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This paragraph has to be re-written by a specialist. This excerpt gives a false impression that the concept of the condition number revolves around the concept of derivative. Although there is an obvious connection between the two, neither is generalisation or the consequence of the other. I placed ,
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frequently applied to questions in linear algebra, 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.
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The phrase 'ill-conditioned' redirects here. I'm willing to wager that nearly anyone looking up 'ill-conditioned' is eager to find strategies for solving problems that have ill-conditioned matrices. I do not see anything useful presented here in that regard. I recommend that either (1) strategies for
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The error in the approximate solution is no greater than the (condition number) x (relative error in the initial solution). If the new approximate solution is not within the desired precision of the actual solution, then the system would be "ill-conditioned." Also, to obtain a higher precision in an
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I think it would be very helpful for more general readers if the introduction (in particular the first paragraph---currently 2 sentences) mentioned that the condition number is generally used to evaluate how large the error/uncertainty of a function will be, given a certain error/uncertainty in the
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I'm trying to show that in a bunch of cases (200 or so) two 2D vectors are "approximately" multiples of one another. So I make a 2x2 matrix out of these two vectors, and I find the condition number. On average the condition number is around 2e6, and the minimum is 9e4. Does this make the vectors
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The condition number is an application of the derivative, and is formally defined as the value of the asymptotic 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
381:"wheras a large condition number will enhance error in b." I think this is confusing because we are talking about the bad conditioning increasing the amount of error in the calculated x due do a small error in b (and not really doing anything to b).
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If you use exact arithmetic (infinite precision), and an iterative method to compute an approximation to your solution (whose rate of convergence degrades with increasing condition number), then the condition number is of great relevance.
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I think it would be useful in the introduction to mention the condition number rule of thumb. It was mentioned to me in a
Numerical Analysis class, and found that it cleared up some questions I had. The rule of thumb is if
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It depends on how accurately you know these vectors. The amount of significant digits you lose is equal to log(condition number), so if you only have the vector components 3 digits accurate, then a condition number:
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Can someone help me with an estimate of how big a condition number actually has to be before a matrix is considered "ill conditioned?" I'm sure it's dependent on many things but I'd like to get a grasp on it.
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I agree. The next step is that you think of a better formulation and edit the article. These articles don't write themselves ;) Don't worry too much about making mistakes; they will be corrected. --
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Seconded. I'm mathematically literate enough to be interested in this article, but haven't been able to guess the principles that ought to be invoked to find this maximum. GRB, 15 August 2011.
471:"if the condition number is large, even a small error in b may cause a large error in x", is it just opposite? i think a large condition number means a small error in x cause a large error in b.
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I am not sure if the relevance of finite/infinite precision is clear. If you have infinite precision, and do not assume any kind of errors in the input, the condition number is of no relevance.
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2072:- (2b) Specialisation to the matrix spectral norm (i.e., **the** matrix condition number; maybe also add the generalisation to rectangular matrices, see Trefethen&Bau p130 or Meyer p426)
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What is the symbol for condition number? I recall being told—this recollection is not a high-quality citation—that, at least for a matrix of double derivatives, it is written |||
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And then they proceed to say that $ k$ digits of precision lost resulting from the base ten logarithm of the aforementioned computation of the condition number.
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digits of accuracy on top of what would be lost to the numerical method due to loss of precision from arithmetic methods. This can be seen in
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If you are free to choose e and b such that the quantities are maximum, then ||A^(-1) e||/||e||=||A^(-1)|| and ||b||/||A^(-1) b||=||A||
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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data/parameters. As it stands, the introduction is somewhat vague---while it is quite clear, *once* I know what its talking about.
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Does Cheney and
Kincaids rules of thumb apply in computations outside of the 2 norm? In the book they have it listed as
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because the statement is meaningless (if understood in a broad sense) or technically false (if understood narrowly).
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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I added a plausible derivation of the condition number for operator (induced) matrix norms. I hope it helps.
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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Here is a good possible lead for a future contribution, I quote Golub & Van Loan 4th ed. p.88:
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coping with ill-conditioned matrices be included in this page or (2) this issue get its own page.
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The maximum value (for nonzero b and e) is easily seen to be the product of the two operator norms:
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You are absolutely right. I think these (subtle?) issues could/should be addressed in the article.
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1552:"If the condition number is not too much larger than one (but it can still be a multiple of one)"
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Yes, the "General
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If you found an error with any archives or the URLs themselves, you can fix them with
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I'll move this to the reference desk in a day or two if someone isn't watching here.
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I don't find this so easy - please could somebody give more details about this line.
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The definition at the very top ("General
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at all, which makes the ratio diverge in the limit. To fix, please specify how
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http://numericalmethods.eng.usf.edu/mws/gen/04sle/mws_gen_sle_spe_adequacy.pdf
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ill-conditioned system, a much more accurate initial condition is required.
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Both are true. If the condition number is large, then a small error in
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that redirects to this article. So there is a need to explain what
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There needs to be more clarification if this is not the case.
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for additional information. I made the following changes:
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Numerical
Mathematics and Computing, by Cheney and Kincaid
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Mention 'error' or 'uncertainty' in early introduction
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1811:{\displaystyle \varepsilon \to 0}
1592:. Please take a moment to review
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865:{\displaystyle \|A^{-1}b\|/\|b\|}
812:{\displaystyle \|b\|/\|A^{-1}b\|}
719:{\displaystyle \|A^{-1}e\|/\|e\|}
284:Knowledge:WikiProject Mathematics
2108:Start-Class mathematics articles
673:It's obvious for the first part
407:I tried to change it myself. --
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278:and see a list of open tasks.
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1984:{\displaystyle \varepsilon }
1944:{\displaystyle {\tilde {f}}}
1915:{\displaystyle \varepsilon }
1863:{\displaystyle \varepsilon }
1364:. What do you guys think?
1125:Make the change of variable
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953:{\displaystyle A:x\mapsto b}
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1843:{\displaystyle \delta f(x)}
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50:New to Knowledge? Welcome!
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1585:Hello fellow Wikipedians,
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755:{\displaystyle \|A^{-1}\|}
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362:1000 will produce garbage.
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1508:All Clues Key
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383:Richard Giuly
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123:Find sources:
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110:Verifiability
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52:Learn to edit
49:
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41:
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37:
36:
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26:
22:
18:
17:
1993:— Preceding
1791:
1772:
1754:
1750:
1746:
1731:
1726:
1704:
1700:
1696:
1694:
1691:What symbol?
1672:
1669:
1644:source check
1623:
1617:
1614:
1587:
1584:
1557:— Preceding
1554:
1551:
1528:— Preceding
1524:
1504:
1488:Faraz.yashar
1485:
1482:
1377:
1296:
1124:
1113:138.246.2.58
1036:, and hence
672:
658:
640:— Preceding
637:
623:
605:
598:
596:
589:
585:
581:
575:
558:Jitse Niesen
499:
495:
491:
487:
470:
437:
409:Jitse Niesen
393:Jitse Niesen
380:
359:
348:
340:
336:
333:
325:
306:Mid-priority
305:
265:
231:Mid‑priority
209:WikiProjects
171:
165:
157:
150:
144:
138:
132:
122:
94:
19:This is the
1951:depends on
1850:depends on
1366:Mikesmith00
646:99.24.63.46
281:Mathematics
272:mathematics
228:Mathematics
199:Start-class
148:free images
31:not a forum
2102:Categories
1709:JDAWiseman
1681:Report bug
1664:this tool
1657:this tool
1563:Clauariel
1293:Additions
425:Perdelsky
377:Confusing
88:if needed
71:Be polite
21:talk page
1995:unsigned
1670:Cheers.—
1571:contribs
1559:unsigned
1530:unsigned
642:unsigned
607:TomyDuby
473:Shangkun
56:get help
29:This is
27:article.
2083:AVM2019
1775:AVM2019
1756:AVM2019
1735:AVM2019
1594:my edit
1164:. Then
626:Nlinton
460:Berland
440:Berland
353:Jaboles
308:on the
154:WP refs
142:scholar
205:scale.
126:Google
2036:is.
556:. --
451:Lunch
344:moink
169:JSTOR
130:books
84:Seek
2087:talk
2042:talk
2003:talk
1971:and
1779:talk
1760:talk
1739:talk
1713:talk
1567:talk
1538:talk
1512:talk
1492:talk
1370:talk
1284:talk
1117:talk
1111:. --
665:talk
650:talk
630:talk
611:talk
588:and
562:talk
477:talk
413:talk
397:talk
368:talk
162:FENS
136:news
73:and
1991:.
1638:RfC
1608:to
1278:.
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1173:max
1045:max
361:-->
300:Mid
176:TWL
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2081:.
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2021:δ
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1979:ε
1936:~
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1878:δ
1858:ε
1826:δ
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1800:ε
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1099:‖
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910:‖
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