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Some properties of the conjugate-gradient approach
are well known but hard to explain.
D. G. Luenberger's book,
*Introduction to Linear and Nonlinear Programming*,
is a good place to look for formal explanations of this magic.
(His book also provides other forms of the conjugate-gradient algorithm.)
Another helpful book is Strang's
*Introduction to Applied Mathematics*.
Known properties follow:
- 1.
- The
**conjugate-gradient method** gets the exact answer
(assuming exact arithmetic) in *n* descent steps (or less),
where *n* is the number of unknowns.
- 2.
- Since it is helpful to use the previous step,
you might wonder why not use the previous two steps,
since it is not hard to solve a three-by-three set
of simultaneous linear equations.
It turns out that the third direction does not help:
the distance moved in the extra direction is zero.

** Next:** Conjugate-gradient theory for programmers
** Up:** ITERATIVE METHODS
** Previous:** Conjugate gradient
Stanford Exploration Project

10/21/1998