Re: SVD, EIG and CHOL of a matrix

[Carlo de Falco]

On 22 Jan 2010, at 14:38, Alberto Frigerio wrote:

> Yeah Jaroslav, you were completely right. I've seen the description  
> of the
> command SIGN but I misunderstood it ... and obviously the one I made  
> was
> completely non-sense. Hence I succeed in making simple programs about
> eigenvalues and SVD, but I got a mathematical problem about Cholesky
> factorization.
>
> If I get a positive definite matrix I've found how to calculate its  
> Cholesky
> factorization and luckily it is the same of the Octave's one. But I  
> have
> some problems when the input matrix A is not positive definite,  
> beacuse I
> succeded only in finding a representation like A=L*D*L', where L is
> triangular and D is diagonal, and I'd like to have a representation  
> like the
> Octave one (I cannot take sqrt(D) because it has also negative  
> values).

You just discovered by yourself why the Cholesky decomposition of a  
matrix A is possible
if and only if A is symmetric and positive definite. You cannot  
compute a Cholesky
factorization of a non-SDP matrix.

> By the way, I noted something strange in the Octave CHOL function.  
> In the
> help command I read : "With two or more output arguments P flags  
> whether the
> matrix was positive definite and `chol' does not fail. A zero value
> indicated that the matrix was positive definite and the R gives the
> factorization, and P will have a positive value otherwise."
>
> But if I try to run [R,P]=chol(A) with the matrix A=magic(4) I got  
> P=-1.
> Obviously it is not a big problem, BUT if I try to valuate R'*R I  
> got a
> completely different result from A. Hence, what algorithm is Octave  
> using in
> the not positive-definite case?

I guess in this case Octave returns the partially computed  
factorization at the point where the algorithm failed, maybe someone  
else can be more precise about this.
Anyway I see that the help string in chol.cc can use some improvement,  
a changeset is attached.

> Thanks,
>   Alberto

c.

patch.txt
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