Are there any matrices for which the Gaussian method yields wrong/ or most inaccurate results? I've implemented a full choice algorythm, where i switch rows and columns so that the current element is biggest.
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1If you mean Gaussian Elimination (http://en.wikipedia.org/wiki/Gaussian_elimination), then it can proven to always reduce a matrix to a reduced row echelon form. – Dan Rust Apr 08 '13 at 17:49
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Gaussian elimination with complete pivoting is quite stable. – J. M. ain't a mathematician Apr 08 '13 at 17:51
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1The main issue with Gaussian Elimination is 'element growth'. In practice, this is not an issue. In fact, just partial pivoting is pretty good, in practice. – copper.hat Apr 08 '13 at 18:00
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With respect to @copper.hat's comment, you will want to read the book by Golub and Van Loan (which I linked to in my previous comment). – J. M. ain't a mathematician Apr 08 '13 at 18:14
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@J.M.: Useful link. I need to get a newer edition. I just read about rank pivoting in link. – copper.hat Apr 08 '13 at 18:17
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If you are looking for matrices which will give inaccurate results, when solving numerically, you might want to look at the Hilbert matrix which has a condition number that grows exponentially with its size.
J. M. ain't a mathematician
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Vincent Pfenninger
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right condition number for matrix should be close to $1$,+1 for Hilbert matrix – dato datuashvili Apr 08 '13 at 17:57
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what about the last column? Since the Hilbert Matrix is square, should i fill that with random values? – Bartlomiej Lewandowski Apr 08 '13 at 18:08
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2The funny thing is that since the Hilbert matrix is symmetric positive definite, you don't even need to pivot! In any event, see this paper for things on the decomposition of the Hilbert matrix. – J. M. ain't a mathematician Apr 08 '13 at 18:13
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http://kalandarson.blogspot.com/2012/03/naive-gaussian-elimination-for-solving.html – dato datuashvili Apr 08 '13 at 18:17
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as wikipedia says it always works,but sometimes it needs minor changes, so you can read this document http://www.personal.psu.edu/bmw5075/360notes.pdf
here is given advantages and disadvantages of this method ,their comparison,please see it.
EDITED:
for matrix ,on last page is given formulas and this statement
Matrix Norms Matrix norms are natural extensions of vector norms, but are not as clearly defined. They have to be tested over the whole area of the matrix:
we can's simply apply on matrix vector's norm definition,just test on whole matrix
use wikipedia http://en.wikipedia.org/wiki/Matrix_norm
dato datuashvili
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looks really nice, but could you explain the condition with the norms a bit more? Should i check for norm 1 or norm infinity? – Bartlomiej Lewandowski Apr 08 '13 at 18:02
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