Krylov Subspace Methods

A Julia implementation of iterative solvers for linear systems Ax = b using Krylov subspace methods.

Overview

This package provides implementations of commonly used iterative methods for solving sparse linear systems:

• CG - Conjugate Gradient (for symmetric positive-definite systems)
• MINRES - Minimum Residual (for symmetric indefinite systems)
• GMRES - Generalized Minimum Residual (for non-symmetric systems)
• CGW - Conjugate Gradient on the Normal Equations (for non-symmetric systems)

Usage

CLI Tool

Solve a linear system using the command-line interface:

julia bin/solve.jl <matrix> [options]

Arguments

• matrix - Path to the matrix file in Matrix Market format (.mtx)

Options

• -m, --method - Solver method: cg, cgw, minres, gmres (default: cg)
• --tol - Convergence tolerance (default: 1e-6)
• --maxiter - Maximum iterations (default: 1000)
• --rhs - Path to right-hand side vector file (one number per line). If not provided, a synthetic system is generated with x_exact = 1

Example

julia bin/solve.jl example/test_matrix.mtx --method cg --tol 1e-6 --maxiter 500
julia bin/solve.jl example/test_matrix.mtx --method gmres --rhs example/rhs.txt

Roadmap

• Benchmark against standard Julia linear solvers
• Add BiCG-STAB (BiConjugate Gradient Stabilized)
• Add preconditioning support
• Add convergence history plotting
• Parallel implementation for large-scale systems