M.A. Wolfe

On interval enclosures using slope arithmetic

This paper describes how a combination of interval derivative arithmetic and interval slope arithmetic can be used with the programming language Pascal-SC to obtain an enclosure of the range of a factorable function @?:R^n -> R^1 with @? @? C^2 (D) which is often narrower than that which is obtained by using slope arithmetic alone and which is always narrower than that which is obtained by using interval derivative arithmetic alone.

Interval methods for global optimization

This paper contains a brief survey of the fundamental ideas that underlie interval methods for global optimization. Some recent work is also surveyed and possible future developments are suggested.

An interval maximum entropy method for a discrete minimax problem

An interval method for a discrete minimax problem is described, in which the constituent objective functions are Lipschitz continuous but not differentiable. A pseudocode Fortran 90 algorithm is described, and numerical results from a Fortran 90 implementation of the algorithm are presented.

An algorithm for the simultaneous inclusion of real polynomial zeros

The repeated symmetric single-step (RSS) method for the simultaneous inclusion of real polynomial zeros, which is based on the symmetric single-step (SS) idea of Alefeld, is described. It is shown that the RSS method converges under the same hypotheses as the total-step (T) and single-step (S) methods. Computational results indicate that RSS is more efficient than T, S, and SS for bounding the real zeros of real polynomials, and preliminary experiments with complex rectangular interval arithmetic [8, 9] suggest that the same is true for complex polynomial zeros.

An interval algorithm for nondifferentiable global optimization

An interval algorithm for constrained nondifferentiable global optimization in which an exact penalty function is used is described, and the determination of the penalty function parameter is discussed. Numerical results are presented.

A note on the comparison of the Kantorovich and Moore theorems

F(x) = 0 (1.1) where F: D E R” --+ R” is continuously differentiable on the open convex set D. A theoretical comparison by Rall [6] of the theorems of Kantorovich [2] and of Moore [3, 41 which contain sufficient conditions for the existence a unique zero of F in a given subset of D shows that the Kantorovich theorem has only a slight advantage over the Moore theorem with regard to sensitivity and precision, while the latter requires ‘far less computational labour than the former. Rall’s comparison is based on the assumption that the interval extension F’: Z(D) + Z(R” ’ “) of the derivative F’: D -+ R” ’ n is defined by

Interval versions of some procedures for the simultaneous estimation of complex polynomial zeros

An idea due to Neumaier is used to construct interval versions of point iterative procedures for the estimation of simple zeros of analytic functions. In particular, interval versions of some point iterative procedures for the simultaneous estimation of simple complex polynomial zeros are described. Some numerical results are presented.

Slope tests for newton-type methods

It is shown that the computable test for the existence of a solution of a system of nonlinear algebraic equations in a given region due to Pandian [8] may be weakened, and that sharp componentwise error bounds can be obtained if derivatives are replaced with slopes [4], [12]. Illustrative numerical examples are presented.

On first zero crossing points

Three algorithms, FZ1, FZ2 and FZ3 for bounding the first zero crossing point of a set of univariate functions on a bounded closed interval are described. Extended interval arithmetic is used in both FZ2 and FZ3. Automatic derivative arithmetic is used in FZ2 and automatic slope arithmetic is used in FZ3. Numerical results are presented.

Interval enclosures for a certain class of multiple integrals

An interval algorithm for bounding multiple integrals of integrands f: R^n x R^m -> R^1 over compact rectangular regions in R^m is described. It is supposed that the limits of integration in R^m are constant, that x varies over a rectangular compact subset of R^n and that f(x, y) is continuously differentiable with respect to x and is either once, twice, or four times continuously differentiable with respect to y. A pseudocode Fortran 90 implementation of the algorithm is described and numerical results from Fortran 90 implementations of the algorithm are presented.