# Accession Number:

## AD0765748

# Title:

## The Convergence of the Method of Conjugate Gradients at Isolated Extreme Points of the Spectrum,

# Descriptive Note:

# Corporate Author:

## CARNEGIE-MELLON UNIV PITTSBURGH PA DEPT OF COMPUTER SCIENCE

# Personal Author(s):

# Report Date:

## 1973-08-01

# Pagination or Media Count:

## 19.0

# Abstract:

Let A be a positive definite matrix with a simple eigenvalue lambda sub 1 that lies outside an interval alpha,beta containing the remaining eigenvalues. Let the method of conjugate gradients be applied to the solution of the linear system Az b producing a sequence of iterates z sub 0,z sub 1,... and an associated sequence of error vectors e sub i z - z sub i. In the paper bounds are obtained for the component of the error vector lying along the eigenvector associated with lambda sub 1. The bounds imply that, provided lambda sub 1 is well separated from alpha,beta, this component will decrease rapidly even when the matrix A is moderately ill conditioned. Author

# Subject Categories:

- Theoretical Mathematics