Preconditioners for the interval Gauss-Seidel method

Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box ${\bf X} \subset {\bf R}^n $ of a system of nonlinear equations $F(X) = 0$with mathematical certainty, even in finite-precision arithmetic. In such methods, the system $F(X) = 0$ is transformed into a linear interval system $0 = F(M) + {\bf F'}({\bf X})({\bf \bar X} - M)$ ; if interval arithmetic is then used to bound the solutions of this system, the resulting box ${{\bf \bar X}}$ contains all roots of the nonlinear system. The interval Gauss–Seidel method is a reasonable way of finding such solution bounds.For the overall interval Newton/bisection algorithm to be efficient, the image box ${{\bf \bar X}}$ should be as small as possible. To do this, the linear interval system is multiplied by a preconditioner matrix Y before the interval Gauss–Seidel method is applied. In this paper, a technique for computing such preconditioner matrices Y is described. This te...