Preconditioner

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 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 P {displaystyle P} of a matrix A {displaystyle A} is a matrix such that P − 1 A {displaystyle P^{-1}A} has a smaller condition number than A {displaystyle A} . It is also common to call T = P − 1 {displaystyle T=P^{-1}} the preconditioner, rather than P {displaystyle P} , since P {displaystyle P} itself is rarely explicitly available. In modern preconditioning, the application of T = P − 1 {displaystyle T=P^{-1}} , i.e., multiplication of a column vector, or a block of column vectors, by T = P − 1 {displaystyle T=P^{-1}} , is commonly performed by rather sophisticated computer software packages in a matrix-free fashion, i.e., where neither P {displaystyle P} , nor T = P − 1 {displaystyle T=P^{-1}} (and often not even A {displaystyle A} ) are explicitly available in a matrix form. Preconditioners are useful in iterative methods to solve a linear system A x = b {displaystyle Ax=b} for x {displaystyle x} since the rate of convergence for most iterative linear solvers increases because 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 A {displaystyle A} is not stored explicitly, but is accessed by evaluating matrix-vector products.