Approximation Techniques for Engineers: Second Edition by Louis Komzsik

Author:Louis Komzsik [Komzsik, Louis]
Language: eng
Format: azw3, pdf
Publisher: CRC Press
Published: 2017-04-21T04:00:00+00:00

9.4 Conjugate gradient method

The exploitation of the tridiagonal form in the case of positive definite matrices yields the method of conjugate gradients [3]. Positive definiteness implies that the tridiagonal matrix may be factored without pivoting into

Ti = LiUi.

Here the factor matrices are a lower bidiagonal matrix of

and a unit upper bidiagonal matrix of

We use this factorization in order to produce the inverse

The following clever algorithm, invented by Hestenes and Stiefel [3], avoids the explicit computations of the terms of the tridiagonal matrix, as well as its factorization. Instead the following process is executed. Initialize as

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