TOTALLY NONNEGATIVE INTERVAL MATRICES

Abstract

This chapter describes the totally nonnegative interval matrices. A real matrix is totally nonnegative (t.n.) if all its minors are nonnegative. T.n. matrices arise in some interpolation processes such as the determination of the B-spline coefficients of an interpolating spline function, cf. If there is an interpolation problem where only bounds are known for the data points, then to compute bounds for the interpolating function, a system of linear equations with interval coefficients has to be solved. The coefficient matrix of this system does, however, not necessarily contains only nonsingular t.n. real matrices. An interval matrix is t.n.—that is, the matrix contains only t.n. real matrices if two special boundary matrices of this matrix are t.n. matrices. The chapter presents two applications where the first application is the determination of the interval hull of the solution set of a system of linear equations with a t.n. nonsingular coefficient matrix and an oscillating right-hand side, and the second application is the determination of bounds for the eigenvalues of a real matrix being contained in a special t.n. interval matrix.