# Preconditioning

Preconditioning of interval linear system (4) is multiplying both the matrix A and the right-hand side vector b by a point matrix, with the intension to improve the properties of the system. So the system `Ax = b` is substituted by the following system
`(CA)x = Cb`

where C is some square point matrix. Preconditioning is widely used in classical computational linear algebra, and many interval solver algorithms (for example, interval Gauss method, interval Gauss-Seidel method and some others) also require a suitable preconditioning prior to their use.

One of the widely used preconditioning methods for the interval linear systems is preconditioning done by the inverse of the midpoint matrix, often called midpoint-inverse preconditioning. In Intel MKL, the midpoint inverse preconditioning is implemented in the routine ?gemip.

Parent topic: Interval Linear Systems