Concept

# Preconditioner

Summary
In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Preconditioning is typically related to reducing a condition number of the problem. The preconditioned problem is then usually solved by an iterative method. Preconditioning for linear systems In linear algebra and numerical analysis, a preconditioner P of a matrix A is a matrix such that P^{-1}A has a smaller condition number than A. It is also common to call T=P^{-1} the preconditioner, rather than P, since P itself is rarely explicitly available. In modern preconditioning, the application of T = P^{-1}, i.e., multiplication of a column vector, or a block of column vectors, by T = P^{-1}, is commonly performed in a matrix-free fashion, i.e., where neither
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