Predicting the behaviour of finite precision Lanczos and conjugate gradient computations

• Published in: SIAM journal on matrix analysis and applications Vol. 13; no. 1; pp. 121 - 137
• Main Authors: GREENBAUM, A, STRAKOS, Z
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 1992
• Subjects: Algorithms; Eigenvalues; Error analysis; Exact sciences and technology; Mathematics; Nonlinear equations; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; Conjugate gradient method; Computers arithmetic; Equation system solving; Symmetric matrix; Positive definite matrix; Eigenvalue; Lanczos method
• DOI: 10.1137/0613011
• ISSN: 0895-4798

It is demonstrated that finite precision Lanczos and conjugate gradient computations for solving a symmetric positive definite linear system $Ax = b$ or computing the eigenvalues of $A$ behave very similarly to the exact algorithms applied to any of a certain class of larger matrices. This class consists of matrices $\hat{A} $ which have many eigenvalues spread throughout tiny intervals about the eigenvalues of $A$. The width of these intervals is a modest multiple of the machine precision times the norm of $A$. This analogy appears to hold, provided only that the algorithms are not run for huge numbers of steps. Numerical examples are given to show that many of the phenomena observed in finite precision computations with $A$ can also be observed in the exact algorithms applied to such a matrix $\hat{A} $.