Analysis of the finite precision bi-conjugate gradient algorithm for nonsymmetric linear systems

In this paper we analyze the bi-conjugate gradient algorithm in finite precision arithmetic, and suggest reasons for its often observed robustness. By using a tridiagonal structure, which is preserved by the finite precision bi-conjugate gradient iteration, we are able to bound its residual norm by a minimum polynomial of a perturbed matrix (i.e. the residual norm of the exact GMRES applied to a perturbed matrix) multiplied by an amplification factor. This shows that occurrence of near-breakdowns or loss of biorthogonality does not necessarily deter convergence of the residuals provided that the amplification factor remains bounded. Numerical examples are given to gain insights into these bounds.

• Type: Journal
• Year: 2000
• Where: MOC
• Authors: Charles H. Tong, Qiang Ye
• Keywords: Finite Precision, MOC 2000, Perturbed Matrix, Residual Norm