Linear Solvers

Goal

Provide a universal workflow to select a solver, assess conditioning, and diagnose convergence for linear systems arising in numerical simulations.

Requirements

• Python 3.8+
• NumPy, SciPy (for matrix operations)
• See individual scripts for dependencies

Inputs to Gather

Input	Description	Example
Matrix size	Dimension of system	n = 1000000
Sparsity	Fraction of nonzeros	0.01%
Symmetry	Is A = Aᵀ?	yes
Definiteness	Is A positive definite?	yes (SPD)
Conditioning	Estimated condition number	10⁶

Decision Guidance

Solver Selection Flowchart

Is matrix small (n < 5000) and dense?
├── YES → Use direct solver (LU, Cholesky)
└── NO → Is matrix symmetric?
    ├── YES → Is it positive definite?
    │   ├── YES → Use CG with AMG/IC preconditioner
    │   └── NO → Use MINRES
    └── NO → Is it nearly symmetric?
        ├── YES → Use BiCGSTAB
        └── NO → Use GMRES with ILU/AMG

Quick Reference

Matrix Type	Solver	Preconditioner
SPD, sparse	CG	AMG, IC
Symmetric indefinite	MINRES	ILU
Nonsymmetric	GMRES, BiCGSTAB	ILU, AMG
Dense