2854:
4221:
can lead to sensitivity to apparently arbitrary choices of measurement units. For example, if smoothing with respect to distance and time an isotropic smoother will give different results if distance is measure in metres and time in seconds, to what will occur if we change the units to centimetres and hours.
4220:
co-ordinate system the estimate will not change, but also that we are assuming that the same level of smoothing is appropriate in all directions. This is often considered reasonable when smoothing with respect to spatial location, but in many other cases isotropy is not an appropriate assumption and
4190:
The thin plate spline approach can be generalized to smoothing with respect to more than two dimensions and to other orders of differentiation in the penalty. As the dimension increases there are some restrictions on the smallest order of differential that can be used, but actually Duchon's original
4224:
The second class of generalizations to multi-dimensional smoothing deals directly with this scale invariance issue using tensor product spline constructions. Such splines have smoothing penalties with multiple smoothing parameters, which is the price that must be paid for not assuming that the same
752:
is a smoothing parameter, controlling the trade-off between fidelity to the data and roughness of the function estimate. This is often estimated by generalized cross-validation, or by restricted marginal likelihood (REML) which exploits the link between spline smoothing and
Bayesian estimation (the
4185:
2657:
718:
3490:
2370:
1564:
1183:
913:
objective in which the sum of squares terms is replaced by another log-likelihood based measure of fidelity to the data. The sum of squares term corresponds to penalized likelihood with a
Gaussian assumption on the
1423:
1641:
2576:
1243:
1948:
2849:{\displaystyle p\sum _{i=1}^{n}\left({\frac {Y_{i}-{\hat {f}}\left(x_{i}\right)}{\delta _{i}}}\right)^{2}+\left(1-p\right)\int \left({\hat {f}}^{\left(m\right)}\left(x\right)\right)^{2}\,dx}
2642:
1021:
4554:
J. Duchon, 1976, Splines minimizing rotation invariant semi-norms in
Sobolev spaces. pp 85–100, In: Constructive Theory of Functions of Several Variables, Oberwolfach 1976, W. Schempp and
3860:
455:
393:
2438:
2114:
1791:
574:
2249:
2014:
1311:
2960:
2190:
3377:
1858:
863:
190:
2995:
3852:
750:
1062:
939:
830:
482:
255:
78:
3566:
295:
4246:. In this method, the data is fitted to a set of spline basis functions with a reduced set of knots, typically by least squares. No roughness penalty is used. (See also
3369:
3260:
3191:
2499:
2470:
1728:
511:
566:
1462:
141:
4276:
penalty for approximation error with the bending and stretching penalty of the approximating manifold and uses the coarse discretization of the optimization problem.
3804:
3675:
3593:
3022:
1699:
1672:
904:
801:
217:
105:
4218:
3701:
3519:
3122:
3092:
3769:
3749:
3721:
3613:
3539:
3340:
3320:
3300:
3280:
3231:
3211:
3162:
3142:
3066:
3046:
2877:
771:
531:
315:
2909:
2260:
4247:
1078:
4651:
4607:
4504:
4355:
1319:
1470:
3771:. The first approach simply generalizes the spline smoothing penalty to the multidimensional setting. For example, if trying to estimate
876:
is the most common in modern statistics literature, although the method can easily be adapted to penalties based on other derivatives.
297:
data. The most familiar example is the cubic smoothing spline, but there are many other possibilities, including for the case where
2651:
De Boor's approach exploits the same idea, of finding a balance between having a smooth curve and being close to the given data.
2504:
4706:
4701:
4476:
1863:
776:
The integral is often evaluated over the whole real line although it is also possible to restrict the range to that of
4180:{\displaystyle \sum _{i=1}^{n}\{y_{i}-{\hat {f}}(x_{i},z_{i})\}^{2}+\lambda \int \left{\textrm {d}}x\,{\textrm {d}}z.}
2581:
957:
713:{\displaystyle \sum _{i=1}^{n}\{Y_{i}-{\hat {f}}(x_{i})\}^{2}+\lambda \int {\hat {f}}^{\prime \prime }(x)^{2}\,dx.}
1188:
398:
328:
2378:
2034:
1576:
1747:
3485:{\displaystyle \sum _{i=1}^{n}\left({\frac {Y_{i}-{\hat {f}}\left(x_{i}\right)}{\delta _{i}}}\right)^{2}\leq S}
2195:
1953:
1252:
2914:
2119:
1812:
866:
4256:. This combines the reduced knots of regression splines, with the roughness penalty of smoothing splines.
842:
1185:
of fitted values, the sum-of-squares part of the spline criterion is fixed. It remains only to minimize
150:
2965:
4301:
4286:
3813:
1246:
833:
33:
729:
1029:
917:
809:
460:
222:
45:
4555:
4260:
3544:
260:
3345:
3236:
3167:
2475:
2446:
1704:
487:
4400:"A Correspondence between Bayesian Estimation on Stochastic Processes and Smoothing by Splines"
539:
4647:
4603:
4525:
4500:
4470:
4351:
4269:
3807:
3731:
There are two main classes of method for generalizing from smoothing with respect to a scalar
3095:
1431:
873:
865:(infinite smoothing), the roughness penalty becomes paramount and the estimate converges to a
110:
4537:
4411:
4380:
3774:
3618:
3571:
3000:
1677:
1650:
882:
779:
195:
83:
4290:
144:
4197:
3680:
3498:
3101:
3071:
3754:
3734:
3706:
3598:
3524:
3325:
3305:
3285:
3265:
3216:
3196:
3147:
3127:
3051:
3031:
2862:
2365:{\displaystyle \{Y-{\hat {m}}\}^{T}\{Y-{\hat {m}}\}+\lambda {\hat {m}}^{T}A{\hat {m}},}
756:
516:
300:
4623:
Eilers, P.H.C. and Marx B. (1996). "Flexible smoothing with B-splines and penalties".
3262:
converges to a straight line (the smoothest curve). Since finding a suitable value of
2882:
906:, second or third-order differences were used in the penalty, rather than derivatives.
4695:
4273:
4234:
3025:
1464:
are a set of spline basis functions. As a result, the roughness penalty has the form
534:
4451:
4333:
Nonparametric
Regression and Generalized Linear Models: A roughness penalty approach
4191:
paper, gives slightly more complicated penalties that can avoid this restriction.
2962:
are the quantities controlling the extent of smoothing (they represent the weight
17:
1313:. This interpolating spline is a linear operator, and can be written in the form
484:
are independent, zero mean random variables. The cubic smoothing spline estimate
4265:
4416:
4399:
4371:
Craven, P.; Wahba, G. (1979). "Smoothing noisy data with spline functions".
1178:{\displaystyle {\hat {m}}=({\hat {f}}(x_{1}),\ldots ,{\hat {f}}(x_{n}))^{T}}
2254:
Now back to the first step. The penalized sum-of-squares can be written as
4304:. The updated sources are available also on Carl de Boor's official site
3723:
means we obtain a smoother curve by getting farther from the given data.
4541:
4384:
4298:
909:
The penalized sum of squares smoothing objective can be replaced by a
1418:{\displaystyle {\hat {f}}(x)=\sum _{i=1}^{n}{\hat {f}}(x_{i})f_{i}(x)}
4194:
The thin plate splines are isotropic, meaning that if we rotate the
1559:{\displaystyle \int {\hat {f}}''(x)^{2}dx={\hat {m}}^{T}A{\hat {m}}.}
4305:
3193:
converges to the "natural" spline interpolant to the given data. As
3703:
means the solution is the "natural" spline interpolant. Increasing
753:
smoothing penalty can be viewed as being induced by a prior on the
950:
It is useful to think of fitting a smoothing spline in two steps:
4558:, eds., Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977
2879:
is a parameter called smooth factor and belongs to the interval
4600:
Generalized
Additive Models: An Introduction with R (2nd ed)
679:
676:
4431:
Whittaker, E.T. (1922). "On a new method of graduation".
2571:{\displaystyle -2\{Y-{\hat {m}}\}+2\lambda A{\hat {m}}=0}
1647:, depend on the configuration of the predictor variables
2251:, the distances between successive knots (or x values).
4225:
degree of smoothness is appropriate in all directions.
4679:
Nonparametric
Regression and Generalized Linear Models
832:(no smoothing), the smoothing spline converges to the
4239:
Smoothing splines are related to, but distinct from:
4200:
3863:
3816:
3777:
3757:
3737:
3709:
3683:
3621:
3601:
3574:
3547:
3527:
3501:
3380:
3348:
3328:
3308:
3288:
3268:
3239:
3219:
3199:
3170:
3150:
3130:
3104:
3074:
3054:
3034:
3003:
2968:
2917:
2885:
2865:
2660:
2584:
2507:
2478:
2449:
2381:
2263:
2198:
2122:
2037:
1956:
1866:
1815:
1750:
1707:
1680:
1653:
1579:
1473:
1434:
1322:
1255:
1191:
1081:
1032:
960:
920:
885:
845:
812:
782:
759:
732:
577:
542:
519:
490:
463:
401:
331:
303:
263:
225:
219:
with a derivative based measure of the smoothness of
198:
153:
113:
86:
48:
4642:
Ruppert, David; Wand, M. P.; Carroll, R. J. (2003).
3282:is a task of trial and error, a redundant constant
4212:
4179:
3846:
3798:
3763:
3743:
3715:
3695:
3669:
3607:
3587:
3560:
3533:
3513:
3484:
3363:
3334:
3314:
3294:
3274:
3254:
3225:
3205:
3185:
3156:
3136:
3116:
3086:
3060:
3040:
3016:
2989:
2954:
2903:
2871:
2848:
2636:
2570:
2493:
2464:
2432:
2364:
2243:
2184:
2108:
2008:
1943:{\displaystyle \Delta _{i,i+1}=-1/h_{i}-1/h_{i+1}}
1942:
1852:
1785:
1722:
1693:
1666:
1635:
1558:
1456:
1417:
1305:
1237:
1177:
1056:
1015:
933:
898:
857:
824:
795:
765:
744:
712:
560:
525:
505:
476:
449:
395:be a set of observations, modeled by the relation
387:
309:
289:
249:
211:
184:
135:
99:
72:
4433:Proceedings of the Edinburgh Mathematical Society
879:In early literature, with equally-spaced ordered
3568:is an estimation of the standard deviation for
3495:The algorithm described by de Boor starts with
2637:{\displaystyle {\hat {m}}=(I+\lambda A)^{-1}Y.}
1016:{\displaystyle {\hat {f}}(x_{i});i=1,\ldots ,n}
4686:A Practical Guide to Splines (Revised Edition)
4497:A Practical Guide to Splines (Revised Edition)
3322:is used to numerically determine the value of
533:is defined to be the unique minimizer, in the
4520:
4518:
4516:
2029:symmetric tri-diagonal matrix with elements:
8:
4490:
4488:
4486:
4289:smoothing can be found in the examples from
3943:
3885:
3615:is recommended to be chosen in the interval
2535:
2514:
2316:
2295:
2286:
2264:
1807:matrix of second differences with elements:
1643:. The basis functions, and hence the matrix
1238:{\displaystyle \int {\hat {f}}''(x)^{2}\,dx}
644:
599:
450:{\displaystyle Y_{i}=f(x_{i})+\epsilon _{i}}
388:{\displaystyle \{x_{i},Y_{i}:i=1,\dots ,n\}}
382:
332:
80:, obtained from a set of noisy observations
2433:{\displaystyle Y=(Y_{1},\ldots ,Y_{n})^{T}}
2109:{\displaystyle W_{i-1,i}=W_{i,i-1}=h_{i}/6}
1636:{\displaystyle \int f_{i}''(x)f_{j}''(x)dx}
257:. They provide a means for smoothing noisy
27:Method of smoothing using a spline function
4677:Green, P. J. and Silverman, B. W. (1994).
1786:{\displaystyle A=\Delta ^{T}W^{-1}\Delta }
4528:(1967). "Smoothing by Spline Functions".
4452:"Smoothing and Non-Parametric Regression"
4415:
4346:Hastie, T. J.; Tibshirani, R. J. (1990).
4199:
4165:
4164:
4163:
4154:
4153:
4142:
4129:
4109:
4108:
4102:
4095:
4081:
4049:
4048:
4042:
4035:
4018:
4005:
3985:
3984:
3978:
3971:
3946:
3933:
3920:
3902:
3901:
3892:
3879:
3868:
3862:
3818:
3817:
3815:
3776:
3756:
3736:
3708:
3682:
3652:
3633:
3620:
3600:
3579:
3573:
3552:
3546:
3526:
3500:
3470:
3458:
3443:
3424:
3423:
3414:
3407:
3396:
3385:
3379:
3350:
3349:
3347:
3327:
3307:
3287:
3267:
3241:
3240:
3238:
3218:
3198:
3172:
3171:
3169:
3149:
3129:
3103:
3073:
3053:
3033:
3008:
3002:
2978:
2973:
2967:
2922:
2916:
2884:
2864:
2839:
2833:
2803:
2792:
2791:
2753:
2741:
2726:
2707:
2706:
2697:
2690:
2679:
2668:
2659:
2619:
2586:
2585:
2583:
2551:
2550:
2524:
2523:
2506:
2480:
2479:
2477:
2451:
2450:
2448:
2424:
2414:
2395:
2380:
2348:
2347:
2338:
2327:
2326:
2305:
2304:
2289:
2274:
2273:
2262:
2244:{\displaystyle h_{i}=\xi _{i+1}-\xi _{i}}
2235:
2216:
2203:
2197:
2174:
2159:
2146:
2127:
2121:
2098:
2092:
2067:
2042:
2036:
2009:{\displaystyle \Delta _{i,i+2}=1/h_{i+1}}
1994:
1985:
1961:
1955:
1928:
1919:
1907:
1898:
1871:
1865:
1844:
1835:
1820:
1814:
1771:
1761:
1749:
1709:
1708:
1706:
1685:
1679:
1658:
1652:
1609:
1587:
1578:
1542:
1541:
1532:
1521:
1520:
1504:
1479:
1478:
1472:
1439:
1433:
1400:
1387:
1369:
1368:
1362:
1351:
1324:
1323:
1321:
1306:{\displaystyle (x_{i},{\hat {f}}(x_{i}))}
1291:
1273:
1272:
1263:
1254:
1228:
1222:
1197:
1196:
1190:
1169:
1156:
1138:
1137:
1119:
1101:
1100:
1083:
1082:
1080:
1034:
1033:
1031:
980:
962:
961:
959:
925:
919:
890:
884:
844:
811:
787:
781:
758:
731:
700:
694:
675:
664:
663:
647:
634:
616:
615:
606:
593:
582:
576:
552:
547:
541:
518:
492:
491:
489:
468:
462:
441:
425:
406:
400:
352:
339:
330:
302:
281:
268:
262:
227:
226:
224:
203:
197:
173:
155:
154:
152:
124:
112:
91:
85:
50:
49:
47:
4248:multivariate adaptive regression splines
2955:{\displaystyle \delta _{i};i=1,\dots ,n}
2185:{\displaystyle W_{ii}=(h_{i}+h_{i+1})/3}
946:Derivation of the cubic smoothing spline
32:For broader coverage of this topic, see
4585:Smoothing Spline ANOVA Models (2nd ed.)
4316:
3751:to smoothing with respect to a vector
1245:, and the minimizer is a natural cubic
4468:
4331:Green, P. J.; Silverman, B.W. (1994).
4404:The Annals of Mathematical Statistics
7:
4672:Spline Models for Observational Data
4570:Spline Models for Observational Data
4326:
4324:
4322:
4320:
1853:{\displaystyle \Delta _{ii}=1/h_{i}}
4398:Kimeldorf, G.S.; Wahba, G. (1970).
872:The roughness penalty based on the
858:{\displaystyle \lambda \to \infty }
143:, in order to balance a measure of
4122:
4099:
4068:
4062:
4039:
3998:
3975:
1958:
1868:
1817:
1780:
1758:
1072:Now, treat the second step first.
852:
25:
4450:Rodriguez, German (Spring 2001).
185:{\displaystyle {\hat {f}}(x_{i})}
3302:was introduced for convenience.
2990:{\displaystyle \delta _{i}^{-2}}
4457:. 2.3.1 Computation. p. 12
3847:{\displaystyle {\hat {f}}(x,z)}
3541:until the condition is met. If
3371:meets the following condition:
4646:. Cambridge University Press.
4499:. Springer. pp. 207–214.
4114:
4054:
3990:
3939:
3913:
3907:
3841:
3829:
3823:
3793:
3781:
3429:
3355:
3246:
3177:
2898:
2886:
2797:
2712:
2616:
2600:
2591:
2556:
2529:
2485:
2456:
2421:
2388:
2353:
2332:
2310:
2279:
2171:
2139:
1714:
1624:
1618:
1602:
1596:
1547:
1526:
1501:
1494:
1484:
1451:
1445:
1412:
1406:
1393:
1380:
1374:
1341:
1335:
1329:
1300:
1297:
1284:
1278:
1256:
1219:
1212:
1202:
1166:
1162:
1149:
1143:
1125:
1112:
1106:
1097:
1088:
1051:
1045:
1039:
986:
973:
967:
849:
816:
745:{\displaystyle \lambda \geq 0}
691:
684:
669:
640:
627:
621:
497:
431:
418:
244:
238:
232:
179:
166:
160:
130:
117:
67:
61:
55:
1:
1249:that interpolates the points
1057:{\displaystyle {\hat {f}}(x)}
934:{\displaystyle \epsilon _{i}}
825:{\displaystyle \lambda \to 0}
477:{\displaystyle \epsilon _{i}}
250:{\displaystyle {\hat {f}}(x)}
73:{\displaystyle {\hat {f}}(x)}
4295:A Practical Guide to Splines
4348:Generalized Additive Models
4272:. This method combines the
3561:{\displaystyle \delta _{i}}
2472:by differentiating against
1674:, but not on the responses
290:{\displaystyle x_{i},y_{i}}
4723:
4602:. Chapman & Hall/CRC.
4297:. The examples are in the
4232:
3364:{\displaystyle {\hat {f}}}
3255:{\displaystyle {\hat {f}}}
3186:{\displaystyle {\hat {f}}}
2494:{\displaystyle {\hat {m}}}
2465:{\displaystyle {\hat {m}}}
1723:{\displaystyle {\hat {m}}}
1026:From these values, derive
568:on a compact interval, of
506:{\displaystyle {\hat {f}}}
31:
4644:Semiparametric Regression
954:First, derive the values
561:{\displaystyle W_{2}^{2}}
4475:: CS1 maint: location (
3727:Multidimensional splines
2646:
1457:{\displaystyle f_{i}(x)}
136:{\displaystyle f(x_{i})}
42:are function estimates,
4417:10.1214/aoms/1177697089
321:Cubic spline definition
4214:
4181:
3884:
3848:
3800:
3799:{\displaystyle f(x,z)}
3765:
3745:
3717:
3697:
3671:
3609:
3589:
3562:
3535:
3515:
3486:
3401:
3365:
3336:
3316:
3296:
3276:
3256:
3227:
3207:
3187:
3158:
3138:
3118:
3088:
3062:
3042:
3024:). In practice, since
3018:
2991:
2956:
2905:
2873:
2850:
2684:
2638:
2572:
2495:
2466:
2434:
2366:
2245:
2186:
2110:
2010:
1944:
1854:
1787:
1724:
1695:
1668:
1637:
1569:where the elements of
1560:
1458:
1419:
1367:
1307:
1239:
1179:
1058:
1017:
935:
900:
859:
826:
797:
767:
746:
714:
598:
562:
527:
507:
478:
451:
389:
317:is a vector quantity.
311:
291:
251:
213:
186:
137:
101:
74:
4707:Splines (mathematics)
4674:. SIAM, Philadelphia.
4530:Numerische Mathematik
4373:Numerische Mathematik
4215:
4182:
3864:
3849:
3810:penalty and find the
3801:
3766:
3746:
3718:
3698:
3672:
3670:{\displaystyle \left}
3610:
3590:
3588:{\displaystyle Y_{i}}
3563:
3536:
3516:
3487:
3381:
3366:
3342:so that the function
3337:
3317:
3297:
3277:
3257:
3228:
3208:
3188:
3159:
3139:
3119:
3089:
3063:
3043:
3019:
3017:{\displaystyle Y_{i}}
2992:
2957:
2906:
2874:
2851:
2664:
2639:
2573:
2496:
2467:
2435:
2367:
2246:
2187:
2111:
2011:
1945:
1855:
1788:
1725:
1696:
1694:{\displaystyle Y_{i}}
1669:
1667:{\displaystyle x_{i}}
1638:
1561:
1459:
1420:
1347:
1308:
1240:
1180:
1059:
1018:
936:
901:
899:{\displaystyle x_{i}}
860:
827:
798:
796:{\displaystyle x_{i}}
768:
747:
715:
578:
563:
528:
508:
479:
452:
390:
312:
292:
252:
214:
212:{\displaystyle y_{i}}
187:
138:
102:
100:{\displaystyle y_{i}}
75:
4684:De Boor, C. (2001).
4598:Wood, S. N. (2017).
4526:Reinsch, Christian H
4495:De Boor, C. (2001).
4350:. Chapman and Hall.
4302:programming language
4198:
3861:
3814:
3775:
3755:
3735:
3707:
3681:
3619:
3599:
3572:
3545:
3525:
3499:
3378:
3346:
3326:
3306:
3286:
3266:
3237:
3217:
3197:
3168:
3148:
3128:
3102:
3072:
3052:
3032:
3001:
2966:
2915:
2883:
2863:
2658:
2582:
2505:
2476:
2447:
2379:
2261:
2196:
2120:
2035:
1954:
1864:
1813:
1748:
1705:
1678:
1651:
1577:
1471:
1432:
1320:
1253:
1189:
1079:
1030:
958:
918:
911:penalized likelihood
883:
867:linear least squares
843:
834:interpolating spline
810:
780:
757:
730:
575:
540:
517:
488:
461:
399:
329:
301:
261:
223:
196:
151:
111:
84:
46:
34:Spline (mathematics)
4702:Regression analysis
4625:Statistical Science
4335:. Chapman and Hall.
4213:{\displaystyle x,z}
3696:{\displaystyle S=0}
3514:{\displaystyle p=0}
3117:{\displaystyle m=2}
3087:{\displaystyle m=2}
3068:. The solution for
2986:
2501:. This results in:
1617:
1595:
557:
4670:Wahba, G. (1990).
4583:Gu, Chong (2013).
4542:10.1007/BF02162161
4385:10.1007/bf01404567
4261:Thin plate splines
4244:Regression splines
4210:
4177:
3844:
3796:
3761:
3741:
3713:
3693:
3667:
3605:
3585:
3558:
3531:
3511:
3482:
3361:
3332:
3312:
3292:
3272:
3252:
3223:
3203:
3183:
3154:
3134:
3114:
3084:
3058:
3038:
3014:
2987:
2969:
2952:
2901:
2869:
2846:
2647:De Boor's approach
2634:
2568:
2491:
2462:
2430:
2362:
2241:
2182:
2106:
2006:
1940:
1850:
1783:
1720:
1691:
1664:
1633:
1605:
1583:
1556:
1454:
1415:
1303:
1235:
1175:
1054:
1013:
931:
896:
855:
822:
793:
763:
742:
710:
558:
543:
523:
503:
474:
447:
385:
307:
287:
247:
209:
182:
133:
97:
70:
18:Regression splines
4653:978-0-521-78050-6
4609:978-1-58488-474-3
4506:978-0-387-90356-9
4357:978-0-412-34390-2
4270:manifold learning
4254:Penalized splines
4168:
4157:
4136:
4117:
4075:
4057:
4012:
3993:
3910:
3826:
3808:Thin plate spline
3806:we might use the
3764:{\displaystyle x}
3744:{\displaystyle x}
3716:{\displaystyle S}
3660:
3641:
3608:{\displaystyle S}
3534:{\displaystyle p}
3464:
3432:
3358:
3335:{\displaystyle p}
3315:{\displaystyle S}
3295:{\displaystyle S}
3275:{\displaystyle p}
3249:
3226:{\displaystyle 0}
3206:{\displaystyle p}
3180:
3157:{\displaystyle 1}
3137:{\displaystyle p}
3096:Christian Reinsch
3061:{\displaystyle 2}
3041:{\displaystyle m}
3028:are mostly used,
2872:{\displaystyle p}
2800:
2747:
2715:
2594:
2559:
2532:
2488:
2459:
2356:
2335:
2313:
2282:
1717:
1550:
1529:
1487:
1377:
1332:
1281:
1205:
1146:
1109:
1091:
1075:Given the vector
1042:
970:
874:second derivative
766:{\displaystyle f}
672:
624:
526:{\displaystyle f}
500:
310:{\displaystyle x}
235:
163:
58:
40:Smoothing splines
16:(Redirected from
4714:
4658:
4657:
4639:
4633:
4632:
4620:
4614:
4613:
4595:
4589:
4588:
4580:
4574:
4573:
4565:
4559:
4552:
4546:
4545:
4522:
4511:
4510:
4492:
4481:
4480:
4474:
4466:
4464:
4462:
4456:
4447:
4441:
4440:
4428:
4422:
4421:
4419:
4395:
4389:
4388:
4368:
4362:
4361:
4343:
4337:
4336:
4328:
4285:Source code for
4219:
4217:
4216:
4211:
4186:
4184:
4183:
4178:
4170:
4169:
4166:
4159:
4158:
4155:
4152:
4148:
4147:
4146:
4141:
4137:
4135:
4134:
4133:
4120:
4119:
4118:
4110:
4107:
4106:
4096:
4086:
4085:
4080:
4076:
4074:
4060:
4059:
4058:
4050:
4047:
4046:
4036:
4023:
4022:
4017:
4013:
4011:
4010:
4009:
3996:
3995:
3994:
3986:
3983:
3982:
3972:
3951:
3950:
3938:
3937:
3925:
3924:
3912:
3911:
3903:
3897:
3896:
3883:
3878:
3853:
3851:
3850:
3845:
3828:
3827:
3819:
3805:
3803:
3802:
3797:
3770:
3768:
3767:
3762:
3750:
3748:
3747:
3742:
3722:
3720:
3719:
3714:
3702:
3700:
3699:
3694:
3676:
3674:
3673:
3668:
3666:
3662:
3661:
3653:
3642:
3634:
3614:
3612:
3611:
3606:
3594:
3592:
3591:
3586:
3584:
3583:
3567:
3565:
3564:
3559:
3557:
3556:
3540:
3538:
3537:
3532:
3520:
3518:
3517:
3512:
3491:
3489:
3488:
3483:
3475:
3474:
3469:
3465:
3463:
3462:
3453:
3452:
3448:
3447:
3434:
3433:
3425:
3419:
3418:
3408:
3400:
3395:
3370:
3368:
3367:
3362:
3360:
3359:
3351:
3341:
3339:
3338:
3333:
3321:
3319:
3318:
3313:
3301:
3299:
3298:
3293:
3281:
3279:
3278:
3273:
3261:
3259:
3258:
3253:
3251:
3250:
3242:
3232:
3230:
3229:
3224:
3212:
3210:
3209:
3204:
3192:
3190:
3189:
3184:
3182:
3181:
3173:
3163:
3161:
3160:
3155:
3143:
3141:
3140:
3135:
3123:
3121:
3120:
3115:
3094:was proposed by
3093:
3091:
3090:
3085:
3067:
3065:
3064:
3059:
3047:
3045:
3044:
3039:
3023:
3021:
3020:
3015:
3013:
3012:
2996:
2994:
2993:
2988:
2985:
2977:
2961:
2959:
2958:
2953:
2927:
2926:
2910:
2908:
2907:
2904:{\displaystyle }
2902:
2878:
2876:
2875:
2870:
2855:
2853:
2852:
2847:
2838:
2837:
2832:
2828:
2827:
2816:
2815:
2814:
2802:
2801:
2793:
2780:
2776:
2758:
2757:
2752:
2748:
2746:
2745:
2736:
2735:
2731:
2730:
2717:
2716:
2708:
2702:
2701:
2691:
2683:
2678:
2643:
2641:
2640:
2635:
2627:
2626:
2596:
2595:
2587:
2577:
2575:
2574:
2569:
2561:
2560:
2552:
2534:
2533:
2525:
2500:
2498:
2497:
2492:
2490:
2489:
2481:
2471:
2469:
2468:
2463:
2461:
2460:
2452:
2443:Minimizing over
2439:
2437:
2436:
2431:
2429:
2428:
2419:
2418:
2400:
2399:
2371:
2369:
2368:
2363:
2358:
2357:
2349:
2343:
2342:
2337:
2336:
2328:
2315:
2314:
2306:
2294:
2293:
2284:
2283:
2275:
2250:
2248:
2247:
2242:
2240:
2239:
2227:
2226:
2208:
2207:
2191:
2189:
2188:
2183:
2178:
2170:
2169:
2151:
2150:
2135:
2134:
2115:
2113:
2112:
2107:
2102:
2097:
2096:
2084:
2083:
2059:
2058:
2015:
2013:
2012:
2007:
2005:
2004:
1989:
1978:
1977:
1949:
1947:
1946:
1941:
1939:
1938:
1923:
1912:
1911:
1902:
1888:
1887:
1859:
1857:
1856:
1851:
1849:
1848:
1839:
1828:
1827:
1792:
1790:
1789:
1784:
1779:
1778:
1766:
1765:
1744:matrix given by
1729:
1727:
1726:
1721:
1719:
1718:
1710:
1700:
1698:
1697:
1692:
1690:
1689:
1673:
1671:
1670:
1665:
1663:
1662:
1642:
1640:
1639:
1634:
1613:
1591:
1565:
1563:
1562:
1557:
1552:
1551:
1543:
1537:
1536:
1531:
1530:
1522:
1509:
1508:
1493:
1489:
1488:
1480:
1463:
1461:
1460:
1455:
1444:
1443:
1424:
1422:
1421:
1416:
1405:
1404:
1392:
1391:
1379:
1378:
1370:
1366:
1361:
1334:
1333:
1325:
1312:
1310:
1309:
1304:
1296:
1295:
1283:
1282:
1274:
1268:
1267:
1244:
1242:
1241:
1236:
1227:
1226:
1211:
1207:
1206:
1198:
1184:
1182:
1181:
1176:
1174:
1173:
1161:
1160:
1148:
1147:
1139:
1124:
1123:
1111:
1110:
1102:
1093:
1092:
1084:
1063:
1061:
1060:
1055:
1044:
1043:
1035:
1022:
1020:
1019:
1014:
985:
984:
972:
971:
963:
940:
938:
937:
932:
930:
929:
905:
903:
902:
897:
895:
894:
864:
862:
861:
856:
831:
829:
828:
823:
802:
800:
799:
794:
792:
791:
772:
770:
769:
764:
751:
749:
748:
743:
719:
717:
716:
711:
699:
698:
683:
682:
674:
673:
665:
652:
651:
639:
638:
626:
625:
617:
611:
610:
597:
592:
567:
565:
564:
559:
556:
551:
532:
530:
529:
524:
513:of the function
512:
510:
509:
504:
502:
501:
493:
483:
481:
480:
475:
473:
472:
456:
454:
453:
448:
446:
445:
430:
429:
411:
410:
394:
392:
391:
386:
357:
356:
344:
343:
316:
314:
313:
308:
296:
294:
293:
288:
286:
285:
273:
272:
256:
254:
253:
248:
237:
236:
228:
218:
216:
215:
210:
208:
207:
191:
189:
188:
183:
178:
177:
165:
164:
156:
142:
140:
139:
134:
129:
128:
106:
104:
103:
98:
96:
95:
79:
77:
76:
71:
60:
59:
51:
21:
4722:
4721:
4717:
4716:
4715:
4713:
4712:
4711:
4692:
4691:
4667:
4665:Further reading
4662:
4661:
4654:
4641:
4640:
4636:
4622:
4621:
4617:
4610:
4597:
4596:
4592:
4582:
4581:
4577:
4567:
4566:
4562:
4553:
4549:
4524:
4523:
4514:
4507:
4494:
4493:
4484:
4467:
4460:
4458:
4454:
4449:
4448:
4444:
4430:
4429:
4425:
4397:
4396:
4392:
4370:
4369:
4365:
4358:
4345:
4344:
4340:
4330:
4329:
4318:
4313:
4283:
4237:
4231:
4229:Related methods
4196:
4195:
4125:
4121:
4098:
4097:
4091:
4090:
4061:
4038:
4037:
4031:
4030:
4001:
3997:
3974:
3973:
3967:
3966:
3965:
3961:
3942:
3929:
3916:
3888:
3859:
3858:
3812:
3811:
3773:
3772:
3753:
3752:
3733:
3732:
3729:
3705:
3704:
3679:
3678:
3626:
3622:
3617:
3616:
3597:
3596:
3595:, the constant
3575:
3570:
3569:
3548:
3543:
3542:
3523:
3522:
3497:
3496:
3454:
3439:
3435:
3410:
3409:
3403:
3402:
3376:
3375:
3344:
3343:
3324:
3323:
3304:
3303:
3284:
3283:
3264:
3263:
3235:
3234:
3215:
3214:
3195:
3194:
3166:
3165:
3146:
3145:
3126:
3125:
3100:
3099:
3070:
3069:
3050:
3049:
3030:
3029:
3004:
2999:
2998:
2964:
2963:
2918:
2913:
2912:
2881:
2880:
2861:
2860:
2817:
2804:
2790:
2789:
2785:
2784:
2766:
2762:
2737:
2722:
2718:
2693:
2692:
2686:
2685:
2656:
2655:
2649:
2615:
2580:
2579:
2503:
2502:
2474:
2473:
2445:
2444:
2420:
2410:
2391:
2377:
2376:
2325:
2285:
2259:
2258:
2231:
2212:
2199:
2194:
2193:
2155:
2142:
2123:
2118:
2117:
2088:
2063:
2038:
2033:
2032:
1990:
1957:
1952:
1951:
1924:
1903:
1867:
1862:
1861:
1840:
1816:
1811:
1810:
1767:
1757:
1746:
1745:
1703:
1702:
1681:
1676:
1675:
1654:
1649:
1648:
1575:
1574:
1519:
1500:
1477:
1469:
1468:
1435:
1430:
1429:
1396:
1383:
1318:
1317:
1287:
1259:
1251:
1250:
1218:
1195:
1187:
1186:
1165:
1152:
1115:
1077:
1076:
1028:
1027:
976:
956:
955:
948:
921:
916:
915:
886:
881:
880:
841:
840:
808:
807:
783:
778:
777:
755:
754:
728:
727:
690:
662:
643:
630:
602:
573:
572:
538:
537:
515:
514:
486:
485:
464:
459:
458:
437:
421:
402:
397:
396:
348:
335:
327:
326:
323:
299:
298:
277:
264:
259:
258:
221:
220:
199:
194:
193:
169:
149:
148:
145:goodness of fit
120:
109:
108:
87:
82:
81:
44:
43:
37:
28:
23:
22:
15:
12:
11:
5:
4720:
4718:
4710:
4709:
4704:
4694:
4693:
4690:
4689:
4682:
4675:
4666:
4663:
4660:
4659:
4652:
4634:
4615:
4608:
4590:
4575:
4568:Wahba, Grace.
4560:
4547:
4536:(3): 177–183.
4512:
4505:
4482:
4442:
4423:
4410:(2): 495–502.
4390:
4379:(4): 377–403.
4363:
4356:
4338:
4315:
4314:
4312:
4309:
4291:Carl de Boor's
4282:
4279:
4278:
4277:
4257:
4251:
4230:
4227:
4209:
4206:
4203:
4188:
4187:
4176:
4173:
4162:
4151:
4145:
4140:
4132:
4128:
4124:
4116:
4113:
4105:
4101:
4094:
4089:
4084:
4079:
4073:
4070:
4067:
4064:
4056:
4053:
4045:
4041:
4034:
4029:
4026:
4021:
4016:
4008:
4004:
4000:
3992:
3989:
3981:
3977:
3970:
3964:
3960:
3957:
3954:
3949:
3945:
3941:
3936:
3932:
3928:
3923:
3919:
3915:
3909:
3906:
3900:
3895:
3891:
3887:
3882:
3877:
3874:
3871:
3867:
3843:
3840:
3837:
3834:
3831:
3825:
3822:
3795:
3792:
3789:
3786:
3783:
3780:
3760:
3740:
3728:
3725:
3712:
3692:
3689:
3686:
3665:
3659:
3656:
3651:
3648:
3645:
3640:
3637:
3632:
3629:
3625:
3604:
3582:
3578:
3555:
3551:
3530:
3521:and increases
3510:
3507:
3504:
3493:
3492:
3481:
3478:
3473:
3468:
3461:
3457:
3451:
3446:
3442:
3438:
3431:
3428:
3422:
3417:
3413:
3406:
3399:
3394:
3391:
3388:
3384:
3357:
3354:
3331:
3311:
3291:
3271:
3248:
3245:
3222:
3202:
3179:
3176:
3153:
3133:
3113:
3110:
3107:
3083:
3080:
3077:
3057:
3037:
3011:
3007:
2997:of each point
2984:
2981:
2976:
2972:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2925:
2921:
2900:
2897:
2894:
2891:
2888:
2868:
2857:
2856:
2845:
2842:
2836:
2831:
2826:
2823:
2820:
2813:
2810:
2807:
2799:
2796:
2788:
2783:
2779:
2775:
2772:
2769:
2765:
2761:
2756:
2751:
2744:
2740:
2734:
2729:
2725:
2721:
2714:
2711:
2705:
2700:
2696:
2689:
2682:
2677:
2674:
2671:
2667:
2663:
2648:
2645:
2633:
2630:
2625:
2622:
2618:
2614:
2611:
2608:
2605:
2602:
2599:
2593:
2590:
2567:
2564:
2558:
2555:
2549:
2546:
2543:
2540:
2537:
2531:
2528:
2522:
2519:
2516:
2513:
2510:
2487:
2484:
2458:
2455:
2427:
2423:
2417:
2413:
2409:
2406:
2403:
2398:
2394:
2390:
2387:
2384:
2373:
2372:
2361:
2355:
2352:
2346:
2341:
2334:
2331:
2324:
2321:
2318:
2312:
2309:
2303:
2300:
2297:
2292:
2288:
2281:
2278:
2272:
2269:
2266:
2238:
2234:
2230:
2225:
2222:
2219:
2215:
2211:
2206:
2202:
2181:
2177:
2173:
2168:
2165:
2162:
2158:
2154:
2149:
2145:
2141:
2138:
2133:
2130:
2126:
2105:
2101:
2095:
2091:
2087:
2082:
2079:
2076:
2073:
2070:
2066:
2062:
2057:
2054:
2051:
2048:
2045:
2041:
2003:
2000:
1997:
1993:
1988:
1984:
1981:
1976:
1973:
1970:
1967:
1964:
1960:
1937:
1934:
1931:
1927:
1922:
1918:
1915:
1910:
1906:
1901:
1897:
1894:
1891:
1886:
1883:
1880:
1877:
1874:
1870:
1847:
1843:
1838:
1834:
1831:
1826:
1823:
1819:
1782:
1777:
1774:
1770:
1764:
1760:
1756:
1753:
1716:
1713:
1688:
1684:
1661:
1657:
1632:
1629:
1626:
1623:
1620:
1616:
1612:
1608:
1604:
1601:
1598:
1594:
1590:
1586:
1582:
1567:
1566:
1555:
1549:
1546:
1540:
1535:
1528:
1525:
1518:
1515:
1512:
1507:
1503:
1499:
1496:
1492:
1486:
1483:
1476:
1453:
1450:
1447:
1442:
1438:
1426:
1425:
1414:
1411:
1408:
1403:
1399:
1395:
1390:
1386:
1382:
1376:
1373:
1365:
1360:
1357:
1354:
1350:
1346:
1343:
1340:
1337:
1331:
1328:
1302:
1299:
1294:
1290:
1286:
1280:
1277:
1271:
1266:
1262:
1258:
1234:
1231:
1225:
1221:
1217:
1214:
1210:
1204:
1201:
1194:
1172:
1168:
1164:
1159:
1155:
1151:
1145:
1142:
1136:
1133:
1130:
1127:
1122:
1118:
1114:
1108:
1105:
1099:
1096:
1090:
1087:
1070:
1069:
1053:
1050:
1047:
1041:
1038:
1024:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
983:
979:
975:
969:
966:
947:
944:
943:
942:
928:
924:
907:
893:
889:
877:
870:
854:
851:
848:
837:
821:
818:
815:
804:
790:
786:
774:
762:
741:
738:
735:
721:
720:
709:
706:
703:
697:
693:
689:
686:
681:
678:
671:
668:
661:
658:
655:
650:
646:
642:
637:
633:
629:
623:
620:
614:
609:
605:
601:
596:
591:
588:
585:
581:
555:
550:
546:
522:
499:
496:
471:
467:
444:
440:
436:
433:
428:
424:
420:
417:
414:
409:
405:
384:
381:
378:
375:
372:
369:
366:
363:
360:
355:
351:
347:
342:
338:
334:
322:
319:
306:
284:
280:
276:
271:
267:
246:
243:
240:
234:
231:
206:
202:
181:
176:
172:
168:
162:
159:
132:
127:
123:
119:
116:
107:of the target
94:
90:
69:
66:
63:
57:
54:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4719:
4708:
4705:
4703:
4700:
4699:
4697:
4687:
4683:
4680:
4676:
4673:
4669:
4668:
4664:
4655:
4649:
4645:
4638:
4635:
4630:
4626:
4619:
4616:
4611:
4605:
4601:
4594:
4591:
4586:
4579:
4576:
4571:
4564:
4561:
4557:
4551:
4548:
4543:
4539:
4535:
4531:
4527:
4521:
4519:
4517:
4513:
4508:
4502:
4498:
4491:
4489:
4487:
4483:
4478:
4472:
4453:
4446:
4443:
4438:
4434:
4427:
4424:
4418:
4413:
4409:
4405:
4401:
4394:
4391:
4386:
4382:
4378:
4374:
4367:
4364:
4359:
4353:
4349:
4342:
4339:
4334:
4327:
4325:
4323:
4321:
4317:
4310:
4308:
4306:
4303:
4300:
4296:
4292:
4288:
4280:
4275:
4274:least squares
4271:
4267:
4263:
4262:
4258:
4255:
4252:
4249:
4245:
4242:
4241:
4240:
4236:
4235:Curve fitting
4228:
4226:
4222:
4207:
4204:
4201:
4192:
4174:
4171:
4160:
4149:
4143:
4138:
4130:
4126:
4111:
4103:
4092:
4087:
4082:
4077:
4071:
4065:
4051:
4043:
4032:
4027:
4024:
4019:
4014:
4006:
4002:
3987:
3979:
3968:
3962:
3958:
3955:
3952:
3947:
3934:
3930:
3926:
3921:
3917:
3904:
3898:
3893:
3889:
3880:
3875:
3872:
3869:
3865:
3857:
3856:
3855:
3838:
3835:
3832:
3820:
3809:
3790:
3787:
3784:
3778:
3758:
3738:
3726:
3724:
3710:
3690:
3687:
3684:
3663:
3657:
3654:
3649:
3646:
3643:
3638:
3635:
3630:
3627:
3623:
3602:
3580:
3576:
3553:
3549:
3528:
3508:
3505:
3502:
3479:
3476:
3471:
3466:
3459:
3455:
3449:
3444:
3440:
3436:
3426:
3420:
3415:
3411:
3404:
3397:
3392:
3389:
3386:
3382:
3374:
3373:
3372:
3352:
3329:
3309:
3289:
3269:
3243:
3220:
3200:
3174:
3151:
3131:
3111:
3108:
3105:
3098:in 1967. For
3097:
3081:
3078:
3075:
3055:
3035:
3027:
3026:cubic splines
3009:
3005:
2982:
2979:
2974:
2970:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2923:
2919:
2895:
2892:
2889:
2866:
2843:
2840:
2834:
2829:
2824:
2821:
2818:
2811:
2808:
2805:
2794:
2786:
2781:
2777:
2773:
2770:
2767:
2763:
2759:
2754:
2749:
2742:
2738:
2732:
2727:
2723:
2719:
2709:
2703:
2698:
2694:
2687:
2680:
2675:
2672:
2669:
2665:
2661:
2654:
2653:
2652:
2644:
2631:
2628:
2623:
2620:
2612:
2609:
2606:
2603:
2597:
2588:
2565:
2562:
2553:
2547:
2544:
2541:
2538:
2526:
2520:
2517:
2511:
2508:
2482:
2453:
2441:
2425:
2415:
2411:
2407:
2404:
2401:
2396:
2392:
2385:
2382:
2359:
2350:
2344:
2339:
2329:
2322:
2319:
2307:
2301:
2298:
2290:
2276:
2270:
2267:
2257:
2256:
2255:
2252:
2236:
2232:
2228:
2223:
2220:
2217:
2213:
2209:
2204:
2200:
2179:
2175:
2166:
2163:
2160:
2156:
2152:
2147:
2143:
2136:
2131:
2128:
2124:
2103:
2099:
2093:
2089:
2085:
2080:
2077:
2074:
2071:
2068:
2064:
2060:
2055:
2052:
2049:
2046:
2043:
2039:
2030:
2028:
2024:
2020:
2016:
2001:
1998:
1995:
1991:
1986:
1982:
1979:
1974:
1971:
1968:
1965:
1962:
1935:
1932:
1929:
1925:
1920:
1916:
1913:
1908:
1904:
1899:
1895:
1892:
1889:
1884:
1881:
1878:
1875:
1872:
1845:
1841:
1836:
1832:
1829:
1824:
1821:
1808:
1806:
1802:
1798:
1794:
1775:
1772:
1768:
1762:
1754:
1751:
1743:
1739:
1735:
1731:
1711:
1686:
1682:
1659:
1655:
1646:
1630:
1627:
1621:
1614:
1610:
1606:
1599:
1592:
1588:
1584:
1580:
1572:
1553:
1544:
1538:
1533:
1523:
1516:
1513:
1510:
1505:
1497:
1490:
1481:
1474:
1467:
1466:
1465:
1448:
1440:
1436:
1409:
1401:
1397:
1388:
1384:
1371:
1363:
1358:
1355:
1352:
1348:
1344:
1338:
1326:
1316:
1315:
1314:
1292:
1288:
1275:
1269:
1264:
1260:
1248:
1232:
1229:
1223:
1215:
1208:
1199:
1192:
1170:
1157:
1153:
1140:
1134:
1131:
1128:
1120:
1116:
1103:
1094:
1085:
1073:
1067:
1048:
1036:
1025:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
981:
977:
964:
953:
952:
951:
945:
926:
922:
912:
908:
891:
887:
878:
875:
871:
868:
846:
838:
835:
819:
813:
805:
788:
784:
775:
760:
739:
736:
733:
726:
725:
724:
707:
704:
701:
695:
687:
666:
659:
656:
653:
648:
635:
631:
618:
612:
607:
603:
594:
589:
586:
583:
579:
571:
570:
569:
553:
548:
544:
536:
535:Sobolev space
520:
494:
469:
465:
442:
438:
434:
426:
422:
415:
412:
407:
403:
379:
376:
373:
370:
367:
364:
361:
358:
353:
349:
345:
340:
336:
320:
318:
304:
282:
278:
274:
269:
265:
241:
229:
204:
200:
174:
170:
157:
146:
125:
121:
114:
92:
88:
64:
52:
41:
35:
30:
19:
4685:
4681:. CRC Press.
4678:
4671:
4643:
4637:
4631:(2): 89–121.
4628:
4624:
4618:
4599:
4593:
4584:
4578:
4569:
4563:
4550:
4533:
4529:
4496:
4459:. Retrieved
4445:
4436:
4432:
4426:
4407:
4403:
4393:
4376:
4372:
4366:
4347:
4341:
4332:
4294:
4284:
4266:Elastic maps
4259:
4253:
4243:
4238:
4223:
4193:
4189:
3730:
3494:
2858:
2650:
2442:
2374:
2253:
2031:
2026:
2022:
2018:
2017:
1809:
1804:
1800:
1796:
1795:
1741:
1737:
1733:
1732:
1644:
1570:
1568:
1427:
1074:
1071:
1065:
949:
910:
722:
324:
39:
38:
29:
4688:. Springer.
4587:. Springer.
4281:Source code
4268:method for
3854:minimizing
3213:approaches
3144:approaches
3048:is usually
457:where the
4696:Categories
4311:References
4233:See also:
4556:K. Zeller
4123:∂
4115:^
4100:∂
4069:∂
4063:∂
4055:^
4040:∂
3999:∂
3991:^
3976:∂
3959:∫
3956:λ
3908:^
3899:−
3866:∑
3824:^
3677:. Having
3631:−
3550:δ
3477:≤
3456:δ
3430:^
3421:−
3383:∑
3356:^
3247:^
3178:^
2980:−
2971:δ
2944:…
2920:δ
2798:^
2782:∫
2771:−
2739:δ
2713:^
2704:−
2666:∑
2621:−
2610:λ
2592:^
2557:^
2545:λ
2530:^
2521:−
2509:−
2486:^
2457:^
2405:…
2354:^
2333:^
2323:λ
2311:^
2302:−
2280:^
2271:−
2233:ξ
2229:−
2214:ξ
2078:−
2047:−
1959:Δ
1914:−
1893:−
1869:Δ
1818:Δ
1781:Δ
1773:−
1759:Δ
1715:^
1581:∫
1548:^
1527:^
1485:^
1475:∫
1375:^
1349:∑
1330:^
1279:^
1203:^
1193:∫
1144:^
1132:…
1107:^
1089:^
1040:^
1005:…
968:^
923:ϵ
869:estimate.
853:∞
850:→
847:λ
817:→
814:λ
737:≥
734:λ
723:Remarks:
680:′
677:′
670:^
660:∫
657:λ
622:^
613:−
580:∑
498:^
466:ϵ
439:ϵ
374:…
233:^
161:^
56:^
4471:cite web
4461:28 April
4439:: 63–75.
1615:″
1593:″
1491:″
1209:″
1064:for all
4572:. SIAM.
4299:Fortran
3124:, when
4650:
4606:
4503:
4354:
4287:spline
2911:, and
2859:where
2375:where
2021:is an
1799:is an
1797:Δ
1736:is an
1428:where
1247:spline
4455:(PDF)
4293:book
2027:(n-2)
2023:(n-2)
1801:(n-2)
4648:ISBN
4604:ISBN
4501:ISBN
4477:link
4463:2024
4352:ISBN
4264:and
2578:and
2192:and
1573:are
325:Let
4538:doi
4412:doi
4381:doi
1701:or
839:As
806:As
192:to
147:of
4698::
4629:11
4627:.
4534:10
4532:.
4515:^
4485:^
4473:}}
4469:{{
4437:41
4435:.
4408:41
4406:.
4402:.
4377:31
4375:.
4319:^
4307:.
4250:.)
3233:,
3164:,
2440:.
2116:,
1950:,
1860:,
1793:.
1730:.
773:).
4656:.
4612:.
4544:.
4540::
4509:.
4479:)
4465:.
4420:.
4414::
4387:.
4383::
4360:.
4208:z
4205:,
4202:x
4175:.
4172:z
4167:d
4161:x
4156:d
4150:]
4144:2
4139:)
4131:2
4127:z
4112:f
4104:2
4093:(
4088:+
4083:2
4078:)
4072:z
4066:x
4052:f
4044:2
4033:(
4028:2
4025:+
4020:2
4015:)
4007:2
4003:x
3988:f
3980:2
3969:(
3963:[
3953:+
3948:2
3944:}
3940:)
3935:i
3931:z
3927:,
3922:i
3918:x
3914:(
3905:f
3894:i
3890:y
3886:{
3881:n
3876:1
3873:=
3870:i
3842:)
3839:z
3836:,
3833:x
3830:(
3821:f
3794:)
3791:z
3788:,
3785:x
3782:(
3779:f
3759:x
3739:x
3711:S
3691:0
3688:=
3685:S
3664:]
3658:n
3655:2
3650:+
3647:n
3644:,
3639:n
3636:2
3628:n
3624:[
3603:S
3581:i
3577:Y
3554:i
3529:p
3509:0
3506:=
3503:p
3480:S
3472:2
3467:)
3460:i
3450:)
3445:i
3441:x
3437:(
3427:f
3416:i
3412:Y
3405:(
3398:n
3393:1
3390:=
3387:i
3353:f
3330:p
3310:S
3290:S
3270:p
3244:f
3221:0
3201:p
3175:f
3152:1
3132:p
3112:2
3109:=
3106:m
3082:2
3079:=
3076:m
3056:2
3036:m
3010:i
3006:Y
2983:2
2975:i
2950:n
2947:,
2941:,
2938:1
2935:=
2932:i
2929:;
2924:i
2899:]
2896:1
2893:,
2890:0
2887:[
2867:p
2844:x
2841:d
2835:2
2830:)
2825:)
2822:x
2819:(
2812:)
2809:m
2806:(
2795:f
2787:(
2778:)
2774:p
2768:1
2764:(
2760:+
2755:2
2750:)
2743:i
2733:)
2728:i
2724:x
2720:(
2710:f
2699:i
2695:Y
2688:(
2681:n
2676:1
2673:=
2670:i
2662:p
2632:.
2629:Y
2624:1
2617:)
2613:A
2607:+
2604:I
2601:(
2598:=
2589:m
2566:0
2563:=
2554:m
2548:A
2542:2
2539:+
2536:}
2527:m
2518:Y
2515:{
2512:2
2483:m
2454:m
2426:T
2422:)
2416:n
2412:Y
2408:,
2402:,
2397:1
2393:Y
2389:(
2386:=
2383:Y
2360:,
2351:m
2345:A
2340:T
2330:m
2320:+
2317:}
2308:m
2299:Y
2296:{
2291:T
2287:}
2277:m
2268:Y
2265:{
2237:i
2224:1
2221:+
2218:i
2210:=
2205:i
2201:h
2180:3
2176:/
2172:)
2167:1
2164:+
2161:i
2157:h
2153:+
2148:i
2144:h
2140:(
2137:=
2132:i
2129:i
2125:W
2104:6
2100:/
2094:i
2090:h
2086:=
2081:1
2075:i
2072:,
2069:i
2065:W
2061:=
2056:i
2053:,
2050:1
2044:i
2040:W
2025:×
2019:W
2002:1
1999:+
1996:i
1992:h
1987:/
1983:1
1980:=
1975:2
1972:+
1969:i
1966:,
1963:i
1936:1
1933:+
1930:i
1926:h
1921:/
1917:1
1909:i
1905:h
1900:/
1896:1
1890:=
1885:1
1882:+
1879:i
1876:,
1873:i
1846:i
1842:h
1837:/
1833:1
1830:=
1825:i
1822:i
1805:n
1803:×
1776:1
1769:W
1763:T
1755:=
1752:A
1742:n
1740:×
1738:n
1734:A
1712:m
1687:i
1683:Y
1660:i
1656:x
1645:A
1631:x
1628:d
1625:)
1622:x
1619:(
1611:j
1607:f
1603:)
1600:x
1597:(
1589:i
1585:f
1571:A
1554:.
1545:m
1539:A
1534:T
1524:m
1517:=
1514:x
1511:d
1506:2
1502:)
1498:x
1495:(
1482:f
1452:)
1449:x
1446:(
1441:i
1437:f
1413:)
1410:x
1407:(
1402:i
1398:f
1394:)
1389:i
1385:x
1381:(
1372:f
1364:n
1359:1
1356:=
1353:i
1345:=
1342:)
1339:x
1336:(
1327:f
1301:)
1298:)
1293:i
1289:x
1285:(
1276:f
1270:,
1265:i
1261:x
1257:(
1233:x
1230:d
1224:2
1220:)
1216:x
1213:(
1200:f
1171:T
1167:)
1163:)
1158:n
1154:x
1150:(
1141:f
1135:,
1129:,
1126:)
1121:1
1117:x
1113:(
1104:f
1098:(
1095:=
1086:m
1068:.
1066:x
1052:)
1049:x
1046:(
1037:f
1023:.
1011:n
1008:,
1002:,
999:1
996:=
993:i
990:;
987:)
982:i
978:x
974:(
965:f
941:.
927:i
892:i
888:x
836:.
820:0
803:.
789:i
785:x
761:f
740:0
708:.
705:x
702:d
696:2
692:)
688:x
685:(
667:f
654:+
649:2
645:}
641:)
636:i
632:x
628:(
619:f
608:i
604:Y
600:{
595:n
590:1
587:=
584:i
554:2
549:2
545:W
521:f
495:f
470:i
443:i
435:+
432:)
427:i
423:x
419:(
416:f
413:=
408:i
404:Y
383:}
380:n
377:,
371:,
368:1
365:=
362:i
359::
354:i
350:Y
346:,
341:i
337:x
333:{
305:x
283:i
279:y
275:,
270:i
266:x
245:)
242:x
239:(
230:f
205:i
201:y
180:)
175:i
171:x
167:(
158:f
131:)
126:i
122:x
118:(
115:f
93:i
89:y
68:)
65:x
62:(
53:f
36:.
20:)
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