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GMRES

GMRES uses minimizing residuals over a growing Krylov subspace for linear system solving.

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Quick Facts

Compact metadata for implementation and discovery.

Category
Numerical Methods
Difficulty
Intermediate
Family
Iterative Linear Solvers
Tier
Core
Status
Verified
Review
Source Checked
Sources
2
Formulas
0
Relationships
6
Code example
Ready
Visualization
Ready
Aliases
Generalized Minimal Residual

Overview

GMRES applies minimizing residuals over a growing Krylov subspace to handle linear system solving with a named, inspectable procedure.

Core Concepts

Krylov subspaceArnoldi iterationresidual minimizationleast-squares projection

Quick facts

Complexity

Runtime and memory behavior at a glance.

Best

O(n^2)

Average

O(n^3)

Worst

O(n^3)

Space

O(n^2)

When to Use

  • Use GMRES when solving linear system solving and you want to benefit from GMRES uses minimizing residuals over a growing Krylov subspace for linear system solving.
  • Choose it when O(n^3) average/worst-case time and O(n^2) memory match your workload profile.
  • Prefer it over MINRES or BiCGSTAB when the problem structure favors Krylov subspace.

When Not to Use

  • Avoid when the input is ill-conditioned and the method lacks a stability safeguard.
  • Avoid when exact symbolic results are required instead of numeric approximations.

What It Does

  • GMRES uses minimizing residuals over a growing Krylov subspace for linear system solving
  • Organizes the work around Krylov subspace and Arnoldi iteration.

Use Cases

  • Production and research workloads requiring linear system solving via GMRES uses minimizing residuals over a growing Krylov subspace for linear system solving.
  • Iterative Linear Solvers pipelines where O(n^3) time and O(n^2) space fit system constraints.
  • Engineering comparisons against MINRES and BiCGSTAB when tuning linear system solving.

Inputs

  • Numbers, vectors, matrices, or functions
  • Tolerance or precision settings

Outputs

  • Computed numeric result
  • Residual, factorization, or approximation

Pseudocode

  1. 1.Validate input dimensions, conditioning, and numerical tolerance parameters for GMRES.
  2. 2.Initialize the working buffers, basis vectors, or decomposition factors (Krylov subspace).
  3. 3.Execute the core numerical recurrence by GMRES uses minimizing residuals over a growing Krylov subspace for linear system solving.
  4. 4.Monitor residual error, stability bounds, or step convergence across iterations.
  5. 5.Return the computed linear system solving coefficients, factors, or transformed values.

Canonical Example

Input
Numbers, vectors, matrices, or functions configured for a representative linear system solving test case
Output
Computed numeric result satisfying GMRES invariants
Explanation
GMRES processes the input by GMRES uses minimizing residuals over a growing Krylov subspace for linear system solving, tracking Krylov subspace and Arnoldi iteration to produce the verified linear system solving result.

Implementation Notes

  • Document parameter meanings such as tolerance, precision, and iteration limit.
  • Use residual or error checks rather than trusting iteration count alone.
  • Prefer stable formulations when inputs are ill-conditioned.

Common Pitfalls

  • Ignoring overflow, cancellation, or loss of precision.
  • Applying the method outside its stability assumptions.
  • Testing only well-conditioned inputs.

Interactive Mechanism Walkthrough

Iterative Convergence & State Trajectory

Show minimizing residuals over a growing Krylov subspace on a small linear system solving example.

GMRES mechanism walkthrough step 1 of 5x₀x₁x₂x*
Step 1 of 5: Validate input dimensions, conditioning, and numerical tolerance parameters for GMRES.
numerical-methods-mechanism

Notes

GMRES operates in O(n^3) time and O(n^2) space within the Iterative Linear Solvers family. Often compared with MINRES and BiCGSTAB.

Sources (2)

What each reference supports for this entry.

Saad-Schultz GMRES

Paper

Original GMRES reference for residual minimization over a Krylov subspace.

Open source

Templates for the Solution of Linear Systems

Book

Iterative solver reference for GMRES's sparse linear-system behavior and convergence tradeoffs.

Open source

Explore Next

Good next jumps from the relationship graph, ranked for useful discovery.

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Alternative toBiCGSTAB

Numerical Methods · Extended

GMRES and BiCGSTAB are common alternatives for nonsymmetric linear systems.

Compare with this
Alternative toMINRES

Numerical Methods · Extended

GMRES and MINRES are Krylov solvers with different matrix assumptions and residual objectives.

Compare with this
Used byHouseholder QR

Numerical Methods · Core

Householder QR uses GMRES as a subroutine, representation, or supporting idea while reflecting columns to zero out subdiagonal entries.

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UsesArnoldi Iteration

Numerical Methods · Specialized

GMRES builds an orthonormal Krylov basis with Arnoldi iteration before solving a projected least-squares problem.

Compare with this

Relationships

View full relationship page
UsesArnoldi Iteration

GMRES builds an orthonormal Krylov basis with Arnoldi iteration before solving a projected least-squares problem.

Alternative toMINRES

GMRES and MINRES are Krylov solvers with different matrix assumptions and residual objectives.

Alternative toBiCGSTAB

GMRES and BiCGSTAB are common alternatives for nonsymmetric linear systems.

Related Algorithms

MINRESBiCGSTABArnoldi IterationConjugate Gradient

Tags

Numerical Methods