Solving A*x=b when A is full rank but numerically rank deficient

[CdeMills]

Hello,

I have to solve a problem implying number of molecules in various states.
The issue is the the smallest number is around 10^4 and the biggest 10^34.
And the interesting phenomena occur with the smallest fraction.

Here is a pseudo-problem generation:
# generate a symmetric non-singular matrix
A=randn(5,5); A=A.'*A;
 AA=A*kron(logspace(34, 4, 5), ones(5,1));
X=rand(5,1); B=AA*X; 
[AA\B X]

The solution clearly originates from a minimum-norm solver: only one
component of X is OK. The main issue is that A singular values span the
original range of 10^4 to 10^34. Any idea on how to solve this kind of
problem overcoming the numerical resolution issue ?

Regards

Pascal





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