       Re: LUDecomposition output?

• Subject: [mg2623] Re: LUDecomposition output?
• From: danl (Daniel Lichtblau)
• Date: Thu, 30 Nov 1995 20:59:20 -0500
• Approved: usenet@wri.com
• Distribution: local
• Newsgroups: wri.mathgroup
• Organization: Wolfram Research, Inc.

```In article <49dojj\$8a6 at dragonfly.wri.com> buchan at metronet.com (John S.
Buchan) writes:
> Can someone tell me how, exactly, is the LUDecomposition output
formatted?
> I have a symmetric matrix which I need to decompose into a lower and an
upper
> diagonal matrix, but I cannot find any reference to the command itself.
> Executing LUDecomposition[ ] gives me a single table, but I do not know
which
> coefficents belong to which matrix.  Any help will be appreciated.
>

The first element returned is a matrix containing the factorization
itself, and the second element is the list of permutations, that is, row
exchanges, used during the Gaussian elimination process. They are
relative, not absolute. Specifically, the value k in the jth entry means
that what was row k at the time row j was processed (not necessarily
initially) became row j.
For the factorization, we record the upper triangular matrix as the
diagonal-and-above part. Below the diagonal we store the multipliers
comprising the lower diagonal matrix less its main diagonal, which
consists of ones.
This is spelled out in some detail in the LINPACK User's Guide by
Dongarra et al, in the first chapter, around p. 1.10. Our code uses the
LINPACK technology. With a bit of patience you can work out the conversion
details involving the explicit construction of the permuted lower
triangular matrix.

Daniel Lichtblau
Wolfram Research, Inc.

```

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