Preconditioner - Preconditioning For Linear Systems

Preconditioning For Linear Systems

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

Preconditioners are useful in iterative methods to solve a linear system for since the rate of convergence for most iterative linear solvers increases as the condition number of a matrix decreases as a result of preconditioning. Preconditioned iterative solvers typically outperform direct solvers, e.g., Gaussian elimination, for large, especially for sparse, matrices. Iterative solvers can be used as matrix-free methods, i.e. become the only choice if the coefficient matrix is not stored explicitly, but is accessed by evaluating matrix-vector products.