Preconditioning

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.

In linear algebra and numerical analysis, a preconditioner ${\displaystyle P}$ of a matrix ${\displaystyle A}$ is a matrix such that ${\displaystyle P^{-1}A}$ has a smaller condition number than ${\displaystyle A}$. It is also common to call ${\displaystyle T=P^{-1}}$ the preconditioner, rather than ${\displaystyle P}$, since ${\displaystyle P}$ itself is rarely explicitly available. In modern preconditioning, the application of ${\displaystyle T=P^{-1}}$, i.e., multiplication of a column vector, or a block of column vectors, by ${\displaystyle T=P^{-1}}$, is commonly performed by rather sophisticated computer software packages in a matrix-free fashion, i.e., where neither ${\displaystyle P}$, nor ${\displaystyle T=P^{-1}}$ (and often not even ${\displaystyle A}$) are explicitly available in a matrix form.

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