Predicting the behaviour of finite precision Lanczos and conjugate gradient computations

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 con...

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Bibliographic Details
Published inSIAM journal on matrix analysis and applications Vol. 13; no. 1; pp. 121 - 137
Main Authors GREENBAUM, A, STRAKOS, Z
Format Journal Article
LanguageEnglish
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
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Summary: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} $.
ISSN:0895-4798
1095-7162
DOI:10.1137/0613011