1705:
1260:
1700:{\displaystyle {\begin{aligned}\max _{e,b\neq 0}\left\{{\frac {\left\|A^{-1}e\right\|}{\|e\|}}{\frac {\|b\|}{\left\|A^{-1}b\right\|}}\right\}&=\max _{e\neq 0}\left\{{\frac {\left\|A^{-1}e\right\|}{\|e\|}}\right\}\,\max _{b\neq 0}\left\{{\frac {\|b\|}{\left\|A^{-1}b\right\|}}\right\}\\&=\max _{e\neq 0}\left\{{\frac {\left\|A^{-1}e\right\|}{\|e\|}}\right\}\,\max _{x\neq 0}\left\{{\frac {\|Ax\|}{\|x\|}}\right\}\\&=\left\|A^{-1}\right\|\,\|A\|.\end{aligned}}}
1237:
3842:
127:. This means that the correct solution/answer to the equation becomes hard to find. The condition number is a property of the problem. Paired with the problem are any number of algorithms that can be used to solve the problem, that is, to calculate the solution. Some algorithms have a property called
198:
digits of accuracy on top of what would be lost to the numerical method due to loss of precision from arithmetic methods. However, the condition number does not give the exact value of the maximum inaccuracy that may occur in the algorithm. It generally just bounds it with an estimate (whose computed
1831:
However, it does not mean that the algorithm will converge rapidly to this solution, just that it will not diverge arbitrarily because of inaccuracy on the source data (backward error), provided that the forward error introduced by the algorithm does not diverge as well because of accumulating
927:
768:
2226:
3043:
104:, in which case the derivative is straightforward but the error could be in many different directions, and is thus computed from the geometry of the matrix. More generally, condition numbers can be defined for non-linear functions in several variables.
2676:
1050:
1815:
3574:
3318:
at a point, its condition number at the point is infinite, as infinitesimal changes in the input can change the output from zero to positive or negative, yielding a ratio with zero in the denominator, hence infinite relative change.
133:; in general, a backward stable algorithm can be expected to accurately solve well-conditioned problems. Numerical analysis textbooks give formulas for the condition numbers of problems and identify known backward stable algorithms.
5343:
5011:. They express how sensitive that function is to small changes (or small errors) in its arguments. This is crucial in assessing the sensitivity and potential accuracy difficulties of numerous computational problems, for example,
2002:
630:
783:
99:
worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to questions in
2798:. For square matrices, this unfortunately makes the condition number discontinuous, but it is a useful definition for rectangular matrices, which are never invertible but are still used to define systems of equations.
487:
664:
4773:
2134:
4661:
1828:), then a solution algorithm can find (in principle, meaning if the algorithm introduces no errors of its own) an approximation of the solution whose precision is no worse than that of the data.
5500:
1265:
393:
4875:
4285:
2765:
3957:
3227:
3137:
2298:
2262:
2893:
2077:
2041:
4561:
2547:
46:
a function is to changes or errors in the input, and how much error in the output results from an error in the input. Very frequently, one is solving the inverse problem: given
968:
accuracy of the computer used to solve the corresponding system. In particular, one should think of the condition number as being (very roughly) the rate at which the solution
176:
1232:{\displaystyle {\frac {\left\|A^{-1}e\right\|}{\left\|A^{-1}b\right\|}}/{\frac {\|e\|}{\|b\|}}={\frac {\left\|A^{-1}e\right\|}{\|e\|}}{\frac {\|b\|}{\left\|A^{-1}b\right\|}}.}
315:
3461:
1918:
2384:
2792:
2519:
254:
2457:
5372:
4811:
4699:
4599:
4453:
4369:
3263:
2428:
1877:
5619:
5136:
3343:
4491:
4407:
4323:
3292:
2921:
2707:. Practically, such a matrix is almost singular, and the computation of its inverse, or solution of a linear system of equations is prone to large numerical errors.
1720:
82:
4235:
4163:
4097:
3837:{\displaystyle {\frac {/f(x)}{(\Delta x)/x}}={\frac {x}{f(x)}}{\frac {f(x+\Delta x)-f(x)}{(x+\Delta x)-x}}={\frac {x}{f(x)}}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}.}
3566:
5534:
5093:
5005:
3490:
4194:
4128:
4041:
3913:
3986:
5584:
5564:
5419:
5396:
5156:
5113:
5060:
5040:
4976:
4952:
4929:
4905:
4062:
4007:
3383:
3363:
3312:
3157:
3081:
2913:
2847:
2539:
2479:
2345:
2322:
2125:
2101:
656:
274:
225:
196:
2818:
of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest.
5164:
5835:
1926:
499:
922:{\displaystyle \lim _{\varepsilon \rightarrow 0^{+}}\,\sup _{\|\delta x\|\,\leq \,\varepsilon }{\frac {\|\delta f(x)\|/\|f(x)\|}{\|\delta x\|/\|x\|}}.}
2688:, but it can be evaluated more easily (and this is often the only practicably computable condition number, when the problem to solve involves a
405:
2703:, which means that its inverse can be computed with good accuracy. If the condition number is very large, then the matrix is said to be
5634:
763:{\displaystyle \lim _{\varepsilon \rightarrow 0^{+}}\,\sup _{\|\delta x\|\,\leq \,\varepsilon }{\frac {\|\delta f(x)\|}{\|\delta x\|}}}
42:
measures how much the output value of the function can change for a small change in the input argument. This is used to measure how
2710:
A matrix that is not invertible is often said to have a condition number equal to infinity. Alternatively, it can be defined as
4705:
4605:
5787:
5755:
5728:
5427:
324:
5012:
4817:
4241:
2713:
5819:
5696:
2684:
The condition number computed with this norm is generally larger than the condition number computed relative to the
3919:
2221:{\displaystyle \kappa (A)={\frac {\left|\lambda _{\text{max}}(A)\right|}{\left|\lambda _{\text{min}}(A)\right|}},}
5810:(1990). "Nearest Defective Matrices and the Geometry of Ill-conditioning". In Cox, M. G.; Hammarling, S. (eds.).
17:
2431:
2267:
2231:
1880:
2671:{\displaystyle \kappa (A)\geq {\frac {\max _{i}{\big (}|a_{ii}|{\big )}}{\min _{i}{\big (}|a_{ii}|{\big )}}}}
2046:
2010:
119:. In non-mathematical terms, an ill-conditioned problem is one where, for a small change in the inputs (the
5854:
4497:
5869:
3162:
2852:
139:
92:
5777:
1839:(does not possess a unique, well-defined solution for each choice of data; that is, the matrix is not
279:
3388:
3086:
1890:
2394:
2354:
2770:
2488:
230:
2827:
2693:
2435:
5849:
5351:
4784:
4672:
4572:
4413:
4329:
3038:{\displaystyle \left|{\frac {xf'(x)}{f(x)}}\right|=\left|{\frac {(\log f)'}{(\log x)'}}\right|.}
2407:
1856:
1810:{\displaystyle \kappa (A)=\left\|A^{-1}\right\|\,\left\|A\right\|\geq \left\|A^{-1}A\right\|=1.}
3266:
3060:
3049:
39:
5745:
5684:
5589:
5118:
3325:
5716:
4464:
4380:
4296:
120:
49:
4208:
4908:
4141:
4134:
4075:
3870:
and can be computed immediately from the derivative. A few important ones are given below:
3495:
3232:
2815:
957:
43:
5510:
5069:
4981:
3466:
8:
5737:
5639:
4169:
4103:
4018:
3892:
3867:
3863:
3856:
3848:
2806:
Condition numbers can also be defined for nonlinear functions, and can be computed using
2681:
recalling that the eigenvalues of any triangular matrix are simply the diagonal entries.
129:
96:
5338:{\displaystyle \lim _{\varepsilon \to 0^{+}}\sup _{\|\delta x\|\leq \varepsilon }\left,}
3968:
3272:
2810:. The condition number varies with the point; in some cases one can use the maximum (or
5569:
5549:
5543:
5404:
5381:
5375:
5141:
5098:
5045:
5025:
4961:
4937:
4914:
4890:
4047:
3992:
3368:
3348:
3315:
3297:
3142:
3066:
2898:
2832:
2524:
2464:
2330:
2307:
2110:
2086:
1847:
1836:
1023:
641:
259:
210:
181:
124:
31:
1997:{\displaystyle \kappa (A)={\frac {\sigma _{\text{max}}(A)}{\sigma _{\text{min}}(A)}},}
625:{\displaystyle \|\delta f(x)\|/\|f(x)\|=\left\|f(x)-{\tilde {f}}(x)\right\|/\|f(x)\|.}
5874:
5844:
5815:
5783:
5751:
5724:
5712:
5692:
5649:
4200:
2482:
1840:
2700:
1825:
5539:
2704:
953:
938:
5689:
Regression
Diagnostics: Identifying Influential Data and Sources of Collinearity
1835:
The condition number may also be infinite, but this implies that the problem is
5654:
5644:
3056:
2685:
2348:
2080:
996:
965:
490:
396:
101:
995:
The condition number is defined more precisely to be the maximum ratio of the
5863:
5659:
2795:
2128:
1251:
2699:
If the condition number is not significantly larger than one, the matrix is
5807:
5622:
5008:
1040:. The ratio of the relative error in the solution to the relative error in
4932:
3852:
1711:
984:. On the other hand, if the condition number is small, then the error in
5016:
4068:
2301:
5066:) is then defined to be the maximum ratio of the fractional change in
202:
5680:
961:
952:
will be after approximation. Note that this is before the effects of
199:
value depends on the choice of the norm to measure the inaccuracy).
4955:
2811:
2807:
1820:
When the condition number is exactly one (which can only happen if
482:{\displaystyle \|\delta f(x)\|=\left\|f(x)-{\tilde {f}}(x)\right\|}
88:
and thus the condition number of the (local) inverse must be used.
1843:), and no algorithm can be expected to reliably find a solution.
976:. Thus, if the condition number is large, even a small error in
1846:
The definition of the condition number depends on the choice of
113:, while a problem with a high condition number is said to be
3269:
rate of relative change in a function: it is the derivative
5294:
5222:
4768:{\displaystyle {\frac {|x|}{{\sqrt {1-x^{2}}}\arccos(x)}}}
956:
are taken into account; conditioning is a property of the
5855:
Who
Invented the Matrix Condition Number? by Nick Higham
4656:{\displaystyle {\frac {x}{{\sqrt {1-x^{2}}}\arcsin(x)}}}
5723:. New York: Oxford University Press. pp. 67–72 .
937:
For example, the condition number associated with the
5845:
MATLAB library function to determine condition number
5691:. New York: John Wiley & Sons. pp. 100–104.
5592:
5572:
5552:
5513:
5430:
5407:
5384:
5354:
5167:
5144:
5121:
5101:
5072:
5048:
5028:
4984:
4964:
4940:
4917:
4893:
4820:
4787:
4708:
4675:
4608:
4575:
4500:
4467:
4416:
4383:
4332:
4299:
4244:
4211:
4172:
4144:
4106:
4078:
4050:
4021:
3995:
3971:
3922:
3895:
3577:
3498:
3469:
3391:
3371:
3351:
3328:
3300:
3275:
3235:
3165:
3145:
3089:
3069:
2924:
2901:
2855:
2835:
2773:
2716:
2550:
2527:
2491:
2467:
2438:
2410:
2357:
2333:
2310:
2270:
2234:
2137:
2113:
2089:
2049:
2013:
1929:
1893:
1859:
1723:
1263:
1053:
786:
667:
644:
502:
408:
327:
282:
262:
233:
213:
184:
142:
52:
3265:. This is because the logarithmic derivative is the
107:
A problem with a low condition number is said to be
5495:{\displaystyle {\frac {\|J(x)\|}{\|f(x)\|/\|x\|}},}
203:
General definition in the context of error analysis
91:The condition number is derived from the theory of
5613:
5578:
5558:
5528:
5494:
5413:
5390:
5366:
5337:
5150:
5130:
5107:
5087:
5054:
5034:
4999:
4970:
4946:
4923:
4899:
4887:Condition numbers can be defined for any function
4869:
4805:
4767:
4693:
4655:
4593:
4555:
4485:
4447:
4401:
4363:
4317:
4279:
4229:
4188:
4157:
4122:
4091:
4056:
4035:
4001:
3980:
3951:
3907:
3836:
3560:
3484:
3455:
3377:
3357:
3337:
3306:
3286:
3257:
3221:
3151:
3131:
3075:
3037:
2907:
2887:
2841:
2786:
2759:
2670:
2533:
2513:
2473:
2451:
2422:
2378:
2339:
2316:
2292:
2256:
2220:
2119:
2095:
2071:
2035:
1996:
1912:
1881:matrix norm induced by the (vector) Euclidean norm
1871:
1809:
1699:
1231:
921:
762:
650:
624:
481:
388:{\displaystyle \delta f(x):=f(x)-{\tilde {f}}(x),}
387:
309:
268:
248:
219:
190:
170:
76:
5861:
5678:
5192:
5169:
4870:{\displaystyle {\frac {x}{(1+x^{2})\arctan(x)}}}
4280:{\displaystyle \left|{\frac {1}{\ln(x)}}\right|}
2760:{\displaystyle \kappa (A)=\|A\|\|A^{\dagger }\|}
2692:, for example when approximating irrational and
2619:
2570:
1597:
1531:
1456:
1390:
1269:
812:
788:
693:
669:
3055:Most elegantly, this can be understood as (the
1710:The same definition is used for any consistent
5850:Condition number – Encyclopedia of Mathematics
5771:
5769:
5767:
2696:functions or numbers with numerical methods).
95:, and is formally defined as the value of the
5743:
3952:{\displaystyle \left|{\frac {x}{x+a}}\right|}
2660:
2630:
2611:
2581:
948:gives a bound on how inaccurate the solution
5775:
5608:
5593:
5483:
5477:
5469:
5454:
5449:
5434:
5361:
5355:
5321:
5315:
5310:
5301:
5288:
5273:
5205:
5196:
3048:Note that this is the absolute value of the
2754:
2741:
2738:
2732:
2417:
2411:
1901:
1894:
1866:
1860:
1687:
1681:
1639:
1633:
1628:
1619:
1585:
1579:
1484:
1478:
1444:
1438:
1345:
1339:
1330:
1324:
1250:) is then seen to be the product of the two
1194:
1188:
1179:
1173:
1135:
1129:
1124:
1118:
910:
904:
896:
887:
882:
867:
859:
841:
825:
816:
754:
745:
740:
722:
706:
697:
616:
601:
544:
529:
521:
503:
427:
409:
136:As a rule of thumb, if the condition number
5814:. Oxford: Clarendon Press. pp. 35–55.
5764:
123:) there is a large change in the answer or
5635:Numerical methods for linear least squares
5421:is differentiable, this is equivalent to:
5007:), where both the domain and codomain are
988:will not be much bigger than the error in
1850:, as can be illustrated by two examples.
1760:
1680:
1595:
1454:
832:
828:
810:
713:
709:
691:
27:Function's sensitivity to argument change
3866:are particularly important in computing
2293:{\displaystyle \lambda _{\text{min}}(A)}
2257:{\displaystyle \lambda _{\text{max}}(A)}
972:will change with respect to a change in
5721:Time Series and Panel Data Econometrics
5711:
2072:{\displaystyle \sigma _{\text{min}}(A)}
2036:{\displaystyle \sigma _{\text{max}}(A)}
14:
5862:
5806:
2389:The condition number with respect to
5840:Holistic Numerical Methods Institute
4882:
4556:{\displaystyle |x(\tan(x)+\cot(x))|}
3322:More directly, given a small change
3139:, and the logarithmic derivative of
2300:are maximal and minimal (by moduli)
5747:Numerical Mathematics and Computing
24:
5800:
5776:Trefethen, L. N.; Bau, D. (1997).
3822:
3796:
3739:
3704:
3645:
3596:
3514:
3436:
3404:
3329:
3222:{\displaystyle (\log x)'=x'/x=1/x}
2888:{\displaystyle \left|xf'/f\right|}
2444:
1026:matrix, the error in the solution
25:
5886:
5829:
2849:in one variable as a function is
171:{\displaystyle \kappa (A)=10^{k}}
5115:, in the limit where the change
3314:. Note that if a function has a
310:{\displaystyle {\tilde {f}}(x),}
5717:"The Multicollinearity Problem"
5158:becomes infinitesimally small:
3463:, while the relative change in
3456:{\displaystyle /x=(\Delta x)/x}
2821:
1242:The maximum value (for nonzero
5812:Reliable Numerical Computation
5705:
5672:
5605:
5599:
5523:
5517:
5466:
5460:
5446:
5440:
5285:
5279:
5268:
5264:
5258:
5249:
5234:
5227:
5176:
5082:
5076:
4994:
4988:
4861:
4855:
4846:
4827:
4800:
4794:
4759:
4753:
4721:
4713:
4688:
4682:
4647:
4641:
4588:
4582:
4549:
4545:
4542:
4536:
4524:
4518:
4509:
4502:
4480:
4474:
4441:
4437:
4431:
4418:
4396:
4390:
4357:
4353:
4347:
4334:
4312:
4306:
4267:
4261:
4224:
4218:
4182:
4174:
4116:
4108:
3817:
3811:
3802:
3787:
3775:
3769:
3745:
3730:
3725:
3719:
3710:
3695:
3683:
3677:
3651:
3642:
3637:
3631:
3620:
3617:
3611:
3602:
3587:
3581:
3555:
3549:
3538:
3535:
3529:
3520:
3505:
3499:
3479:
3473:
3442:
3433:
3419:
3410:
3395:
3392:
3179:
3166:
3132:{\displaystyle (\log f)'=f'/f}
3103:
3090:
3018:
3005:
2996:
2983:
2963:
2957:
2949:
2943:
2726:
2720:
2654:
2636:
2605:
2587:
2560:
2554:
2367:
2361:
2287:
2281:
2251:
2245:
2205:
2199:
2176:
2170:
2147:
2141:
2066:
2060:
2030:
2024:
1985:
1979:
1964:
1958:
1939:
1933:
1913:{\displaystyle \|\cdot \|_{2}}
1887:norm and typically denoted as
1832:intermediate rounding errors.
1797:
1776:
1768:
1762:
1756:
1740:
1733:
1727:
1676:
1660:
1574:
1553:
1510:
1489:
1433:
1412:
1371:
1350:
1319:
1298:
1220:
1199:
1168:
1147:
1104:
1083:
1078:
1057:
879:
873:
856:
850:
795:
737:
731:
676:
638:condition number of a problem
613:
607:
592:
588:
582:
576:
564:
558:
551:
541:
535:
518:
512:
475:
471:
465:
459:
447:
441:
434:
424:
418:
379:
373:
367:
355:
349:
340:
334:
301:
295:
289:
240:
152:
146:
62:
56:
13:
1:
5665:
2397:that it is given a name, the
2379:{\displaystyle \kappa (A)=1.}
5836:Condition Number of a Matrix
5095:to any fractional change in
3862:Condition numbers of common
3052:of a function in economics.
2814:) condition number over the
2801:
2787:{\displaystyle A^{\dagger }}
2514:{\displaystyle a_{ii}\neq 0}
2432:matrix norm induced by the
2399:condition number of a matrix
249:{\displaystyle {\tilde {f}}}
7:
5628:
4907:mapping its data from some
2452:{\displaystyle L^{\infty }}
980:may cause a large error in
932:
10:
5891:
5378:on the domain/codomain of
5367:{\displaystyle \|\cdot \|}
4806:{\displaystyle \arctan(x)}
4694:{\displaystyle \arccos(x)}
4594:{\displaystyle \arcsin(x)}
4448:{\displaystyle |x\tan(x)|}
4364:{\displaystyle |x\cot(x)|}
3568:. Taking the ratio yields
2826:The condition number of a
2423:{\displaystyle \|\cdot \|}
1872:{\displaystyle \|\cdot \|}
1824:is a scalar multiple of a
1714:, i.e. one that satisfies
178:, then you may lose up to
93:propagation of uncertainty
5683:; Welsch, Roy E. (1980).
5064:relative condition number
3365:, the relative change in
1003:to the relative error in
5779:Numerical Linear Algebra
5744:Cheney; Kincaid (2008).
5614:{\displaystyle \|J(x)\|}
5131:{\displaystyle \delta x}
5022:The condition number of
4779:Inverse tangent function
3338:{\displaystyle \Delta x}
2395:numerical linear algebra
2079:are maximal and minimal
1883:(sometimes known as the
5013:polynomial root finding
4978:-tuple of real numbers
4667:Inverse cosine function
4486:{\displaystyle \tan(x)}
4402:{\displaystyle \cos(x)}
4318:{\displaystyle \sin(x)}
3859:yields the derivative.
3294:scaled by the value of
2895:. Evaluated at a point
2828:differentiable function
77:{\displaystyle f(x)=y,}
5685:"The Condition Number"
5615:
5580:
5560:
5530:
5496:
5415:
5392:
5368:
5339:
5152:
5132:
5109:
5089:
5056:
5036:
5001:
4972:
4948:
4925:
4901:
4871:
4807:
4769:
4695:
4657:
4595:
4557:
4487:
4449:
4403:
4365:
4319:
4281:
4231:
4230:{\displaystyle \ln(x)}
4190:
4159:
4124:
4093:
4058:
4037:
4003:
3982:
3953:
3909:
3887:Addition / subtraction
3838:
3562:
3486:
3457:
3379:
3359:
3339:
3308:
3288:
3259:
3229:, yielding a ratio of
3223:
3153:
3133:
3077:
3061:logarithmic derivative
3039:
2909:
2889:
2843:
2788:
2761:
2672:
2535:
2515:
2475:
2453:
2424:
2380:
2341:
2318:
2294:
2258:
2222:
2121:
2097:
2073:
2037:
1998:
1914:
1873:
1811:
1701:
1233:
923:
764:
652:
626:
483:
389:
311:
270:
250:
221:
192:
172:
78:
5616:
5581:
5561:
5531:
5497:
5416:
5393:
5369:
5340:
5153:
5133:
5110:
5090:
5057:
5037:
5002:
4973:
4949:
4926:
4902:
4872:
4808:
4770:
4696:
4658:
4596:
4567:Inverse sine function
4558:
4488:
4450:
4404:
4366:
4320:
4282:
4232:
4191:
4160:
4158:{\displaystyle e^{x}}
4125:
4094:
4092:{\displaystyle x^{n}}
4059:
4038:
4004:
3983:
3963:Scalar multiplication
3954:
3910:
3847:The last term is the
3839:
3563:
3561:{\displaystyle /f(x)}
3487:
3458:
3380:
3360:
3340:
3309:
3289:
3260:
3258:{\displaystyle xf'/f}
3224:
3154:
3134:
3078:
3059:of) the ratio of the
3040:
2910:
2890:
2844:
2794:is the Moore-Penrose
2789:
2762:
2673:
2536:
2516:
2476:
2454:
2425:
2381:
2342:
2319:
2295:
2259:
2223:
2122:
2103:respectively. Hence:
2098:
2074:
2038:
1999:
1915:
1874:
1812:
1702:
1234:
924:
765:
653:
634:In this context, the
627:
484:
390:
312:
271:
251:
222:
193:
173:
121:independent variables
79:
5590:
5570:
5550:
5529:{\displaystyle J(x)}
5511:
5428:
5405:
5382:
5352:
5165:
5142:
5119:
5099:
5088:{\displaystyle f(x)}
5070:
5046:
5026:
5000:{\displaystyle f(x)}
4982:
4962:
4938:
4915:
4891:
4818:
4785:
4706:
4673:
4606:
4573:
4498:
4465:
4414:
4381:
4330:
4297:
4242:
4209:
4170:
4142:
4135:Exponential function
4104:
4076:
4048:
4019:
3993:
3969:
3920:
3893:
3864:elementary functions
3575:
3496:
3485:{\displaystyle f(x)}
3467:
3389:
3369:
3349:
3326:
3298:
3273:
3233:
3163:
3143:
3087:
3067:
2922:
2899:
2853:
2833:
2771:
2714:
2548:
2525:
2489:
2465:
2436:
2408:
2355:
2331:
2308:
2268:
2232:
2135:
2111:
2087:
2047:
2011:
1927:
1891:
1857:
1721:
1261:
1051:
784:
777:condition number is
665:
642:
500:
406:
325:
280:
260:
231:
211:
182:
140:
50:
5679:Belsley, David A.;
5640:Numerical stability
5544:partial derivatives
5062:(specifically, its
4189:{\displaystyle |x|}
4123:{\displaystyle |n|}
4036:{\displaystyle 1/x}
3908:{\displaystyle x+a}
3868:significant figures
3849:difference quotient
2485:non-singular (i.e.
2393:arises so often in
84:one is solving for
5870:Numerical analysis
5713:Pesaran, M. Hashem
5611:
5576:
5556:
5526:
5492:
5411:
5388:
5364:
5335:
5215:
5190:
5148:
5128:
5105:
5085:
5052:
5032:
4997:
4968:
4944:
4921:
4897:
4867:
4803:
4765:
4691:
4653:
4591:
4553:
4483:
4445:
4399:
4361:
4315:
4277:
4227:
4186:
4155:
4120:
4089:
4054:
4033:
3999:
3981:{\displaystyle ax}
3978:
3949:
3905:
3855:), and taking the
3851:(the slope of the
3834:
3558:
3482:
3453:
3375:
3355:
3335:
3304:
3287:{\displaystyle f'}
3284:
3255:
3219:
3149:
3129:
3073:
3035:
2905:
2885:
2839:
2784:
2757:
2690:non-linear algebra
2668:
2627:
2578:
2531:
2511:
2471:
2449:
2420:
2376:
2337:
2314:
2290:
2254:
2218:
2117:
2093:
2069:
2033:
1994:
1910:
1869:
1807:
1697:
1695:
1611:
1545:
1470:
1404:
1289:
1229:
919:
837:
809:
760:
718:
690:
648:
622:
479:
385:
307:
266:
246:
217:
188:
168:
130:backward stability
125:dependent variable
74:
32:numerical analysis
5789:978-0-89871-361-9
5757:978-0-495-11475-8
5730:978-0-19-875998-0
5650:Ill-posed problem
5579:{\displaystyle x}
5559:{\displaystyle f}
5487:
5414:{\displaystyle f}
5391:{\displaystyle f}
5325:
5292:
5191:
5168:
5151:{\displaystyle x}
5108:{\displaystyle x}
5055:{\displaystyle x}
5035:{\displaystyle f}
4971:{\displaystyle n}
4947:{\displaystyle x}
4924:{\displaystyle m}
4900:{\displaystyle f}
4883:Several variables
4880:
4879:
4865:
4763:
4745:
4651:
4633:
4271:
4201:Natural logarithm
4057:{\displaystyle 1}
4002:{\displaystyle 1}
3943:
3882:Condition number
3829:
3779:
3755:
3687:
3663:
3378:{\displaystyle x}
3358:{\displaystyle x}
3307:{\displaystyle f}
3152:{\displaystyle x}
3076:{\displaystyle f}
3026:
2967:
2908:{\displaystyle x}
2842:{\displaystyle f}
2666:
2618:
2569:
2534:{\displaystyle i}
2474:{\displaystyle A}
2340:{\displaystyle A}
2317:{\displaystyle A}
2278:
2242:
2213:
2196:
2167:
2120:{\displaystyle A}
2096:{\displaystyle A}
2057:
2021:
1989:
1976:
1955:
1643:
1596:
1589:
1530:
1514:
1455:
1448:
1389:
1375:
1334:
1268:
1224:
1183:
1139:
1108:
914:
811:
787:
758:
692:
668:
651:{\displaystyle f}
579:
462:
370:
292:
269:{\displaystyle x}
243:
227:and an algorithm
220:{\displaystyle f}
191:{\displaystyle k}
16:(Redirected from
5882:
5825:
5794:
5793:
5773:
5762:
5761:
5741:
5735:
5734:
5709:
5703:
5702:
5676:
5620:
5618:
5617:
5612:
5585:
5583:
5582:
5577:
5565:
5563:
5562:
5557:
5537:
5535:
5533:
5532:
5527:
5501:
5499:
5498:
5493:
5488:
5486:
5476:
5452:
5432:
5420:
5418:
5417:
5412:
5397:
5395:
5394:
5389:
5373:
5371:
5370:
5365:
5344:
5342:
5341:
5336:
5331:
5327:
5326:
5324:
5313:
5299:
5297:
5293:
5291:
5271:
5267:
5225:
5214:
5189:
5188:
5187:
5157:
5155:
5154:
5149:
5137:
5135:
5134:
5129:
5114:
5112:
5111:
5106:
5094:
5092:
5091:
5086:
5061:
5059:
5058:
5053:
5041:
5039:
5038:
5033:
5006:
5004:
5003:
4998:
4977:
4975:
4974:
4969:
4953:
4951:
4950:
4945:
4930:
4928:
4927:
4922:
4906:
4904:
4903:
4898:
4876:
4874:
4873:
4868:
4866:
4864:
4845:
4844:
4822:
4812:
4810:
4809:
4804:
4774:
4772:
4771:
4766:
4764:
4762:
4746:
4744:
4743:
4728:
4725:
4724:
4716:
4710:
4700:
4698:
4697:
4692:
4662:
4660:
4659:
4654:
4652:
4650:
4634:
4632:
4631:
4616:
4610:
4600:
4598:
4597:
4592:
4562:
4560:
4559:
4554:
4552:
4505:
4492:
4490:
4489:
4484:
4459:Tangent function
4454:
4452:
4451:
4446:
4444:
4421:
4408:
4406:
4405:
4400:
4370:
4368:
4367:
4362:
4360:
4337:
4324:
4322:
4321:
4316:
4286:
4284:
4283:
4278:
4276:
4272:
4270:
4250:
4236:
4234:
4233:
4228:
4195:
4193:
4192:
4187:
4185:
4177:
4164:
4162:
4161:
4156:
4154:
4153:
4129:
4127:
4126:
4121:
4119:
4111:
4098:
4096:
4095:
4090:
4088:
4087:
4063:
4061:
4060:
4055:
4042:
4040:
4039:
4034:
4029:
4008:
4006:
4005:
4000:
3987:
3985:
3984:
3979:
3958:
3956:
3955:
3950:
3948:
3944:
3942:
3928:
3914:
3912:
3911:
3906:
3873:
3872:
3843:
3841:
3840:
3835:
3830:
3828:
3820:
3782:
3780:
3778:
3761:
3756:
3754:
3728:
3690:
3688:
3686:
3669:
3664:
3662:
3658:
3640:
3627:
3579:
3567:
3565:
3564:
3559:
3545:
3491:
3489:
3488:
3483:
3462:
3460:
3459:
3454:
3449:
3426:
3384:
3382:
3381:
3376:
3364:
3362:
3361:
3356:
3344:
3342:
3341:
3336:
3313:
3311:
3310:
3305:
3293:
3291:
3290:
3285:
3283:
3264:
3262:
3261:
3256:
3251:
3246:
3228:
3226:
3225:
3220:
3215:
3201:
3196:
3185:
3158:
3156:
3155:
3150:
3138:
3136:
3135:
3130:
3125:
3120:
3109:
3082:
3080:
3079:
3074:
3044:
3042:
3041:
3036:
3031:
3027:
3025:
3024:
3003:
3002:
2981:
2972:
2968:
2966:
2952:
2942:
2930:
2914:
2912:
2911:
2906:
2894:
2892:
2891:
2886:
2884:
2880:
2876:
2871:
2848:
2846:
2845:
2840:
2793:
2791:
2790:
2785:
2783:
2782:
2766:
2764:
2763:
2758:
2753:
2752:
2701:well-conditioned
2677:
2675:
2674:
2669:
2667:
2665:
2664:
2663:
2657:
2652:
2651:
2639:
2634:
2633:
2626:
2616:
2615:
2614:
2608:
2603:
2602:
2590:
2585:
2584:
2577:
2567:
2540:
2538:
2537:
2532:
2520:
2518:
2517:
2512:
2504:
2503:
2483:lower triangular
2480:
2478:
2477:
2472:
2458:
2456:
2455:
2450:
2448:
2447:
2429:
2427:
2426:
2421:
2385:
2383:
2382:
2377:
2346:
2344:
2343:
2338:
2323:
2321:
2320:
2315:
2299:
2297:
2296:
2291:
2280:
2279:
2276:
2263:
2261:
2260:
2255:
2244:
2243:
2240:
2227:
2225:
2224:
2219:
2214:
2212:
2208:
2198:
2197:
2194:
2183:
2179:
2169:
2168:
2165:
2154:
2126:
2124:
2123:
2118:
2102:
2100:
2099:
2094:
2078:
2076:
2075:
2070:
2059:
2058:
2055:
2042:
2040:
2039:
2034:
2023:
2022:
2019:
2003:
2001:
2000:
1995:
1990:
1988:
1978:
1977:
1974:
1967:
1957:
1956:
1953:
1946:
1919:
1917:
1916:
1911:
1909:
1908:
1878:
1876:
1875:
1870:
1816:
1814:
1813:
1808:
1800:
1796:
1792:
1791:
1771:
1759:
1755:
1754:
1706:
1704:
1703:
1698:
1696:
1679:
1675:
1674:
1652:
1648:
1644:
1642:
1631:
1617:
1610:
1594:
1590:
1588:
1577:
1573:
1569:
1568:
1551:
1544:
1523:
1519:
1515:
1513:
1509:
1505:
1504:
1487:
1476:
1469:
1453:
1449:
1447:
1436:
1432:
1428:
1427:
1410:
1403:
1381:
1377:
1376:
1374:
1370:
1366:
1365:
1348:
1337:
1335:
1333:
1322:
1318:
1314:
1313:
1296:
1288:
1238:
1236:
1235:
1230:
1225:
1223:
1219:
1215:
1214:
1197:
1186:
1184:
1182:
1171:
1167:
1163:
1162:
1145:
1140:
1138:
1127:
1116:
1114:
1109:
1107:
1103:
1099:
1098:
1081:
1077:
1073:
1072:
1055:
1018:. Assuming that
1014:be the error in
928:
926:
925:
920:
915:
913:
903:
885:
866:
839:
836:
808:
807:
806:
769:
767:
766:
761:
759:
757:
743:
720:
717:
689:
688:
687:
657:
655:
654:
649:
631:
629:
628:
623:
600:
595:
591:
581:
580:
572:
528:
488:
486:
485:
480:
478:
474:
464:
463:
455:
394:
392:
391:
386:
372:
371:
363:
316:
314:
313:
308:
294:
293:
285:
275:
273:
272:
267:
255:
253:
252:
247:
245:
244:
236:
226:
224:
223:
218:
207:Given a problem
197:
195:
194:
189:
177:
175:
174:
169:
167:
166:
110:well-conditioned
83:
81:
80:
75:
36:condition number
21:
5890:
5889:
5885:
5884:
5883:
5881:
5880:
5879:
5860:
5859:
5832:
5822:
5803:
5801:Further reading
5798:
5797:
5790:
5774:
5765:
5758:
5750:. p. 321.
5742:
5738:
5731:
5710:
5706:
5699:
5677:
5673:
5668:
5631:
5625:on the matrix.
5591:
5588:
5587:
5571:
5568:
5567:
5551:
5548:
5547:
5540:Jacobian matrix
5512:
5509:
5508:
5506:
5472:
5453:
5433:
5431:
5429:
5426:
5425:
5406:
5403:
5402:
5383:
5380:
5379:
5353:
5350:
5349:
5314:
5300:
5298:
5272:
5230:
5226:
5224:
5221:
5220:
5216:
5195:
5183:
5179:
5172:
5166:
5163:
5162:
5143:
5140:
5139:
5120:
5117:
5116:
5100:
5097:
5096:
5071:
5068:
5067:
5047:
5044:
5043:
5027:
5024:
5023:
4983:
4980:
4979:
4963:
4960:
4959:
4939:
4936:
4935:
4916:
4913:
4912:
4892:
4889:
4888:
4885:
4840:
4836:
4826:
4821:
4819:
4816:
4815:
4786:
4783:
4782:
4739:
4735:
4727:
4726:
4720:
4712:
4711:
4709:
4707:
4704:
4703:
4674:
4671:
4670:
4627:
4623:
4615:
4614:
4609:
4607:
4604:
4603:
4574:
4571:
4570:
4548:
4501:
4499:
4496:
4495:
4466:
4463:
4462:
4440:
4417:
4415:
4412:
4411:
4382:
4379:
4378:
4375:Cosine function
4356:
4333:
4331:
4328:
4327:
4298:
4295:
4294:
4254:
4249:
4245:
4243:
4240:
4239:
4210:
4207:
4206:
4181:
4173:
4171:
4168:
4167:
4149:
4145:
4143:
4140:
4139:
4115:
4107:
4105:
4102:
4101:
4083:
4079:
4077:
4074:
4073:
4049:
4046:
4045:
4025:
4020:
4017:
4016:
3994:
3991:
3990:
3970:
3967:
3966:
3932:
3927:
3923:
3921:
3918:
3917:
3894:
3891:
3890:
3821:
3783:
3781:
3765:
3760:
3729:
3691:
3689:
3673:
3668:
3654:
3641:
3623:
3580:
3578:
3576:
3573:
3572:
3541:
3497:
3494:
3493:
3468:
3465:
3464:
3445:
3422:
3390:
3387:
3386:
3370:
3367:
3366:
3350:
3347:
3346:
3327:
3324:
3323:
3299:
3296:
3295:
3276:
3274:
3271:
3270:
3247:
3239:
3234:
3231:
3230:
3211:
3197:
3189:
3178:
3164:
3161:
3160:
3144:
3141:
3140:
3121:
3113:
3102:
3088:
3085:
3084:
3068:
3065:
3064:
3017:
3004:
2995:
2982:
2980:
2976:
2953:
2935:
2931:
2929:
2925:
2923:
2920:
2919:
2900:
2897:
2896:
2872:
2864:
2860:
2856:
2854:
2851:
2850:
2834:
2831:
2830:
2824:
2804:
2778:
2774:
2772:
2769:
2768:
2748:
2744:
2715:
2712:
2711:
2705:ill-conditioned
2659:
2658:
2653:
2644:
2640:
2635:
2629:
2628:
2622:
2617:
2610:
2609:
2604:
2595:
2591:
2586:
2580:
2579:
2573:
2568:
2566:
2549:
2546:
2545:
2526:
2523:
2522:
2496:
2492:
2490:
2487:
2486:
2466:
2463:
2462:
2443:
2439:
2437:
2434:
2433:
2409:
2406:
2405:
2356:
2353:
2352:
2332:
2329:
2328:
2309:
2306:
2305:
2275:
2271:
2269:
2266:
2265:
2239:
2235:
2233:
2230:
2229:
2193:
2189:
2188:
2184:
2164:
2160:
2159:
2155:
2153:
2136:
2133:
2132:
2112:
2109:
2108:
2088:
2085:
2084:
2081:singular values
2054:
2050:
2048:
2045:
2044:
2018:
2014:
2012:
2009:
2008:
1973:
1969:
1968:
1952:
1948:
1947:
1945:
1928:
1925:
1924:
1904:
1900:
1892:
1889:
1888:
1858:
1855:
1854:
1826:linear isometry
1784:
1780:
1779:
1775:
1761:
1747:
1743:
1739:
1722:
1719:
1718:
1694:
1693:
1667:
1663:
1659:
1650:
1649:
1632:
1618:
1616:
1612:
1600:
1578:
1561:
1557:
1556:
1552:
1550:
1546:
1534:
1521:
1520:
1497:
1493:
1492:
1488:
1477:
1475:
1471:
1459:
1437:
1420:
1416:
1415:
1411:
1409:
1405:
1393:
1382:
1358:
1354:
1353:
1349:
1338:
1336:
1323:
1306:
1302:
1301:
1297:
1295:
1294:
1290:
1272:
1264:
1262:
1259:
1258:
1207:
1203:
1202:
1198:
1187:
1185:
1172:
1155:
1151:
1150:
1146:
1144:
1128:
1117:
1115:
1110:
1091:
1087:
1086:
1082:
1065:
1061:
1060:
1056:
1054:
1052:
1049:
1048:
954:round-off error
939:linear equation
935:
899:
886:
862:
840:
838:
815:
802:
798:
791:
785:
782:
781:
744:
721:
719:
696:
683:
679:
672:
666:
663:
662:
643:
640:
639:
596:
571:
570:
554:
550:
524:
501:
498:
497:
454:
453:
437:
433:
407:
404:
403:
362:
361:
326:
323:
322:
284:
283:
281:
278:
277:
261:
258:
257:
235:
234:
232:
229:
228:
212:
209:
208:
205:
183:
180:
179:
162:
158:
141:
138:
137:
116:ill-conditioned
51:
48:
47:
28:
23:
22:
18:Ill-conditioned
15:
12:
11:
5:
5888:
5878:
5877:
5872:
5858:
5857:
5852:
5847:
5842:
5831:
5830:External links
5828:
5827:
5826:
5820:
5802:
5799:
5796:
5795:
5788:
5763:
5756:
5736:
5729:
5704:
5697:
5670:
5669:
5667:
5664:
5663:
5662:
5657:
5655:Singular value
5652:
5647:
5645:Hilbert matrix
5642:
5637:
5630:
5627:
5610:
5607:
5604:
5601:
5598:
5595:
5575:
5555:
5525:
5522:
5519:
5516:
5503:
5502:
5491:
5485:
5482:
5479:
5475:
5471:
5468:
5465:
5462:
5459:
5456:
5451:
5448:
5445:
5442:
5439:
5436:
5410:
5387:
5363:
5360:
5357:
5346:
5345:
5334:
5330:
5323:
5320:
5317:
5312:
5309:
5306:
5303:
5296:
5290:
5287:
5284:
5281:
5278:
5275:
5270:
5266:
5263:
5260:
5257:
5254:
5251:
5248:
5245:
5242:
5239:
5236:
5233:
5229:
5223:
5219:
5213:
5210:
5207:
5204:
5201:
5198:
5194:
5186:
5182:
5178:
5175:
5171:
5147:
5127:
5124:
5104:
5084:
5081:
5078:
5075:
5051:
5031:
4996:
4993:
4990:
4987:
4967:
4943:
4920:
4896:
4884:
4881:
4878:
4877:
4863:
4860:
4857:
4854:
4851:
4848:
4843:
4839:
4835:
4832:
4829:
4825:
4813:
4802:
4799:
4796:
4793:
4790:
4780:
4776:
4775:
4761:
4758:
4755:
4752:
4749:
4742:
4738:
4734:
4731:
4723:
4719:
4715:
4701:
4690:
4687:
4684:
4681:
4678:
4668:
4664:
4663:
4649:
4646:
4643:
4640:
4637:
4630:
4626:
4622:
4619:
4613:
4601:
4590:
4587:
4584:
4581:
4578:
4568:
4564:
4563:
4551:
4547:
4544:
4541:
4538:
4535:
4532:
4529:
4526:
4523:
4520:
4517:
4514:
4511:
4508:
4504:
4493:
4482:
4479:
4476:
4473:
4470:
4460:
4456:
4455:
4443:
4439:
4436:
4433:
4430:
4427:
4424:
4420:
4409:
4398:
4395:
4392:
4389:
4386:
4376:
4372:
4371:
4359:
4355:
4352:
4349:
4346:
4343:
4340:
4336:
4325:
4314:
4311:
4308:
4305:
4302:
4292:
4288:
4287:
4275:
4269:
4266:
4263:
4260:
4257:
4253:
4248:
4237:
4226:
4223:
4220:
4217:
4214:
4204:
4197:
4196:
4184:
4180:
4176:
4165:
4152:
4148:
4137:
4131:
4130:
4118:
4114:
4110:
4099:
4086:
4082:
4071:
4065:
4064:
4053:
4043:
4032:
4028:
4024:
4014:
4010:
4009:
3998:
3988:
3977:
3974:
3964:
3960:
3959:
3947:
3941:
3938:
3935:
3931:
3926:
3915:
3904:
3901:
3898:
3888:
3884:
3883:
3880:
3877:
3845:
3844:
3833:
3827:
3824:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3777:
3774:
3771:
3768:
3764:
3759:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3703:
3700:
3697:
3694:
3685:
3682:
3679:
3676:
3672:
3667:
3661:
3657:
3653:
3650:
3647:
3644:
3639:
3636:
3633:
3630:
3626:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3557:
3554:
3551:
3548:
3544:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3504:
3501:
3481:
3478:
3475:
3472:
3452:
3448:
3444:
3441:
3438:
3435:
3432:
3429:
3425:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3374:
3354:
3334:
3331:
3303:
3282:
3279:
3254:
3250:
3245:
3242:
3238:
3218:
3214:
3210:
3207:
3204:
3200:
3195:
3192:
3188:
3184:
3181:
3177:
3174:
3171:
3168:
3148:
3128:
3124:
3119:
3116:
3112:
3108:
3105:
3101:
3098:
3095:
3092:
3072:
3057:absolute value
3046:
3045:
3034:
3030:
3023:
3020:
3016:
3013:
3010:
3007:
3001:
2998:
2994:
2991:
2988:
2985:
2979:
2975:
2971:
2965:
2962:
2959:
2956:
2951:
2948:
2945:
2941:
2938:
2934:
2928:
2904:
2883:
2879:
2875:
2870:
2867:
2863:
2859:
2838:
2823:
2820:
2803:
2800:
2781:
2777:
2756:
2751:
2747:
2743:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2694:transcendental
2686:Euclidean norm
2679:
2678:
2662:
2656:
2650:
2647:
2643:
2638:
2632:
2625:
2621:
2613:
2607:
2601:
2598:
2594:
2589:
2583:
2576:
2572:
2565:
2562:
2559:
2556:
2553:
2530:
2510:
2507:
2502:
2499:
2495:
2470:
2446:
2442:
2419:
2416:
2413:
2387:
2386:
2375:
2372:
2369:
2366:
2363:
2360:
2336:
2325:
2313:
2289:
2286:
2283:
2274:
2253:
2250:
2247:
2238:
2217:
2211:
2207:
2204:
2201:
2192:
2187:
2182:
2178:
2175:
2172:
2163:
2158:
2152:
2149:
2146:
2143:
2140:
2116:
2092:
2068:
2065:
2062:
2053:
2032:
2029:
2026:
2017:
2005:
2004:
1993:
1987:
1984:
1981:
1972:
1966:
1963:
1960:
1951:
1944:
1941:
1938:
1935:
1932:
1907:
1903:
1899:
1896:
1868:
1865:
1862:
1818:
1817:
1806:
1803:
1799:
1795:
1790:
1787:
1783:
1778:
1774:
1770:
1767:
1764:
1758:
1753:
1750:
1746:
1742:
1738:
1735:
1732:
1729:
1726:
1708:
1707:
1692:
1689:
1686:
1683:
1678:
1673:
1670:
1666:
1662:
1658:
1655:
1653:
1651:
1647:
1641:
1638:
1635:
1630:
1627:
1624:
1621:
1615:
1609:
1606:
1603:
1599:
1593:
1587:
1584:
1581:
1576:
1572:
1567:
1564:
1560:
1555:
1549:
1543:
1540:
1537:
1533:
1529:
1526:
1524:
1522:
1518:
1512:
1508:
1503:
1500:
1496:
1491:
1486:
1483:
1480:
1474:
1468:
1465:
1462:
1458:
1452:
1446:
1443:
1440:
1435:
1431:
1426:
1423:
1419:
1414:
1408:
1402:
1399:
1396:
1392:
1388:
1385:
1383:
1380:
1373:
1369:
1364:
1361:
1357:
1352:
1347:
1344:
1341:
1332:
1329:
1326:
1321:
1317:
1312:
1309:
1305:
1300:
1293:
1287:
1284:
1281:
1278:
1275:
1271:
1267:
1266:
1252:operator norms
1240:
1239:
1228:
1222:
1218:
1213:
1210:
1206:
1201:
1196:
1193:
1190:
1181:
1178:
1175:
1170:
1166:
1161:
1158:
1154:
1149:
1143:
1137:
1134:
1131:
1126:
1123:
1120:
1113:
1106:
1102:
1097:
1094:
1090:
1085:
1080:
1076:
1071:
1068:
1064:
1059:
997:relative error
966:floating-point
934:
931:
930:
929:
918:
912:
909:
906:
902:
898:
895:
892:
889:
884:
881:
878:
875:
872:
869:
865:
861:
858:
855:
852:
849:
846:
843:
835:
831:
827:
824:
821:
818:
814:
805:
801:
797:
794:
790:
771:
770:
756:
753:
750:
747:
742:
739:
736:
733:
730:
727:
724:
716:
712:
708:
705:
702:
699:
695:
686:
682:
678:
675:
671:
647:
621:
618:
615:
612:
609:
606:
603:
599:
594:
590:
587:
584:
578:
575:
569:
566:
563:
560:
557:
553:
549:
546:
543:
540:
537:
534:
531:
527:
523:
520:
517:
514:
511:
508:
505:
477:
473:
470:
467:
461:
458:
452:
449:
446:
443:
440:
436:
432:
429:
426:
423:
420:
417:
414:
411:
384:
381:
378:
375:
369:
366:
360:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
306:
303:
300:
297:
291:
288:
265:
256:with an input
242:
239:
216:
204:
201:
187:
165:
161:
157:
154:
151:
148:
145:
102:linear algebra
73:
70:
67:
64:
61:
58:
55:
26:
9:
6:
4:
3:
2:
5887:
5876:
5873:
5871:
5868:
5867:
5865:
5856:
5853:
5851:
5848:
5846:
5843:
5841:
5837:
5834:
5833:
5823:
5821:0-19-853564-3
5817:
5813:
5809:
5808:Demmel, James
5805:
5804:
5791:
5785:
5781:
5780:
5772:
5770:
5768:
5759:
5753:
5749:
5748:
5740:
5732:
5726:
5722:
5718:
5714:
5708:
5700:
5698:0-471-05856-4
5694:
5690:
5686:
5682:
5675:
5671:
5661:
5660:Wilson matrix
5658:
5656:
5653:
5651:
5648:
5646:
5643:
5641:
5638:
5636:
5633:
5632:
5626:
5624:
5602:
5596:
5573:
5553:
5545:
5541:
5520:
5514:
5489:
5480:
5473:
5463:
5457:
5443:
5437:
5424:
5423:
5422:
5408:
5399:
5385:
5377:
5358:
5332:
5328:
5318:
5307:
5304:
5282:
5276:
5261:
5255:
5252:
5246:
5243:
5240:
5237:
5231:
5217:
5211:
5208:
5202:
5199:
5184:
5180:
5173:
5161:
5160:
5159:
5145:
5125:
5122:
5102:
5079:
5073:
5065:
5049:
5029:
5020:
5018:
5015:or computing
5014:
5010:
5009:Banach spaces
4991:
4985:
4965:
4957:
4941:
4934:
4918:
4910:
4894:
4858:
4852:
4849:
4841:
4837:
4833:
4830:
4823:
4814:
4797:
4791:
4788:
4781:
4778:
4777:
4756:
4750:
4747:
4740:
4736:
4732:
4729:
4717:
4702:
4685:
4679:
4676:
4669:
4666:
4665:
4644:
4638:
4635:
4628:
4624:
4620:
4617:
4611:
4602:
4585:
4579:
4576:
4569:
4566:
4565:
4539:
4533:
4530:
4527:
4521:
4515:
4512:
4506:
4494:
4477:
4471:
4468:
4461:
4458:
4457:
4434:
4428:
4425:
4422:
4410:
4393:
4387:
4384:
4377:
4374:
4373:
4350:
4344:
4341:
4338:
4326:
4309:
4303:
4300:
4293:
4291:Sine function
4290:
4289:
4273:
4264:
4258:
4255:
4251:
4246:
4238:
4221:
4215:
4212:
4205:
4202:
4199:
4198:
4178:
4166:
4150:
4146:
4138:
4136:
4133:
4132:
4112:
4100:
4084:
4080:
4072:
4070:
4067:
4066:
4051:
4044:
4030:
4026:
4022:
4015:
4012:
4011:
3996:
3989:
3975:
3972:
3965:
3962:
3961:
3945:
3939:
3936:
3933:
3929:
3924:
3916:
3902:
3899:
3896:
3889:
3886:
3885:
3881:
3878:
3875:
3874:
3871:
3869:
3865:
3860:
3858:
3854:
3850:
3831:
3825:
3814:
3808:
3805:
3799:
3793:
3790:
3784:
3772:
3766:
3762:
3757:
3751:
3748:
3742:
3736:
3733:
3722:
3716:
3713:
3707:
3701:
3698:
3692:
3680:
3674:
3670:
3665:
3659:
3655:
3648:
3634:
3628:
3624:
3614:
3608:
3605:
3599:
3593:
3590:
3584:
3571:
3570:
3569:
3552:
3546:
3542:
3532:
3526:
3523:
3517:
3511:
3508:
3502:
3476:
3470:
3450:
3446:
3439:
3430:
3427:
3423:
3416:
3413:
3407:
3401:
3398:
3372:
3352:
3332:
3320:
3317:
3301:
3280:
3277:
3268:
3267:infinitesimal
3252:
3248:
3243:
3240:
3236:
3216:
3212:
3208:
3205:
3202:
3198:
3193:
3190:
3186:
3182:
3175:
3172:
3169:
3146:
3126:
3122:
3117:
3114:
3110:
3106:
3099:
3096:
3093:
3070:
3062:
3058:
3053:
3051:
3032:
3028:
3021:
3014:
3011:
3008:
2999:
2992:
2989:
2986:
2977:
2973:
2969:
2960:
2954:
2946:
2939:
2936:
2932:
2926:
2918:
2917:
2916:
2902:
2881:
2877:
2873:
2868:
2865:
2861:
2857:
2836:
2829:
2819:
2817:
2813:
2809:
2799:
2797:
2796:pseudoinverse
2779:
2775:
2749:
2745:
2735:
2729:
2723:
2717:
2708:
2706:
2702:
2697:
2695:
2691:
2687:
2682:
2648:
2645:
2641:
2623:
2599:
2596:
2592:
2574:
2563:
2557:
2551:
2544:
2543:
2542:
2528:
2508:
2505:
2500:
2497:
2493:
2484:
2468:
2460:
2459:(vector) norm
2440:
2414:
2402:
2400:
2396:
2392:
2373:
2370:
2364:
2358:
2350:
2334:
2326:
2324:respectively.
2311:
2303:
2284:
2272:
2248:
2236:
2215:
2209:
2202:
2190:
2185:
2180:
2173:
2161:
2156:
2150:
2144:
2138:
2130:
2114:
2106:
2105:
2104:
2090:
2082:
2063:
2051:
2027:
2015:
1991:
1982:
1970:
1961:
1949:
1942:
1936:
1930:
1923:
1922:
1921:
1905:
1897:
1886:
1882:
1863:
1851:
1849:
1844:
1842:
1838:
1833:
1829:
1827:
1823:
1804:
1801:
1793:
1788:
1785:
1781:
1772:
1765:
1751:
1748:
1744:
1736:
1730:
1724:
1717:
1716:
1715:
1713:
1690:
1684:
1671:
1668:
1664:
1656:
1654:
1645:
1636:
1625:
1622:
1613:
1607:
1604:
1601:
1591:
1582:
1570:
1565:
1562:
1558:
1547:
1541:
1538:
1535:
1527:
1525:
1516:
1506:
1501:
1498:
1494:
1481:
1472:
1466:
1463:
1460:
1450:
1441:
1429:
1424:
1421:
1417:
1406:
1400:
1397:
1394:
1386:
1384:
1378:
1367:
1362:
1359:
1355:
1342:
1327:
1315:
1310:
1307:
1303:
1291:
1285:
1282:
1279:
1276:
1273:
1257:
1256:
1255:
1253:
1249:
1245:
1226:
1216:
1211:
1208:
1204:
1191:
1176:
1164:
1159:
1156:
1152:
1141:
1132:
1121:
1111:
1100:
1095:
1092:
1088:
1074:
1069:
1066:
1062:
1047:
1046:
1045:
1043:
1039:
1036:
1032:
1029:
1025:
1021:
1017:
1013:
1008:
1006:
1002:
998:
993:
991:
987:
983:
979:
975:
971:
967:
963:
959:
955:
951:
947:
944: =
943:
940:
916:
907:
900:
893:
890:
876:
870:
863:
853:
847:
844:
833:
829:
822:
819:
803:
799:
792:
780:
779:
778:
776:
751:
748:
734:
728:
725:
714:
710:
703:
700:
684:
680:
673:
661:
660:
659:
645:
637:
632:
619:
610:
604:
597:
585:
573:
567:
561:
555:
547:
538:
532:
525:
515:
509:
506:
495:
493:
468:
456:
450:
444:
438:
430:
421:
415:
412:
401:
399:
382:
376:
364:
358:
352:
346:
343:
337:
331:
328:
320:
304:
298:
286:
263:
237:
214:
200:
185:
163:
159:
155:
149:
143:
134:
132:
131:
126:
122:
118:
117:
112:
111:
105:
103:
98:
94:
89:
87:
71:
68:
65:
59:
53:
45:
41:
37:
33:
19:
5839:
5811:
5778:
5746:
5739:
5720:
5707:
5688:
5674:
5623:induced norm
5538:denotes the
5504:
5400:
5347:
5063:
5021:
4954:) into some
4933:real numbers
4886:
3861:
3846:
3321:
3054:
3047:
2825:
2822:One variable
2805:
2709:
2698:
2689:
2683:
2680:
2403:
2398:
2390:
2388:
2006:
1884:
1852:
1845:
1834:
1830:
1821:
1819:
1709:
1254:as follows:
1247:
1243:
1241:
1041:
1037:
1034:
1030:
1027:
1019:
1015:
1011:
1009:
1004:
1000:
994:
989:
985:
981:
977:
973:
969:
949:
945:
941:
936:
774:
772:
635:
633:
491:
397:
318:
206:
135:
128:
115:
114:
109:
108:
106:
90:
85:
35:
29:
5042:at a point
5017:eigenvalues
3853:secant line
3159:, which is
3083:, which is
2302:eigenvalues
1024:nonsingular
276:and output
5864:Categories
5681:Kuh, Edwin
5666:References
4931:-tuple of
4069:Polynomial
3050:elasticity
2915:, this is
1841:invertible
960:, not the
97:asymptotic
5609:‖
5594:‖
5484:‖
5478:‖
5470:‖
5455:‖
5450:‖
5435:‖
5362:‖
5359:⋅
5356:‖
5322:‖
5316:‖
5311:‖
5305:δ
5302:‖
5289:‖
5274:‖
5253:−
5244:δ
5212:ε
5209:≤
5206:‖
5200:δ
5197:‖
5177:→
5174:ε
5123:δ
4958:(e.g. an
4911:(e.g. an
4853:
4792:
4751:
4733:−
4680:
4639:
4621:−
4580:
4534:
4516:
4472:
4429:
4388:
4345:
4304:
4259:
4216:
3823:Δ
3806:−
3797:Δ
3749:−
3740:Δ
3714:−
3705:Δ
3646:Δ
3606:−
3597:Δ
3524:−
3515:Δ
3437:Δ
3414:−
3405:Δ
3330:Δ
3173:
3097:
3012:
2990:
2802:Nonlinear
2780:†
2755:‖
2750:†
2742:‖
2739:‖
2733:‖
2718:κ
2564:≥
2552:κ
2506:≠
2445:∞
2418:‖
2415:⋅
2412:‖
2359:κ
2273:λ
2237:λ
2191:λ
2162:λ
2139:κ
2052:σ
2016:σ
1971:σ
1950:σ
1931:κ
1902:‖
1898:⋅
1895:‖
1867:‖
1864:⋅
1861:‖
1837:ill-posed
1786:−
1773:≥
1749:−
1725:κ
1688:‖
1682:‖
1669:−
1640:‖
1634:‖
1629:‖
1620:‖
1605:≠
1586:‖
1580:‖
1563:−
1539:≠
1499:−
1485:‖
1479:‖
1464:≠
1445:‖
1439:‖
1422:−
1398:≠
1360:−
1346:‖
1340:‖
1331:‖
1325:‖
1308:−
1283:≠
1209:−
1195:‖
1189:‖
1180:‖
1174:‖
1157:−
1136:‖
1130:‖
1125:‖
1119:‖
1093:−
1067:−
962:algorithm
911:‖
905:‖
897:‖
891:δ
888:‖
883:‖
868:‖
860:‖
845:δ
842:‖
834:ε
830:≤
826:‖
820:δ
817:‖
796:→
793:ε
755:‖
749:δ
746:‖
741:‖
726:δ
723:‖
715:ε
711:≤
707:‖
701:δ
698:‖
677:→
674:ε
617:‖
602:‖
577:~
568:−
545:‖
530:‖
522:‖
507:δ
504:‖
460:~
451:−
428:‖
413:δ
410:‖
368:~
359:−
329:δ
290:~
241:~
144:κ
44:sensitive
5875:Matrices
5782:. SIAM.
5715:(2015).
5629:See also
5269:‖
5228:‖
4956:codomain
4203:function
4013:Division
3281:′
3244:′
3194:′
3183:′
3118:′
3107:′
3022:′
3000:′
2940:′
2869:′
2812:supremum
2808:calculus
2767:, where
2541:), then
2521:for all
1920:), then
1798:‖
1777:‖
1769:‖
1763:‖
1757:‖
1741:‖
1677:‖
1661:‖
1575:‖
1554:‖
1511:‖
1490:‖
1434:‖
1413:‖
1372:‖
1351:‖
1320:‖
1299:‖
1221:‖
1200:‖
1169:‖
1148:‖
1105:‖
1084:‖
1079:‖
1058:‖
933:Matrices
775:relative
773:and the
636:absolute
593:‖
552:‖
492:relative
489:and the
476:‖
435:‖
398:absolute
40:function
5621:is the
5536:
5507:
2430:is the
2351:, then
2349:unitary
2131:, then
1879:is the
5818:
5786:
5754:
5727:
5695:
5586:, and
5505:where
5348:where
4909:domain
4850:arctan
4789:arctan
4748:arccos
4677:arccos
4636:arcsin
4577:arcsin
3879:Symbol
2816:domain
2228:where
2129:normal
2007:where
958:matrix
34:, the
5374:is a
3857:limit
1022:is a
494:error
400:error
319:error
38:of a
5816:ISBN
5784:ISBN
5752:ISBN
5725:ISBN
5693:ISBN
5376:norm
3876:Name
3316:zero
2461:and
2264:and
2043:and
1848:norm
1712:norm
1246:and
1010:Let
395:the
317:the
5838:at
5566:at
5546:of
5542:of
5401:If
5193:sup
5170:lim
5138:in
4531:cot
4513:tan
4469:tan
4426:tan
4385:cos
4342:cot
4301:sin
3492:is
3385:is
3345:in
3170:log
3094:log
3063:of
3009:log
2987:log
2620:min
2571:max
2481:is
2404:If
2347:is
2327:If
2304:of
2277:min
2241:max
2195:min
2166:max
2127:is
2107:If
2083:of
2056:min
2020:max
1975:min
1954:max
1853:If
1598:max
1532:max
1457:max
1391:max
1270:max
1044:is
1033:is
999:in
964:or
813:sup
789:lim
694:sup
670:lim
658:is
496:is
402:is
321:is
30:In
5866::
5766:^
5719:.
5687:.
5398:.
5019:.
4256:ln
4213:ln
2401:.
2374:1.
1805:1.
1007:.
992:.
942:Ax
344::=
160:10
86:x,
5824:.
5792:.
5760:.
5733:.
5701:.
5606:)
5603:x
5600:(
5597:J
5574:x
5554:f
5524:)
5521:x
5518:(
5515:J
5490:,
5481:x
5474:/
5467:)
5464:x
5461:(
5458:f
5447:)
5444:x
5441:(
5438:J
5409:f
5386:f
5333:,
5329:]
5319:x
5308:x
5295:/
5286:)
5283:x
5280:(
5277:f
5265:)
5262:x
5259:(
5256:f
5250:)
5247:x
5241:+
5238:x
5235:(
5232:f
5218:[
5203:x
5185:+
5181:0
5146:x
5126:x
5103:x
5083:)
5080:x
5077:(
5074:f
5050:x
5030:f
4995:)
4992:x
4989:(
4986:f
4966:n
4942:x
4919:m
4895:f
4862:)
4859:x
4856:(
4847:)
4842:2
4838:x
4834:+
4831:1
4828:(
4824:x
4801:)
4798:x
4795:(
4760:)
4757:x
4754:(
4741:2
4737:x
4730:1
4722:|
4718:x
4714:|
4689:)
4686:x
4683:(
4648:)
4645:x
4642:(
4629:2
4625:x
4618:1
4612:x
4589:)
4586:x
4583:(
4550:|
4546:)
4543:)
4540:x
4537:(
4528:+
4525:)
4522:x
4519:(
4510:(
4507:x
4503:|
4481:)
4478:x
4475:(
4442:|
4438:)
4435:x
4432:(
4423:x
4419:|
4397:)
4394:x
4391:(
4358:|
4354:)
4351:x
4348:(
4339:x
4335:|
4313:)
4310:x
4307:(
4274:|
4268:)
4265:x
4262:(
4252:1
4247:|
4225:)
4222:x
4219:(
4183:|
4179:x
4175:|
4151:x
4147:e
4117:|
4113:n
4109:|
4085:n
4081:x
4052:1
4031:x
4027:/
4023:1
3997:1
3976:x
3973:a
3946:|
3940:a
3937:+
3934:x
3930:x
3925:|
3903:a
3900:+
3897:x
3832:.
3826:x
3818:)
3815:x
3812:(
3809:f
3803:)
3800:x
3794:+
3791:x
3788:(
3785:f
3776:)
3773:x
3770:(
3767:f
3763:x
3758:=
3752:x
3746:)
3743:x
3737:+
3734:x
3731:(
3726:)
3723:x
3720:(
3717:f
3711:)
3708:x
3702:+
3699:x
3696:(
3693:f
3684:)
3681:x
3678:(
3675:f
3671:x
3666:=
3660:x
3656:/
3652:)
3649:x
3643:(
3638:)
3635:x
3632:(
3629:f
3625:/
3621:]
3618:)
3615:x
3612:(
3609:f
3603:)
3600:x
3594:+
3591:x
3588:(
3585:f
3582:[
3556:)
3553:x
3550:(
3547:f
3543:/
3539:]
3536:)
3533:x
3530:(
3527:f
3521:)
3518:x
3512:+
3509:x
3506:(
3503:f
3500:[
3480:)
3477:x
3474:(
3471:f
3451:x
3447:/
3443:)
3440:x
3434:(
3431:=
3428:x
3424:/
3420:]
3417:x
3411:)
3408:x
3402:+
3399:x
3396:(
3393:[
3373:x
3353:x
3333:x
3302:f
3278:f
3253:f
3249:/
3241:f
3237:x
3217:x
3213:/
3209:1
3206:=
3203:x
3199:/
3191:x
3187:=
3180:)
3176:x
3167:(
3147:x
3127:f
3123:/
3115:f
3111:=
3104:)
3100:f
3091:(
3071:f
3033:.
3029:|
3019:)
3015:x
3006:(
2997:)
2993:f
2984:(
2978:|
2974:=
2970:|
2964:)
2961:x
2958:(
2955:f
2950:)
2947:x
2944:(
2937:f
2933:x
2927:|
2903:x
2882:|
2878:f
2874:/
2866:f
2862:x
2858:|
2837:f
2776:A
2746:A
2736:A
2730:=
2727:)
2724:A
2721:(
2661:)
2655:|
2649:i
2646:i
2642:a
2637:|
2631:(
2624:i
2612:)
2606:|
2600:i
2597:i
2593:a
2588:|
2582:(
2575:i
2561:)
2558:A
2555:(
2529:i
2509:0
2501:i
2498:i
2494:a
2469:A
2441:L
2391:L
2371:=
2368:)
2365:A
2362:(
2335:A
2312:A
2288:)
2285:A
2282:(
2252:)
2249:A
2246:(
2216:,
2210:|
2206:)
2203:A
2200:(
2186:|
2181:|
2177:)
2174:A
2171:(
2157:|
2151:=
2148:)
2145:A
2142:(
2115:A
2091:A
2067:)
2064:A
2061:(
2031:)
2028:A
2025:(
1992:,
1986:)
1983:A
1980:(
1965:)
1962:A
1959:(
1943:=
1940:)
1937:A
1934:(
1906:2
1885:L
1822:A
1802:=
1794:A
1789:1
1782:A
1766:A
1752:1
1745:A
1737:=
1734:)
1731:A
1728:(
1691:.
1685:A
1672:1
1665:A
1657:=
1646:}
1637:x
1626:x
1623:A
1614:{
1608:0
1602:x
1592:}
1583:e
1571:e
1566:1
1559:A
1548:{
1542:0
1536:e
1528:=
1517:}
1507:b
1502:1
1495:A
1482:b
1473:{
1467:0
1461:b
1451:}
1442:e
1430:e
1425:1
1418:A
1407:{
1401:0
1395:e
1387:=
1379:}
1368:b
1363:1
1356:A
1343:b
1328:e
1316:e
1311:1
1304:A
1292:{
1286:0
1280:b
1277:,
1274:e
1248:e
1244:b
1227:.
1217:b
1212:1
1205:A
1192:b
1177:e
1165:e
1160:1
1153:A
1142:=
1133:b
1122:e
1112:/
1101:b
1096:1
1089:A
1075:e
1070:1
1063:A
1042:b
1038:e
1035:A
1031:b
1028:A
1020:A
1016:b
1012:e
1005:b
1001:x
990:b
986:x
982:x
978:b
974:b
970:x
950:x
946:b
917:.
908:x
901:/
894:x
880:)
877:x
874:(
871:f
864:/
857:)
854:x
851:(
848:f
823:x
804:+
800:0
752:x
738:)
735:x
732:(
729:f
704:x
685:+
681:0
646:f
620:.
614:)
611:x
608:(
605:f
598:/
589:)
586:x
583:(
574:f
565:)
562:x
559:(
556:f
548:=
542:)
539:x
536:(
533:f
526:/
519:)
516:x
513:(
510:f
472:)
469:x
466:(
457:f
448:)
445:x
442:(
439:f
431:=
425:)
422:x
419:(
416:f
383:,
380:)
377:x
374:(
365:f
356:)
353:x
350:(
347:f
341:)
338:x
335:(
332:f
305:,
302:)
299:x
296:(
287:f
264:x
238:f
215:f
186:k
164:k
156:=
153:)
150:A
147:(
72:,
69:y
66:=
63:)
60:x
57:(
54:f
20:)
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