3869:
3887:
3945:
5943:
2571:
2041:
The definitions of Q-convergence rates have a shortcoming in that they do not naturally capture the convergence behavior of sequences that do not converge with an asymptotically constant rate with every step, such as the staggered geometric progression below that gets closer to its limit only every
2927:
3481:
converges R-linearly to 0 with rate 1/2, but it does not converge Q-linearly. The defining Q-linear convergence limits do not exist for this sequence because one subsequence of error quotients (the sequence of quotients taken from odd steps) has a different limit than another subsequence (the
3427:
1010:
methods. Ideally the solution of a discretized problem will converge to the solution of the continuous problem as the grid size goes to zero, and the speed of that convergence is one important characterization of the efficiency of a discretization method. However, the terminology of "rates of
5622:
3723:
3863:
959:
Where greater methodological precision is required, these rates and orders of convergence are known specifically as the rates and orders of Q-convergence, short for quotient-convergence, since the limit in question is a quotient of error terms. The rate of convergence
2723:
2372:
5490:
4857:
954:
1432:â 1.618. This is technically called Q-convergence, short for quotient-convergence, and the rates and orders are called rates and orders of Q-convergence in certain technical settings where alternative rate definitions are more appropriate; see
2756:
3582:
6077:
2032:
3284:
1268:
3731:
1848:
1695:
2198:
5938:{\displaystyle f(x_{n})=f(nh)=y_{0}\exp(-\kappa nh)=y_{0}\left^{n}=y_{0}\left(1-h\kappa +{\frac {h^{2}\kappa ^{2}}{2}}+....\right)^{n}=y_{0}\left(1-nh\kappa +{\frac {n^{2}h^{2}\kappa ^{2}}{2}}+...\right).}
4536:
3277:
5324:
5197:
4957:
4624:
2589:
6358:. It is said to converge exponentially using the convention for discretization methods. However, it only converges linearly (that is, with order 1) using the convention for iterative methods.
999:
for calculating numerical approximations. If the order of convergence is higher, then typically fewer iterations are necessary to yield a useful approximation. Strictly speaking, however, the
1017:
is a collection of techniques for improving the rate of convergence of a series discretization and possibly its order of convergence, also. These accelerations are commonly accomplished with
4161:
3054:
5561:
5270:
5068:
3530:
6880:
6741:. These methods in general (and in particular Aitken's method) do not increase the order of convergence, and are useful only if initially the convergence is not faster than linear: if
2339:
2303:
6647:
2566:{\displaystyle q\approx {\frac {\log \left|\displaystyle {\frac {x_{k+1}-x_{k}}{x_{k}-x_{k-1}}}\right|}{\log \left|\displaystyle {\frac {x_{k}-x_{k-1}}{x_{k-1}-x_{k-2}}}\right|}}.}
3180:
4754:
4252:
2349:
A practical method to calculate the order of convergence for a sequence generated by a fixed point iteration is to calculate the following sequence, which converges to the order
1372:
848:
2267:
2231:
2133:
6697:
6516:
6419:
2045:
In such cases, a closely related but more technical definition of rate of convergence called R-convergence is more appropriate; The "R-" prefix stands for "root." A sequence
6597:
4746:
4719:
4692:
4665:
3456:
6269:
5617:
1308:
6348:
5315:
5113:
3574:
6149:
5588:
4439:
3090:
2996:
1912:
6805:
6772:
6305:
6216:
6113:
4403:
2962:
2076:
1883:
1068:
817:
768:
6466:
4327:
4073:
1585:
1398:
6543:
4188:
1161:
1154:
978:
840:
6175:
3210:
3120:
2749:
1552:
1521:
1466:
1334:
7386:
3479:
5004:
4980:
4367:
4347:
4292:
4272:
4003:
3983:
2367:
2096:
1904:
1742:
1719:
1492:
1418:
1130:
1110:
1088:
788:
5953:
1748:
1592:
2922:{\displaystyle \lim _{k\to \infty }{\frac {\left|1/2^{k+1}-0\right|}{\left|1/2^{k}-0\right|}}=\lim _{k\to \infty }{\frac {2^{k}}{2^{k+1}}}={\frac {1}{2}}.}
3422:{\textstyle (x_{k})=1,1,{\frac {1}{4}},{\frac {1}{4}},{\frac {1}{16}},{\frac {1}{16}},\ldots ,1/4^{\left\lfloor {\frac {k}{2}}\right\rfloor },\ldots ,}
2233:
converges Q-linearly to zero; analogous definitions hold for R-superlinear convergence, R-sublinear convergence, R-quadratic convergence, and so on.
7781:
7786:
3718:{\displaystyle (x_{k})={\frac {1}{2}},{\frac {1}{4}},{\frac {1}{16}},{\frac {1}{256}},{\frac {1}{65,\!536}},\ldots ,{\frac {1}{2^{2^{k}}}},\ldots }
708:
7649:
449:
7776:
2341:
provides a lower bound on the rate and order of R-convergence and the greatest lower bound gives the exact rate and order of R-convergence.
7379:
7093:
2138:
7436:
192:
4080:
3858:{\displaystyle (d_{k})=1,{\frac {1}{2}},{\frac {1}{3}},{\frac {1}{4}},{\frac {1}{5}},{\frac {1}{6}},\ldots ,{\frac {1}{k+1}},\ldots }
2236:
In order to define the rates and orders of R-convergence, one uses the rate and order of Q-convergence of am error-bounding sequence
4455:
4349:, inversely proportional to the number of grid points, i.e. the number of points in the sequence required to reach a given value of
7549:
3218:
323:
7372:
5125:
4868:
7593:
364:
7487:
7354:
7153:
6944:
254:
4544:
2718:{\textstyle (x_{k})=1,{\frac {1}{2}},{\frac {1}{4}},{\frac {1}{8}},{\frac {1}{16}},{\frac {1}{32}},\ldots ,1/{2^{k}},\dots }
7771:
4009:
3904:
701:
272:
156:
7681:
7696:
7477:
737:
are quantities that represent how quickly the sequence approaches its limit. In the most common applications, a sequence
575:
229:
17:
7482:
7446:
7322:
7302:
7284:
7264:
6738:
4027:
3926:
250:
202:
7544:
7456:
6271:
was introduced above. This sequence converges with order 1 according to the convention for discretization methods.
5485:{\displaystyle y_{n}=y_{0}(1-h\kappa )^{n}=y_{0}\left(1-nh\kappa +n(n-1){\frac {h^{2}\kappa ^{2}}{2}}+....\right).}
4086:
3725:
converges to zero Q-superlinearly. In fact, it is quadratically convergent with a quadratic convergence rate of 1.
3004:
318:
237:
212:
6882:. On the other hand, if the convergence is already of order ≥ 2, Aitken's method will bring no improvement.
7512:
5497:
4005:. Section should be modified for consistency and include an explanation of alternative (equivalent?) definitions.
694:
5208:
5024:
3485:
7827:
7426:
6810:
6807:
that still converges linearly (except for pathologically designed special cases), but faster in the sense that
3908:
629:
182:
7583:
2751:. Plugging the sequence into the definition of Q-linear convergence (i.e., order of convergence 1) shows that
354:
7598:
7441:
7431:
369:
197:
187:
4852:{\displaystyle q\approx {\frac {\log(e_{\text{new}}/e_{\text{old}})}{\log(h_{\text{new}}/h_{\text{old}})}},}
949:{\displaystyle \lim _{n\rightarrow \infty }{\frac {\left|x_{n+1}-L\right|}{\left|x_{n}-L\right|^{q}}}=\mu .}
7451:
7421:
7223:
644:
495:
398:
285:
207:
6729:
into a second one that converges more quickly to the same limit. Such techniques are in general known as "
7893:
7686:
7556:
7517:
4193:
3430:
3128:
2308:
2272:
1339:
330:
245:
2239:
2203:
2105:
6989:
6369:
4640:
A practical method to estimate the order of convergence for a discretization method is pick step sizes
535:
403:
7522:
7807:
7726:
3960:
506:
484:
7862:
7832:
4724:
4697:
4670:
4643:
1425:
649:
7721:
6602:
6221:
5593:
3435:
1275:
500:
408:
6652:
6471:
6310:
7731:
7701:
7691:
7588:
6726:
4983:
3897:
3482:
sequence of quotients taken from even steps). Generally, for any staggered geometric progression
1018:
580:
570:
562:
518:
359:
7195:
6961:
6737:
of approximating the limits of the transformed sequences. One example of series acceleration is
5278:
5076:
7017:
6552:
5007:
3535:
393:
6118:
5570:
4408:
3059:
2967:
7847:
7842:
7736:
7660:
7639:
7507:
7502:
7497:
7492:
7416:
7395:
6777:
6744:
6700:
6277:
6188:
6085:
4375:
2934:
2584:
2048:
1855:
1040:
796:
740:
639:
624:
513:
456:
438:
277:
33:
6436:
4297:
4043:
1564:
1377:
7741:
7665:
7634:
7170:
6521:
6426:
4634:
4166:
4076:
3212:. The same holds also for geometric progressions and geometric series parameterized by any
525:
461:
434:
1139:
1003:
of a sequence does not give conclusive information about any finite part of the sequence.
963:
825:
8:
7751:
7538:
6900:
6730:
6720:
6154:
4040:
A similar situation exists for discretization methods designed to approximate a function
3185:
3095:
2728:
2575:
For numerical approximation of an exact value through a numerical method of order q see.
1531:
1500:
1445:
1313:
1014:
1000:
734:
557:
542:
307:
146:
113:
104:
7065:
6072:{\displaystyle e=|y_{n}-f(x_{n})|={\frac {nh^{2}\kappa ^{2}}{2}}={\mathcal {O}}(h^{2}),}
3461:
7872:
7711:
7706:
7629:
7461:
7252:
7116:
7040:
6734:
4989:
4965:
4352:
4332:
4277:
4257:
3988:
3968:
2352:
2081:
1889:
1727:
1704:
1477:
1403:
1115:
1095:
1073:
773:
722:
552:
547:
430:
7350:
7318:
7298:
7280:
7260:
7149:
7044:
6940:
6725:
Many methods exist to increase the rate of convergence of a given sequence, i.e., to
6422:
3955:
609:
388:
123:
4862:
which comes from writing the truncation error, at the old and new grid spacings, as
664:
7857:
7837:
7108:
7032:
5564:
5318:
3532:, the sequence will not converge Q-linearly but will converge R-linearly with rate
3123:
1011:
convergence" in this case is different from the terminology for iterative methods.
996:
674:
659:
3965:
There appears to be a mixture of defining convergence with regards to grid points
995:
In practice, the rate and order of convergence provide useful insights when using
7817:
7746:
7346:
7145:
7094:"Acceleration of convergence of a family of logarithmically convergent sequences"
3576:
this example highlights why the "R" in R-linear convergence is short for "root."
2027:{\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-x_{k}|}{|x_{k}-x_{k-1}|}}=1.}
614:
530:
57:
7822:
669:
7898:
7812:
7619:
4627:
3213:
1007:
634:
619:
425:
413:
132:
7887:
2305:
could have been chosen that would converge with a faster rate and order; any
1421:
1263:{\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|^{q}}}=\mu }
7290:
7867:
7802:
7716:
5116:
3872:
Linear, linear, superlinear (quadratic), and sublinear rates of convergence
1429:
984:. Note that this terminology is not standardized and some authors will use
654:
604:
490:
118:
4748:. The order of convergence is then approximated by the following formula:
7624:
7364:
62:
7852:
7120:
7036:
3911: in this section. Unsourced material may be challenged and removed.
1843:{\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=1.}
1690:{\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=0,}
679:
7614:
6429:
is of particular interest. Assuming that the relevant derivatives of
4083:(see example below). The discretization method generates a sequence
420:
141:
84:
74:
7112:
3886:
7654:
3868:
2193:{\textstyle |x_{k}-L|\leq \varepsilon _{k}\quad {\text{for all }}k}
1035:
4986:(GTE), in that it represents a sum of errors accumulated over all
95:
90:
79:
3876:
4531:{\displaystyle |y_{n}-f(x_{n})|<Ch^{q}{\text{ for all }}n.}
6649:, then one has at least quadratic convergence, and so on. If
6518:, one has at least linear convergence for any starting value
6433:
are continuous, one can (easily) show that for a fixed point
4294:. The important parameter here for the convergence speed to
3458:
that gives the largest integer that is less than or equal to
3272:{\displaystyle a\in \mathbb {C} ,r\in \mathbb {C} ,|r|<1.}
6703:
and no starting value will produce a sequence converging to
4633:
This is the relevant definition when discussing methods for
4637:
or the solution of ordinary differential equations (ODEs).
1024:
5192:{\displaystyle {\frac {y_{n+1}-y_{n}}{h}}=-\kappa y_{n},}
4952:{\displaystyle e=|y_{n}-f(x_{n})|={\mathcal {O}}(h^{q}).}
6939:(1st ed.). New York, NY: Springer. pp. 28â29.
6350:, which was also introduced above, converges with order
6361:
4619:{\displaystyle |y_{n}-f(x_{n})|={\mathcal {O}}(h^{q})}
4274:
between successive values of the independent variable
3538:
3438:
3287:
3131:
2592:
2141:
6813:
6780:
6747:
6655:
6605:
6555:
6524:
6474:
6439:
6372:
6313:
6280:
6224:
6191:
6157:
6121:
6088:
5956:
5625:
5596:
5573:
5500:
5327:
5281:
5211:
5128:
5079:
5027:
4992:
4968:
4871:
4757:
4727:
4700:
4673:
4646:
4547:
4458:
4411:
4378:
4355:
4335:
4300:
4280:
4260:
4196:
4169:
4089:
4046:
3991:
3971:
3865:
converges to zero Q-sublinearly and logarithmically.
3734:
3585:
3488:
3464:
3221:
3188:
3098:
3062:
3007:
2970:
2937:
2759:
2731:
2479:
2395:
2375:
2355:
2311:
2275:
2242:
2206:
2108:
2084:
2051:
1915:
1892:
1858:
1751:
1730:
1707:
1595:
1567:
1534:
1503:
1480:
1448:
1406:
1380:
1342:
1316:
1278:
1164:
1142:
1118:
1098:
1076:
1043:
966:
851:
828:
799:
776:
743:
5013:
1722:(i.e., faster than linearly). A sequence is said to
4075:, which might be an integral being approximated by
7196:"Computing and Estimating the Rate of Convergence"
6962:"Computing and Estimating the Rate of Convergence"
6874:
6799:
6766:
6691:
6641:
6591:
6537:
6510:
6460:
6413:
6342:
6299:
6263:
6210:
6169:
6143:
6107:
6071:
5937:
5611:
5582:
5555:
5484:
5309:
5264:
5191:
5107:
5062:
4998:
4974:
4951:
4851:
4740:
4713:
4686:
4659:
4618:
4530:
4433:
4397:
4361:
4341:
4321:
4286:
4266:
4246:
4182:
4155:
4067:
3997:
3977:
3857:
3717:
3568:
3524:
3473:
3450:
3421:
3271:
3204:
3174:
3114:
3084:
3048:
2990:
2956:
2921:
2743:
2717:
2565:
2361:
2333:
2297:
2269:chosen such that no other error-bounding sequence
2261:
2225:
2192:
2127:
2090:
2070:
2026:
1898:
1877:
1842:
1736:
1713:
1689:
1579:
1546:
1515:
1486:
1460:
1412:
1392:
1366:
1328:
1302:
1262:
1148:
1124:
1104:
1082:
1062:
972:
948:
834:
811:
782:
762:
7782:List of nonlinear ordinary differential equations
5117:Forward Euler scheme for numerical discretization
3669:
7885:
7787:List of nonlinear partial differential equations
7313:Richard L. Burden and J. Douglas Faires (2001),
7133:
6814:
2964:converges Q-linearly with a convergence rate of
2858:
2761:
1917:
1753:
1745:(i.e., slower than linearly) if it converges and
1597:
1166:
853:
7257:Numerical analysis: a mathematical introduction
7025:Journal of Optimization Theory and Applications
1029:
27:Speed of convergence of a mathematical sequence
7777:List of linear ordinary differential equations
6714:
7380:
7340:
7139:
6934:
4156:{\displaystyle {y_{0},y_{1},y_{2},y_{3},...}}
4081:solution of an ordinary differential equation
3049:{\displaystyle a\in \mathbb {R} ,r\in (-1,1)}
1587:for a sequence or for any sequence such that
702:
7162:
5018:Consider the ordinary differential equation
3877:Convergence speed for discretization methods
3514:
3500:
3445:
3439:
7341:Nocedal, Jorge; Wright, Stephen J. (2006).
7140:Nocedal, Jorge; Wright, Stephen J. (2006).
6935:Nocedal, Jorge; Wright, Stephen J. (1999).
5556:{\displaystyle y=f(x)=y_{0}\exp(-\kappa x)}
2102:if there exists an error-bounding sequence
1907:if the sequence converges sublinearly and
7394:
7387:
7373:
5265:{\displaystyle y_{n+1}=y_{n}(1-h\kappa ).}
5063:{\displaystyle {\frac {dy}{dx}}=-\kappa y}
3525:{\displaystyle (ar^{\lfloor k/m\rfloor })}
709:
695:
7168:
6875:{\displaystyle \lim(a_{n}-L)/(x_{n}-L)=0}
4028:Learn how and when to remove this message
3927:Learn how and when to remove this message
3243:
3229:
3015:
7091:
6774:converges linearly, one gets a sequence
6707:(unless one directly jumps to the point
5317:, this sequence is as follows, from the
5115:. We can solve this equation using the
3867:
7224:"Verifying Numerical Convergence Rates"
7018:"On Q-Order and R-Order of Convergence"
6180:
1025:Convergence speed for iterative methods
14:
7886:
7295:An introduction to numerical analysis,
5947:In this case, the truncation error is
3122:and the sequence of partial sums of a
7368:
7015:
7772:List of named differential equations
7277:Numerical analysis: an introduction,
6930:
6928:
6926:
6924:
6922:
6920:
6898:
6362:Recurrent sequences and fixed points
4405:is said to converge to the sequence
3938:
3909:adding citations to reliable sources
3880:
3281:The staggered geometric progression
3175:{\textstyle (\sum _{n=0}^{k}ar^{n})}
157:List of named differential equations
7697:Method of undetermined coefficients
7478:Dependent and independent variables
7271:The extended definition is used in
7221:
7193:
6987:
6959:
4694:and calculate the resulting errors
4247:{\displaystyle y_{j-1},y_{j-2},...}
2344:
2334:{\displaystyle (\varepsilon '_{k})}
2298:{\displaystyle (\varepsilon '_{k})}
1367:{\displaystyle \mu \in (0,\infty )}
230:Dependent and independent variables
24:
7345:(2nd ed.). Berlin, New York:
7144:(2nd ed.). Berlin, New York:
7063:
6045:
5494:The exact solution to this ODE is
4925:
4595:
3182:also converges linearly with rate
2868:
2771:
2262:{\displaystyle (\varepsilon _{k})}
2226:{\displaystyle (\varepsilon _{k})}
2128:{\displaystyle (\varepsilon _{k})}
1927:
1763:
1607:
1433:
1358:
1176:
863:
25:
7910:
7248:The simple definition is used in
6917:
6414:{\displaystyle x_{n+1}:=f(x_{n})}
5563:, corresponding to the following
5014:Example of discretization methods
7594:Carathéodory's existence theorem
7309:The Big O definition is used in
6366:The case of recurrent sequences
3943:
3885:
2036:
1424:, when converging to a regular,
1420:be an integer. For example, the
365:Carathéodory's existence theorem
7215:
7187:
6901:"Order and rate of convergence"
5010:(LTE) over just one iteration.
3896:needs additional citations for
2181:
7085:
7057:
7009:
6981:
6953:
6892:
6863:
6844:
6836:
6817:
6794:
6781:
6761:
6748:
6739:Aitken's delta-squared process
6679:
6675:
6669:
6657:
6629:
6625:
6619:
6607:
6579:
6575:
6569:
6557:
6498:
6494:
6488:
6476:
6449:
6443:
6425:and in the context of various
6408:
6395:
6294:
6281:
6258:
6246:
6205:
6192:
6138:
6125:
6102:
6089:
6063:
6050:
6001:
5997:
5984:
5964:
5737:
5725:
5697:
5682:
5660:
5651:
5642:
5629:
5550:
5538:
5516:
5510:
5427:
5415:
5367:
5351:
5291:
5285:
5256:
5241:
5089:
5083:
4943:
4930:
4916:
4912:
4899:
4879:
4840:
4812:
4801:
4773:
4741:{\displaystyle e_{\text{old}}}
4714:{\displaystyle e_{\text{new}}}
4687:{\displaystyle h_{\text{old}}}
4660:{\displaystyle h_{\text{new}}}
4613:
4600:
4586:
4582:
4569:
4549:
4497:
4493:
4480:
4460:
4428:
4415:
4392:
4379:
4316:
4310:
4062:
4056:
3748:
3735:
3599:
3586:
3551:
3543:
3519:
3489:
3451:{\textstyle \lfloor x\rfloor }
3301:
3288:
3259:
3251:
3198:
3190:
3169:
3132:
3108:
3100:
3079:
3063:
3043:
3028:
2951:
2938:
2865:
2768:
2606:
2593:
2328:
2312:
2292:
2276:
2256:
2243:
2220:
2207:
2164:
2143:
2122:
2109:
2065:
2052:
2011:
1977:
1970:
1936:
1924:
1872:
1859:
1827:
1806:
1799:
1772:
1760:
1671:
1650:
1643:
1616:
1604:
1361:
1349:
1297:
1285:
1241:
1219:
1212:
1185:
1173:
1057:
1044:
1006:Similar concepts are used for
860:
757:
744:
452: / Integral solutions
13:
1:
7243:
6885:
6642:{\displaystyle |f''(p)|<1}
6264:{\displaystyle d_{k}=1/(k+1)}
5612:{\displaystyle h\kappa \ll 1}
5202:which generates the sequence
3092:converges linearly with rate
1886:converges logarithmically to
1303:{\displaystyle \mu \in (0,1)}
7422:Notation for differentiation
7297:Cambridge University Press.
7092:Van Tuyl, Andrew H. (1994).
6733:" methods. These reduce the
6727:transform one given sequence
6692:{\displaystyle |f'(p)|>1}
6511:{\displaystyle |f'(p)|<1}
6343:{\displaystyle a_{k}=2^{-k}}
5006:iterations, as opposed to a
4254:along with the grid spacing
2100:converge at least R-linearly
1472:and the sequence is said to
1030:Convergence rate definitions
496:Exponential response formula
242:Coupled / Decoupled
7:
7518:Exact differential equation
7259:, Clarendon Press, Oxford.
6715:Acceleration of convergence
4445:if there exists a constant
4372:In this case, the sequence
3963:. The specific problem is:
2578:
1400:. It is not necessary that
1272:for some positive constant
10:
7915:
7101:Mathematics of Computation
6718:
5310:{\displaystyle y(0)=y_{0}}
5108:{\displaystyle y(0)=y_{0}}
3569:{\textstyle {\sqrt{|r|}};}
3056:, a geometric progression
1701:converge superlinearly to
1090:. The sequence is said to
7828:JĂłzef Maria Hoene-WroĆski
7808:Gottfried Wilhelm Leibniz
7795:
7764:
7674:
7607:
7599:CauchyâKowalevski theorem
7576:
7569:
7531:
7470:
7409:
7402:
7293:and David Mayers (2003),
7169:Bockelman, Brian (2005).
6899:Ruye, Wang (2015-02-12).
6592:{\displaystyle |f'(p)|=0}
4982:is, more specifically, a
1699:that sequence is said to
982:asymptotic error constant
630:JĂłzef Maria Hoene-WroĆski
576:Undetermined coefficients
485:Method of characteristics
370:CauchyâKowalevski theorem
7722:Finite difference method
7317:(7th ed.), Brooks/Cole.
7275:Walter Gautschi (1997),
6151:with a convergence rate
6144:{\displaystyle f(x_{n})}
5583:{\displaystyle h\kappa }
4434:{\displaystyle f(x_{n})}
4163:, where each successive
3085:{\displaystyle (ar^{k})}
3001:More generally, for any
2991:{\displaystyle \mu =1/2}
1724:converge sublinearly to
1070:converges to the number
1019:sequence transformations
988:where this article uses
355:PicardâLindelöf theorem
349:Existence and uniqueness
7702:Variation of parameters
7692:Separation of variables
7589:Peano existence theorem
7584:PicardâLindelöf theorem
7471:Attributes of variables
6800:{\displaystyle (a_{n})}
6767:{\displaystyle (x_{n})}
6300:{\displaystyle (a_{k})}
6211:{\displaystyle (d_{k})}
6108:{\displaystyle (y_{n})}
5073:with initial condition
5008:local truncation error
4984:global truncation error
4398:{\displaystyle (y_{n})}
3728:Finally, the sequence
2957:{\displaystyle (t_{k})}
2071:{\displaystyle (x_{k})}
1878:{\displaystyle (x_{k})}
1439:Convergence with order
1063:{\displaystyle (x_{k})}
812:{\displaystyle q\geq 1}
763:{\displaystyle (x_{n})}
581:Variation of parameters
571:Separation of variables
360:Peano existence theorem
7863:Carl David Tolmé Runge
7437:Differential-algebraic
7396:Differential equations
7343:Numerical Optimization
7171:"Rates of Convergence"
7142:Numerical Optimization
7073:University of Arkansas
7066:"Order of Convergence"
6937:Numerical Optimization
6876:
6801:
6768:
6693:
6643:
6593:
6545:sufficiently close to
6539:
6512:
6462:
6461:{\displaystyle f(p)=p}
6415:
6344:
6301:
6265:
6212:
6171:
6145:
6109:
6073:
5939:
5613:
5584:
5557:
5486:
5311:
5266:
5193:
5109:
5064:
5000:
4976:
4953:
4853:
4742:
4715:
4688:
4661:
4620:
4532:
4435:
4399:
4363:
4343:
4323:
4322:{\displaystyle y=f(x)}
4288:
4268:
4248:
4184:
4157:
4069:
4068:{\displaystyle y=f(x)}
3999:
3979:
3873:
3859:
3719:
3570:
3526:
3475:
3452:
3423:
3273:
3206:
3176:
3155:
3116:
3086:
3050:
2992:
2958:
2923:
2745:
2719:
2567:
2363:
2335:
2299:
2263:
2227:
2194:
2129:
2092:
2072:
2028:
1900:
1879:
1844:
1738:
1715:
1691:
1581:
1580:{\displaystyle q>1}
1548:
1525:quadratic convergence.
1517:
1488:
1462:
1414:
1394:
1393:{\displaystyle q>1}
1368:
1330:
1304:
1264:
1150:
1126:
1106:
1084:
1064:
974:
950:
836:
813:
784:
764:
650:Carl David Tolmé Runge
193:Differential-algebraic
34:Differential equations
7848:Augustin-Louis Cauchy
7843:Joseph-Louis Lagrange
7737:Finite element method
7727:CrankâNicolson method
7661:Numerical integration
7640:Exponential stability
7532:Relation to processes
7417:Differential operator
7222:Senning, Jonathan R.
7194:Senning, Jonathan R.
7016:Porta, F. A. (1989).
6990:"Rate of Convergence"
6960:Senning, Jonathan R.
6877:
6802:
6769:
6701:repulsive fixed point
6694:
6644:
6594:
6540:
6538:{\displaystyle x_{0}}
6513:
6463:
6416:
6345:
6302:
6266:
6213:
6172:
6146:
6110:
6074:
5940:
5614:
5585:
5558:
5487:
5312:
5267:
5194:
5110:
5065:
5001:
4977:
4954:
4854:
4743:
4716:
4689:
4662:
4621:
4533:
4436:
4400:
4364:
4344:
4324:
4289:
4269:
4249:
4185:
4183:{\displaystyle y_{j}}
4158:
4070:
4000:
3980:
3871:
3860:
3720:
3571:
3527:
3476:
3453:
3424:
3274:
3207:
3177:
3135:
3117:
3087:
3051:
2993:
2959:
2924:
2746:
2720:
2585:geometric progression
2568:
2364:
2336:
2300:
2264:
2228:
2195:
2130:
2093:
2073:
2029:
1901:
1880:
1845:
1739:
1716:
1692:
1582:
1549:
1518:
1489:
1474:converge linearly to
1463:
1415:
1395:
1369:
1331:
1305:
1265:
1151:
1127:
1107:
1085:
1065:
975:
951:
837:
814:
785:
765:
640:Augustin-Louis Cauchy
625:Joseph-Louis Lagrange
457:Numerical integration
439:Exponential stability
302:Relation to processes
7742:Finite volume method
7666:Dirac delta function
7635:Asymptotic stability
7577:Existence/uniqueness
7442:Integro-differential
7349:. pp. 619+620.
7279:BirkhÀuser, Boston.
6811:
6778:
6745:
6653:
6603:
6553:
6522:
6472:
6437:
6427:fixed-point theorems
6370:
6311:
6278:
6222:
6189:
6181:Examples (continued)
6155:
6119:
6086:
5954:
5623:
5594:
5571:
5498:
5325:
5279:
5209:
5126:
5077:
5025:
4990:
4966:
4869:
4755:
4725:
4698:
4671:
4644:
4635:numerical quadrature
4545:
4456:
4409:
4376:
4353:
4333:
4329:is the grid spacing
4298:
4278:
4258:
4194:
4167:
4087:
4077:numerical quadrature
4044:
4010:improve this section
3989:
3969:
3959:to meet Knowledge's
3905:improve this article
3732:
3583:
3536:
3486:
3462:
3436:
3285:
3219:
3186:
3129:
3096:
3060:
3005:
2968:
2935:
2757:
2729:
2590:
2373:
2353:
2309:
2273:
2240:
2204:
2139:
2106:
2082:
2049:
1913:
1890:
1856:
1749:
1728:
1705:
1593:
1565:
1532:
1501:
1478:
1446:
1434:§ R-convergence
1404:
1378:
1340:
1314:
1276:
1162:
1149:{\displaystyle \mu }
1140:
1116:
1096:
1092:converge with order
1074:
1041:
973:{\displaystyle \mu }
964:
849:
835:{\displaystyle \mu }
826:
797:
792:order of convergence
774:
741:
727:order of convergence
462:Dirac delta function
198:Integro-differential
7752:Perturbation theory
7732:RungeâKutta methods
7712:Integral transforms
7645:Rate of convergence
7541:(discrete analogue)
6735:computational costs
6731:series acceleration
6721:Series acceleration
6170:{\displaystyle q=2}
4541:This is written as
4519: for all
3985:and with step size
3205:{\displaystyle |r|}
3115:{\displaystyle |r|}
2744:{\displaystyle L=0}
2327:
2291:
1547:{\displaystyle q=3}
1516:{\displaystyle q=2}
1461:{\displaystyle q=1}
1329:{\displaystyle q=1}
1135:rate of convergence
1015:Series acceleration
1001:asymptotic behavior
980:is also called the
821:rate of convergence
735:convergent sequence
731:rate of convergence
558:Perturbation theory
553:Integral transforms
444:Rate of convergence
310:(discrete analogue)
147:Population dynamics
114:Continuum mechanics
105:Applied mathematics
7894:Numerical analysis
7873:Sofya Kovalevskaya
7707:Integrating factor
7630:Lyapunov stability
7550:Stochastic partial
7315:Numerical Analysis
7253:Michelle Schatzman
7037:10.1007/BF00939805
6988:Hundley, Douglas.
6872:
6797:
6764:
6689:
6639:
6589:
6535:
6508:
6458:
6411:
6340:
6297:
6261:
6208:
6167:
6141:
6105:
6069:
5935:
5609:
5580:
5553:
5482:
5307:
5262:
5189:
5105:
5060:
4996:
4972:
4949:
4849:
4738:
4711:
4684:
4657:
4616:
4528:
4431:
4395:
4359:
4339:
4319:
4284:
4264:
4244:
4180:
4153:
4065:
3995:
3975:
3874:
3855:
3715:
3566:
3522:
3474:{\displaystyle x,}
3471:
3448:
3419:
3269:
3202:
3172:
3112:
3082:
3046:
2988:
2954:
2919:
2872:
2775:
2741:
2715:
2563:
2552:
2462:
2359:
2331:
2315:
2295:
2279:
2259:
2223:
2190:
2125:
2088:
2078:that converges to
2068:
2024:
1931:
1896:
1875:
1840:
1767:
1734:
1711:
1687:
1611:
1577:
1561:In addition, when
1556:cubic convergence.
1544:
1513:
1484:
1470:linear convergence
1458:
1428:, has an order of
1410:
1390:
1364:
1326:
1300:
1260:
1180:
1146:
1122:
1102:
1080:
1060:
970:
946:
867:
832:
809:
780:
770:that converges to
760:
723:numerical analysis
548:Integrating factor
389:Initial conditions
324:Stochastic partial
18:Linear convergence
7881:
7880:
7760:
7759:
7565:
7564:
7356:978-0-387-30303-1
7155:978-0-387-30303-1
6946:978-0-387-98793-4
6699:, then one has a
6423:dynamical systems
6354:for every number
6038:
5913:
5812:
5457:
5165:
5046:
4999:{\displaystyle n}
4975:{\displaystyle e}
4844:
4837:
4822:
4798:
4783:
4735:
4708:
4681:
4654:
4520:
4362:{\displaystyle x}
4342:{\displaystyle h}
4287:{\displaystyle x}
4267:{\displaystyle h}
4190:is a function of
4038:
4037:
4030:
3998:{\displaystyle h}
3978:{\displaystyle n}
3961:quality standards
3952:This section may
3937:
3936:
3929:
3847:
3820:
3807:
3794:
3781:
3768:
3707:
3674:
3652:
3639:
3626:
3613:
3561:
3402:
3366:
3353:
3340:
3327:
2914:
2901:
2857:
2852:
2760:
2678:
2665:
2652:
2639:
2626:
2558:
2550:
2460:
2362:{\displaystyle q}
2185:
2091:{\displaystyle L}
2016:
1916:
1899:{\displaystyle L}
1832:
1752:
1737:{\displaystyle L}
1714:{\displaystyle L}
1676:
1596:
1487:{\displaystyle L}
1413:{\displaystyle q}
1252:
1165:
1125:{\displaystyle L}
1105:{\displaystyle q}
1083:{\displaystyle L}
1034:Suppose that the
997:iterative methods
935:
852:
783:{\displaystyle L}
719:
718:
610:Gottfried Leibniz
501:Finite difference
293:
292:
154:
153:
124:Dynamical systems
16:(Redirected from
7906:
7858:Phyllis Nicolson
7838:Rudolf Lipschitz
7675:Solution methods
7650:Series solutions
7574:
7573:
7407:
7406:
7389:
7382:
7375:
7366:
7365:
7360:
7237:
7236:
7234:
7233:
7228:
7219:
7213:
7212:
7210:
7209:
7200:
7191:
7185:
7184:
7182:
7181:
7166:
7160:
7159:
7137:
7131:
7130:
7128:
7127:
7107:(207): 229â246.
7098:
7089:
7083:
7082:
7080:
7079:
7070:
7061:
7055:
7054:
7052:
7051:
7022:
7013:
7007:
7006:
7004:
7003:
6994:
6985:
6979:
6978:
6976:
6975:
6966:
6957:
6951:
6950:
6932:
6915:
6914:
6912:
6911:
6896:
6881:
6879:
6878:
6873:
6856:
6855:
6843:
6829:
6828:
6806:
6804:
6803:
6798:
6793:
6792:
6773:
6771:
6770:
6765:
6760:
6759:
6698:
6696:
6695:
6690:
6682:
6668:
6660:
6648:
6646:
6645:
6640:
6632:
6618:
6610:
6598:
6596:
6595:
6590:
6582:
6568:
6560:
6544:
6542:
6541:
6536:
6534:
6533:
6517:
6515:
6514:
6509:
6501:
6487:
6479:
6467:
6465:
6464:
6459:
6421:which occurs in
6420:
6418:
6417:
6412:
6407:
6406:
6388:
6387:
6349:
6347:
6346:
6341:
6339:
6338:
6323:
6322:
6306:
6304:
6303:
6298:
6293:
6292:
6270:
6268:
6267:
6262:
6245:
6234:
6233:
6217:
6215:
6214:
6209:
6204:
6203:
6176:
6174:
6173:
6168:
6150:
6148:
6147:
6142:
6137:
6136:
6114:
6112:
6111:
6106:
6101:
6100:
6078:
6076:
6075:
6070:
6062:
6061:
6049:
6048:
6039:
6034:
6033:
6032:
6023:
6022:
6009:
6004:
5996:
5995:
5977:
5976:
5967:
5944:
5942:
5941:
5936:
5931:
5927:
5914:
5909:
5908:
5907:
5898:
5897:
5888:
5887:
5877:
5852:
5851:
5839:
5838:
5833:
5829:
5813:
5808:
5807:
5806:
5797:
5796:
5786:
5763:
5762:
5750:
5749:
5744:
5740:
5712:
5711:
5675:
5674:
5641:
5640:
5618:
5616:
5615:
5610:
5589:
5587:
5586:
5581:
5565:Taylor expansion
5562:
5560:
5559:
5554:
5531:
5530:
5491:
5489:
5488:
5483:
5478:
5474:
5458:
5453:
5452:
5451:
5442:
5441:
5431:
5388:
5387:
5375:
5374:
5350:
5349:
5337:
5336:
5319:Binomial theorem
5316:
5314:
5313:
5308:
5306:
5305:
5271:
5269:
5268:
5263:
5240:
5239:
5227:
5226:
5198:
5196:
5195:
5190:
5185:
5184:
5166:
5161:
5160:
5159:
5147:
5146:
5130:
5114:
5112:
5111:
5106:
5104:
5103:
5069:
5067:
5066:
5061:
5047:
5045:
5037:
5029:
5005:
5003:
5002:
4997:
4981:
4979:
4978:
4973:
4958:
4956:
4955:
4950:
4942:
4941:
4929:
4928:
4919:
4911:
4910:
4892:
4891:
4882:
4858:
4856:
4855:
4850:
4845:
4843:
4839:
4838:
4835:
4829:
4824:
4823:
4820:
4804:
4800:
4799:
4796:
4790:
4785:
4784:
4781:
4765:
4747:
4745:
4744:
4739:
4737:
4736:
4733:
4720:
4718:
4717:
4712:
4710:
4709:
4706:
4693:
4691:
4690:
4685:
4683:
4682:
4679:
4666:
4664:
4663:
4658:
4656:
4655:
4652:
4625:
4623:
4622:
4617:
4612:
4611:
4599:
4598:
4589:
4581:
4580:
4562:
4561:
4552:
4537:
4535:
4534:
4529:
4521:
4518:
4516:
4515:
4500:
4492:
4491:
4473:
4472:
4463:
4440:
4438:
4437:
4432:
4427:
4426:
4404:
4402:
4401:
4396:
4391:
4390:
4368:
4366:
4365:
4360:
4348:
4346:
4345:
4340:
4328:
4326:
4325:
4320:
4293:
4291:
4290:
4285:
4273:
4271:
4270:
4265:
4253:
4251:
4250:
4245:
4231:
4230:
4212:
4211:
4189:
4187:
4186:
4181:
4179:
4178:
4162:
4160:
4159:
4154:
4152:
4139:
4138:
4126:
4125:
4113:
4112:
4100:
4099:
4074:
4072:
4071:
4066:
4033:
4026:
4022:
4019:
4013:
4004:
4002:
4001:
3996:
3984:
3982:
3981:
3976:
3947:
3946:
3939:
3932:
3925:
3921:
3918:
3912:
3889:
3881:
3864:
3862:
3861:
3856:
3848:
3846:
3832:
3821:
3813:
3808:
3800:
3795:
3787:
3782:
3774:
3769:
3761:
3747:
3746:
3724:
3722:
3721:
3716:
3708:
3706:
3705:
3704:
3703:
3686:
3675:
3673:
3658:
3653:
3645:
3640:
3632:
3627:
3619:
3614:
3606:
3598:
3597:
3575:
3573:
3572:
3567:
3562:
3560:
3555:
3554:
3546:
3540:
3531:
3529:
3528:
3523:
3518:
3517:
3510:
3480:
3478:
3477:
3472:
3457:
3455:
3454:
3449:
3428:
3426:
3425:
3420:
3409:
3408:
3407:
3403:
3395:
3384:
3367:
3359:
3354:
3346:
3341:
3333:
3328:
3320:
3300:
3299:
3278:
3276:
3275:
3270:
3262:
3254:
3246:
3232:
3211:
3209:
3208:
3203:
3201:
3193:
3181:
3179:
3178:
3173:
3168:
3167:
3154:
3149:
3124:geometric series
3121:
3119:
3118:
3113:
3111:
3103:
3091:
3089:
3088:
3083:
3078:
3077:
3055:
3053:
3052:
3047:
3018:
2997:
2995:
2994:
2989:
2984:
2963:
2961:
2960:
2955:
2950:
2949:
2928:
2926:
2925:
2920:
2915:
2907:
2902:
2900:
2899:
2884:
2883:
2874:
2871:
2853:
2851:
2847:
2840:
2839:
2830:
2817:
2813:
2806:
2805:
2790:
2777:
2774:
2750:
2748:
2747:
2742:
2724:
2722:
2721:
2716:
2708:
2707:
2706:
2696:
2679:
2671:
2666:
2658:
2653:
2645:
2640:
2632:
2627:
2619:
2605:
2604:
2572:
2570:
2569:
2564:
2559:
2557:
2556:
2551:
2549:
2548:
2547:
2529:
2528:
2512:
2511:
2510:
2492:
2491:
2481:
2467:
2466:
2461:
2459:
2458:
2457:
2439:
2438:
2428:
2427:
2426:
2414:
2413:
2397:
2383:
2368:
2366:
2365:
2360:
2345:Order estimation
2340:
2338:
2337:
2332:
2323:
2304:
2302:
2301:
2296:
2287:
2268:
2266:
2265:
2260:
2255:
2254:
2232:
2230:
2229:
2224:
2219:
2218:
2199:
2197:
2196:
2191:
2186:
2183:
2180:
2179:
2167:
2156:
2155:
2146:
2134:
2132:
2131:
2126:
2121:
2120:
2097:
2095:
2094:
2089:
2077:
2075:
2074:
2069:
2064:
2063:
2033:
2031:
2030:
2025:
2017:
2015:
2014:
2009:
2008:
1990:
1989:
1980:
1974:
1973:
1968:
1967:
1955:
1954:
1939:
1933:
1930:
1905:
1903:
1902:
1897:
1884:
1882:
1881:
1876:
1871:
1870:
1849:
1847:
1846:
1841:
1833:
1831:
1830:
1819:
1818:
1809:
1803:
1802:
1791:
1790:
1775:
1769:
1766:
1743:
1741:
1740:
1735:
1720:
1718:
1717:
1712:
1696:
1694:
1693:
1688:
1677:
1675:
1674:
1663:
1662:
1653:
1647:
1646:
1635:
1634:
1619:
1613:
1610:
1586:
1584:
1583:
1578:
1553:
1551:
1550:
1545:
1522:
1520:
1519:
1514:
1493:
1491:
1490:
1485:
1467:
1465:
1464:
1459:
1419:
1417:
1416:
1411:
1399:
1397:
1396:
1391:
1373:
1371:
1370:
1365:
1335:
1333:
1332:
1327:
1309:
1307:
1306:
1301:
1269:
1267:
1266:
1261:
1253:
1251:
1250:
1249:
1244:
1232:
1231:
1222:
1216:
1215:
1204:
1203:
1188:
1182:
1179:
1155:
1153:
1152:
1147:
1131:
1129:
1128:
1123:
1111:
1109:
1108:
1103:
1089:
1087:
1086:
1081:
1069:
1067:
1066:
1061:
1056:
1055:
979:
977:
976:
971:
955:
953:
952:
947:
936:
934:
933:
928:
924:
917:
916:
901:
897:
890:
889:
869:
866:
841:
839:
838:
833:
818:
816:
815:
810:
790:is said to have
789:
787:
786:
781:
769:
767:
766:
761:
756:
755:
711:
704:
697:
675:Phyllis Nicolson
660:Rudolf Lipschitz
543:Green's function
519:Infinite element
510:
475:Solution methods
453:
311:
222:By variable type
176:
175:
58:Natural sciences
51:
50:
30:
29:
21:
7914:
7913:
7909:
7908:
7907:
7905:
7904:
7903:
7884:
7883:
7882:
7877:
7818:Jacob Bernoulli
7791:
7756:
7747:Galerkin method
7670:
7608:Solution topics
7603:
7561:
7527:
7466:
7398:
7393:
7357:
7347:Springer-Verlag
7246:
7241:
7240:
7231:
7229:
7226:
7220:
7216:
7207:
7205:
7198:
7192:
7188:
7179:
7177:
7167:
7163:
7156:
7146:Springer-Verlag
7138:
7134:
7125:
7123:
7113:10.2307/2153571
7096:
7090:
7086:
7077:
7075:
7068:
7062:
7058:
7049:
7047:
7020:
7014:
7010:
7001:
6999:
6997:Whitman College
6992:
6986:
6982:
6973:
6971:
6964:
6958:
6954:
6947:
6933:
6918:
6909:
6907:
6897:
6893:
6888:
6851:
6847:
6839:
6824:
6820:
6812:
6809:
6808:
6788:
6784:
6779:
6776:
6775:
6755:
6751:
6746:
6743:
6742:
6723:
6717:
6678:
6661:
6656:
6654:
6651:
6650:
6628:
6611:
6606:
6604:
6601:
6600:
6578:
6561:
6556:
6554:
6551:
6550:
6529:
6525:
6523:
6520:
6519:
6497:
6480:
6475:
6473:
6470:
6469:
6438:
6435:
6434:
6402:
6398:
6377:
6373:
6371:
6368:
6367:
6364:
6331:
6327:
6318:
6314:
6312:
6309:
6308:
6288:
6284:
6279:
6276:
6275:
6241:
6229:
6225:
6223:
6220:
6219:
6199:
6195:
6190:
6187:
6186:
6183:
6156:
6153:
6152:
6132:
6128:
6120:
6117:
6116:
6096:
6092:
6087:
6084:
6083:
6057:
6053:
6044:
6043:
6028:
6024:
6018:
6014:
6010:
6008:
6000:
5991:
5987:
5972:
5968:
5963:
5955:
5952:
5951:
5903:
5899:
5893:
5889:
5883:
5879:
5878:
5876:
5857:
5853:
5847:
5843:
5834:
5802:
5798:
5792:
5788:
5787:
5785:
5769:
5765:
5764:
5758:
5754:
5745:
5718:
5714:
5713:
5707:
5703:
5670:
5666:
5636:
5632:
5624:
5621:
5620:
5595:
5592:
5591:
5572:
5569:
5568:
5526:
5522:
5499:
5496:
5495:
5447:
5443:
5437:
5433:
5432:
5430:
5393:
5389:
5383:
5379:
5370:
5366:
5345:
5341:
5332:
5328:
5326:
5323:
5322:
5301:
5297:
5280:
5277:
5276:
5235:
5231:
5216:
5212:
5210:
5207:
5206:
5180:
5176:
5155:
5151:
5136:
5132:
5131:
5129:
5127:
5124:
5123:
5099:
5095:
5078:
5075:
5074:
5038:
5030:
5028:
5026:
5023:
5022:
5016:
4991:
4988:
4987:
4967:
4964:
4963:
4937:
4933:
4924:
4923:
4915:
4906:
4902:
4887:
4883:
4878:
4870:
4867:
4866:
4834:
4830:
4825:
4819:
4815:
4805:
4795:
4791:
4786:
4780:
4776:
4766:
4764:
4756:
4753:
4752:
4732:
4728:
4726:
4723:
4722:
4705:
4701:
4699:
4696:
4695:
4678:
4674:
4672:
4669:
4668:
4651:
4647:
4645:
4642:
4641:
4607:
4603:
4594:
4593:
4585:
4576:
4572:
4557:
4553:
4548:
4546:
4543:
4542:
4517:
4511:
4507:
4496:
4487:
4483:
4468:
4464:
4459:
4457:
4454:
4453:
4422:
4418:
4410:
4407:
4406:
4386:
4382:
4377:
4374:
4373:
4354:
4351:
4350:
4334:
4331:
4330:
4299:
4296:
4295:
4279:
4276:
4275:
4259:
4256:
4255:
4220:
4216:
4201:
4197:
4195:
4192:
4191:
4174:
4170:
4168:
4165:
4164:
4134:
4130:
4121:
4117:
4108:
4104:
4095:
4091:
4090:
4088:
4085:
4084:
4045:
4042:
4041:
4034:
4023:
4017:
4014:
4007:
3990:
3987:
3986:
3970:
3967:
3966:
3948:
3944:
3933:
3922:
3916:
3913:
3902:
3890:
3879:
3836:
3831:
3812:
3799:
3786:
3773:
3760:
3742:
3738:
3733:
3730:
3729:
3699:
3695:
3694:
3690:
3685:
3662:
3657:
3644:
3631:
3618:
3605:
3593:
3589:
3584:
3581:
3580:
3556:
3550:
3542:
3541:
3539:
3537:
3534:
3533:
3506:
3499:
3495:
3487:
3484:
3483:
3463:
3460:
3459:
3437:
3434:
3433:
3394:
3390:
3389:
3385:
3380:
3358:
3345:
3332:
3319:
3295:
3291:
3286:
3283:
3282:
3258:
3250:
3242:
3228:
3220:
3217:
3216:
3214:complex numbers
3197:
3189:
3187:
3184:
3183:
3163:
3159:
3150:
3139:
3130:
3127:
3126:
3107:
3099:
3097:
3094:
3093:
3073:
3069:
3061:
3058:
3057:
3014:
3006:
3003:
3002:
2980:
2969:
2966:
2965:
2945:
2941:
2936:
2933:
2932:
2906:
2889:
2885:
2879:
2875:
2873:
2861:
2835:
2831:
2826:
2822:
2818:
2795:
2791:
2786:
2782:
2778:
2776:
2764:
2758:
2755:
2754:
2730:
2727:
2726:
2702:
2698:
2697:
2692:
2670:
2657:
2644:
2631:
2618:
2600:
2596:
2591:
2588:
2587:
2581:
2537:
2533:
2518:
2514:
2513:
2500:
2496:
2487:
2483:
2482:
2480:
2475:
2468:
2447:
2443:
2434:
2430:
2429:
2422:
2418:
2403:
2399:
2398:
2396:
2391:
2384:
2382:
2374:
2371:
2370:
2354:
2351:
2350:
2347:
2319:
2310:
2307:
2306:
2283:
2274:
2271:
2270:
2250:
2246:
2241:
2238:
2237:
2214:
2210:
2205:
2202:
2201:
2182:
2175:
2171:
2163:
2151:
2147:
2142:
2140:
2137:
2136:
2116:
2112:
2107:
2104:
2103:
2083:
2080:
2079:
2059:
2055:
2050:
2047:
2046:
2039:
2010:
1998:
1994:
1985:
1981:
1976:
1975:
1969:
1963:
1959:
1944:
1940:
1935:
1934:
1932:
1920:
1914:
1911:
1910:
1891:
1888:
1887:
1866:
1862:
1857:
1854:
1853:
1826:
1814:
1810:
1805:
1804:
1798:
1780:
1776:
1771:
1770:
1768:
1756:
1750:
1747:
1746:
1729:
1726:
1725:
1706:
1703:
1702:
1670:
1658:
1654:
1649:
1648:
1642:
1624:
1620:
1615:
1614:
1612:
1600:
1594:
1591:
1590:
1566:
1563:
1562:
1533:
1530:
1529:
1502:
1499:
1498:
1479:
1476:
1475:
1447:
1444:
1443:
1405:
1402:
1401:
1379:
1376:
1375:
1341:
1338:
1337:
1315:
1312:
1311:
1277:
1274:
1273:
1245:
1240:
1239:
1227:
1223:
1218:
1217:
1211:
1193:
1189:
1184:
1183:
1181:
1169:
1163:
1160:
1159:
1141:
1138:
1137:
1117:
1114:
1113:
1097:
1094:
1093:
1075:
1072:
1071:
1051:
1047:
1042:
1039:
1038:
1032:
1027:
965:
962:
961:
929:
912:
908:
907:
903:
902:
879:
875:
874:
870:
868:
856:
850:
847:
846:
827:
824:
823:
798:
795:
794:
775:
772:
771:
751:
747:
742:
739:
738:
715:
686:
685:
684:
615:Jacob Bernoulli
599:
586:
585:
567:
536:PetrovâGalerkin
504:
489:
476:
468:
467:
466:
448:
394:Boundary values
383:
375:
374:
350:
337:
336:
335:
309:
303:
295:
294:
282:
259:
217:
173:
160:
159:
155:
133:Social sciences
89:
67:
48:
28:
23:
22:
15:
12:
11:
5:
7912:
7902:
7901:
7896:
7879:
7878:
7876:
7875:
7870:
7865:
7860:
7855:
7850:
7845:
7840:
7835:
7833:Ernst Lindelöf
7830:
7825:
7820:
7815:
7813:Leonhard Euler
7810:
7805:
7799:
7797:
7796:Mathematicians
7793:
7792:
7790:
7789:
7784:
7779:
7774:
7768:
7766:
7762:
7761:
7758:
7757:
7755:
7754:
7749:
7744:
7739:
7734:
7729:
7724:
7719:
7714:
7709:
7704:
7699:
7694:
7689:
7684:
7678:
7676:
7672:
7671:
7669:
7668:
7663:
7658:
7652:
7647:
7642:
7637:
7632:
7627:
7622:
7620:Phase portrait
7617:
7611:
7609:
7605:
7604:
7602:
7601:
7596:
7591:
7586:
7580:
7578:
7571:
7567:
7566:
7563:
7562:
7560:
7559:
7554:
7553:
7552:
7542:
7535:
7533:
7529:
7528:
7526:
7525:
7523:On jet bundles
7520:
7515:
7510:
7505:
7500:
7495:
7490:
7488:Nonhomogeneous
7485:
7480:
7474:
7472:
7468:
7467:
7465:
7464:
7459:
7454:
7449:
7444:
7439:
7434:
7429:
7424:
7419:
7413:
7411:
7404:
7403:Classification
7400:
7399:
7392:
7391:
7384:
7377:
7369:
7363:
7362:
7355:
7326:
7325:
7307:
7306:
7288:
7269:
7268:
7245:
7242:
7239:
7238:
7214:
7186:
7161:
7154:
7132:
7084:
7064:Arnold, Mark.
7056:
7031:(3): 415â431.
7008:
6980:
6952:
6945:
6916:
6890:
6889:
6887:
6884:
6871:
6868:
6865:
6862:
6859:
6854:
6850:
6846:
6842:
6838:
6835:
6832:
6827:
6823:
6819:
6816:
6796:
6791:
6787:
6783:
6763:
6758:
6754:
6750:
6719:Main article:
6716:
6713:
6688:
6685:
6681:
6677:
6674:
6671:
6667:
6664:
6659:
6638:
6635:
6631:
6627:
6624:
6621:
6617:
6614:
6609:
6588:
6585:
6581:
6577:
6574:
6571:
6567:
6564:
6559:
6532:
6528:
6507:
6504:
6500:
6496:
6493:
6490:
6486:
6483:
6478:
6457:
6454:
6451:
6448:
6445:
6442:
6410:
6405:
6401:
6397:
6394:
6391:
6386:
6383:
6380:
6376:
6363:
6360:
6337:
6334:
6330:
6326:
6321:
6317:
6296:
6291:
6287:
6283:
6260:
6257:
6254:
6251:
6248:
6244:
6240:
6237:
6232:
6228:
6207:
6202:
6198:
6194:
6182:
6179:
6166:
6163:
6160:
6140:
6135:
6131:
6127:
6124:
6104:
6099:
6095:
6091:
6080:
6079:
6068:
6065:
6060:
6056:
6052:
6047:
6042:
6037:
6031:
6027:
6021:
6017:
6013:
6007:
6003:
5999:
5994:
5990:
5986:
5983:
5980:
5975:
5971:
5966:
5962:
5959:
5934:
5930:
5926:
5923:
5920:
5917:
5912:
5906:
5902:
5896:
5892:
5886:
5882:
5875:
5872:
5869:
5866:
5863:
5860:
5856:
5850:
5846:
5842:
5837:
5832:
5828:
5825:
5822:
5819:
5816:
5811:
5805:
5801:
5795:
5791:
5784:
5781:
5778:
5775:
5772:
5768:
5761:
5757:
5753:
5748:
5743:
5739:
5736:
5733:
5730:
5727:
5724:
5721:
5717:
5710:
5706:
5702:
5699:
5696:
5693:
5690:
5687:
5684:
5681:
5678:
5673:
5669:
5665:
5662:
5659:
5656:
5653:
5650:
5647:
5644:
5639:
5635:
5631:
5628:
5608:
5605:
5602:
5599:
5579:
5576:
5552:
5549:
5546:
5543:
5540:
5537:
5534:
5529:
5525:
5521:
5518:
5515:
5512:
5509:
5506:
5503:
5481:
5477:
5473:
5470:
5467:
5464:
5461:
5456:
5450:
5446:
5440:
5436:
5429:
5426:
5423:
5420:
5417:
5414:
5411:
5408:
5405:
5402:
5399:
5396:
5392:
5386:
5382:
5378:
5373:
5369:
5365:
5362:
5359:
5356:
5353:
5348:
5344:
5340:
5335:
5331:
5304:
5300:
5296:
5293:
5290:
5287:
5284:
5273:
5272:
5261:
5258:
5255:
5252:
5249:
5246:
5243:
5238:
5234:
5230:
5225:
5222:
5219:
5215:
5200:
5199:
5188:
5183:
5179:
5175:
5172:
5169:
5164:
5158:
5154:
5150:
5145:
5142:
5139:
5135:
5102:
5098:
5094:
5091:
5088:
5085:
5082:
5071:
5070:
5059:
5056:
5053:
5050:
5044:
5041:
5036:
5033:
5015:
5012:
4995:
4971:
4960:
4959:
4948:
4945:
4940:
4936:
4932:
4927:
4922:
4918:
4914:
4909:
4905:
4901:
4898:
4895:
4890:
4886:
4881:
4877:
4874:
4860:
4859:
4848:
4842:
4833:
4828:
4818:
4814:
4811:
4808:
4803:
4794:
4789:
4779:
4775:
4772:
4769:
4763:
4760:
4731:
4704:
4677:
4650:
4628:big O notation
4615:
4610:
4606:
4602:
4597:
4592:
4588:
4584:
4579:
4575:
4571:
4568:
4565:
4560:
4556:
4551:
4539:
4538:
4527:
4524:
4514:
4510:
4506:
4503:
4499:
4495:
4490:
4486:
4482:
4479:
4476:
4471:
4467:
4462:
4430:
4425:
4421:
4417:
4414:
4394:
4389:
4385:
4381:
4358:
4338:
4318:
4315:
4312:
4309:
4306:
4303:
4283:
4263:
4243:
4240:
4237:
4234:
4229:
4226:
4223:
4219:
4215:
4210:
4207:
4204:
4200:
4177:
4173:
4151:
4148:
4145:
4142:
4137:
4133:
4129:
4124:
4120:
4116:
4111:
4107:
4103:
4098:
4094:
4064:
4061:
4058:
4055:
4052:
4049:
4036:
4035:
3994:
3974:
3951:
3949:
3942:
3935:
3934:
3893:
3891:
3884:
3878:
3875:
3854:
3851:
3845:
3842:
3839:
3835:
3830:
3827:
3824:
3819:
3816:
3811:
3806:
3803:
3798:
3793:
3790:
3785:
3780:
3777:
3772:
3767:
3764:
3759:
3756:
3753:
3750:
3745:
3741:
3737:
3714:
3711:
3702:
3698:
3693:
3689:
3684:
3681:
3678:
3672:
3668:
3665:
3661:
3656:
3651:
3648:
3643:
3638:
3635:
3630:
3625:
3622:
3617:
3612:
3609:
3604:
3601:
3596:
3592:
3588:
3579:The sequence
3565:
3559:
3553:
3549:
3545:
3521:
3516:
3513:
3509:
3505:
3502:
3498:
3494:
3491:
3470:
3467:
3447:
3444:
3441:
3431:floor function
3418:
3415:
3412:
3406:
3401:
3398:
3393:
3388:
3383:
3379:
3376:
3373:
3370:
3365:
3362:
3357:
3352:
3349:
3344:
3339:
3336:
3331:
3326:
3323:
3318:
3315:
3312:
3309:
3306:
3303:
3298:
3294:
3290:
3268:
3265:
3261:
3257:
3253:
3249:
3245:
3241:
3238:
3235:
3231:
3227:
3224:
3200:
3196:
3192:
3171:
3166:
3162:
3158:
3153:
3148:
3145:
3142:
3138:
3134:
3110:
3106:
3102:
3081:
3076:
3072:
3068:
3065:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3017:
3013:
3010:
2987:
2983:
2979:
2976:
2973:
2953:
2948:
2944:
2940:
2918:
2913:
2910:
2905:
2898:
2895:
2892:
2888:
2882:
2878:
2870:
2867:
2864:
2860:
2856:
2850:
2846:
2843:
2838:
2834:
2829:
2825:
2821:
2816:
2812:
2809:
2804:
2801:
2798:
2794:
2789:
2785:
2781:
2773:
2770:
2767:
2763:
2740:
2737:
2734:
2714:
2711:
2705:
2701:
2695:
2691:
2688:
2685:
2682:
2677:
2674:
2669:
2664:
2661:
2656:
2651:
2648:
2643:
2638:
2635:
2630:
2625:
2622:
2617:
2614:
2611:
2608:
2603:
2599:
2595:
2580:
2577:
2562:
2555:
2546:
2543:
2540:
2536:
2532:
2527:
2524:
2521:
2517:
2509:
2506:
2503:
2499:
2495:
2490:
2486:
2478:
2474:
2471:
2465:
2456:
2453:
2450:
2446:
2442:
2437:
2433:
2425:
2421:
2417:
2412:
2409:
2406:
2402:
2394:
2390:
2387:
2381:
2378:
2358:
2346:
2343:
2330:
2326:
2322:
2318:
2314:
2294:
2290:
2286:
2282:
2278:
2258:
2253:
2249:
2245:
2222:
2217:
2213:
2209:
2189:
2178:
2174:
2170:
2166:
2162:
2159:
2154:
2150:
2145:
2124:
2119:
2115:
2111:
2087:
2067:
2062:
2058:
2054:
2038:
2035:
2023:
2020:
2013:
2007:
2004:
2001:
1997:
1993:
1988:
1984:
1979:
1972:
1966:
1962:
1958:
1953:
1950:
1947:
1943:
1938:
1929:
1926:
1923:
1919:
1895:
1874:
1869:
1865:
1861:
1839:
1836:
1829:
1825:
1822:
1817:
1813:
1808:
1801:
1797:
1794:
1789:
1786:
1783:
1779:
1774:
1765:
1762:
1759:
1755:
1733:
1710:
1686:
1683:
1680:
1673:
1669:
1666:
1661:
1657:
1652:
1645:
1641:
1638:
1633:
1630:
1627:
1623:
1618:
1609:
1606:
1603:
1599:
1576:
1573:
1570:
1559:
1558:
1543:
1540:
1537:
1527:
1512:
1509:
1506:
1496:
1483:
1457:
1454:
1451:
1409:
1389:
1386:
1383:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1325:
1322:
1319:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1259:
1256:
1248:
1243:
1238:
1235:
1230:
1226:
1221:
1214:
1210:
1207:
1202:
1199:
1196:
1192:
1187:
1178:
1175:
1172:
1168:
1145:
1121:
1101:
1079:
1059:
1054:
1050:
1046:
1031:
1028:
1026:
1023:
1008:discretization
969:
957:
956:
945:
942:
939:
932:
927:
923:
920:
915:
911:
906:
900:
896:
893:
888:
885:
882:
878:
873:
865:
862:
859:
855:
831:
808:
805:
802:
779:
759:
754:
750:
746:
717:
716:
714:
713:
706:
699:
691:
688:
687:
683:
682:
677:
672:
667:
665:Ernst Lindelöf
662:
657:
652:
647:
642:
637:
635:Joseph Fourier
632:
627:
622:
620:Leonhard Euler
617:
612:
607:
601:
600:
597:
596:
593:
592:
588:
587:
584:
583:
578:
573:
566:
565:
560:
555:
550:
545:
540:
539:
538:
528:
523:
522:
521:
514:Finite element
511:
507:CrankâNicolson
498:
493:
487:
482:
478:
477:
474:
473:
470:
469:
465:
464:
459:
454:
446:
441:
428:
426:Phase portrait
423:
418:
417:
416:
414:Cauchy problem
411:
406:
401:
391:
385:
384:
382:General topics
381:
380:
377:
376:
373:
372:
367:
362:
357:
351:
348:
347:
344:
343:
339:
338:
334:
333:
328:
327:
326:
315:
314:
313:
304:
301:
300:
297:
296:
291:
290:
289:
288:
281:
280:
275:
269:
266:
265:
261:
260:
258:
257:
255:Nonhomogeneous
248:
243:
240:
234:
233:
232:
224:
223:
219:
218:
216:
215:
210:
205:
200:
195:
190:
185:
179:
174:
171:
170:
167:
166:
165:Classification
162:
161:
152:
151:
150:
149:
144:
136:
135:
129:
128:
127:
126:
121:
116:
108:
107:
101:
100:
99:
98:
93:
87:
82:
77:
69:
68:
66:
65:
60:
54:
49:
46:
45:
42:
41:
37:
36:
26:
9:
6:
4:
3:
2:
7911:
7900:
7897:
7895:
7892:
7891:
7889:
7874:
7871:
7869:
7866:
7864:
7861:
7859:
7856:
7854:
7851:
7849:
7846:
7844:
7841:
7839:
7836:
7834:
7831:
7829:
7826:
7824:
7821:
7819:
7816:
7814:
7811:
7809:
7806:
7804:
7801:
7800:
7798:
7794:
7788:
7785:
7783:
7780:
7778:
7775:
7773:
7770:
7769:
7767:
7763:
7753:
7750:
7748:
7745:
7743:
7740:
7738:
7735:
7733:
7730:
7728:
7725:
7723:
7720:
7718:
7715:
7713:
7710:
7708:
7705:
7703:
7700:
7698:
7695:
7693:
7690:
7688:
7685:
7683:
7680:
7679:
7677:
7673:
7667:
7664:
7662:
7659:
7656:
7653:
7651:
7648:
7646:
7643:
7641:
7638:
7636:
7633:
7631:
7628:
7626:
7623:
7621:
7618:
7616:
7613:
7612:
7610:
7606:
7600:
7597:
7595:
7592:
7590:
7587:
7585:
7582:
7581:
7579:
7575:
7572:
7568:
7558:
7555:
7551:
7548:
7547:
7546:
7543:
7540:
7537:
7536:
7534:
7530:
7524:
7521:
7519:
7516:
7514:
7511:
7509:
7506:
7504:
7501:
7499:
7496:
7494:
7491:
7489:
7486:
7484:
7481:
7479:
7476:
7475:
7473:
7469:
7463:
7460:
7458:
7455:
7453:
7450:
7448:
7445:
7443:
7440:
7438:
7435:
7433:
7430:
7428:
7425:
7423:
7420:
7418:
7415:
7414:
7412:
7408:
7405:
7401:
7397:
7390:
7385:
7383:
7378:
7376:
7371:
7370:
7367:
7358:
7352:
7348:
7344:
7339:
7338:
7337:
7335:
7331:
7324:
7323:0-534-38216-9
7320:
7316:
7312:
7311:
7310:
7304:
7303:0-521-00794-1
7300:
7296:
7292:
7289:
7286:
7285:0-8176-3895-4
7282:
7278:
7274:
7273:
7272:
7266:
7265:0-19-850279-6
7262:
7258:
7254:
7251:
7250:
7249:
7225:
7218:
7204:
7197:
7190:
7176:
7172:
7165:
7157:
7151:
7147:
7143:
7136:
7122:
7118:
7114:
7110:
7106:
7102:
7095:
7088:
7074:
7067:
7060:
7046:
7042:
7038:
7034:
7030:
7026:
7019:
7012:
6998:
6991:
6984:
6970:
6963:
6956:
6948:
6942:
6938:
6931:
6929:
6927:
6925:
6923:
6921:
6906:
6902:
6895:
6891:
6883:
6869:
6866:
6860:
6857:
6852:
6848:
6840:
6833:
6830:
6825:
6821:
6789:
6785:
6756:
6752:
6740:
6736:
6732:
6728:
6722:
6712:
6710:
6706:
6702:
6686:
6683:
6672:
6665:
6662:
6636:
6633:
6622:
6615:
6612:
6586:
6583:
6572:
6565:
6562:
6548:
6530:
6526:
6505:
6502:
6491:
6484:
6481:
6455:
6452:
6446:
6440:
6432:
6428:
6424:
6403:
6399:
6392:
6389:
6384:
6381:
6378:
6374:
6359:
6357:
6353:
6335:
6332:
6328:
6324:
6319:
6315:
6289:
6285:
6274:The sequence
6272:
6255:
6252:
6249:
6242:
6238:
6235:
6230:
6226:
6200:
6196:
6185:The sequence
6178:
6164:
6161:
6158:
6133:
6129:
6122:
6115:converges to
6097:
6093:
6066:
6058:
6054:
6040:
6035:
6029:
6025:
6019:
6015:
6011:
6005:
5992:
5988:
5981:
5978:
5973:
5969:
5960:
5957:
5950:
5949:
5948:
5945:
5932:
5928:
5924:
5921:
5918:
5915:
5910:
5904:
5900:
5894:
5890:
5884:
5880:
5873:
5870:
5867:
5864:
5861:
5858:
5854:
5848:
5844:
5840:
5835:
5830:
5826:
5823:
5820:
5817:
5814:
5809:
5803:
5799:
5793:
5789:
5782:
5779:
5776:
5773:
5770:
5766:
5759:
5755:
5751:
5746:
5741:
5734:
5731:
5728:
5722:
5719:
5715:
5708:
5704:
5700:
5694:
5691:
5688:
5685:
5679:
5676:
5671:
5667:
5663:
5657:
5654:
5648:
5645:
5637:
5633:
5626:
5606:
5603:
5600:
5597:
5577:
5574:
5566:
5547:
5544:
5541:
5535:
5532:
5527:
5523:
5519:
5513:
5507:
5504:
5501:
5492:
5479:
5475:
5471:
5468:
5465:
5462:
5459:
5454:
5448:
5444:
5438:
5434:
5424:
5421:
5418:
5412:
5409:
5406:
5403:
5400:
5397:
5394:
5390:
5384:
5380:
5376:
5371:
5363:
5360:
5357:
5354:
5346:
5342:
5338:
5333:
5329:
5320:
5302:
5298:
5294:
5288:
5282:
5259:
5253:
5250:
5247:
5244:
5236:
5232:
5228:
5223:
5220:
5217:
5213:
5205:
5204:
5203:
5186:
5181:
5177:
5173:
5170:
5167:
5162:
5156:
5152:
5148:
5143:
5140:
5137:
5133:
5122:
5121:
5120:
5118:
5100:
5096:
5092:
5086:
5080:
5057:
5054:
5051:
5048:
5042:
5039:
5034:
5031:
5021:
5020:
5019:
5011:
5009:
4993:
4985:
4969:
4946:
4938:
4934:
4920:
4907:
4903:
4896:
4893:
4888:
4884:
4875:
4872:
4865:
4864:
4863:
4846:
4831:
4826:
4816:
4809:
4806:
4792:
4787:
4777:
4770:
4767:
4761:
4758:
4751:
4750:
4749:
4729:
4702:
4675:
4648:
4638:
4636:
4631:
4629:
4608:
4604:
4590:
4577:
4573:
4566:
4563:
4558:
4554:
4525:
4522:
4512:
4508:
4504:
4501:
4488:
4484:
4477:
4474:
4469:
4465:
4452:
4451:
4450:
4448:
4444:
4423:
4419:
4412:
4387:
4383:
4370:
4356:
4336:
4313:
4307:
4304:
4301:
4281:
4261:
4241:
4238:
4235:
4232:
4227:
4224:
4221:
4217:
4213:
4208:
4205:
4202:
4198:
4175:
4171:
4149:
4146:
4143:
4140:
4135:
4131:
4127:
4122:
4118:
4114:
4109:
4105:
4101:
4096:
4092:
4082:
4078:
4059:
4053:
4050:
4047:
4032:
4029:
4021:
4011:
4006:
3992:
3972:
3962:
3958:
3957:
3950:
3941:
3940:
3931:
3928:
3920:
3910:
3906:
3900:
3899:
3894:This section
3892:
3888:
3883:
3882:
3870:
3866:
3852:
3849:
3843:
3840:
3837:
3833:
3828:
3825:
3822:
3817:
3814:
3809:
3804:
3801:
3796:
3791:
3788:
3783:
3778:
3775:
3770:
3765:
3762:
3757:
3754:
3751:
3743:
3739:
3726:
3712:
3709:
3700:
3696:
3691:
3687:
3682:
3679:
3676:
3670:
3666:
3663:
3659:
3654:
3649:
3646:
3641:
3636:
3633:
3628:
3623:
3620:
3615:
3610:
3607:
3602:
3594:
3590:
3577:
3563:
3557:
3547:
3511:
3507:
3503:
3496:
3492:
3468:
3465:
3442:
3432:
3416:
3413:
3410:
3404:
3399:
3396:
3391:
3386:
3381:
3377:
3374:
3371:
3368:
3363:
3360:
3355:
3350:
3347:
3342:
3337:
3334:
3329:
3324:
3321:
3316:
3313:
3310:
3307:
3304:
3296:
3292:
3279:
3266:
3263:
3255:
3247:
3239:
3236:
3233:
3225:
3222:
3215:
3194:
3164:
3160:
3156:
3151:
3146:
3143:
3140:
3136:
3125:
3104:
3074:
3070:
3066:
3040:
3037:
3034:
3031:
3025:
3022:
3019:
3011:
3008:
2999:
2985:
2981:
2977:
2974:
2971:
2946:
2942:
2929:
2916:
2911:
2908:
2903:
2896:
2893:
2890:
2886:
2880:
2876:
2862:
2854:
2848:
2844:
2841:
2836:
2832:
2827:
2823:
2819:
2814:
2810:
2807:
2802:
2799:
2796:
2792:
2787:
2783:
2779:
2765:
2752:
2738:
2735:
2732:
2725:converges to
2712:
2709:
2703:
2699:
2693:
2689:
2686:
2683:
2680:
2675:
2672:
2667:
2662:
2659:
2654:
2649:
2646:
2641:
2636:
2633:
2628:
2623:
2620:
2615:
2612:
2609:
2601:
2597:
2586:
2576:
2573:
2560:
2553:
2544:
2541:
2538:
2534:
2530:
2525:
2522:
2519:
2515:
2507:
2504:
2501:
2497:
2493:
2488:
2484:
2476:
2472:
2469:
2463:
2454:
2451:
2448:
2444:
2440:
2435:
2431:
2423:
2419:
2415:
2410:
2407:
2404:
2400:
2392:
2388:
2385:
2379:
2376:
2356:
2342:
2324:
2320:
2316:
2288:
2284:
2280:
2251:
2247:
2234:
2215:
2211:
2187:
2184:for all
2176:
2172:
2168:
2160:
2157:
2152:
2148:
2117:
2113:
2101:
2085:
2060:
2056:
2043:
2042:other step.
2037:R-convergence
2034:
2021:
2018:
2005:
2002:
1999:
1995:
1991:
1986:
1982:
1964:
1960:
1956:
1951:
1948:
1945:
1941:
1921:
1908:
1906:
1893:
1867:
1863:
1850:
1837:
1834:
1823:
1820:
1815:
1811:
1795:
1792:
1787:
1784:
1781:
1777:
1757:
1744:
1731:
1721:
1708:
1697:
1684:
1681:
1678:
1667:
1664:
1659:
1655:
1639:
1636:
1631:
1628:
1625:
1621:
1601:
1588:
1574:
1571:
1568:
1557:
1541:
1538:
1535:
1528:
1526:
1510:
1507:
1504:
1497:
1494:
1481:
1471:
1455:
1452:
1449:
1442:
1441:
1440:
1437:
1435:
1431:
1427:
1423:
1422:secant method
1407:
1387:
1384:
1381:
1355:
1352:
1346:
1343:
1323:
1320:
1317:
1294:
1291:
1288:
1282:
1279:
1270:
1257:
1254:
1246:
1236:
1233:
1228:
1224:
1208:
1205:
1200:
1197:
1194:
1190:
1170:
1157:
1143:
1136:
1133:, and with a
1132:
1119:
1099:
1077:
1052:
1048:
1037:
1022:
1020:
1016:
1012:
1009:
1004:
1002:
998:
993:
991:
987:
983:
967:
943:
940:
937:
930:
925:
921:
918:
913:
909:
904:
898:
894:
891:
886:
883:
880:
876:
871:
857:
845:
844:
843:
829:
822:
806:
803:
800:
793:
777:
752:
748:
736:
732:
728:
724:
712:
707:
705:
700:
698:
693:
692:
690:
689:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
616:
613:
611:
608:
606:
603:
602:
595:
594:
590:
589:
582:
579:
577:
574:
572:
569:
568:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
537:
534:
533:
532:
529:
527:
526:Finite volume
524:
520:
517:
516:
515:
512:
508:
502:
499:
497:
494:
492:
488:
486:
483:
480:
479:
472:
471:
463:
460:
458:
455:
451:
447:
445:
442:
440:
436:
432:
429:
427:
424:
422:
419:
415:
412:
410:
407:
405:
402:
400:
397:
396:
395:
392:
390:
387:
386:
379:
378:
371:
368:
366:
363:
361:
358:
356:
353:
352:
346:
345:
341:
340:
332:
329:
325:
322:
321:
320:
317:
316:
312:
306:
305:
299:
298:
287:
284:
283:
279:
276:
274:
271:
270:
268:
267:
263:
262:
256:
252:
249:
247:
244:
241:
239:
236:
235:
231:
228:
227:
226:
225:
221:
220:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
180:
178:
177:
169:
168:
164:
163:
158:
148:
145:
143:
140:
139:
138:
137:
134:
131:
130:
125:
122:
120:
117:
115:
112:
111:
110:
109:
106:
103:
102:
97:
94:
92:
88:
86:
83:
81:
78:
76:
73:
72:
71:
70:
64:
61:
59:
56:
55:
53:
52:
44:
43:
39:
38:
35:
32:
31:
19:
7868:Martin Kutta
7823:Ămile Picard
7803:Isaac Newton
7717:Euler method
7687:Substitution
7644:
7342:
7336:are used in
7333:
7329:
7327:
7314:
7308:
7294:
7276:
7270:
7256:
7247:
7230:. Retrieved
7217:
7206:. Retrieved
7202:
7189:
7178:. Retrieved
7175:math.unl.edu
7174:
7164:
7141:
7135:
7124:. Retrieved
7104:
7100:
7087:
7076:. Retrieved
7072:
7059:
7048:. Retrieved
7028:
7024:
7011:
7000:. Retrieved
6996:
6983:
6972:. Retrieved
6968:
6955:
6936:
6908:. Retrieved
6904:
6894:
6724:
6708:
6704:
6546:
6430:
6365:
6355:
6351:
6273:
6184:
6081:
5946:
5493:
5275:In terms of
5274:
5201:
5072:
5017:
4961:
4861:
4639:
4632:
4540:
4446:
4442:
4371:
4039:
4024:
4015:
4008:Please help
3964:
3953:
3923:
3914:
3903:Please help
3898:verification
3895:
3727:
3578:
3280:
3000:
2930:
2753:
2582:
2574:
2348:
2235:
2099:
2044:
2040:
1909:
1885:
1851:
1723:
1700:
1698:
1589:
1560:
1555:
1524:
1473:
1469:
1438:
1271:
1158:
1134:
1091:
1033:
1013:
1005:
994:
989:
985:
981:
958:
820:
791:
730:
726:
720:
670:Ămile Picard
655:Martin Kutta
645:George Green
605:Isaac Newton
443:
437: /
433: /
253: /
119:Chaos theory
7625:Phase space
7483:Homogeneous
4441:with order
4018:August 2020
4012:if you can.
3917:August 2020
2098:is said to
1852:A sequence
1426:simple root
563:RungeâKutta
308:Difference
251:Homogeneous
63:Engineering
7888:Categories
7853:John Crank
7682:Inspection
7545:Stochastic
7539:Difference
7513:Autonomous
7457:Non-linear
7447:Fractional
7410:Operations
7328:The terms
7291:Endre SĂŒli
7244:Literature
7232:2024-02-09
7208:2020-08-07
7203:gordon.edu
7180:2020-07-31
7126:2020-08-02
7078:2022-12-13
7050:2020-07-31
7002:2020-12-13
6974:2020-08-07
6969:gordon.edu
6910:2020-07-31
6886:References
6468:such that
4962:The error
4449:such that
3429:using the
2135:such that
1554:is called
1523:is called
1468:is called
992:(e.g., ).
680:John Crank
481:Inspection
435:Asymptotic
319:Stochastic
238:Autonomous
213:Non-linear
203:Fractional
7657:solutions
7615:Wronskian
7570:Solutions
7498:Decoupled
7462:Holonomic
7045:116192710
6858:−
6831:−
6711:itself).
6333:−
6026:κ
5979:−
5901:κ
5871:κ
5862:−
5800:κ
5780:κ
5774:−
5732:κ
5729:−
5723:
5689:κ
5686:−
5680:
5604:≪
5601:κ
5578:κ
5545:κ
5542:−
5536:
5445:κ
5422:−
5407:κ
5398:−
5364:κ
5358:−
5254:κ
5248:−
5174:κ
5171:−
5149:−
5055:κ
5052:−
4894:−
4810:
4771:
4762:≈
4564:−
4475:−
4225:−
4206:−
4079:, or the
3853:…
3826:…
3713:…
3680:…
3515:⌋
3501:⌊
3446:⌋
3440:⌊
3414:…
3372:…
3240:∈
3226:∈
3137:∑
3032:−
3026:∈
3012:∈
2972:μ
2869:∞
2866:→
2842:−
2808:−
2772:∞
2769:→
2713:…
2684:…
2542:−
2531:−
2523:−
2505:−
2494:−
2473:
2452:−
2441:−
2416:−
2389:
2380:≈
2317:ε
2281:ε
2248:ε
2212:ε
2173:ε
2169:≤
2158:−
2114:ε
2003:−
1992:−
1957:−
1928:∞
1925:→
1821:−
1793:−
1764:∞
1761:→
1665:−
1637:−
1608:∞
1605:→
1359:∞
1347:∈
1344:μ
1283:∈
1280:μ
1258:μ
1234:−
1206:−
1177:∞
1174:→
1144:μ
968:μ
941:μ
919:−
892:−
864:∞
861:→
830:μ
804:≥
421:Wronskian
399:Dirichlet
142:Economics
85:Chemistry
75:Astronomy
7765:Examples
7655:Integral
7427:Ordinary
7334:R-linear
7330:Q-linear
7255:(2002),
6666:′
6616:″
6566:′
6485:′
3954:require
3405:⌋
3392:⌊
2579:Examples
2325:′
2289:′
1036:sequence
729:and the
531:Galerkin
431:Lyapunov
342:Solution
286:Notation
278:Operator
264:Features
183:Ordinary
7493:Coupled
7432:Partial
7121:2153571
6905:hmc.edu
3956:cleanup
1436:below.
404:Neumann
188:Partial
96:Geology
91:Biology
80:Physics
7508:Degree
7452:Linear
7353:
7321:
7301:
7283:
7263:
7152:
7119:
7043:
6943:
6549:. If
4626:using
725:, the
591:People
503:
450:Series
208:Linear
47:Fields
7899:Rates
7557:Delay
7503:Order
7227:(PDF)
7199:(PDF)
7117:JSTOR
7097:(PDF)
7069:(PDF)
7041:S2CID
7021:(PDF)
6993:(PDF)
6965:(PDF)
6599:and
6307:with
6218:with
2931:Thus
1156:, if
990:order
733:of a
491:Euler
409:Robin
331:Delay
273:Order
246:Exact
172:Types
40:Scope
7351:ISBN
7332:and
7319:ISBN
7299:ISBN
7281:ISBN
7261:ISBN
7150:ISBN
6941:ISBN
6684:>
6634:<
6503:<
5590:for
4721:and
4667:and
4502:<
3264:<
2583:The
2200:and
1572:>
1385:>
1336:and
986:rate
842:if
819:and
598:List
7109:doi
7033:doi
6815:lim
6082:so
5720:exp
5677:exp
5567:in
5533:exp
4836:old
4821:new
4807:log
4797:old
4782:new
4768:log
4734:old
4707:new
4680:old
4653:new
3907:by
3671:536
3650:256
2859:lim
2762:lim
2470:log
2386:log
1918:lim
1754:lim
1598:lim
1374:if
1310:if
1167:lim
1112:to
854:lim
721:In
7890::
7201:.
7173:.
7148:.
7115:.
7105:63
7103:.
7099:.
7071:.
7039:.
7029:63
7027:.
7023:.
6995:.
6967:.
6919:^
6903:.
6390::=
6177:.
5619::
5321::
5119::
4630:.
4369:.
3664:65
3637:16
3364:16
3351:16
3267:1.
2998:.
2676:32
2663:16
2369::
2022:1.
1838:1.
1021:.
7388:e
7381:t
7374:v
7361:.
7359:.
7305:.
7287:.
7267:.
7235:.
7211:.
7183:.
7158:.
7129:.
7111::
7081:.
7053:.
7035::
7005:.
6977:.
6949:.
6913:.
6870:0
6867:=
6864:)
6861:L
6853:n
6849:x
6845:(
6841:/
6837:)
6834:L
6826:n
6822:a
6818:(
6795:)
6790:n
6786:a
6782:(
6762:)
6757:n
6753:x
6749:(
6709:p
6705:p
6687:1
6680:|
6676:)
6673:p
6670:(
6663:f
6658:|
6637:1
6630:|
6626:)
6623:p
6620:(
6613:f
6608:|
6587:0
6584:=
6580:|
6576:)
6573:p
6570:(
6563:f
6558:|
6547:p
6531:0
6527:x
6506:1
6499:|
6495:)
6492:p
6489:(
6482:f
6477:|
6456:p
6453:=
6450:)
6447:p
6444:(
6441:f
6431:f
6409:)
6404:n
6400:x
6396:(
6393:f
6385:1
6382:+
6379:n
6375:x
6356:q
6352:q
6336:k
6329:2
6325:=
6320:k
6316:a
6295:)
6290:k
6286:a
6282:(
6259:)
6256:1
6253:+
6250:k
6247:(
6243:/
6239:1
6236:=
6231:k
6227:d
6206:)
6201:k
6197:d
6193:(
6165:2
6162:=
6159:q
6139:)
6134:n
6130:x
6126:(
6123:f
6103:)
6098:n
6094:y
6090:(
6067:,
6064:)
6059:2
6055:h
6051:(
6046:O
6041:=
6036:2
6030:2
6020:2
6016:h
6012:n
6006:=
6002:|
5998:)
5993:n
5989:x
5985:(
5982:f
5974:n
5970:y
5965:|
5961:=
5958:e
5933:.
5929:)
5925:.
5922:.
5919:.
5916:+
5911:2
5905:2
5895:2
5891:h
5885:2
5881:n
5874:+
5868:h
5865:n
5859:1
5855:(
5849:0
5845:y
5841:=
5836:n
5831:)
5827:.
5824:.
5821:.
5818:.
5815:+
5810:2
5804:2
5794:2
5790:h
5783:+
5777:h
5771:1
5767:(
5760:0
5756:y
5752:=
5747:n
5742:]
5738:)
5735:h
5726:(
5716:[
5709:0
5705:y
5701:=
5698:)
5695:h
5692:n
5683:(
5672:0
5668:y
5664:=
5661:)
5658:h
5655:n
5652:(
5649:f
5646:=
5643:)
5638:n
5634:x
5630:(
5627:f
5607:1
5598:h
5575:h
5551:)
5548:x
5539:(
5528:0
5524:y
5520:=
5517:)
5514:x
5511:(
5508:f
5505:=
5502:y
5480:.
5476:)
5472:.
5469:.
5466:.
5463:.
5460:+
5455:2
5449:2
5439:2
5435:h
5428:)
5425:1
5419:n
5416:(
5413:n
5410:+
5404:h
5401:n
5395:1
5391:(
5385:0
5381:y
5377:=
5372:n
5368:)
5361:h
5355:1
5352:(
5347:0
5343:y
5339:=
5334:n
5330:y
5303:0
5299:y
5295:=
5292:)
5289:0
5286:(
5283:y
5260:.
5257:)
5251:h
5245:1
5242:(
5237:n
5233:y
5229:=
5224:1
5221:+
5218:n
5214:y
5187:,
5182:n
5178:y
5168:=
5163:h
5157:n
5153:y
5144:1
5141:+
5138:n
5134:y
5101:0
5097:y
5093:=
5090:)
5087:0
5084:(
5081:y
5058:y
5049:=
5043:x
5040:d
5035:y
5032:d
4994:n
4970:e
4947:.
4944:)
4939:q
4935:h
4931:(
4926:O
4921:=
4917:|
4913:)
4908:n
4904:x
4900:(
4897:f
4889:n
4885:y
4880:|
4876:=
4873:e
4847:,
4841:)
4832:h
4827:/
4817:h
4813:(
4802:)
4793:e
4788:/
4778:e
4774:(
4759:q
4730:e
4703:e
4676:h
4649:h
4614:)
4609:q
4605:h
4601:(
4596:O
4591:=
4587:|
4583:)
4578:n
4574:x
4570:(
4567:f
4559:n
4555:y
4550:|
4526:.
4523:n
4513:q
4509:h
4505:C
4498:|
4494:)
4489:n
4485:x
4481:(
4478:f
4470:n
4466:y
4461:|
4447:C
4443:q
4429:)
4424:n
4420:x
4416:(
4413:f
4393:)
4388:n
4384:y
4380:(
4357:x
4337:h
4317:)
4314:x
4311:(
4308:f
4305:=
4302:y
4282:x
4262:h
4242:.
4239:.
4236:.
4233:,
4228:2
4222:j
4218:y
4214:,
4209:1
4203:j
4199:y
4176:j
4172:y
4150:.
4147:.
4144:.
4141:,
4136:3
4132:y
4128:,
4123:2
4119:y
4115:,
4110:1
4106:y
4102:,
4097:0
4093:y
4063:)
4060:x
4057:(
4054:f
4051:=
4048:y
4031:)
4025:(
4020:)
4016:(
3993:h
3973:n
3930:)
3924:(
3919:)
3915:(
3901:.
3850:,
3844:1
3841:+
3838:k
3834:1
3829:,
3823:,
3818:6
3815:1
3810:,
3805:5
3802:1
3797:,
3792:4
3789:1
3784:,
3779:3
3776:1
3771:,
3766:2
3763:1
3758:,
3755:1
3752:=
3749:)
3744:k
3740:d
3736:(
3710:,
3701:k
3697:2
3692:2
3688:1
3683:,
3677:,
3667:,
3660:1
3655:,
3647:1
3642:,
3634:1
3629:,
3624:4
3621:1
3616:,
3611:2
3608:1
3603:=
3600:)
3595:k
3591:x
3587:(
3564:;
3558:m
3552:|
3548:r
3544:|
3520:)
3512:m
3508:/
3504:k
3497:r
3493:a
3490:(
3469:,
3466:x
3443:x
3417:,
3411:,
3400:2
3397:k
3387:4
3382:/
3378:1
3375:,
3369:,
3361:1
3356:,
3348:1
3343:,
3338:4
3335:1
3330:,
3325:4
3322:1
3317:,
3314:1
3311:,
3308:1
3305:=
3302:)
3297:k
3293:x
3289:(
3260:|
3256:r
3252:|
3248:,
3244:C
3237:r
3234:,
3230:C
3223:a
3199:|
3195:r
3191:|
3170:)
3165:n
3161:r
3157:a
3152:k
3147:0
3144:=
3141:n
3133:(
3109:|
3105:r
3101:|
3080:)
3075:k
3071:r
3067:a
3064:(
3044:)
3041:1
3038:,
3035:1
3029:(
3023:r
3020:,
3016:R
3009:a
2986:2
2982:/
2978:1
2975:=
2952:)
2947:k
2943:t
2939:(
2917:.
2912:2
2909:1
2904:=
2897:1
2894:+
2891:k
2887:2
2881:k
2877:2
2863:k
2855:=
2849:|
2845:0
2837:k
2833:2
2828:/
2824:1
2820:|
2815:|
2811:0
2803:1
2800:+
2797:k
2793:2
2788:/
2784:1
2780:|
2766:k
2739:0
2736:=
2733:L
2710:,
2704:k
2700:2
2694:/
2690:1
2687:,
2681:,
2673:1
2668:,
2660:1
2655:,
2650:8
2647:1
2642:,
2637:4
2634:1
2629:,
2624:2
2621:1
2616:,
2613:1
2610:=
2607:)
2602:k
2598:x
2594:(
2561:.
2554:|
2545:2
2539:k
2535:x
2526:1
2520:k
2516:x
2508:1
2502:k
2498:x
2489:k
2485:x
2477:|
2464:|
2455:1
2449:k
2445:x
2436:k
2432:x
2424:k
2420:x
2411:1
2408:+
2405:k
2401:x
2393:|
2377:q
2357:q
2329:)
2321:k
2313:(
2293:)
2285:k
2277:(
2257:)
2252:k
2244:(
2221:)
2216:k
2208:(
2188:k
2177:k
2165:|
2161:L
2153:k
2149:x
2144:|
2123:)
2118:k
2110:(
2086:L
2066:)
2061:k
2057:x
2053:(
2019:=
2012:|
2006:1
2000:k
1996:x
1987:k
1983:x
1978:|
1971:|
1965:k
1961:x
1952:1
1949:+
1946:k
1942:x
1937:|
1922:k
1894:L
1873:)
1868:k
1864:x
1860:(
1835:=
1828:|
1824:L
1816:k
1812:x
1807:|
1800:|
1796:L
1788:1
1785:+
1782:k
1778:x
1773:|
1758:k
1732:L
1709:L
1685:,
1682:0
1679:=
1672:|
1668:L
1660:k
1656:x
1651:|
1644:|
1640:L
1632:1
1629:+
1626:k
1622:x
1617:|
1602:k
1575:1
1569:q
1542:3
1539:=
1536:q
1511:2
1508:=
1505:q
1495:.
1482:L
1456:1
1453:=
1450:q
1430:Ï
1408:q
1388:1
1382:q
1362:)
1356:,
1353:0
1350:(
1324:1
1321:=
1318:q
1298:)
1295:1
1292:,
1289:0
1286:(
1255:=
1247:q
1242:|
1237:L
1229:k
1225:x
1220:|
1213:|
1209:L
1201:1
1198:+
1195:k
1191:x
1186:|
1171:k
1120:L
1100:q
1078:L
1058:)
1053:k
1049:x
1045:(
944:.
938:=
931:q
926:|
922:L
914:n
910:x
905:|
899:|
895:L
887:1
884:+
881:n
877:x
872:|
858:n
807:1
801:q
778:L
758:)
753:n
749:x
745:(
710:e
703:t
696:v
509:)
505:(
20:)
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