Conjugate Gradient Methods

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


In the following, A ∈ ℝ I x I and b ∈ ℝ I are real. We consider a system
$$ Ax\, = \,b $$
and assume that
$$ A\,is\,positive\,definite. $$
System (1) is associated with the function
$$ F\left( x \right): = \,\frac{1}{2}\left\langle {Ax,\,x} \right\rangle \, - \,\left\langle {b,\,x} \right\rangle . $$
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
$$ F{\text{'}}'\left( x \right)\, = \,grad\,F\left( x \right)\, = \,Ax\, - \,b. $$


Gradient Method Search Direction Conjugate Gradient Method Conjugate Direction Basic Iteration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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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