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Sparse Matrix Algorithms |
The conjugate gradient algorithm can be interpreted as the following optimization problem: minimize defined by
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where and
are symmetric and positive definite.
At each iteration is minimized along an
-conjugate direction, constructing orthogonal residuals:
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where is a Krylov subspace:
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Minimum residual algorithms work by minimizing the Euclidean norm over
. At each iteration,
is the vector in
that gives the smallest residual.
The biconjugate gradient algorithm belongs to a more general class of Petrov-Galerkin methods, where orthogonality is enforced in a different -dimensional subspace (
remains in
):
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