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1861:Runge–Kutta methods
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1762:First-order methods
1933:Numerical analysis
1818:Beeman's algorithm
1803:Verlet integration
1700:Weisstein, Eric W.
1592:Beeman's algorithm
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18:numerical analysis
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1785:Exponential Euler
1683:978-0-521-88068-8
1660:978-0-471-96758-3
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192:{\displaystyle h}
48:subsequent point.
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63:explicit method
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1775:Backward Euler
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391:
388:
384:
376:
375:
374:
357:
349:
345:
341:
336:
332:
325:
322:
319:
314:
310:
306:
301:
298:
295:
285:
274:
273:
272:
256:
253:
250:
240:
214:
210:
200:
186:
163:
158:
154:
150:
142:
138:
131:
127:
121:
118:
115:
109:
106:
102:
99:
91:
90:
89:
86:
84:
80:
76:
75:Heun's method
68:
66:
64:
60:
52:
47:
43:
39:
35:
31:
30:
29:
27:
23:
19:
1770:Euler method
1706:
1673:
1649:, New York:
1646:
1626:Butcher 2003
1621:
1574:
1572:
1569:
1126:
832:
830:
555:
547:
372:
201:
178:
87:
79:Euler method
72:
56:
45:
41:
37:
21:
15:
1928:Algorithms
1922:Categories
1636:References
26:algorithms
1708:MathWorld
1523:^
1363:~
1236:^
1149:~
1080:~
1030:~
924:~
857:~
784:~
579:~
505:~
289:~
244:~
38:predicted
1645:(2003),
1581:See also
103:′
1886:Yoshida
1900:Theory
1680:
1657:
33:point.
1613:Notes
1678:ISBN
1655:ISBN
46:same
16:In
1924::
1705:.
1672:.
1653:,
199:.
20:,
1746:e
1739:t
1732:v
1711:.
1686:.
1664:.
1575:k
1551:.
1546:)
1541:)
1536:1
1533:+
1530:i
1520:y
1513:,
1508:1
1505:+
1502:i
1498:t
1494:(
1491:f
1488:+
1485:)
1480:i
1476:y
1472:,
1467:i
1463:t
1459:(
1456:f
1451:(
1446:h
1440:2
1437:1
1431:+
1426:i
1422:y
1418:=
1409:1
1406:+
1403:i
1399:y
1391:,
1386:)
1381:)
1376:1
1373:+
1370:i
1360:y
1353:,
1348:1
1345:+
1342:i
1338:t
1334:(
1331:f
1328:+
1325:)
1320:i
1316:y
1312:,
1307:i
1303:t
1299:(
1296:f
1291:(
1286:h
1280:2
1277:1
1271:+
1266:i
1262:y
1258:=
1249:1
1246:+
1243:i
1233:y
1222:,
1219:)
1214:i
1210:y
1206:,
1201:i
1197:t
1193:(
1190:f
1187:h
1184:+
1179:i
1175:y
1171:=
1162:1
1159:+
1156:i
1146:y
1108:.
1103:)
1098:)
1093:1
1090:+
1087:i
1077:y
1070:,
1065:1
1062:+
1059:i
1055:t
1051:(
1048:f
1045:+
1042:)
1037:i
1027:y
1020:,
1015:i
1011:t
1007:(
1004:f
999:(
994:h
988:2
985:1
979:+
974:i
970:y
966:=
957:1
954:+
951:i
947:y
939:,
936:)
931:i
921:y
914:,
909:i
905:t
901:(
898:f
895:h
892:+
887:i
883:y
879:=
870:1
867:+
864:i
854:y
833:f
812:.
807:)
802:)
797:1
794:+
791:i
781:y
774:,
769:1
766:+
763:i
759:t
755:(
752:f
749:+
746:)
741:i
737:y
733:,
728:i
724:t
720:(
717:f
712:(
707:h
701:2
698:1
692:+
687:i
683:y
679:=
670:1
667:+
664:i
660:y
652:,
649:)
644:i
640:y
636:,
631:i
627:t
623:(
620:f
617:h
614:+
609:i
605:y
601:=
592:1
589:+
586:i
576:y
533:.
528:)
523:)
518:1
515:+
512:i
502:y
495:,
490:1
487:+
484:i
480:t
476:(
473:f
470:+
467:)
462:i
458:y
454:,
449:i
445:t
441:(
438:f
433:(
428:h
422:2
419:1
413:+
408:i
404:y
400:=
395:1
392:+
389:i
385:y
358:.
355:)
350:i
346:y
342:,
337:i
333:t
329:(
326:f
323:h
320:+
315:i
311:y
307:=
302:1
299:+
296:i
286:y
257:1
254:+
251:i
241:y
215:i
211:y
187:h
164:,
159:0
155:y
151:=
148:)
143:0
139:t
135:(
132:y
128:,
125:)
122:y
119:,
116:t
113:(
110:f
107:=
100:y
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