Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices |

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*To*: eigen@xxxxxxxxxxxxxxxxxxx*Subject*: Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices*From*: Benoit Jacob <jacob.benoit.1@xxxxxxxxx>*Date*: Wed, 24 Feb 2010 06:32:46 -0500*Dkim-signature*: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=gamma; h=domainkey-signature:mime-version:received:in-reply-to:references :date:message-id:subject:from:to:content-type :content-transfer-encoding; bh=PBTRFyO2s+gMVGScKOv4NyZ7JOro2FKJyWv9aPvfE80=; b=p5sO9BzwPAHcnFyQbsfWDaxLzl+sL0TLM1tljW/JqMb/SEF0FLrlWqXNDlzpBgxi90 3/qO6CwyywKmfDT6cc0tJ8cdCBnkvKgZTA3BxyrL3B2Wg41TbdK0loF6+X+u3I8G5q0d RWwUrwRsGngpLbuR2SPJcQobCGKi3Drh7ERMA=*Domainkey-signature*: a=rsa-sha1; c=nofws; d=gmail.com; s=gamma; h=mime-version:in-reply-to:references:date:message-id:subject:from:to :content-type:content-transfer-encoding; b=Ey9lYNEAIcvDG0/ZkN9xpdtgeJFonWXfd2edoIC37uWkmIYUELMiGkq1RdFMyWxrx0 /0xU0jXnn0rMMSwb8F1yH826ghcZiVgbe1Rf3VxIuB6w1w2my5264JzjjiXKNWT7dTnu TUe0kwWZMhF7d+lNfSoKsOb2ugtmbaDIqWxIs=

2010/2/24 Jitse Niesen <jitse@xxxxxxxxxxxxxxxxx>: > On Wed, 24 Feb 2010, Gael Guennebaud wrote: > >> note that the LDLT decomp does pivoting, so it is normal that you don't >> get the same diagonal matrix. If you reconstruct the matrix from the decomp >> you will see that in this case the decomp is correct. > > To further clarify, because this also threw me off for a while: I think the > important thing here is that the matrix A is singular. Great observation :) For a detailed analysis of the stability of LDLt with pivoting for singular matrices, see http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.3360 Formulas are given on page 12, suggesting that it's not very stable, AFAIU. In particular, LDLt, even with pivoting, is not a rank-revealing decomposition, which means that the data "how many zeros in vectorD()" should be considered meaningless. Benoit > The diagonal D is > unique (up to reordering) for positive-definite matrices. However, there is > no such uniqueness for singular matrices. Simple example > > A = [ 0 0; 0 1 ] (these are 2-by-2 matrices in Matlab notation) > L = [ 0 0; sqrt(3) 1 ] > D = [ 1/4 0; 0 1/4 ] > > The matrix A has two LDL^T decompositions, A = A * I * A^T = L * D * L^T, > but the diagonal parts are not the same. > > Cheers, > Jitse > > >

**Follow-Ups**:**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Benoit Jacob

**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*FMDSPAM

**References**:**[eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Ben Goodrich

**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Benoit Jacob

**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Ben Goodrich

**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Benoit Jacob

**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Ben Goodrich

**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Gael Guennebaud

**Re: [eigen] [patch] LDLt decomposition with rank-deficient matrices***From:*Jitse Niesen

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