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Linear Algebra, Optimisation and Numerical Methods / Iterative Methods and Sparse Matrices

GMRES

The Arnoldi iteration built an orthonormal basis for a growing sequence of Krylov subspaces, using only matrix-vector products. This note puts that basis to work: GMRES (Generalised Minimal RESidual) finds the best approximate solution to Ax=bA\mathbf{x}=\mathbf{b} within the Krylov subspace Kk(A,b)\mathcal{K}_k(A,\mathbf{b}) — turning the Arnoldi machinery from a basis-construction tool into a genuine, general-purpose iterative linear solver.
Learning objectives. By the end of this note you should be able to:
  • state the GMRES minimisation problem and explain why it is restricted to the Krylov subspace;
  • explain how the Arnoldi relation reduces the large problem to a small least-squares problem;
  • explain the monotone-decrease property of GMRES's residual;
  • know when GMRES is the appropriate method versus a direct solver.

In this lesson

  1. The GMRES minimisation problem
  2. Monotone convergence

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