## Abstract

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.

Original language | English |
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Pages (from-to) | 1559-1575 |

Number of pages | 17 |

Journal | Mathematics of Computation |

Volume | 69 |

Issue number | 232 |

DOIs | |

State | Published - Oct 2000 |

## Keywords

- Bi-conjugate gradient algorithm
- Convergence analysis
- Error analysis
- Nonsymmetric linear systems

## ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics