linear-solvers

Direct solvers work well for dense or small matrices but become memory-intensive at scale; iterative solvers (CG, GMRES, BiCGSTAB) suit large sparse systems. This Skill guides you through the tradeoff based on matrix size, sparsity, and structure to pick the right method for Ax=b problems.

Stagnation typically stems from poor matrix conditioning or inappropriate solver choice. This Skill diagnoses convergence issues by analyzing residual trends and conditioning estimates, then recommends preconditioners like ILU, IC, or AMG to accelerate convergence.

For SPD sparse matrices, IC (incomplete Cholesky) and AMG (algebraic multigrid) preconditioners are effective choices. This Skill evaluates matrix properties and recommends the preconditioner that balances robustness and computational cost for your problem scale.

Yes. This Skill handles both symmetric and nonsymmetric cases, recommending solvers like BiCGSTAB or GMRES with appropriate preconditioners such as ILU to address convergence and conditioning challenges in nonsymmetric Ax=b problems.

Poor conditioning slows convergence and amplifies numerical errors. This Skill estimates conditioning, suggests scaling or equilibration steps to improve it, and recommends preconditioners to mitigate ill-conditioning effects in dense and sparse matrices.

You need Python 3.8+, NumPy, SciPy, and the provided scripts. The Skill works with sparse and dense matrices of any size and applies to SPD and nonsymmetric Ax=b problems without additional dependencies beyond NumPy.