Preconditioning

Updated 01 Oct 2026

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What it is
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  transforms a linear-system problem to imrpove the behavior of an iterative solver while preserving the desired solution
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Mechanism
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  choose a matrix $M$ that approximates useful features of $A$ and is inexpensive to solve with
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  implement solves involving $M$ , rather than explicitly forming $M^{-1}$
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Example
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  Jacobi preconditioning uses the diagonal of $A$
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Limitations
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  diagonal scaling does not generally reduce the condition number to approximately $1$
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  a useful preconditioner depends on matrix structure; CG requires an appropriate symmetry/positive-definiteness-compatible formulation
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Related
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  Condition Number
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  Conjugate Gradient Method

Linked references 3