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1 results

  • 7 - Variational and Krylov Subspace Methods

  • from Part I - Numerical Linear Algebra
  • Abner J. Salgado, University of Tennessee, Knoxville, Steven M. Wise, University of Tennessee, Knoxville
  • Book:
    Classical Numerical Analysis
    Published online:
    29 September 2022
    Print publication:
    20 October 2022, pp 156-196

  • Summary

    We prove the equivalence of the solution to a linear system of equations with an HPD matrix to the problem of quadratic minimization. With the help of this equivalence, we study the minimization of a quadratic energy and introduce gradient descent methods with exact and approximate line search, we study the preconditioned steepest descent method. We introduce the conjugate gradient method, with preconditioning, as a Galerkin approximation over Krylov subspaces and show its convergence. For systems with non HPD matrices we discuss the GMRES method.