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For $A$ and $B$ known matrices which are not square matrices, I have the following equation sistem i would like to solve numerically \begin{equation} A'Ax=A'Bz \end{equation} I want to know which is a numerically stable and fast algorithm that can be used avoiding the computation of $A'A$ which is a full rank matrix. I know i can use Cholesky method if I compute $A'A$, but I guess i could get Cholesky descomposition from $A$ or may be something better.

Thanks a lot

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A slow but numerically stable method would be to use the QR decomposition. –  copper.hat Feb 18 at 17:13
    
Singular value decomposition, see also here –  Jean-Claude Arbaut Feb 18 at 17:14
    
QR fits my needs. Thanks! –  Manuel Feb 18 at 17:45
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2 Answers

up vote 1 down vote accepted

Assuming you are solving for $x$.

Apply the least squares algorithm to the problem $Ax = Bz$. This is the most efficient way.

Numerical Recipes recommends as follows:

We need to warn you that the solution of a least-squares problem directly from the normal equations is rather susceptible to roundoff error. An alternative, and preferred, technique involves QR decomposition

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Interesting, Do i need any assumption over $A$ to do this? Do you know which would be the alternative technique? –  Manuel Feb 18 at 18:10
    
@Manuel what do you mean by alternative technique - alternative to least squares or an alternative to QR decomposition implementation of least squares? –  gt6989b Feb 18 at 23:43
    
I missread the warning. I thought that QR descomposition was a different technique to solve the problem, not that It was the method to calculate least squares –  Manuel Feb 19 at 22:21
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If $A'$ is invertible, your system becomes $Ax=Bz$. Then you can use Gauss-Jordan algorithm, or Gauss-Seidel iterative algorithm, if you want respectively an exact or approximated solution.

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$A'$ is not a squared matrix –  Manuel Feb 18 at 17:26
    
Oh sorry, I didn't read carefully. I will think about it. –  7raiden7 Feb 18 at 17:33
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