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Beginner
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## Matrix Inversion of ill conditioned Matrix

Hello,
We need to find the inverse of Matrix of complex numbers.
The problem is sometime ill conditioned.
Which MKL function is suggested?
Is there any algorithm in order to refine the solution?

BR
Gianluca

4 Replies
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Employee
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Hi,

The simplest way to compute inverse matrix is to solve the system with initial matrix and unit right hand side. So you can call sequence of zgetrf and zgetri for dense matrix or pardiso for sparse

Thanks,

Alex

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Beginner
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Dear Alex,

the question is documented in the attached paper:

The document is because some formula.

Thank you

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Employee
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Hi,

Without taking into account fact of ill-condition system - if sizes of all matrices are about the same than you don't need to calculate inverse matrix, you need to solve system with matrix A and rhs B:

Y = A^{-1}*B <=> AY=B

That can be done by combination of zgetrf  and zgetrs functions

About ill-condition system - this approach can dramatically increase condition number of the system, for example if you matrix has following form:

| 0 B^t |

|B A    |

then with several conditions on matrices B and A it's condition number can be quite good, but your approach produce absolutely incorrect answer. So it can be used only if matrix A in your term is well-condition.

Thanks,

Alex

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Beginner
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Alex

Y = A^{-1}*B <=> AY=B

Of couse if A size is lower tha 4GB we don't invert A and we use zgetsv.

The problem is when we have a large matrix over 4GB, in this case we follow a trick as already documented.

This require a matrix inversion, and we want to avoid it for the reason we told! ill conditioned.

Do you know an alternative of our approach?.

Thks

Gian