Preconditioner
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In linear algebra and numerical analysis, a preconditioner P of a matrix A is a matrix such that P−1A has a smaller condition number than A. A preconditioner as defined here is also known as a left preconditioner; there are also right preconditioners, but they are not discussed in this article.
Preconditioners are useful when using an iterative method to solve a large, sparse linear system
for x. Solving the preconditioned system
is equivalent to solving the original system. However, the preconditioned system has a smaller condition number; it is better conditioned. The number of iterations used to solve the system increases with the condition number so by reducing the number of needed iterations (if the cost — computing time — of applying P-1 is small), a gain in efficiency is achieved.
The matrix P−1A is never explicitly formed. In using an iterative method, only the action of applying P−1 to a given vector need be computed.
