How to reduce the large condition number of matrix

Recently, I encounter a problem. I need to solve a equation with the form of Ax=b, where A is a matrix with a large condition number. The calculated value of x deviate from the actual value a lot. Under this condition, how to effectively reduce the condition number of M or how to solve this illed problem well? Thank you very much.

Answer 1

[BhaTTa]

@Mathis, When dealing with a system of linear equations ( Ax = b ) where ( A ) has a large condition number, the solution can be sensitive to small perturbations in ( A ) or ( b ). This is known as an ill-conditioned problem. Here are some strategies to address this issue:Strategies to Solve Ill-Conditioned Systems

1. Regularization:

• Tikhonov Regularization (Ridge Regression): Add a regularization term to the matrix equation to stabilize the solution.

2. Preconditioning:

• Use a preconditioner matrix ( M ) to transform the system into one with a lower condition number. Solve ( MAx = Mb ) instead, where ( M ) is chosen such that ( MA ) is better conditioned than ( A ).

3. Iterative Refinement:

• Use an initial solution and refine it iteratively. This technique can improve the accuracy of the solution when using floating-point arithmetic.

4. Scaling:

• Scale the matrix ( A ) and vector ( b ) to reduce the condition number. This involves normalizing the rows or columns of ( A ) to have similar magnitudes.

5. Use of More Accurate Arithmetic:

• If possible, use arbitrary precision arithmetic or double precision to reduce numerical errors.