4372:
5063:
5121:
7121:
5025:
1410:
In practice, the asymptotic rate and order of convergence of a sequence of iterates or of approximations provide useful insights when using iterative methods and discretization methods for calculating numerical approximations. Strictly speaking, however, the asymptotic behavior of a sequence does not
754:
Asymptotic rates and orders of convergence have particular importance both in practical numerics and in formal proof, and they are the primary focus of this article. In practical numerics, asymptotic rates and orders of convergence follow two common conventions for two types of sequences: the first
3984:
converges R-linearly to 0 with rate 1/2, but it does not converge Q-linearly; see the second plot of the figure below. 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
2583:
The definitions of Q-convergence rates have a shortcoming in that they do not naturally capture the convergence behavior of sequences that do converge, but do not converge with an asymptotically constant rate with every step, so that the Q-convergence limit does not exist. One class of examples is
3430:
3930:
2727:
1119:
will converge to the solution of the continuous differential equation as the grid spacing goes to zero, and if so the asymptotic rate and order of that convergence are an important characterization of the efficiency of the grid discretization method. A sequence of approximate grid solutions
6802:
4226:
4366:
996:
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. Sequences with larger orders
3226:
4826:
1912:
then other more technical rate definitions are called for. This definition is technically called Q-convergence, short for quotient-convergence, and the rates and orders are called rates and orders of Q-convergence when that technical specificity is needed. Compare
6670:
745:
are any of several characterizations of how quickly that sequence approaches its limit. These are broadly divided into rates and orders of convergence that describe how quickly a sequence further approaches its limit once it is already close to it, called
6033:
991:
3259:
7261:
1339:
4085:
750:
rates and orders of convergence, and those that describe how quickly sequences approach their limits from starting points that are not necessarily close to their limits, called non-asymptotic rates and orders of convergence.
2574:
3787:
1672:
4234:
2587:
2394:
2281:
1910:
2935:
7116:{\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}}{2}}h^{2}\kappa ^{2}+...\right).}
5712:
3780:
6504:
6373:
6133:
5800:
3092:
5337:
3557:
6741:
6449:
6244:
4033:
7530:
7391:. 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
8041:
3076:
3040:
5020:{\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|}}.}
4727:
3683:
5930:
5428:
4803:
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
1776:
885:
3004:
2968:
2870:
1257:
4777:
2782:
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
1074:
5922:
5895:
5868:
5841:
4677:
4588:
4473:
3959:
2753:
6797:
2014:
1712:
8436:
6494:
6289:
4077:
1040:
This "smaller rates converge more quickly" behavior among sequences of the same order is standard but it can be counterintuitive. It is therefore also common to define
8441:
7333:
6768:
5615:
3593:
3499:
708:
7455:
7422:
7297:
7131:
5579:
4228:
converges to zero Q-superlinearly. In fact, it is quadratically convergent with a quadratic convergence rate of 1. It is shown in the third plot of the figure below.
3465:
2813:
2427:
1472:
1204:
1151:
854:
805:
5503:
5249:
2173:
1802:
8304:
5364:
4623:
449:
4538:
2141:
2087:
1565:
1558:
1385:
1250:
1094:
1035:
877:
8431:
7359:
3713:
3623:
3252:
2121:
2067:
1976:
1738:
3982:
6180:
6156:
5543:
5523:
5468:
5448:
5179:
5159:
5047:
4821:
2833:
2777:
2448:
2040:
1938:
1534:
1514:
1492:
1405:
1227:
1171:
1015:
825:
6379:
8034:
3425:{\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}}.}
8091:
192:
3925:{\textstyle (b_{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 ,}
5256:
2970:
converges Q-linearly to zero; analogous definitions hold for R-superlinear convergence, R-sublinear convergence, R-quadratic convergence, and so on.
2722:{\textstyle (b_{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 }
8204:
323:
4221:{\displaystyle (c_{k})={\frac {1}{2}},{\frac {1}{4}},{\frac {1}{16}},{\frac {1}{256}},{\frac {1}{65,\!536}},\ldots ,{\frac {1}{2^{2^{k}}}},\ldots }
2454:
8248:
8027:
1076:
as the rate; this is the "number of extra decimals of precision per iterate" for sequences that converge with order 1. The rate of convergence
364:
8142:
8009:
7808:
7596:
254:
3078:
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.
2584:
the staggered geometric progressions that get closer to their limits only every other step or every several steps, for instance the example
8426:
5185:
5080:
701:
156:
2294:
2178:
1807:
8351:
8132:
7720:
2875:
575:
229:
4361:{\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 }
3985:
limit than another subsequence (the sequence of quotients taken from even steps). Generally, for any staggered geometric progression
2973:
In order to define the rates and orders of R-convergence, one uses the rate and order of Q-convergence of an error-bounding sequence
8137:
8101:
7977:
7957:
7939:
7919:
5631:
5525:, inversely proportional to the number of grid points, i.e. the number of points in the sequence required to reach a given value of
5203:
5102:
250:
202:
7388:
3721:
8199:
8111:
318:
237:
212:
8167:
6301:
6044:
694:
8482:
8238:
629:
354:
8081:
5084:
272:
182:
8253:
5720:
3221:{\textstyle (a_{k})=1,{\frac {1}{2}},{\frac {1}{4}},{\frac {1}{8}},{\frac {1}{16}},{\frac {1}{32}},\ldots ,1/{2^{k}},\dots }
369:
8096:
8086:
197:
187:
4368:
converges to zero Q-sublinearly and logarithmically and its convergence is shown as the fourth plot of the figure below.
8106:
8076:
644:
495:
398:
285:
207:
8548:
8341:
8336:
8211:
8172:
4788:
4492:
3933:
330:
245:
535:
6665:{\displaystyle y_{n}=y_{0}(1-h\kappa )^{n}=y_{0}\left(1-nh\kappa +{\frac {n(n-1)}{2}}h^{2}\kappa ^{2}+....\right).}
5262:
3507:
403:
8381:
7532:. On the other hand, if the convergence is already of order ≥ 2, Aitken's method will bring no improvement.
506:
8517:
8487:
8462:
6677:
5181:. Section should be modified for consistency and include an explanation of alternative (equivalent?) definitions.
5136:
4496:
649:
484:
6387:
6200:
3988:
8376:
7460:
7457:
that still converges linearly (except for pathologically designed special cases), but faster in the sense that
500:
408:
8386:
3254:. Plugging the sequence into the definition of Q-linear convergence (i.e., order of convergence 1) shows that
562:
1387:
will involve the asymptotic limit of the ratio of some approximation error term above to an asymptotic order
763:
of a target. In formal mathematics, rates of convergence and orders of convergence are often described using
8356:
8346:
8243:
7376:
6159:
6028:{\displaystyle q\approx {\frac {\log(e_{\text{new}}/e_{\text{old}})}{\log(h_{\text{new}}/h_{\text{old}})}},}
5073:
1422:
1364:
986:{\displaystyle \lim _{n\rightarrow \infty }{\frac {\left|x_{n+1}-L\right|}{\left|x_{n}-L\right|^{q}}}=\mu .}
580:
570:
518:
359:
7878:
7379:
into a second one that converges more quickly to the same limit. Such techniques are in general known as "
6183:
4591:
393:
5369:
4411:
that exemplify linear, linear, superlinear (quadratic), and sublinear rates of convergence, respectively.
3631:
3045:
3009:
1743:
8502:
8497:
8391:
8315:
8294:
8162:
8157:
8152:
8147:
8071:
8050:
6497:
4780:
4476:
3087:
2976:
2940:
2842:
1360:
1356:
1112:
722:
639:
624:
513:
456:
438:
277:
33:
7641:
5816:
A practical method to estimate the order of convergence for a discretization method is pick step sizes
8396:
8320:
8289:
6292:
5810:
5252:
4484:
1952:Ï â 1.618. However, the integer powers are common and are given common names. Convergence with order
1418:
1344:
1043:
525:
461:
434:
5900:
5873:
5846:
5819:
1111:
Similar concepts are used for sequences of discretizations. For instance, ideally the solution of a
8406:
8193:
7380:
7370:
6773:
4423:
3938:
2732:
1981:
1679:
1414:
764:
747:
742:
557:
542:
307:
146:
113:
104:
4732:
1417:
methods are techniques for improving the rate of convergence of the sequence of partial sums of a
8527:
8366:
8361:
8284:
8116:
7907:
7743:
7692:
7384:
726:
552:
547:
430:
7850:
7613:
7387:
of approximating the limits of the transformed sequences. One example of series acceleration is
7256:{\displaystyle e=|y_{n}-f(x_{n})|={\frac {nh^{2}\kappa ^{2}}{2}}+\ldots ={\mathcal {O}}(h^{2}),}
6457:
6252:
1421:
and possibly its order of convergence, also. These accelerations are commonly accomplished with
7669:
4682:
4632:
4038:
8005:
7973:
7953:
7935:
7915:
7804:
7592:
7302:
6750:
5584:
5131:
4543:
3562:
3470:
664:
609:
388:
123:
7427:
7394:
7269:
5551:
3437:
2785:
2399:
1444:
1176:
1123:
833:
777:
8512:
8492:
7735:
7684:
6744:
5473:
5219:
4480:
3626:
2152:
1781:
756:
674:
659:
7825:
5342:
4601:
3715:. The same holds also for geometric progressions and geometric series parameterized by any
1334:{\displaystyle \lim _{n\rightarrow \infty }{\frac {\left|y_{n}-S\right|}{h_{n}^{q}}}=\mu ,}
8472:
8401:
8001:
7800:
2126:
2072:
1543:
1370:
1235:
1079:
1020:
862:
614:
530:
57:
8477:
7550:
7338:
5216:
A similar situation exists for discretization methods designed to approximate a function
4505:
3688:
3598:
3231:
2100:
2046:
1955:
1717:
669:
7768:
3964:
8553:
8467:
8274:
6165:
6141:
5803:
5528:
5508:
5453:
5433:
5164:
5144:
5032:
4806:
3716:
2818:
2762:
2756:
2433:
2025:
1923:
1519:
1499:
1477:
1390:
1212:
1156:
1000:
810:
768:
760:
634:
619:
425:
413:
132:
8542:
7945:
7696:
7375:
Many methods exist to increase the rate of convergence of a given sequence, i.e., to
1941:
6038:
which comes from writing the truncation error, at the old and new grid spacings, as
8522:
8457:
8371:
4035:, the sequence will not converge Q-linearly but will converge R-linearly with rate
1949:
1352:
1348:
1116:
654:
604:
490:
118:
5141:
There appears to be a mixture of defining convergence with regards to grid points
5029:
For numerical approximation of an exact value through a numerical method of order
4729:, then the recurrent sequence will converge at least quadratically, and so on. If
4079:
these examples highlight why the "R" in R-linear convergence is short for "root."
1017:
converge more quickly than those with smaller order, and those with smaller rates
17:
7721:"Acceleration of convergence of a family of logarithmically convergent sequences"
1351:. Similar definitions also apply for non-grid discretization schemes such as the
8279:
5062:
62:
4371:
8507:
8177:
2569:{\textstyle \lim _{k\to \infty }{\frac {|x_{k+1}-x_{k}|}{|x_{k}-x_{k-1}|}}=1.}
1945:
679:
3042:
could have been chosen that would converge with a faster rate and order. Any
1667:{\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|^{q}}}=\mu }
8269:
420:
141:
84:
74:
5924:. The order of convergence is then approximated by the following formula:
8309:
8019:
1439:
738:
1037:
converge more quickly than those with larger rates for a given order.
7747:
7688:
5087: in this section. Unsourced material may be challenged and removed.
4483:
and have important general applications in mathematics through various
95:
90:
79:
5259:(see example below). The discretization method generates a sequence
759:
and the second for sequences of successively more accurate numerical
7739:
2389:{\textstyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=1.}
2276:{\textstyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=0,}
1905:{\textstyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=1,}
2930:{\textstyle |x_{k}-L|\leq \varepsilon _{k}\quad {\text{for all }}k}
1411:
give conclusive information about any finite part of the sequence.
6162:(GTE), in that it represents a sum of errors accumulated over all
4370:
771:," which can be used to encompass both of the prior conventions.
1367:: in general, the appropriate definition of the asymptotic rate
8023:
5470:. The important parameter here for the rate of convergence to
6674:
The exact analytical solution to the differential equation is
5707:{\displaystyle |y_{n}-f(x_{n})|<Ch^{q}{\text{ for all }}n.}
5114:
5056:
4594:
and the recurrent sequence will converge at least linearly to
3961:
that gives the largest integer that is less than or equal to
3775:{\displaystyle a\in \mathbb {C} ,r\in \mathbb {C} ,|r|<1.}
7229:
6101:
5809:
This is the relevant definition when discussing methods for
5771:
6291:. We can approximate a solution to this equation using the
5813:
or the solution of ordinary differential equations (ODEs).
6368:{\displaystyle {\frac {y_{n+1}-y_{n}}{h}}=-\kappa y_{n},}
6128:{\displaystyle e=|y_{n}-f(x_{n})|={\mathcal {O}}(h^{q}).}
7591:(1st ed.). New York, NY: Springer. pp. 28â29.
4416:
Convergence rates to fixed points of recurrent sequences
1173:
with a corresponding sequence of regular grid spacings
5795:{\displaystyle |y_{n}-f(x_{n})|={\mathcal {O}}(h^{q})}
5450:
between successive values of the independent variable
4685:
4546:
4508:
4426:
4041:
3941:
3790:
3634:
3095:
2878:
2735:
2590:
2457:
2297:
2181:
1810:
1343:
where the absolute value symbols stand for a function
7463:
7430:
7397:
7341:
7305:
7272:
7134:
6805:
6776:
6753:
6680:
6507:
6460:
6390:
6304:
6255:
6203:
6168:
6144:
6047:
5933:
5903:
5876:
5849:
5822:
5723:
5634:
5587:
5554:
5531:
5511:
5476:
5456:
5436:
5372:
5345:
5265:
5222:
5167:
5147:
5035:
4933:
4849:
4829:
4809:
4735:
4635:
4604:
4237:
4088:
3991:
3967:
3724:
3691:
3601:
3565:
3510:
3473:
3440:
3262:
3234:
3048:
3012:
2979:
2943:
2845:
2821:
2788:
2765:
2436:
2402:
2155:
2129:
2103:
2075:
2049:
2028:
1984:
1958:
1926:
1784:
1746:
1720:
1682:
1568:
1546:
1522:
1502:
1480:
1447:
1393:
1373:
1260:
1238:
1215:
1179:
1159:
1126:
1082:
1046:
1023:
1003:
888:
865:
836:
813:
780:
2287:(i.e., faster than linearly). A sequence is said to
8450:
8419:
8329:
8262:
8231:
8224:
8186:
8125:
8064:
8057:
5251:, which might be an integral being approximated by
2175:for a sequence and for any sequence that satisfies
7851:"Computing and Estimating the Rate of Convergence"
7614:"Computing and Estimating the Rate of Convergence"
7524:
7449:
7416:
7353:
7327:
7291:
7255:
7115:
6791:
6762:
6735:
6664:
6488:
6443:
6367:
6283:
6238:
6174:
6150:
6127:
6027:
5916:
5889:
5862:
5835:
5794:
5706:
5609:
5573:
5537:
5517:
5497:
5462:
5442:
5422:
5358:
5331:
5243:
5173:
5153:
5041:
5019:
4815:
4795:directly from outside of its local neighborhoods.
4771:
4721:
4671:
4617:
4582:
4532:
4467:
4360:
4220:
4071:
4027:
3976:
3953:
3924:
3774:
3707:
3677:
3617:
3587:
3551:
3493:
3459:
3424:
3246:
3220:
3070:
3034:
3006:chosen such that no other error-bounding sequence
2998:
2962:
2929:
2864:
2827:
2807:
2771:
2747:
2721:
2568:
2442:
2421:
2388:
2275:
2167:
2135:
2115:
2081:
2061:
2034:
2008:
1970:
1932:
1904:
1796:
1770:
1732:
1706:
1666:
1552:
1528:
1508:
1486:
1466:
1399:
1379:
1333:
1244:
1221:
1198:
1165:
1153:of some problem that converges to a true solution
1145:
1088:
1068:
1029:
1009:
985:
871:
848:
819:
799:
8437:List of nonlinear ordinary differential equations
6496:, the sequence satisfying that recurrence is the
4172:
2291:(i.e., slower than linearly) if it converges and
1407:power of a discretization scale parameter below.
8442:List of nonlinear partial differential equations
7968:Richard L. Burden and J. Douglas Faires (2001),
7464:
3467:converges Q-linearly with a convergence rate of
3361:
3264:
2459:
2299:
2183:
1812:
1570:
1262:
890:
7912:Numerical analysis: a mathematical introduction
7677:Journal of Optimization Theory and Applications
5053:Rates of convergence for discretization methods
2451:if the sequence converges sublinearly and also
27:Speed of convergence of a mathematical sequence
8432:List of linear ordinary differential equations
1206:that converge to 0 is said to have asymptotic
8035:
5332:{\displaystyle {y_{0},y_{1},y_{2},y_{3},...}}
5257:solution of an ordinary differential equation
3552:{\displaystyle a\in \mathbb {R} ,r\in (-1,1)}
702:
8:
6380:linear recurrence with constant coefficients
6194:Consider the ordinary differential equation
4017:
4003:
3948:
3942:
2742:
2736:
7996:Nocedal, Jorge; Wright, Stephen J. (2006).
7795:Nocedal, Jorge; Wright, Stephen J. (2006).
7587:Nocedal, Jorge; Wright, Stephen J. (1999).
6736:{\displaystyle y=f(x)=y_{0}\exp(-\kappa x)}
2839:if there exists an error-bounding sequence
8228:
8061:
8042:
8028:
8020:
6444:{\displaystyle y_{n+1}=y_{n}(1-h\kappa ).}
6239:{\displaystyle {\frac {dy}{dx}}=-\kappa y}
4375:Log-linear plots of the example sequences
4028:{\displaystyle (ar^{\lfloor k/m\rfloor })}
3501:; see the first plot of the figure below.
1429:Rates of convergence for iterative methods
709:
695:
175:
50:
29:
7525:{\displaystyle \lim(a_{n}-L)/(x_{n}-L)=0}
7501:
7489:
7474:
7462:
7438:
7429:
7405:
7396:
7340:
7316:
7304:
7280:
7271:
7241:
7228:
7227:
7206:
7196:
7186:
7178:
7169:
7150:
7141:
7133:
7087:
7077:
7062:
7056:
7027:
7014:
6982:
6972:
6965:
6938:
6925:
6887:
6850:
6816:
6804:
6775:
6752:
6706:
6679:
6633:
6623:
6592:
6563:
6550:
6525:
6512:
6506:
6480:
6459:
6414:
6395:
6389:
6356:
6331:
6312:
6305:
6303:
6275:
6254:
6204:
6202:
6167:
6143:
6113:
6100:
6099:
6091:
6082:
6063:
6054:
6046:
6010:
6001:
5995:
5971:
5962:
5956:
5940:
5932:
5908:
5902:
5881:
5875:
5854:
5848:
5827:
5821:
5783:
5770:
5769:
5761:
5752:
5733:
5724:
5722:
5693:
5687:
5672:
5663:
5644:
5635:
5633:
5598:
5586:
5562:
5553:
5530:
5510:
5475:
5455:
5435:
5396:
5377:
5371:
5350:
5344:
5310:
5297:
5284:
5271:
5266:
5264:
5221:
5204:Learn how and when to remove this message
5166:
5146:
5103:Learn how and when to remove this message
5034:
4991:
4972:
4954:
4941:
4934:
4901:
4888:
4876:
4857:
4850:
4836:
4828:
4808:
4758:
4736:
4734:
4708:
4686:
4684:
4658:
4636:
4634:
4609:
4603:
4569:
4547:
4545:
4507:
4456:
4431:
4425:
4334:
4315:
4302:
4289:
4276:
4263:
4245:
4236:
4202:
4197:
4188:
4160:
4147:
4134:
4121:
4108:
4096:
4087:
4059:
4053:
4045:
4042:
4040:
4009:
4002:
3990:
3966:
3940:
3897:
3892:
3883:
3861:
3848:
3835:
3822:
3798:
3789:
3761:
3753:
3746:
3745:
3732:
3731:
3723:
3700:
3692:
3690:
3666:
3653:
3642:
3633:
3610:
3602:
3600:
3576:
3564:
3518:
3517:
3509:
3483:
3472:
3448:
3439:
3409:
3392:
3382:
3376:
3364:
3338:
3329:
3298:
3289:
3279:
3267:
3261:
3233:
3205:
3200:
3195:
3173:
3160:
3147:
3134:
3121:
3103:
3094:
3056:
3047:
3020:
3011:
2987:
2978:
2951:
2942:
2919:
2912:
2900:
2888:
2879:
2877:
2853:
2844:
2820:
2796:
2787:
2764:
2734:
2697:
2692:
2683:
2661:
2648:
2635:
2622:
2598:
2589:
2552:
2540:
2527:
2518:
2511:
2505:
2486:
2477:
2474:
2462:
2456:
2435:
2410:
2401:
2372:
2360:
2351:
2344:
2326:
2317:
2314:
2302:
2296:
2256:
2244:
2235:
2228:
2210:
2201:
2198:
2186:
2180:
2154:
2128:
2102:
2074:
2048:
2027:
1983:
1957:
1925:
1885:
1873:
1864:
1857:
1839:
1830:
1827:
1815:
1809:
1783:
1745:
1719:
1681:
1649:
1644:
1631:
1622:
1615:
1597:
1588:
1585:
1573:
1567:
1545:
1521:
1501:
1479:
1455:
1446:
1392:
1372:
1314:
1309:
1288:
1277:
1265:
1259:
1237:
1214:
1187:
1178:
1158:
1134:
1125:
1081:
1054:
1045:
1022:
1002:
966:
949:
916:
905:
893:
887:
864:
835:
812:
788:
779:
7424:converges linearly, one gets a sequence
7879:"Verifying Numerical Convergence Rates"
7670:"On Q-Order and R-Order of Convergence"
7541:
4487:about their convergence behavior. When
1347:for the space of solutions such as the
590:
341:
263:
221:
164:
131:
103:
53:
39:
32:
7950:An introduction to numerical analysis,
7125:In this case, the truncation error is
3625:and the sequence of partial sums of a
7:
8427:List of named differential equations
7932:Numerical analysis: an introduction,
7714:
7712:
7582:
7580:
7578:
7576:
7574:
7572:
7570:
5581:is said to converge to the sequence
5085:adding citations to reliable sources
3784:The staggered geometric progression
3678:{\textstyle (\sum _{n=0}^{k}ar^{n})}
157:List of named differential equations
8352:Method of undetermined coefficients
8133:Dependent and independent variables
7926:The extended definition is used in
5870:and calculate the resulting errors
5423:{\displaystyle y_{j-1},y_{j-2},...}
3071:{\displaystyle (\varepsilon '_{k})}
3035:{\displaystyle (\varepsilon '_{k})}
1771:{\displaystyle \mu \in (0,\infty )}
230:Dependent and independent variables
8000:(2nd ed.). Berlin, New York:
7799:(2nd ed.). Berlin, New York:
4479:, define discrete time autonomous
3685:also converges linearly with rate
3371:
3274:
2999:{\displaystyle (\varepsilon _{k})}
2963:{\displaystyle (\varepsilon _{k})}
2865:{\displaystyle (\varepsilon _{k})}
2469:
2309:
2193:
1914:
1822:
1762:
1580:
1272:
900:
774:For iterative methods, a sequence
755:for sequences of iterations of an
25:
7903:The simple definition is used in
7365:Acceleration of convergence rates
6743:, corresponding to the following
6190:Example of discretization methods
4783:and sequences cannot converge to
8249:Carathéodory's existence theorem
7964:The Big O definition is used in
7383:" methods. These may reduce the
5119:
5061:
4791:, though they may still jump to
1944:, when converging to a regular,
1940:be an integer. For example, the
1804:. If the sequence converges but
365:Carathéodory's existence theorem
7551:"Order and rate of convergence"
6186:(LTE) over just one iteration.
5072:needs additional citations for
2918:
1069:{\displaystyle -\log _{10}\mu }
7513:
7494:
7486:
7467:
7444:
7431:
7411:
7398:
7389:Aitken's delta-squared process
7322:
7309:
7286:
7273:
7247:
7234:
7179:
7175:
7162:
7142:
6917:
6905:
6877:
6862:
6840:
6831:
6822:
6809:
6730:
6718:
6696:
6690:
6610:
6598:
6547:
6531:
6470:
6464:
6435:
6420:
6378:which implies the first-order
6295:for numerical discretization:
6265:
6259:
6119:
6106:
6092:
6088:
6075:
6055:
6016:
5988:
5977:
5949:
5917:{\displaystyle e_{\text{old}}}
5890:{\displaystyle e_{\text{new}}}
5863:{\displaystyle h_{\text{old}}}
5836:{\displaystyle h_{\text{new}}}
5789:
5776:
5762:
5758:
5745:
5725:
5673:
5669:
5656:
5636:
5604:
5591:
5568:
5555:
5492:
5486:
5238:
5232:
4759:
4755:
4749:
4737:
4709:
4705:
4699:
4687:
4659:
4655:
4649:
4637:
4570:
4566:
4560:
4548:
4518:
4512:
4468:{\textstyle x_{n+1}:=f(x_{n})}
4462:
4449:
4251:
4238:
4102:
4089:
4054:
4046:
4022:
3992:
3954:{\textstyle \lfloor x\rfloor }
3804:
3791:
3762:
3754:
3701:
3693:
3672:
3635:
3611:
3603:
3582:
3566:
3546:
3531:
3454:
3441:
3368:
3271:
3109:
3096:
3065:
3049:
3029:
3013:
2993:
2980:
2957:
2944:
2901:
2880:
2859:
2846:
2802:
2789:
2748:{\textstyle \lfloor x\rfloor }
2604:
2591:
2553:
2519:
2512:
2478:
2466:
2416:
2403:
2373:
2352:
2345:
2318:
2306:
2257:
2236:
2229:
2202:
2190:
2003:
1991:
1886:
1865:
1858:
1831:
1819:
1765:
1753:
1701:
1689:
1645:
1623:
1616:
1589:
1577:
1461:
1448:
1269:
1193:
1180:
1140:
1127:
897:
794:
781:
452: / Integral solutions
1:
6792:{\displaystyle h\kappa \ll 1}
3595:converges linearly with rate
2430:converges logarithmically to
2009:{\displaystyle \mu \in (0,1)}
1707:{\displaystyle \mu \in (0,1)}
8077:Notation for differentiation
7952:Cambridge University Press.
7719:Van Tuyl, Andrew H. (1994).
7377:transform one given sequence
6182:iterations, as opposed to a
5430:along with the grid spacing
4779:, then the fixed point is a
4772:{\displaystyle |f'(p)|>1}
2837:converge at least R-linearly
2093:and the sequence is said to
2020:and the sequence is said to
1434:Convergence rate definitions
1100:, and some authors will use
496:Exponential response formula
242:Coupled / Decoupled
8173:Exact differential equation
7914:, Clarendon Press, Oxford.
5621:if there exists a constant
5548:In this case, the sequence
5139:. The specific problem is:
4493:continuously differentiable
1676:for some positive constant
827:is said to have asymptotic
8570:
7728:Mathematics of Computation
7368:
6489:{\displaystyle y(0)=y_{0}}
6284:{\displaystyle y(0)=y_{0}}
4722:{\textstyle |f''(p)|<1}
4072:{\textstyle {\sqrt{|r|}};}
3559:, a geometric progression
1494:. The sequence is said to
757:iterative numerical method
8483:JĂłzef Maria Hoene-WroĆski
8463:Gottfried Wilhelm Leibniz
8254:CauchyâKowalevski theorem
7948:and David Mayers (2003),
7824:Bockelman, Brian (2005).
7549:Ruye, Wang (2015-02-12).
6158:is, more specifically, a
4672:{\displaystyle |f'(p)|=0}
4583:{\textstyle |f'(p)|<1}
2283:that sequence is said to
1920:It is not necessary that
1098:asymptotic error constant
630:JĂłzef Maria Hoene-WroĆski
576:Undetermined coefficients
485:Method of characteristics
370:CauchyâKowalevski theorem
8377:Finite difference method
7972:(7th ed.), Brooks/Cole.
7930:Walter Gautschi (1997),
7335:with a convergence rate
7328:{\displaystyle f(x_{n})}
6763:{\displaystyle h\kappa }
5610:{\displaystyle f(x_{n})}
5339:, where each successive
4590:, the fixed point is an
3588:{\displaystyle (ar^{k})}
3504:More generally, for any
3494:{\displaystyle \mu =1/2}
1474:converges to the number
1423:sequence transformations
1104:where this article uses
355:PicardâLindelöf theorem
349:Existence and uniqueness
8357:Variation of parameters
8347:Separation of variables
8244:Peano existence theorem
8239:PicardâLindelöf theorem
8126:Attributes of variables
7450:{\displaystyle (a_{n})}
7417:{\displaystyle (x_{n})}
7292:{\displaystyle (y_{n})}
6249:with initial condition
6184:local truncation error
6160:global truncation error
5574:{\displaystyle (y_{n})}
4598:for any starting value
4231:Finally, the sequence
3460:{\displaystyle (a_{k})}
2808:{\displaystyle (x_{k})}
2422:{\displaystyle (x_{k})}
1467:{\displaystyle (x_{k})}
1365:computational chemistry
1199:{\displaystyle (h_{n})}
1146:{\displaystyle (y_{n})}
1096:may also be called the
849:{\displaystyle q\geq 1}
800:{\displaystyle (x_{n})}
581:Variation of parameters
571:Separation of variables
360:Peano existence theorem
8518:Carl David Tolmé Runge
8092:Differential-algebraic
8051:Differential equations
7998:Numerical Optimization
7826:"Rates of Convergence"
7797:Numerical Optimization
7776:University of Arkansas
7769:"Order of Convergence"
7589:Numerical Optimization
7526:
7451:
7418:
7355:
7329:
7293:
7257:
7117:
6793:
6764:
6737:
6666:
6490:
6445:
6369:
6285:
6240:
6176:
6152:
6129:
6029:
5918:
5891:
5864:
5837:
5796:
5708:
5611:
5575:
5539:
5519:
5499:
5498:{\displaystyle y=f(x)}
5464:
5444:
5424:
5360:
5333:
5245:
5244:{\displaystyle y=f(x)}
5175:
5155:
5043:
5021:
4817:
4773:
4723:
4673:
4625:sufficiently close to
4619:
4592:attractive fixed point
4584:
4534:
4477:fixed point iterations
4469:
4412:
4362:
4222:
4073:
4029:
3978:
3955:
3926:
3776:
3709:
3679:
3658:
3619:
3589:
3553:
3495:
3461:
3426:
3248:
3222:
3072:
3036:
3000:
2964:
2931:
2866:
2829:
2809:
2773:
2749:
2729:detailed below (where
2723:
2570:
2444:
2423:
2390:
2285:converge superlinearly
2277:
2169:
2168:{\displaystyle q>1}
2137:
2117:
2095:converge quadratically
2083:
2063:
2036:
2010:
1972:
1948:, has an order of the
1934:
1906:
1798:
1797:{\displaystyle q>1}
1772:
1734:
1708:
1668:
1554:
1530:
1510:
1488:
1468:
1401:
1381:
1335:
1246:
1223:
1200:
1167:
1147:
1090:
1070:
1031:
1011:
987:
873:
850:
821:
801:
650:Carl David Tolmé Runge
193:Differential-algebraic
34:Differential equations
8503:Augustin-Louis Cauchy
8498:Joseph-Louis Lagrange
8392:Finite element method
8382:CrankâNicolson method
8316:Numerical integration
8295:Exponential stability
8187:Relation to processes
8072:Differential operator
7877:Senning, Jonathan R.
7849:Senning, Jonathan R.
7668:Porta, F. A. (1989).
7642:"Rate of Convergence"
7612:Senning, Jonathan R.
7527:
7452:
7419:
7356:
7330:
7294:
7258:
7118:
6794:
6765:
6738:
6667:
6498:geometric progression
6491:
6446:
6370:
6286:
6241:
6177:
6153:
6130:
6030:
5919:
5892:
5865:
5838:
5797:
5709:
5612:
5576:
5540:
5520:
5500:
5465:
5445:
5425:
5361:
5359:{\displaystyle y_{j}}
5334:
5246:
5176:
5156:
5044:
5022:
4818:
4781:repulsive fixed point
4774:
4724:
4674:
4620:
4618:{\displaystyle x_{0}}
4585:
4535:
4470:
4374:
4363:
4223:
4074:
4030:
3979:
3956:
3927:
3777:
3710:
3680:
3638:
3620:
3590:
3554:
3496:
3462:
3427:
3249:
3223:
3088:geometric progression
3073:
3037:
3001:
2965:
2932:
2867:
2830:
2810:
2774:
2750:
2724:
2571:
2445:
2424:
2391:
2278:
2170:
2138:
2118:
2091:quadratic convergence
2084:
2064:
2037:
2022:converge linearly to
2011:
1973:
1935:
1907:
1799:
1773:
1735:
1709:
1669:
1555:
1531:
1511:
1489:
1469:
1402:
1382:
1357:finite element method
1336:
1247:
1224:
1201:
1168:
1148:
1113:differential equation
1091:
1071:
1032:
1012:
988:
874:
851:
822:
802:
723:mathematical analysis
640:Augustin-Louis Cauchy
625:Joseph-Louis Lagrange
457:Numerical integration
439:Exponential stability
302:Relation to processes
8397:Finite volume method
8321:Dirac delta function
8290:Asymptotic stability
8232:Existence/uniqueness
8097:Integro-differential
8004:. pp. 619+620.
7934:BirkhÀuser, Boston.
7461:
7428:
7395:
7339:
7303:
7270:
7132:
6803:
6774:
6751:
6678:
6505:
6458:
6388:
6302:
6293:forward Euler method
6253:
6201:
6166:
6142:
6045:
5931:
5901:
5874:
5847:
5820:
5811:numerical quadrature
5721:
5632:
5585:
5552:
5529:
5509:
5505:is the grid spacing
5474:
5454:
5434:
5370:
5343:
5263:
5253:numerical quadrature
5220:
5186:improve this section
5165:
5145:
5135:to meet Knowledge's
5081:improve this article
5033:
4827:
4807:
4733:
4683:
4633:
4602:
4544:
4533:{\textstyle f(p)=p,}
4506:
4485:fixed-point theorems
4424:
4420:Recurrent sequences
4235:
4086:
4039:
3989:
3965:
3939:
3788:
3722:
3689:
3632:
3599:
3563:
3508:
3471:
3438:
3260:
3232:
3093:
3046:
3010:
2977:
2941:
2876:
2843:
2819:
2786:
2763:
2733:
2588:
2455:
2434:
2400:
2295:
2289:converge sublinearly
2179:
2153:
2136:{\displaystyle \mu }
2127:
2101:
2082:{\displaystyle \mu }
2073:
2047:
2026:
1982:
1956:
1924:
1915:§ R-convergence
1808:
1782:
1744:
1718:
1680:
1566:
1553:{\displaystyle \mu }
1544:
1520:
1500:
1496:converge with order
1478:
1445:
1391:
1380:{\displaystyle \mu }
1371:
1258:
1245:{\displaystyle \mu }
1236:
1213:
1208:order of convergence
1177:
1157:
1124:
1089:{\displaystyle \mu }
1080:
1044:
1030:{\displaystyle \mu }
1021:
1001:
886:
872:{\displaystyle \mu }
863:
834:
829:order of convergence
811:
778:
741:that converges to a
735:order of convergence
462:Dirac delta function
198:Integro-differential
8407:Perturbation theory
8387:RungeâKutta methods
8367:Integral transforms
8300:Rate of convergence
8196:(discrete analogue)
7385:computational costs
7381:series acceleration
7371:Series acceleration
7354:{\displaystyle q=2}
5717:This is written as
5695: for all
5161:and with step size
4787:from its immediate
3708:{\displaystyle |r|}
3618:{\displaystyle |r|}
3247:{\displaystyle L=0}
3064:
3028:
2116:{\displaystyle q=3}
2097:. Convergence with
2062:{\displaystyle q=2}
2043:. Convergence with
1971:{\displaystyle q=1}
1733:{\displaystyle q=1}
1539:rate of convergence
1415:Series acceleration
1319:
1231:rate of convergence
858:rate of convergence
765:asymptotic notation
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
8549:Numerical analysis
8528:Sofya Kovalevskaya
8362:Integrating factor
8285:Lyapunov stability
8205:Stochastic partial
7970:Numerical Analysis
7908:Michelle Schatzman
7689:10.1007/BF00939805
7640:Hundley, Douglas.
7522:
7447:
7414:
7351:
7325:
7289:
7253:
7113:
6789:
6760:
6733:
6662:
6486:
6441:
6365:
6281:
6236:
6172:
6148:
6125:
6025:
5914:
5887:
5860:
5833:
5792:
5704:
5607:
5571:
5535:
5515:
5495:
5460:
5440:
5420:
5356:
5329:
5241:
5171:
5151:
5039:
5017:
5006:
4916:
4813:
4769:
4719:
4669:
4615:
4580:
4530:
4465:
4413:
4358:
4218:
4069:
4025:
3977:{\displaystyle x,}
3974:
3951:
3922:
3772:
3705:
3675:
3615:
3585:
3549:
3491:
3457:
3422:
3375:
3278:
3244:
3218:
3068:
3052:
3032:
3016:
2996:
2960:
2927:
2862:
2825:
2815:that converges to
2805:
2769:
2745:
2719:
2566:
2473:
2440:
2419:
2386:
2313:
2273:
2197:
2165:
2145:cubic convergence.
2133:
2113:
2079:
2059:
2032:
2018:linear convergence
2006:
1968:
1930:
1902:
1826:
1794:
1768:
1730:
1704:
1664:
1584:
1550:
1526:
1506:
1484:
1464:
1397:
1377:
1331:
1305:
1276:
1242:
1219:
1196:
1163:
1143:
1115:discretized via a
1086:
1066:
1027:
1007:
983:
904:
869:
846:
817:
807:that converges to
797:
727:numerical analysis
548:Integrating factor
389:Initial conditions
324:Stochastic partial
18:Linear convergence
8536:
8535:
8415:
8414:
8220:
8219:
8011:978-0-387-30303-1
7810:978-0-387-30303-1
7598:978-0-387-98793-4
7216:
7071:
6992:
6617:
6341:
6222:
6175:{\displaystyle n}
6151:{\displaystyle e}
6020:
6013:
5998:
5974:
5959:
5911:
5884:
5857:
5830:
5696:
5538:{\displaystyle x}
5518:{\displaystyle h}
5463:{\displaystyle x}
5443:{\displaystyle h}
5366:is a function of
5214:
5213:
5206:
5174:{\displaystyle h}
5154:{\displaystyle n}
5137:quality standards
5128:This section may
5113:
5112:
5105:
5042:{\displaystyle q}
5012:
5004:
4914:
4816:{\displaystyle q}
4481:dynamical systems
4350:
4323:
4310:
4297:
4284:
4271:
4210:
4177:
4155:
4142:
4129:
4116:
4064:
3905:
3869:
3856:
3843:
3830:
3417:
3404:
3360:
3355:
3263:
3181:
3168:
3155:
3142:
3129:
2922:
2828:{\displaystyle L}
2772:{\displaystyle x}
2705:
2669:
2656:
2643:
2630:
2558:
2458:
2443:{\displaystyle L}
2378:
2298:
2262:
2182:
2149:In general, when
2035:{\displaystyle L}
1933:{\displaystyle q}
1891:
1811:
1656:
1569:
1529:{\displaystyle L}
1509:{\displaystyle q}
1487:{\displaystyle L}
1438:Suppose that the
1400:{\displaystyle q}
1320:
1261:
1222:{\displaystyle q}
1166:{\displaystyle S}
1010:{\displaystyle q}
972:
889:
820:{\displaystyle L}
767:commonly called "
719:
718:
610:Gottfried Leibniz
501:Finite difference
293:
292:
154:
153:
124:Dynamical systems
16:(Redirected from
8561:
8513:Phyllis Nicolson
8493:Rudolf Lipschitz
8330:Solution methods
8305:Series solutions
8229:
8062:
8044:
8037:
8030:
8021:
8015:
7892:
7891:
7889:
7888:
7883:
7874:
7868:
7867:
7865:
7864:
7855:
7846:
7840:
7839:
7837:
7836:
7821:
7815:
7814:
7792:
7786:
7785:
7783:
7782:
7773:
7764:
7758:
7757:
7755:
7754:
7734:(207): 229â246.
7725:
7716:
7707:
7706:
7704:
7703:
7674:
7665:
7659:
7658:
7656:
7655:
7646:
7637:
7631:
7630:
7628:
7627:
7618:
7609:
7603:
7602:
7584:
7565:
7564:
7562:
7561:
7546:
7531:
7529:
7528:
7523:
7506:
7505:
7493:
7479:
7478:
7456:
7454:
7453:
7448:
7443:
7442:
7423:
7421:
7420:
7415:
7410:
7409:
7360:
7358:
7357:
7352:
7334:
7332:
7331:
7326:
7321:
7320:
7298:
7296:
7295:
7290:
7285:
7284:
7262:
7260:
7259:
7254:
7246:
7245:
7233:
7232:
7217:
7212:
7211:
7210:
7201:
7200:
7187:
7182:
7174:
7173:
7155:
7154:
7145:
7122:
7120:
7119:
7114:
7109:
7105:
7092:
7091:
7082:
7081:
7072:
7067:
7066:
7057:
7032:
7031:
7019:
7018:
7013:
7009:
6993:
6988:
6987:
6986:
6977:
6976:
6966:
6943:
6942:
6930:
6929:
6924:
6920:
6892:
6891:
6855:
6854:
6821:
6820:
6798:
6796:
6795:
6790:
6769:
6767:
6766:
6761:
6745:Taylor expansion
6742:
6740:
6739:
6734:
6711:
6710:
6671:
6669:
6668:
6663:
6658:
6654:
6638:
6637:
6628:
6627:
6618:
6613:
6593:
6568:
6567:
6555:
6554:
6530:
6529:
6517:
6516:
6495:
6493:
6492:
6487:
6485:
6484:
6450:
6448:
6447:
6442:
6419:
6418:
6406:
6405:
6374:
6372:
6371:
6366:
6361:
6360:
6342:
6337:
6336:
6335:
6323:
6322:
6306:
6290:
6288:
6287:
6282:
6280:
6279:
6245:
6243:
6242:
6237:
6223:
6221:
6213:
6205:
6181:
6179:
6178:
6173:
6157:
6155:
6154:
6149:
6134:
6132:
6131:
6126:
6118:
6117:
6105:
6104:
6095:
6087:
6086:
6068:
6067:
6058:
6034:
6032:
6031:
6026:
6021:
6019:
6015:
6014:
6011:
6005:
6000:
5999:
5996:
5980:
5976:
5975:
5972:
5966:
5961:
5960:
5957:
5941:
5923:
5921:
5920:
5915:
5913:
5912:
5909:
5896:
5894:
5893:
5888:
5886:
5885:
5882:
5869:
5867:
5866:
5861:
5859:
5858:
5855:
5842:
5840:
5839:
5834:
5832:
5831:
5828:
5801:
5799:
5798:
5793:
5788:
5787:
5775:
5774:
5765:
5757:
5756:
5738:
5737:
5728:
5713:
5711:
5710:
5705:
5697:
5694:
5692:
5691:
5676:
5668:
5667:
5649:
5648:
5639:
5616:
5614:
5613:
5608:
5603:
5602:
5580:
5578:
5577:
5572:
5567:
5566:
5544:
5542:
5541:
5536:
5524:
5522:
5521:
5516:
5504:
5502:
5501:
5496:
5469:
5467:
5466:
5461:
5449:
5447:
5446:
5441:
5429:
5427:
5426:
5421:
5407:
5406:
5388:
5387:
5365:
5363:
5362:
5357:
5355:
5354:
5338:
5336:
5335:
5330:
5328:
5315:
5314:
5302:
5301:
5289:
5288:
5276:
5275:
5250:
5248:
5247:
5242:
5209:
5202:
5198:
5195:
5189:
5180:
5178:
5177:
5172:
5160:
5158:
5157:
5152:
5123:
5122:
5115:
5108:
5101:
5097:
5094:
5088:
5065:
5057:
5048:
5046:
5045:
5040:
5026:
5024:
5023:
5018:
5013:
5011:
5010:
5005:
5003:
5002:
5001:
4983:
4982:
4966:
4965:
4964:
4946:
4945:
4935:
4921:
4920:
4915:
4913:
4912:
4911:
4893:
4892:
4882:
4881:
4880:
4868:
4867:
4851:
4837:
4822:
4820:
4819:
4814:
4799:Order estimation
4778:
4776:
4775:
4770:
4762:
4748:
4740:
4728:
4726:
4725:
4720:
4712:
4698:
4690:
4678:
4676:
4675:
4670:
4662:
4648:
4640:
4624:
4622:
4621:
4616:
4614:
4613:
4589:
4587:
4586:
4581:
4573:
4559:
4551:
4539:
4537:
4536:
4531:
4474:
4472:
4471:
4466:
4461:
4460:
4442:
4441:
4367:
4365:
4364:
4359:
4351:
4349:
4335:
4324:
4316:
4311:
4303:
4298:
4290:
4285:
4277:
4272:
4264:
4250:
4249:
4227:
4225:
4224:
4219:
4211:
4209:
4208:
4207:
4206:
4189:
4178:
4176:
4161:
4156:
4148:
4143:
4135:
4130:
4122:
4117:
4109:
4101:
4100:
4078:
4076:
4075:
4070:
4065:
4063:
4058:
4057:
4049:
4043:
4034:
4032:
4031:
4026:
4021:
4020:
4013:
3983:
3981:
3980:
3975:
3960:
3958:
3957:
3952:
3931:
3929:
3928:
3923:
3912:
3911:
3910:
3906:
3898:
3887:
3870:
3862:
3857:
3849:
3844:
3836:
3831:
3823:
3803:
3802:
3781:
3779:
3778:
3773:
3765:
3757:
3749:
3735:
3714:
3712:
3711:
3706:
3704:
3696:
3684:
3682:
3681:
3676:
3671:
3670:
3657:
3652:
3627:geometric series
3624:
3622:
3621:
3616:
3614:
3606:
3594:
3592:
3591:
3586:
3581:
3580:
3558:
3556:
3555:
3550:
3521:
3500:
3498:
3497:
3492:
3487:
3466:
3464:
3463:
3458:
3453:
3452:
3431:
3429:
3428:
3423:
3418:
3410:
3405:
3403:
3402:
3387:
3386:
3377:
3374:
3356:
3354:
3350:
3343:
3342:
3333:
3320:
3316:
3309:
3308:
3293:
3280:
3277:
3253:
3251:
3250:
3245:
3227:
3225:
3224:
3219:
3211:
3210:
3209:
3199:
3182:
3174:
3169:
3161:
3156:
3148:
3143:
3135:
3130:
3122:
3108:
3107:
3077:
3075:
3074:
3069:
3060:
3041:
3039:
3038:
3033:
3024:
3005:
3003:
3002:
2997:
2992:
2991:
2969:
2967:
2966:
2961:
2956:
2955:
2936:
2934:
2933:
2928:
2923:
2920:
2917:
2916:
2904:
2893:
2892:
2883:
2871:
2869:
2868:
2863:
2858:
2857:
2834:
2832:
2831:
2826:
2814:
2812:
2811:
2806:
2801:
2800:
2778:
2776:
2775:
2770:
2754:
2752:
2751:
2746:
2728:
2726:
2725:
2720:
2712:
2711:
2710:
2706:
2698:
2687:
2670:
2662:
2657:
2649:
2644:
2636:
2631:
2623:
2603:
2602:
2575:
2573:
2572:
2567:
2559:
2557:
2556:
2551:
2550:
2532:
2531:
2522:
2516:
2515:
2510:
2509:
2497:
2496:
2481:
2475:
2472:
2449:
2447:
2446:
2441:
2428:
2426:
2425:
2420:
2415:
2414:
2395:
2393:
2392:
2387:
2379:
2377:
2376:
2365:
2364:
2355:
2349:
2348:
2337:
2336:
2321:
2315:
2312:
2282:
2280:
2279:
2274:
2263:
2261:
2260:
2249:
2248:
2239:
2233:
2232:
2221:
2220:
2205:
2199:
2196:
2174:
2172:
2171:
2166:
2142:
2140:
2139:
2134:
2122:
2120:
2119:
2114:
2088:
2086:
2085:
2080:
2068:
2066:
2065:
2060:
2041:
2039:
2038:
2033:
2015:
2013:
2012:
2007:
1977:
1975:
1974:
1969:
1939:
1937:
1936:
1931:
1911:
1909:
1908:
1903:
1892:
1890:
1889:
1878:
1877:
1868:
1862:
1861:
1850:
1849:
1834:
1828:
1825:
1803:
1801:
1800:
1795:
1777:
1775:
1774:
1769:
1739:
1737:
1736:
1731:
1713:
1711:
1710:
1705:
1673:
1671:
1670:
1665:
1657:
1655:
1654:
1653:
1648:
1636:
1635:
1626:
1620:
1619:
1608:
1607:
1592:
1586:
1583:
1559:
1557:
1556:
1551:
1535:
1533:
1532:
1527:
1515:
1513:
1512:
1507:
1493:
1491:
1490:
1485:
1473:
1471:
1470:
1465:
1460:
1459:
1406:
1404:
1403:
1398:
1386:
1384:
1383:
1378:
1340:
1338:
1337:
1332:
1321:
1318:
1313:
1304:
1300:
1293:
1292:
1278:
1275:
1251:
1249:
1248:
1243:
1228:
1226:
1225:
1220:
1205:
1203:
1202:
1197:
1192:
1191:
1172:
1170:
1169:
1164:
1152:
1150:
1149:
1144:
1139:
1138:
1095:
1093:
1092:
1087:
1075:
1073:
1072:
1067:
1059:
1058:
1036:
1034:
1033:
1028:
1016:
1014:
1013:
1008:
992:
990:
989:
984:
973:
971:
970:
965:
961:
954:
953:
938:
934:
927:
926:
906:
903:
878:
876:
875:
870:
855:
853:
852:
847:
826:
824:
823:
818:
806:
804:
803:
798:
793:
792:
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:
58:Natural sciences
51:
30:
21:
8569:
8568:
8564:
8563:
8562:
8560:
8559:
8558:
8539:
8538:
8537:
8532:
8473:Jacob Bernoulli
8446:
8411:
8402:Galerkin method
8325:
8263:Solution topics
8258:
8216:
8182:
8121:
8053:
8048:
8012:
8002:Springer-Verlag
7995:
7901:
7896:
7895:
7886:
7884:
7881:
7876:
7875:
7871:
7862:
7860:
7853:
7848:
7847:
7843:
7834:
7832:
7823:
7822:
7818:
7811:
7801:Springer-Verlag
7794:
7793:
7789:
7780:
7778:
7771:
7766:
7765:
7761:
7752:
7750:
7740:10.2307/2153571
7723:
7718:
7717:
7710:
7701:
7699:
7672:
7667:
7666:
7662:
7653:
7651:
7649:Whitman College
7644:
7639:
7638:
7634:
7625:
7623:
7616:
7611:
7610:
7606:
7599:
7586:
7585:
7568:
7559:
7557:
7548:
7547:
7543:
7538:
7497:
7470:
7459:
7458:
7434:
7426:
7425:
7401:
7393:
7392:
7373:
7367:
7337:
7336:
7312:
7301:
7300:
7276:
7268:
7267:
7237:
7202:
7192:
7188:
7165:
7146:
7130:
7129:
7083:
7073:
7058:
7037:
7033:
7023:
6978:
6968:
6967:
6949:
6945:
6944:
6934:
6898:
6894:
6893:
6883:
6846:
6812:
6801:
6800:
6772:
6771:
6749:
6748:
6702:
6676:
6675:
6629:
6619:
6594:
6573:
6569:
6559:
6546:
6521:
6508:
6503:
6502:
6476:
6456:
6455:
6410:
6391:
6386:
6385:
6352:
6327:
6308:
6307:
6300:
6299:
6271:
6251:
6250:
6214:
6206:
6199:
6198:
6192:
6164:
6163:
6140:
6139:
6109:
6078:
6059:
6043:
6042:
6006:
5991:
5981:
5967:
5952:
5942:
5929:
5928:
5904:
5899:
5898:
5877:
5872:
5871:
5850:
5845:
5844:
5823:
5818:
5817:
5779:
5748:
5729:
5719:
5718:
5683:
5659:
5640:
5630:
5629:
5594:
5583:
5582:
5558:
5550:
5549:
5527:
5526:
5507:
5506:
5472:
5471:
5452:
5451:
5432:
5431:
5392:
5373:
5368:
5367:
5346:
5341:
5340:
5306:
5293:
5280:
5267:
5261:
5260:
5218:
5217:
5210:
5199:
5193:
5190:
5183:
5163:
5162:
5143:
5142:
5124:
5120:
5109:
5098:
5092:
5089:
5078:
5066:
5055:
5031:
5030:
4987:
4968:
4967:
4950:
4937:
4936:
4929:
4922:
4897:
4884:
4883:
4872:
4853:
4852:
4845:
4838:
4825:
4824:
4805:
4804:
4801:
4741:
4731:
4730:
4691:
4681:
4680:
4641:
4631:
4630:
4605:
4600:
4599:
4552:
4542:
4541:
4504:
4503:
4452:
4427:
4422:
4421:
4418:
4410:
4401:
4392:
4383:
4339:
4241:
4233:
4232:
4198:
4193:
4165:
4092:
4084:
4083:
4044:
4037:
4036:
3998:
3987:
3986:
3963:
3962:
3937:
3936:
3893:
3888:
3794:
3786:
3785:
3720:
3719:
3717:complex numbers
3687:
3686:
3662:
3630:
3629:
3597:
3596:
3572:
3561:
3560:
3506:
3505:
3469:
3468:
3444:
3436:
3435:
3388:
3378:
3334:
3325:
3321:
3294:
3285:
3281:
3258:
3257:
3230:
3229:
3201:
3099:
3091:
3090:
3084:
3044:
3043:
3008:
3007:
2983:
2975:
2974:
2947:
2939:
2938:
2908:
2884:
2874:
2873:
2849:
2841:
2840:
2817:
2816:
2792:
2784:
2783:
2761:
2760:
2731:
2730:
2693:
2688:
2594:
2586:
2585:
2581:
2536:
2523:
2517:
2501:
2482:
2476:
2453:
2452:
2432:
2431:
2406:
2398:
2397:
2356:
2350:
2322:
2316:
2293:
2292:
2240:
2234:
2206:
2200:
2177:
2176:
2151:
2150:
2125:
2124:
2099:
2098:
2071:
2070:
2045:
2044:
2024:
2023:
1980:
1979:
1954:
1953:
1922:
1921:
1869:
1863:
1835:
1829:
1806:
1805:
1780:
1779:
1742:
1741:
1716:
1715:
1678:
1677:
1643:
1627:
1621:
1593:
1587:
1564:
1563:
1542:
1541:
1518:
1517:
1498:
1497:
1476:
1475:
1451:
1443:
1442:
1436:
1431:
1389:
1388:
1369:
1368:
1284:
1283:
1279:
1256:
1255:
1234:
1233:
1229:and asymptotic
1211:
1210:
1183:
1175:
1174:
1155:
1154:
1130:
1122:
1121:
1078:
1077:
1050:
1042:
1041:
1019:
1018:
999:
998:
945:
944:
940:
939:
912:
911:
907:
884:
883:
861:
860:
856:and asymptotic
832:
831:
809:
808:
784:
776:
775:
761:discretizations
725:, particularly
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:
8567:
8565:
8557:
8556:
8551:
8541:
8540:
8534:
8533:
8531:
8530:
8525:
8520:
8515:
8510:
8505:
8500:
8495:
8490:
8488:Ernst Lindelöf
8485:
8480:
8475:
8470:
8468:Leonhard Euler
8465:
8460:
8454:
8452:
8451:Mathematicians
8448:
8447:
8445:
8444:
8439:
8434:
8429:
8423:
8421:
8417:
8416:
8413:
8412:
8410:
8409:
8404:
8399:
8394:
8389:
8384:
8379:
8374:
8369:
8364:
8359:
8354:
8349:
8344:
8339:
8333:
8331:
8327:
8326:
8324:
8323:
8318:
8313:
8307:
8302:
8297:
8292:
8287:
8282:
8277:
8275:Phase portrait
8272:
8266:
8264:
8260:
8259:
8257:
8256:
8251:
8246:
8241:
8235:
8233:
8226:
8222:
8221:
8218:
8217:
8215:
8214:
8209:
8208:
8207:
8197:
8190:
8188:
8184:
8183:
8181:
8180:
8178:On jet bundles
8175:
8170:
8165:
8160:
8155:
8150:
8145:
8143:Nonhomogeneous
8140:
8135:
8129:
8127:
8123:
8122:
8120:
8119:
8114:
8109:
8104:
8099:
8094:
8089:
8084:
8079:
8074:
8068:
8066:
8059:
8058:Classification
8055:
8054:
8049:
8047:
8046:
8039:
8032:
8024:
8018:
8017:
8010:
7981:
7980:
7962:
7961:
7943:
7924:
7923:
7900:
7897:
7894:
7893:
7869:
7841:
7816:
7809:
7787:
7767:Arnold, Mark.
7759:
7708:
7683:(3): 415â431.
7660:
7632:
7604:
7597:
7566:
7540:
7539:
7537:
7534:
7521:
7518:
7515:
7512:
7509:
7504:
7500:
7496:
7492:
7488:
7485:
7482:
7477:
7473:
7469:
7466:
7446:
7441:
7437:
7433:
7413:
7408:
7404:
7400:
7369:Main article:
7366:
7363:
7350:
7347:
7344:
7324:
7319:
7315:
7311:
7308:
7288:
7283:
7279:
7275:
7264:
7263:
7252:
7249:
7244:
7240:
7236:
7231:
7226:
7223:
7220:
7215:
7209:
7205:
7199:
7195:
7191:
7185:
7181:
7177:
7172:
7168:
7164:
7161:
7158:
7153:
7149:
7144:
7140:
7137:
7112:
7108:
7104:
7101:
7098:
7095:
7090:
7086:
7080:
7076:
7070:
7065:
7061:
7055:
7052:
7049:
7046:
7043:
7040:
7036:
7030:
7026:
7022:
7017:
7012:
7008:
7005:
7002:
6999:
6996:
6991:
6985:
6981:
6975:
6971:
6964:
6961:
6958:
6955:
6952:
6948:
6941:
6937:
6933:
6928:
6923:
6919:
6916:
6913:
6910:
6907:
6904:
6901:
6897:
6890:
6886:
6882:
6879:
6876:
6873:
6870:
6867:
6864:
6861:
6858:
6853:
6849:
6845:
6842:
6839:
6836:
6833:
6830:
6827:
6824:
6819:
6815:
6811:
6808:
6788:
6785:
6782:
6779:
6759:
6756:
6732:
6729:
6726:
6723:
6720:
6717:
6714:
6709:
6705:
6701:
6698:
6695:
6692:
6689:
6686:
6683:
6661:
6657:
6653:
6650:
6647:
6644:
6641:
6636:
6632:
6626:
6622:
6616:
6612:
6609:
6606:
6603:
6600:
6597:
6591:
6588:
6585:
6582:
6579:
6576:
6572:
6566:
6562:
6558:
6553:
6549:
6545:
6542:
6539:
6536:
6533:
6528:
6524:
6520:
6515:
6511:
6483:
6479:
6475:
6472:
6469:
6466:
6463:
6452:
6451:
6440:
6437:
6434:
6431:
6428:
6425:
6422:
6417:
6413:
6409:
6404:
6401:
6398:
6394:
6376:
6375:
6364:
6359:
6355:
6351:
6348:
6345:
6340:
6334:
6330:
6326:
6321:
6318:
6315:
6311:
6278:
6274:
6270:
6267:
6264:
6261:
6258:
6247:
6246:
6235:
6232:
6229:
6226:
6220:
6217:
6212:
6209:
6191:
6188:
6171:
6147:
6136:
6135:
6124:
6121:
6116:
6112:
6108:
6103:
6098:
6094:
6090:
6085:
6081:
6077:
6074:
6071:
6066:
6062:
6057:
6053:
6050:
6036:
6035:
6024:
6018:
6009:
6004:
5994:
5990:
5987:
5984:
5979:
5970:
5965:
5955:
5951:
5948:
5945:
5939:
5936:
5907:
5880:
5853:
5826:
5804:big O notation
5791:
5786:
5782:
5778:
5773:
5768:
5764:
5760:
5755:
5751:
5747:
5744:
5741:
5736:
5732:
5727:
5715:
5714:
5703:
5700:
5690:
5686:
5682:
5679:
5675:
5671:
5666:
5662:
5658:
5655:
5652:
5647:
5643:
5638:
5606:
5601:
5597:
5593:
5590:
5570:
5565:
5561:
5557:
5534:
5514:
5494:
5491:
5488:
5485:
5482:
5479:
5459:
5439:
5419:
5416:
5413:
5410:
5405:
5402:
5399:
5395:
5391:
5386:
5383:
5380:
5376:
5353:
5349:
5327:
5324:
5321:
5318:
5313:
5309:
5305:
5300:
5296:
5292:
5287:
5283:
5279:
5274:
5270:
5240:
5237:
5234:
5231:
5228:
5225:
5212:
5211:
5170:
5150:
5127:
5125:
5118:
5111:
5110:
5069:
5067:
5060:
5054:
5051:
5038:
5016:
5009:
5000:
4997:
4994:
4990:
4986:
4981:
4978:
4975:
4971:
4963:
4960:
4957:
4953:
4949:
4944:
4940:
4932:
4928:
4925:
4919:
4910:
4907:
4904:
4900:
4896:
4891:
4887:
4879:
4875:
4871:
4866:
4863:
4860:
4856:
4848:
4844:
4841:
4835:
4832:
4812:
4800:
4797:
4768:
4765:
4761:
4757:
4754:
4751:
4747:
4744:
4739:
4718:
4715:
4711:
4707:
4704:
4701:
4697:
4694:
4689:
4668:
4665:
4661:
4657:
4654:
4651:
4647:
4644:
4639:
4612:
4608:
4579:
4576:
4572:
4568:
4565:
4562:
4558:
4555:
4550:
4529:
4526:
4523:
4520:
4517:
4514:
4511:
4464:
4459:
4455:
4451:
4448:
4445:
4440:
4437:
4434:
4430:
4417:
4414:
4406:
4397:
4388:
4379:
4357:
4354:
4348:
4345:
4342:
4338:
4333:
4330:
4327:
4322:
4319:
4314:
4309:
4306:
4301:
4296:
4293:
4288:
4283:
4280:
4275:
4270:
4267:
4262:
4259:
4256:
4253:
4248:
4244:
4240:
4217:
4214:
4205:
4201:
4196:
4192:
4187:
4184:
4181:
4175:
4171:
4168:
4164:
4159:
4154:
4151:
4146:
4141:
4138:
4133:
4128:
4125:
4120:
4115:
4112:
4107:
4104:
4099:
4095:
4091:
4082:The sequence
4068:
4062:
4056:
4052:
4048:
4024:
4019:
4016:
4012:
4008:
4005:
4001:
3997:
3994:
3973:
3970:
3950:
3947:
3944:
3934:floor function
3921:
3918:
3915:
3909:
3904:
3901:
3896:
3891:
3886:
3882:
3879:
3876:
3873:
3868:
3865:
3860:
3855:
3852:
3847:
3842:
3839:
3834:
3829:
3826:
3821:
3818:
3815:
3812:
3809:
3806:
3801:
3797:
3793:
3771:
3768:
3764:
3760:
3756:
3752:
3748:
3744:
3741:
3738:
3734:
3730:
3727:
3703:
3699:
3695:
3674:
3669:
3665:
3661:
3656:
3651:
3648:
3645:
3641:
3637:
3613:
3609:
3605:
3584:
3579:
3575:
3571:
3568:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3524:
3520:
3516:
3513:
3490:
3486:
3482:
3479:
3476:
3456:
3451:
3447:
3443:
3421:
3416:
3413:
3408:
3401:
3398:
3395:
3391:
3385:
3381:
3373:
3370:
3367:
3363:
3359:
3353:
3349:
3346:
3341:
3337:
3332:
3328:
3324:
3319:
3315:
3312:
3307:
3304:
3301:
3297:
3292:
3288:
3284:
3276:
3273:
3270:
3266:
3243:
3240:
3237:
3217:
3214:
3208:
3204:
3198:
3194:
3191:
3188:
3185:
3180:
3177:
3172:
3167:
3164:
3159:
3154:
3151:
3146:
3141:
3138:
3133:
3128:
3125:
3120:
3117:
3114:
3111:
3106:
3102:
3098:
3083:
3080:
3067:
3063:
3059:
3055:
3051:
3031:
3027:
3023:
3019:
3015:
2995:
2990:
2986:
2982:
2959:
2954:
2950:
2946:
2926:
2915:
2911:
2907:
2903:
2899:
2896:
2891:
2887:
2882:
2861:
2856:
2852:
2848:
2824:
2804:
2799:
2795:
2791:
2768:
2757:floor function
2744:
2741:
2738:
2718:
2715:
2709:
2704:
2701:
2696:
2691:
2686:
2682:
2679:
2676:
2673:
2668:
2665:
2660:
2655:
2652:
2647:
2642:
2639:
2634:
2629:
2626:
2621:
2618:
2615:
2612:
2609:
2606:
2601:
2597:
2593:
2580:
2577:
2565:
2562:
2555:
2549:
2546:
2543:
2539:
2535:
2530:
2526:
2521:
2514:
2508:
2504:
2500:
2495:
2492:
2489:
2485:
2480:
2471:
2468:
2465:
2461:
2439:
2418:
2413:
2409:
2405:
2385:
2382:
2375:
2371:
2368:
2363:
2359:
2354:
2347:
2343:
2340:
2335:
2332:
2329:
2325:
2320:
2311:
2308:
2305:
2301:
2272:
2269:
2266:
2259:
2255:
2252:
2247:
2243:
2238:
2231:
2227:
2224:
2219:
2216:
2213:
2209:
2204:
2195:
2192:
2189:
2185:
2164:
2161:
2158:
2132:
2112:
2109:
2106:
2078:
2058:
2055:
2052:
2031:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1967:
1964:
1961:
1929:
1901:
1898:
1895:
1888:
1884:
1881:
1876:
1872:
1867:
1860:
1856:
1853:
1848:
1845:
1842:
1838:
1833:
1824:
1821:
1818:
1814:
1793:
1790:
1787:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1729:
1726:
1723:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1663:
1660:
1652:
1647:
1642:
1639:
1634:
1630:
1625:
1618:
1614:
1611:
1606:
1603:
1600:
1596:
1591:
1582:
1579:
1576:
1572:
1549:
1525:
1505:
1483:
1463:
1458:
1454:
1450:
1435:
1432:
1430:
1427:
1396:
1376:
1353:polygon meshes
1330:
1327:
1324:
1317:
1312:
1308:
1303:
1299:
1296:
1291:
1287:
1282:
1274:
1271:
1268:
1264:
1241:
1218:
1195:
1190:
1186:
1182:
1162:
1142:
1137:
1133:
1129:
1085:
1065:
1062:
1057:
1053:
1049:
1026:
1006:
994:
993:
982:
979:
976:
969:
964:
960:
957:
952:
948:
943:
937:
933:
930:
925:
922:
919:
915:
910:
902:
899:
896:
892:
868:
845:
842:
839:
816:
796:
791:
787:
783:
769:big O notation
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:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8566:
8555:
8552:
8550:
8547:
8546:
8544:
8529:
8526:
8524:
8521:
8519:
8516:
8514:
8511:
8509:
8506:
8504:
8501:
8499:
8496:
8494:
8491:
8489:
8486:
8484:
8481:
8479:
8476:
8474:
8471:
8469:
8466:
8464:
8461:
8459:
8456:
8455:
8453:
8449:
8443:
8440:
8438:
8435:
8433:
8430:
8428:
8425:
8424:
8422:
8418:
8408:
8405:
8403:
8400:
8398:
8395:
8393:
8390:
8388:
8385:
8383:
8380:
8378:
8375:
8373:
8370:
8368:
8365:
8363:
8360:
8358:
8355:
8353:
8350:
8348:
8345:
8343:
8340:
8338:
8335:
8334:
8332:
8328:
8322:
8319:
8317:
8314:
8311:
8308:
8306:
8303:
8301:
8298:
8296:
8293:
8291:
8288:
8286:
8283:
8281:
8278:
8276:
8273:
8271:
8268:
8267:
8265:
8261:
8255:
8252:
8250:
8247:
8245:
8242:
8240:
8237:
8236:
8234:
8230:
8227:
8223:
8213:
8210:
8206:
8203:
8202:
8201:
8198:
8195:
8192:
8191:
8189:
8185:
8179:
8176:
8174:
8171:
8169:
8166:
8164:
8161:
8159:
8156:
8154:
8151:
8149:
8146:
8144:
8141:
8139:
8136:
8134:
8131:
8130:
8128:
8124:
8118:
8115:
8113:
8110:
8108:
8105:
8103:
8100:
8098:
8095:
8093:
8090:
8088:
8085:
8083:
8080:
8078:
8075:
8073:
8070:
8069:
8067:
8063:
8060:
8056:
8052:
8045:
8040:
8038:
8033:
8031:
8026:
8025:
8022:
8013:
8007:
8003:
7999:
7994:
7993:
7992:
7990:
7986:
7979:
7978:0-534-38216-9
7975:
7971:
7967:
7966:
7965:
7959:
7958:0-521-00794-1
7955:
7951:
7947:
7944:
7941:
7940:0-8176-3895-4
7937:
7933:
7929:
7928:
7927:
7921:
7920:0-19-850279-6
7917:
7913:
7909:
7906:
7905:
7904:
7898:
7880:
7873:
7870:
7859:
7852:
7845:
7842:
7831:
7827:
7820:
7817:
7812:
7806:
7802:
7798:
7791:
7788:
7777:
7770:
7763:
7760:
7749:
7745:
7741:
7737:
7733:
7729:
7722:
7715:
7713:
7709:
7698:
7694:
7690:
7686:
7682:
7678:
7671:
7664:
7661:
7650:
7643:
7636:
7633:
7622:
7615:
7608:
7605:
7600:
7594:
7590:
7583:
7581:
7579:
7577:
7575:
7573:
7571:
7567:
7556:
7552:
7545:
7542:
7535:
7533:
7519:
7516:
7510:
7507:
7502:
7498:
7490:
7483:
7480:
7475:
7471:
7439:
7435:
7406:
7402:
7390:
7386:
7382:
7378:
7372:
7364:
7362:
7348:
7345:
7342:
7317:
7313:
7306:
7299:converges to
7281:
7277:
7250:
7242:
7238:
7224:
7221:
7218:
7213:
7207:
7203:
7197:
7193:
7189:
7183:
7170:
7166:
7159:
7156:
7151:
7147:
7138:
7135:
7128:
7127:
7126:
7123:
7110:
7106:
7102:
7099:
7096:
7093:
7088:
7084:
7078:
7074:
7068:
7063:
7059:
7053:
7050:
7047:
7044:
7041:
7038:
7034:
7028:
7024:
7020:
7015:
7010:
7006:
7003:
7000:
6997:
6994:
6989:
6983:
6979:
6973:
6969:
6962:
6959:
6956:
6953:
6950:
6946:
6939:
6935:
6931:
6926:
6921:
6914:
6911:
6908:
6902:
6899:
6895:
6888:
6884:
6880:
6874:
6871:
6868:
6865:
6859:
6856:
6851:
6847:
6843:
6837:
6834:
6828:
6825:
6817:
6813:
6806:
6786:
6783:
6780:
6777:
6757:
6754:
6746:
6727:
6724:
6721:
6715:
6712:
6707:
6703:
6699:
6693:
6687:
6684:
6681:
6672:
6659:
6655:
6651:
6648:
6645:
6642:
6639:
6634:
6630:
6624:
6620:
6614:
6607:
6604:
6601:
6595:
6589:
6586:
6583:
6580:
6577:
6574:
6570:
6564:
6560:
6556:
6551:
6543:
6540:
6537:
6534:
6526:
6522:
6518:
6513:
6509:
6500:
6499:
6481:
6477:
6473:
6467:
6461:
6438:
6432:
6429:
6426:
6423:
6415:
6411:
6407:
6402:
6399:
6396:
6392:
6384:
6383:
6382:
6381:
6362:
6357:
6353:
6349:
6346:
6343:
6338:
6332:
6328:
6324:
6319:
6316:
6313:
6309:
6298:
6297:
6296:
6294:
6276:
6272:
6268:
6262:
6256:
6233:
6230:
6227:
6224:
6218:
6215:
6210:
6207:
6197:
6196:
6195:
6189:
6187:
6185:
6169:
6161:
6145:
6122:
6114:
6110:
6096:
6083:
6079:
6072:
6069:
6064:
6060:
6051:
6048:
6041:
6040:
6039:
6022:
6007:
6002:
5992:
5985:
5982:
5968:
5963:
5953:
5946:
5943:
5937:
5934:
5927:
5926:
5925:
5905:
5878:
5851:
5824:
5814:
5812:
5807:
5805:
5784:
5780:
5766:
5753:
5749:
5742:
5739:
5734:
5730:
5701:
5698:
5688:
5684:
5680:
5677:
5664:
5660:
5653:
5650:
5645:
5641:
5628:
5627:
5626:
5624:
5620:
5599:
5595:
5588:
5563:
5559:
5546:
5532:
5512:
5489:
5483:
5480:
5477:
5457:
5437:
5417:
5414:
5411:
5408:
5403:
5400:
5397:
5393:
5389:
5384:
5381:
5378:
5374:
5351:
5347:
5325:
5322:
5319:
5316:
5311:
5307:
5303:
5298:
5294:
5290:
5285:
5281:
5277:
5272:
5268:
5258:
5254:
5235:
5229:
5226:
5223:
5208:
5205:
5197:
5187:
5182:
5168:
5148:
5138:
5134:
5133:
5126:
5117:
5116:
5107:
5104:
5096:
5086:
5082:
5076:
5075:
5070:This section
5068:
5064:
5059:
5058:
5052:
5050:
5036:
5027:
5014:
5007:
4998:
4995:
4992:
4988:
4984:
4979:
4976:
4973:
4969:
4961:
4958:
4955:
4951:
4947:
4942:
4938:
4930:
4926:
4923:
4917:
4908:
4905:
4902:
4898:
4894:
4889:
4885:
4877:
4873:
4869:
4864:
4861:
4858:
4854:
4846:
4842:
4839:
4833:
4830:
4810:
4798:
4796:
4794:
4790:
4789:neighborhoods
4786:
4782:
4766:
4763:
4752:
4745:
4742:
4716:
4713:
4702:
4695:
4692:
4666:
4663:
4652:
4645:
4642:
4628:
4610:
4606:
4597:
4593:
4577:
4574:
4563:
4556:
4553:
4527:
4524:
4521:
4515:
4509:
4501:
4498:
4494:
4490:
4486:
4482:
4478:
4457:
4453:
4446:
4443:
4438:
4435:
4432:
4428:
4415:
4409:
4405:
4400:
4396:
4391:
4387:
4382:
4378:
4373:
4369:
4355:
4352:
4346:
4343:
4340:
4336:
4331:
4328:
4325:
4320:
4317:
4312:
4307:
4304:
4299:
4294:
4291:
4286:
4281:
4278:
4273:
4268:
4265:
4260:
4257:
4254:
4246:
4242:
4229:
4215:
4212:
4203:
4199:
4194:
4190:
4185:
4182:
4179:
4173:
4169:
4166:
4162:
4157:
4152:
4149:
4144:
4139:
4136:
4131:
4126:
4123:
4118:
4113:
4110:
4105:
4097:
4093:
4080:
4066:
4060:
4050:
4014:
4010:
4006:
3999:
3995:
3971:
3968:
3945:
3935:
3919:
3916:
3913:
3907:
3902:
3899:
3894:
3889:
3884:
3880:
3877:
3874:
3871:
3866:
3863:
3858:
3853:
3850:
3845:
3840:
3837:
3832:
3827:
3824:
3819:
3816:
3813:
3810:
3807:
3799:
3795:
3782:
3769:
3766:
3758:
3750:
3742:
3739:
3736:
3728:
3725:
3718:
3697:
3667:
3663:
3659:
3654:
3649:
3646:
3643:
3639:
3628:
3607:
3577:
3573:
3569:
3543:
3540:
3537:
3534:
3528:
3525:
3522:
3514:
3511:
3502:
3488:
3484:
3480:
3477:
3474:
3449:
3445:
3432:
3419:
3414:
3411:
3406:
3399:
3396:
3393:
3389:
3383:
3379:
3365:
3357:
3351:
3347:
3344:
3339:
3335:
3330:
3326:
3322:
3317:
3313:
3310:
3305:
3302:
3299:
3295:
3290:
3286:
3282:
3268:
3255:
3241:
3238:
3235:
3228:converges to
3215:
3212:
3206:
3202:
3196:
3192:
3189:
3186:
3183:
3178:
3175:
3170:
3165:
3162:
3157:
3152:
3149:
3144:
3139:
3136:
3131:
3126:
3123:
3118:
3115:
3112:
3104:
3100:
3089:
3081:
3079:
3061:
3057:
3053:
3025:
3021:
3017:
2988:
2984:
2971:
2952:
2948:
2924:
2921:for all
2913:
2909:
2905:
2897:
2894:
2889:
2885:
2854:
2850:
2838:
2822:
2797:
2793:
2780:
2766:
2758:
2739:
2716:
2713:
2707:
2702:
2699:
2694:
2689:
2684:
2680:
2677:
2674:
2671:
2666:
2663:
2658:
2653:
2650:
2645:
2640:
2637:
2632:
2627:
2624:
2619:
2616:
2613:
2610:
2607:
2599:
2595:
2579:R-convergence
2578:
2576:
2563:
2560:
2547:
2544:
2541:
2537:
2533:
2528:
2524:
2506:
2502:
2498:
2493:
2490:
2487:
2483:
2463:
2450:
2437:
2411:
2407:
2383:
2380:
2369:
2366:
2361:
2357:
2341:
2338:
2333:
2330:
2327:
2323:
2303:
2290:
2286:
2270:
2267:
2264:
2253:
2250:
2245:
2241:
2225:
2222:
2217:
2214:
2211:
2207:
2187:
2162:
2159:
2156:
2147:
2146:
2130:
2110:
2107:
2104:
2096:
2092:
2076:
2056:
2053:
2050:
2042:
2029:
2019:
2000:
1997:
1994:
1988:
1985:
1965:
1962:
1959:
1951:
1947:
1943:
1942:secant method
1927:
1918:
1916:
1899:
1896:
1893:
1882:
1879:
1874:
1870:
1854:
1851:
1846:
1843:
1840:
1836:
1816:
1791:
1788:
1785:
1759:
1756:
1750:
1747:
1727:
1724:
1721:
1698:
1695:
1692:
1686:
1683:
1674:
1661:
1658:
1650:
1640:
1637:
1632:
1628:
1612:
1609:
1604:
1601:
1598:
1594:
1574:
1561:
1547:
1540:
1536:
1523:
1503:
1481:
1456:
1452:
1441:
1433:
1428:
1426:
1424:
1420:
1416:
1412:
1408:
1394:
1374:
1366:
1362:
1358:
1354:
1350:
1346:
1341:
1328:
1325:
1322:
1315:
1310:
1306:
1301:
1297:
1294:
1289:
1285:
1280:
1266:
1253:
1239:
1232:
1216:
1209:
1188:
1184:
1160:
1135:
1131:
1118:
1114:
1109:
1107:
1103:
1099:
1083:
1063:
1060:
1055:
1051:
1047:
1038:
1024:
1004:
980:
977:
974:
967:
962:
958:
955:
950:
946:
941:
935:
931:
928:
923:
920:
917:
913:
908:
894:
882:
881:
880:
866:
859:
843:
840:
837:
830:
814:
789:
785:
772:
770:
766:
762:
758:
752:
749:
744:
740:
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:
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:
340:
332:
329:
325:
322:
321:
320:
317:
316:
312:
306:
305:
299:
298:
287:
284:
283:
279:
276:
274:
271:
270:
268:
267:
262:
256:
252:
249:
247:
244:
241:
239:
236:
235:
231:
228:
227:
226:
225:
220:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
180:
178:
177:
169:
168:
163:
158:
148:
145:
143:
140:
139:
138:
137:
134:
130:
125:
122:
120:
117:
115:
112:
111:
110:
109:
106:
102:
97:
94:
92:
88:
86:
83:
81:
78:
76:
73:
72:
71:
70:
64:
61:
59:
56:
55:
52:
44:
43:
38:
35:
31:
19:
8523:Martin Kutta
8478:Ămile Picard
8458:Isaac Newton
8372:Euler method
8342:Substitution
8299:
7997:
7991:are used in
7988:
7984:
7982:
7969:
7963:
7949:
7931:
7925:
7911:
7902:
7885:. Retrieved
7872:
7861:. Retrieved
7857:
7844:
7833:. Retrieved
7830:math.unl.edu
7829:
7819:
7796:
7790:
7779:. Retrieved
7775:
7762:
7751:. Retrieved
7731:
7727:
7700:. Retrieved
7680:
7676:
7663:
7652:. Retrieved
7648:
7635:
7624:. Retrieved
7620:
7607:
7588:
7558:. Retrieved
7554:
7544:
7374:
7265:
7124:
6673:
6501:
6453:
6377:
6248:
6193:
6137:
6037:
5815:
5808:
5716:
5622:
5618:
5547:
5215:
5200:
5191:
5184:Please help
5140:
5129:
5099:
5090:
5079:Please help
5074:verification
5071:
5028:
4802:
4792:
4784:
4626:
4595:
4499:
4488:
4419:
4407:
4403:
4398:
4394:
4389:
4385:
4380:
4376:
4230:
4081:
3783:
3503:
3433:
3256:
3085:
2972:
2836:
2781:
2582:
2429:
2288:
2284:
2148:
2144:
2094:
2090:
2021:
2017:
1950:golden ratio
1919:
1675:
1562:
1538:
1495:
1437:
1413:
1409:
1349:uniform norm
1342:
1254:
1230:
1207:
1117:regular grid
1110:
1105:
1101:
1097:
1039:
995:
857:
828:
773:
753:
734:
730:
720:
670:Ămile Picard
655:Martin Kutta
645:George Green
605:Isaac Newton
443:
437: /
433: /
253: /
119:Chaos theory
8280:Phase space
8138:Homogeneous
5617:with order
5194:August 2020
5188:if you can.
5093:August 2020
4497:fixed point
2835:is said to
2759:applied to
2396:A sequence
1946:simple root
1537:and with a
1108:(e.g., ).
563:RungeâKutta
308:Difference
251:Homogeneous
63:Engineering
8543:Categories
8508:John Crank
8337:Inspection
8200:Stochastic
8194:Difference
8168:Autonomous
8112:Non-linear
8102:Fractional
8065:Operations
7983:The terms
7946:Endre SĂŒli
7899:Literature
7887:2024-02-09
7863:2020-08-07
7858:gordon.edu
7835:2020-07-31
7781:2022-12-13
7753:2020-08-02
7702:2020-07-31
7654:2020-12-13
7626:2020-08-07
7621:gordon.edu
7560:2020-07-31
7536:References
6138:The error
5625:such that
4540:such that
4495:, given a
3932:using the
2872:such that
2143:is called
2089:is called
2016:is called
1361:basis sets
748:asymptotic
680:John Crank
481:Inspection
435:Asymptotic
319:Stochastic
238:Autonomous
213:Non-linear
203:Fractional
8312:solutions
8270:Wronskian
8225:Solutions
8153:Decoupled
8117:Holonomic
7697:116192710
7508:−
7481:−
7222:…
7204:κ
7157:−
7085:κ
7051:κ
7042:−
6980:κ
6960:κ
6954:−
6912:κ
6909:−
6903:
6869:κ
6866:−
6860:
6784:≪
6781:κ
6758:κ
6725:κ
6722:−
6716:
6631:κ
6605:−
6587:κ
6578:−
6544:κ
6538:−
6433:κ
6427:−
6350:κ
6347:−
6325:−
6231:κ
6228:−
6070:−
5986:
5947:
5938:≈
5740:−
5651:−
5401:−
5382:−
5255:, or the
4996:−
4985:−
4977:−
4959:−
4948:−
4927:
4906:−
4895:−
4870:−
4843:
4834:≈
4475:, called
4356:…
4329:…
4216:…
4183:…
4018:⌋
4004:⌊
3949:⌋
3943:⌊
3917:…
3875:…
3743:∈
3729:∈
3640:∑
3535:−
3529:∈
3515:∈
3475:μ
3372:∞
3369:→
3345:−
3311:−
3275:∞
3272:→
3216:…
3187:…
3054:ε
3018:ε
2985:ε
2949:ε
2910:ε
2906:≤
2895:−
2851:ε
2743:⌋
2737:⌊
2717:…
2675:…
2545:−
2534:−
2499:−
2470:∞
2467:→
2367:−
2339:−
2310:∞
2307:→
2251:−
2223:−
2194:∞
2191:→
2131:μ
2077:μ
1989:∈
1986:μ
1880:−
1852:−
1823:∞
1820:→
1763:∞
1751:∈
1748:μ
1687:∈
1684:μ
1662:μ
1638:−
1610:−
1581:∞
1578:→
1548:μ
1375:μ
1326:μ
1295:−
1273:∞
1270:→
1240:μ
1084:μ
1064:μ
1061:
1048:−
1025:μ
978:μ
956:−
929:−
901:∞
898:→
867:μ
841:≥
421:Wronskian
399:Dirichlet
142:Economics
85:Chemistry
75:Astronomy
8420:Examples
8310:Integral
8082:Ordinary
7989:R-linear
7985:Q-linear
7910:(2002),
5130:require
4746:′
4696:″
4646:′
4557:′
3908:⌋
3895:⌊
3082:Examples
3062:′
3026:′
2708:⌋
2695:⌊
2123:and any
2069:and any
1440:sequence
739:sequence
531:Galerkin
431:Lyapunov
342:Solution
286:Notation
278:Operator
264:Features
183:Ordinary
8148:Coupled
8087:Partial
7748:2153571
7555:hmc.edu
5132:cleanup
2755:is the
1917:below.
1359:or the
404:Neumann
188:Partial
96:Geology
91:Biology
80:Physics
8163:Degree
8107:Linear
8008:
7976:
7956:
7938:
7918:
7807:
7746:
7695:
7595:
6454:Given
5802:using
4402:, and
1419:series
1345:metric
729:, the
591:People
503:
450:Series
208:Linear
47:Fields
8554:Rates
8212:Delay
8158:Order
7882:(PDF)
7854:(PDF)
7772:(PDF)
7744:JSTOR
7724:(PDF)
7693:S2CID
7673:(PDF)
7645:(PDF)
7617:(PDF)
5049:see.
4629:. If
3434:Thus
1355:of a
1106:order
743:limit
737:of a
491:Euler
409:Robin
331:Delay
273:Order
246:Exact
172:Types
40:Scope
8006:ISBN
7987:and
7974:ISBN
7954:ISBN
7936:ISBN
7916:ISBN
7805:ISBN
7593:ISBN
6770:for
5897:and
5843:and
5678:<
4764:>
4714:<
4679:and
4575:<
3767:<
3086:The
2937:and
2779:).
2160:>
1978:and
1789:>
1740:and
1102:rate
879:if
733:and
598:List
7736:doi
7685:doi
7465:lim
7266:so
6900:exp
6857:exp
6747:in
6713:exp
6012:old
5997:new
5983:log
5973:old
5958:new
5944:log
5910:old
5883:new
5856:old
5829:new
5083:by
4924:log
4840:log
4491:is
4174:536
4153:256
3362:lim
3265:lim
2460:lim
2300:lim
2184:lim
1813:lim
1778:if
1714:if
1571:lim
1560:if
1516:to
1363:in
1263:lim
1252:if
1052:log
891:lim
721:In
8545::
7856:.
7828:.
7803:.
7774:.
7742:.
7732:63
7730:.
7726:.
7711:^
7691:.
7681:63
7679:.
7675:.
7647:.
7619:.
7569:^
7553:.
7361:.
6799::
5806:.
5545:.
4823::
4502:,
4444::=
4393:,
4384:,
4167:65
4140:16
3867:16
3854:16
3770:1.
3179:32
3166:16
2667:16
2654:16
2564:1.
2384:1.
1425:.
1056:10
8043:e
8036:t
8029:v
8016:.
8014:.
7960:.
7942:.
7922:.
7890:.
7866:.
7838:.
7813:.
7784:.
7756:.
7738::
7705:.
7687::
7657:.
7629:.
7601:.
7563:.
7520:0
7517:=
7514:)
7511:L
7503:n
7499:x
7495:(
7491:/
7487:)
7484:L
7476:n
7472:a
7468:(
7445:)
7440:n
7436:a
7432:(
7412:)
7407:n
7403:x
7399:(
7349:2
7346:=
7343:q
7323:)
7318:n
7314:x
7310:(
7307:f
7287:)
7282:n
7278:y
7274:(
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7248:)
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7239:h
7235:(
7230:O
7225:=
7219:+
7214:2
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7190:n
7184:=
7180:|
7176:)
7171:n
7167:x
7163:(
7160:f
7152:n
7148:y
7143:|
7139:=
7136:e
7111:.
7107:)
7103:.
7100:.
7097:.
7094:+
7089:2
7079:2
7075:h
7069:2
7064:2
7060:n
7054:+
7048:h
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7039:1
7035:(
7029:0
7025:y
7021:=
7016:n
7011:)
7007:.
7004:.
7001:.
6998:.
6995:+
6990:2
6984:2
6974:2
6970:h
6963:+
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6951:1
6947:(
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6936:y
6932:=
6927:n
6922:]
6918:)
6915:h
6906:(
6896:[
6889:0
6885:y
6881:=
6878:)
6875:h
6872:n
6863:(
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6844:=
6841:)
6838:h
6835:n
6832:(
6829:f
6826:=
6823:)
6818:n
6814:x
6810:(
6807:f
6787:1
6778:h
6755:h
6731:)
6728:x
6719:(
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6700:=
6697:)
6694:x
6691:(
6688:f
6685:=
6682:y
6660:.
6656:)
6652:.
6649:.
6646:.
6643:.
6640:+
6635:2
6625:2
6621:h
6615:2
6611:)
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6599:(
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6590:+
6584:h
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6571:(
6565:0
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6557:=
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6548:)
6541:h
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6471:)
6468:0
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6439:.
6436:)
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6408:=
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6339:h
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6320:1
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6314:n
6310:y
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6273:y
6269:=
6266:)
6263:0
6260:(
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6225:=
6219:x
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6211:y
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6123:.
6120:)
6115:q
6111:h
6107:(
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6097:=
6093:|
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6052:=
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6003:/
5993:h
5989:(
5978:)
5969:e
5964:/
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5767:=
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5702:.
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5415:.
5412:.
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5224:y
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5015:.
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4664:=
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