Conjugate Gradient Methods

  • Wolfgang Hackbusch
Part of the Applied Mathematical Sciences book series (AMS, volume 95)

Abstract

In the following, A ∈ ℝ I x I and b ∈ ℝ I are real. We consider a system
$$ Ax\, = \,b $$
(9.1.1)
and assume that
$$ A\,is\,positive\,definite. $$
(9.1.2)
System (1) is associated with the function
$$ F\left( x \right): = \,\frac{1}{2}\left\langle {Ax,\,x} \right\rangle \, - \,\left\langle {b,\,x} \right\rangle . $$
(9.1.3)
The derivative (gradient) of F is \( F'\left( x \right): = \,\frac{1}{2}\left( {A\, + \,{A^T}} \right)x\, - \,b \). Since A = AT by assumption (2), the derivative equals
$$ F{\text{'}}'\left( x \right)\, = \,grad\,F\left( x \right)\, = \,Ax\, - \,b. $$
(9.1.4)

Keywords

Resid Paral Nite Cond Ethod 

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Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • Wolfgang Hackbusch
    • 1
  1. 1.Institut für Informatik und Praktische MathematikChristian-Albrechts-Universität zu KielKielGermany

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