Round-Off Error Analysis of a New Class of Conjugate Gradient Algorithms.
CARNEGIE-MELLON UNIV PITTSBURGH PA DEPT OF COMPUTER SCIENCE
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We perform the rounding error analysis of the conjugate gradient algorithms for the solution of a large system of linear equations Ax b where A is an hermitian and positive definite matrix. We propose a new class of conjugate gradient algorithms and prove that in the spectral norm the relative error of the computed sequence x sub k in floating point arithmetic depends at worst on zeta eta to the 32 power where zeta is the relative computer precision and eta is the condition number of A. We show that the residual vectors r sub k - Ax sub k-b are at worst of order eta A abs. val. x sub k. We point out that with iterative refinement these algorithms are numerically stable. If zeta eta-squared is at most of order unity, then they are also well-behaved. Author
- Theoretical Mathematics