Talk:Conjugate gradient method
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In your algorithm, the formula to calculate pk differs from Jonathan Richard Shewchuk paper. The index of r should be k instead of k-1. Mmmh, sorry, it seams to be correct! ;-) —The preceding unsigned comment was added by 184.108.40.206 (talk • contribs) 01:45, 21 February 2007 (UTC)
Small error in code
Major conceptual flaw
The article gives the impression that you need to keep in memory all vectors calculated in order to find a new one that is conjugate to all the previous ones, but this is not true. The vectors are calculated one by iteration and based just on the previous one, and this is precisely one of the most important aspects of the whole theory.
Yes it's that awesome, it's not a simplification, it's not a crazy new direction conjugate just to the last one, it really is conjuge to all previous in spite of we throwing them away! And it must be made clear that the algorithm is direct _and_ iterative. Sometimes you can have good approximations before the last step, and stop before, but this approximate algorithm is otherwise identical to the direct version. -- NIC1138 (talk) 05:04, 11 April 2010 (UTC)
- It already mentions that, but is simply too vague in doing so. Kxx (talk | contribs) 06:09, 11 April 2010 (UTC)
- I would love to see more discussion of this. I admit, much as I've tried to understand this algorithm, the details of why you only need O(n) memory and O(n) per iteration illude me. It is pretty darn cool "magic" and could really do with more explanation. —Ben FrantzDale (talk) 13:56, 12 April 2010 (UTC)
- Yousef Saad's book has a fairly detailed coverage of CG from the perspective of specialization of the Arnoldi iteration, including derivation of the three-term recurrences, of course. PDF of the first edition is online at Saad's homepage. You can check it out. Kxx (talk | contribs) 20:23, 12 April 2010 (UTC)
Original research in "description" section?
The "description" section has always been painfully confusing to whoever wants to understand the theory behind the conjugate gradient method as the two "as a direct/iterative method" subsections provide nothing close to the actual algorithm, which is only vaguely justified with a cryptic paragraph at the beginning of the "resulting algorithm" subsection.
It now seems even worse to me that a large portion of the section appears to consist of pure original research—although the equations are true, nobody derives the conjugate gradient method that way, i.e., expressing the solution as a linear combination of the search directions and then calculating coefficients. (Let's not mention the "numerical example" subsection, which is probably the most obvious OR offender.) I looked at Hestenes and Stiefel's paper and Jonathan Shewchuk's writing, which are close in the way of thinking. But they describe the method as a special case of the conjugate direction method, which readily gives the expressions for αi and xi, no need for linear expansion.
Probably the best way out is a rewrite. Just follow either or both of the two classical derivations (conjugate direction method and Lanczos iteration). Kxx (talk | contribs) 20:12, 13 April 2010 (UTC)
"Fortunately, the conjugate gradient method can be used as an iterative method as it provides monotonically improving approximations x_k to the exact solution." Is this really true? I find, in practice, that the residual norm(A*x_k - b) is definitely not monotonic w.r.t. k. As an example, I invite you to try the following program with the given octave code. You will see that the residuals, when x_0=[0;0;0], are [1.3776, 1.3986, 1.6422, 1.4e-14].
A := [0.100397715109637, 0.333310651201826, 0.197818584877521; 0.333310651201826, 1.443856474586787, 0.826552007973684; 0.197818584877521, 0.826552007973684, 1.143082020356276];
b := [0.964888535199277, 0.157613081677548, 0.970592781760616]';
Complex (preconditioned) CG
The CG algorithm extends to complex matrices and vectors in both the original and the preconditioned case. The derivation for this can be found in Iterative Methods for Solving Linear Systems, A. Greenbaum. The changes to the algorithms are trivial - the transposes are replaced by the hermitian (conjugate) transpose (which I always denote as superscript H). It's necessary in the preconditioned case to take care whether it is the update for z or r that is conjugated (in the real case, it doesn't matter which is transposed), but the algorithm shown is correct. I suggest a comment should be made somewhere that the complex case is also covered. Finding references to this is _not_ trivial (almost universally the derivations say "For real..."). [edit: claiming ownership or last comment] Hgomersall (talk) 19:19, 4 March 2014 (UTC)