GMRES vs BiCGStab vs MINRES

Three Krylov solvers for large linear systems, each trading memory, robustness, and problem structure differently.

Choosing among nonsymmetric solvers

Most large sparse linear systems in scientific computing are solved by one of three Krylov methods. The right choice depends on whether the matrix is symmetric, how much memory is available, and how smooth convergence needs to be.

GMRES

The Generalized Minimal Residual method builds a full orthonormal basis for the Krylov subspace via Arnoldi iteration and, at each step, chooses the approximation that minimizes the residual norm over the whole subspace. Its residual is monotonically nonincreasing, which makes it very robust. The cost is that memory and orthogonalization work grow with the iteration count, so in practice it is restarted every m steps (GMRES(m)). Restarting can stall on hard problems because it discards accumulated subspace information.

BiCGStab

The Bi-Conjugate Gradient Stabilized method uses short recurrences, so its memory footprint is small and fixed regardless of iteration count. It applies A but conceptually works with both A and its transpose implicitly, combining a BiCG step with a local minimization (the stabilization) to smooth the otherwise erratic BiCG residual. It is cheap per step but can break down or show irregular convergence, and it needs two matrix-vector products per iteration.

MINRES

The Minimal Residual method is the symmetric-indefinite specialist. When A is symmetric (but possibly indefinite), the Lanczos three-term recurrence lets MINRES minimize the residual like GMRES but with constant work and memory per step. It is the method of choice for symmetric saddle-point and shifted systems where conjugate gradient would fail because A is not positive definite.

Quick decision guide

• Symmetric positive definite: use conjugate gradient
• Symmetric indefinite: use MINRES
• Nonsymmetric, memory available, robustness critical: use GMRES
• Nonsymmetric, memory tight, well-behaved spectrum: use BiCGStab

All three benefit enormously from a good preconditioner; the raw method choice matters less than the preconditioner quality on stiff problems like MHD equilibrium solves.