On the optimal properties of the krawczyk-type interval operator

In this paper we discuss a kind of Krawczyk-type interval operator [formula omitted] for solving a system of the nonlinear equations, and obtain that: i) The existence test condition presented in [4] without the interval operations is further studied. Combining the interval operator B(X, A), we obtain an existence test which is easier to apply. ii) Some important properties of the interval operator B(X, A) are discussed. Particularly we prove that the above existence test and the condition B(X,A)⊆X are equivalent. iii) Optimal properties in the same sense of the interval operator B(X, A) are discussed, and the function relationship between the eigenvalues of the matrix P = |I — AL| and the matrix A is given. They provide a basis for the optimal choice of the matrix A. For the Krawczyk-type interval operator, these are new results. Of all these facts, some are an improvement and extension of previous results; some provide useful conditions for constructing more efficient interval algorithms.