Derivation of the conjugate gradient method
where is symmetric positive-definite. The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems.
The intent of this article is to document the important steps in these derivations.
Derivation from the conjugate direction method
|This section requires expansion. (April 2010)|
The conjugate gradient method can be seen as a special case of the conjugate direction method applied to minimization of the quadratic function
The conjugate direction method
In the conjugate direction method for minimizing
one starts with an initial guess and the corresponding residual , and computes the iterate and residual by the formulae
where are a series of mutually conjugate directions, i.e.,
for any .
The conjugate direction method is imprecise in the sense that no formulae are given for selection of the directions . Specific choices lead to various methods including the conjugate gradient method and Gaussian elimination.
Derivation from the Arnoldi/Lanczos iteration
The conjugate gradient method can also be seen as a variant of the Arnoldi/Lanczos iteration applied to solving linear systems.
The general Arnoldi method
by defining where
In other words, for , is found by Gram-Schmidt orthogonalizing against followed by normalization.
Put in matrix form, the iteration is captured by the equation
When applying the Arnoldi iteration to solving linear systems, one starts with , the residual corresponding to an initial guess . After each step of iteration, one computes and the new iterate .
The direct Lanczos method
For the rest of discussion, we assume that is symmetric positive-definite. With symmetry of , the upper Hessenberg matrix becomes symmetric and thus tridiagonal. It then can be more clearly denoted by
This enables a short three-term recurrence for in the iteration, and the Arnoldi iteration is reduced to the Lanczos iteration.
with convenient recurrences for and :
It is now important to observe that
In fact, there are short recurrences for and as well:
With this formulation, we arrive at a simple recurrence for :
The relations above straightforwardly lead to the direct Lanczos method, which turns out to be slightly more complex.
The conjugate gradient method from imposing orthogonality and conjugacy
If we allow to scale and compensate for the scaling in the constant factor, we potentially can have simpler recurrences of the form:
As premises for the simplification, we now derive the orthogonality of and conjugacy of , i.e., for ,
The residuals are mutually orthogonal because is essentially a multiple of since for , , for ,
To see the conjugacy of , it suffices to show that is diagonal:
is symmetric and lower triangular simultaneously and thus must be diagonal.
Now we can derive the constant factors and with respect to the scaled by solely imposing the orthogonality of and conjugacy of .
Due to the orthogonality of , it is necessary that . As a result,
Similarly, due to the conjugacy of , it is necessary that . As a result,
This completes the derivation.
- Hestenes, M. R.; Stiefel, E. (December 1952). "Methods of conjugate gradients for solving linear systems" (PDF). Journal of Research of the National Bureau of Standards 49 (6).
- Saad, Y. (2003). "Chapter 6: Krylov Subspace Methods, Part I". Iterative methods for sparse linear systems (2nd ed.). SIAM. ISBN 978-0-89871-534-7.