Compute inverse of ill-conditioned matrix

Question

I need to compute inverse of a matrix that is highly ill conditioned and nearly singular. I tried using Jacobi preconditioning, a method to add a scalar value to the diagonal entries of the original matrix and SSOR preconditioning but the inverse of the matrix is still 0. Are there any other methods that can be carried to approximate the inverse of this type of matrix.

Answer 1

Is the inverse 0 or can't you inverse the matrix? What do you need the inverse for? If you want to solve a system $$ Ax = b$$ then you could use a normal equation to find a pseudo inverse, e.g. let $A^T$ be the transposed matrix, then solve $$A^T Ax = A^T b$$ You can regularize this expression with a scalar $\lambda$ and identity matrix $$(A^TA+\lambda I)x = A^T b$$ and then invert $(A^TA)$ or $(A^TA+\lambda I)$. This method might be similar to your second approach depending what exactly you did.

Another way to approximate the inverse is singular value decomposition with cut off of small singular values.

If you provide some more information I can give more precise answer.