# Inverting interval matrices

Given an interval square matrix A, an enclosure for the set of all inverse point matrices in A is called the inverse interval matrix `A-1`, that is,
`A-1⊇{A-1|A ∈ A}.`

In classical linear algebra, the solution to a system of linear algebraic equations `Ax = b` with square nonsingular matrix A can be expressed as the product of the inverse A-1 by the right-hand side vector, or `x = A-1b`.

In interval analysis, the similar product `A-1b` also produces an enclosure for the solution set `ΞA,b` of the interval linear system `Ax = b`. However, this method usually causes substantial overestimation and is not recommended. Using specialized procedures for outer estimation of the solution sets is preferable.

Nevertheless, computing tight enclosures for inverse interval matrices is essential in sensitivity-like analysis of equation systems and the like.

Computing the inverse interval matrix may be carried out as finding an enclosure for the solution set of the following interval matrix equation

`AY = I`, where I is the identity matrix,

by applying n times (for every column of the matrix Y ) any method to solve the interval linear systems.

Note also that direct iterative procedures for finding the inverse interval matrix exist, such as Schulz method (see [Herzberger94]), which is included into Intel MKL as ?geszi routine.

Parent topic: Interval Linear Systems