ON THE PROPERTIES OF KRYLOV SUBSPACES IN FINITE PRECISION CG COMPUTATIONS

by · Sep 23, 2014 · 1 027 views ·

Tomáš Gergelits, Charles University in Prague, Faculty of Mathematics and Physics, Department of Numerical Mathematics, Prague, Czech Republic Joint work with Zdenek Strakoš The method of conjugate gradients (CG) for solving linear systems of algebraic equations with a Hermitian and positive definite matrix A is computationally based on short recurrences. Assuming exact arithmetic, they ensure the global orthogonality of the residual vectors, which span at the l-th step the l-dimensional Krylov subspace Kl(A; r0) = span n r0; Ar0;A2r0; : : : ;Al􀀀1r0 o : In practical computations, however, the use of short recurrences inevitably leads to the loss of the global orthogonality and even linear independence among the computed residual vectors. Consequently, the computed Krylov subspaces can be “rank-deficient” which causes a significant delay of convergence in finite precision CG computations. In the contribution we address the question how the Krylov subspaces generated by the CG method in finite precision arithmetic differ from their exact arithmetic counterparts. After recalling the related results published previously we concentrate on the situation with a significant delay of convergence and describe the distance between Krylov subspaces of the appropriately determined equal dimension. We observe that the finite precision Krylov subspaces in this comparison do not substantially deviate from the exact Krylov subspaces.

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