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This paper presents a series of basic computational problems that are mathematically and/or graphically appealing, and provides an idea of places one might go in trying to understand what is happening, integrating mathematics, computation, and graphics. The real point of this paper is to make a case, through those examples, for computation as an early and integral part in the undergraduate mathematics curriculum. By computation we mean the ability to rapidly and easily program exploratory calculations, and as well display the results with a versatile suite of graphics capabilities. The problems used here, and the philosophy behind them, derive from a MATLAB-based course in computation, programming, and mathematical applications that the author has taught for over 10 years. [ABSTRACT FROM AUTHOR]

Iterations of the form x n + 1 = f (x n) are used throughout mathematics, and convergence (or not) of an iteration to a fixed point x * = f (x *) is always a central question. We present an informal graphical technique for resolving whether a fixed point in one dimension is attracting or repelling. Specifically, given a real-valued function f, we are determining whether | f ′ (x *) | < 1 or | f ′ (x *) | > 1 at a point x * ∈ R for which x * = f (x *) . The former condition on f ′ (x *) classifies the fixed point as attracting, and the iteration as locally convergent; the latter corresponds to a repulsive fixed point, in which the iterations get further away from the fixed point no matter how close you start them off. The technique requires only that we be able to compute points (x , f (x)) on the graph of f (x) and not necessarily on an explicit functional form or its derivative. [ABSTRACT FROM AUTHOR]

The article explores the use of differential equations to define exponential and trigonometric functions. It suggests that some functions can be defined through their governing differential equations. Other topics discussed are the basic theorems for differential equations and the definition and properties of the exponential functions, sines and cosines.

On a finite dosed time interval an inspector wishes to detect an event as soon as possible after its occurrence. A loss fid) in incurred if the event remains undetected for a period d. Generally, f is assumed continuous and monotone increasing. Exactly N inspections are allowed. We seek a randomized inspection policy yielding the minimax expected loss. The problem is reformulated as a two-person zero-sum game between the inspector and an instpectee and a pair of equilibrium strategies is sought. An explicit solution for the linear case, ∫(d) = d is obtained and its asymptotic properties (N large)determined. A computational procedure for calculating the optimal policies in the nonlinear case is presented. Two additional related games are briefly treated. The optimal inspection policies each have the salient properly of being a randomization over a one parameter family of pure strategies, [ABSTRACT FROM AUTHOR]

Compares the method of linear interpolation with the differential approximation method. Familiar methods in calculus for approximating the value of differentiable function; Formulation and preliminary observations; Approximation error of an approximation method at a particular value of x; Qualitative results; Numerical considerations.

We present a general framework for the construction of local interpolation methods with a given approximation order. Some applications to multivariate spline spaces are presented. [ABSTRACT FROM AUTHOR]

Let ϕ be a compactly supported function on ℝ and S (ϕ) the linear space with generator ϕ; that is, S (ϕ) is the linear span of the multiinteger translates of ϕ. It is well known that corresponding to a generator ϕ there are infinitely many quasi-interpolation formulas. A characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications, and in addition, provides a clear formulation of the difficult problem of minimally supported quasi-interpolants. We introduce a generalization of interpolation called μ-interpolation and a notion of higher order quasi-interpolation called μ-approximation. A characterization of μ-approximants similar to that of quasi-interpolants is obtained with similar applications. Among these applications are estimating least-squares approximants without matrix inversion, surface fitting to incomplete or semi-scattered discrete data, and constructing generators with one-point quasi-interpolation formulas. It will be seen that the exact values of the generator ϕ at the multi-integers ℤ facilitates the above study. An algorithm to yield this information for box splines is discussed. [ABSTRACT FROM AUTHOR]