@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
- 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.
