LU solve for matrix with zeros on diagonal

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SUMMARY

LU decomposition can be applied to a matrix with zeros on its diagonal by performing row swaps to eliminate the zeros. However, caution is advised when dealing with matrices that have a high condition number, such as 10^8, as this can lead to significant round-off errors in the computed solutions. For improved accuracy, it is recommended to consider alternative methods like singular value decomposition (SVD) in conjunction with LU decomposition to validate the results.

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Is it possible to obtain a solution of the linear system Ax = b with LU decomposition when A contains zeros on its diagonal? I am trying to obtain a solution with LU decomposition and then perform a forward/backward substitution but I get NaN entries in the solution vector x. The condition number of my matrix is 10^8. Appreciate any help/comments.
 
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If the matrix has 0s on the diagonal, you can get rid of those by swapping rows. Since that is a "row operation" it should not give you any trouble with finding the LU decomposition.
 
nawidgc said:
Is it possible to obtain a solution of the linear system Ax = b with LU decomposition when A contains zeros on its diagonal? I am trying to obtain a solution with LU decomposition and then perform a forward/backward substitution but I get NaN entries in the solution vector x. The condition number of my matrix is 10^8. Appreciate any help/comments.

I would be concerned trying to use LU decomp on a matrix with such a high condition number, regardless of whether there are zeroes on the main diagonal. A high condition number means that the solutions obtained from the LU decomp are subject to round-off error during their calculation, so much so that these solutions may be meaningless.

Instead of plain vanilla LU decomp, perhaps you should apply some other techniques to the matrix as well, to check your original solutions. I would recommend you try the singular value decomposition.

http://en.wikipedia.org/wiki/Condition_number

http://en.wikipedia.org/wiki/Singular_value
 

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