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from Part I - Numerical Linear Algebra
Published online by Cambridge University Press: 29 September 2022
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.
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