Abstract. An introduction to both automatic differentiation and object-oriented programming can enrich a numerical analysis course that typically incorporates numerical differentiation and basic MATLAB computation. Automatic differentiation consists of exact algorithms on floating-point arguments. This implementation overloads standard elementary operators and functions in MATLAB with a derivative rule in addition to the function value; for example, sin u will also compute (cos u) * u , where u and u are numerical values. These methods are mostly one-line programs that operate on a class of value-and-derivative objects, providing a simple example of object-oriented programming in MATLAB using the new (as of release 2008a) class definition structure. The resulting powerful tool computes derivative values and multivariable gradients, and is applied to Newton's method for rootfinding in both single and multivariable settings. To compute higher-order derivatives of a single-variable function, another class of series objects keeps Taylor polynomial coefficients up to some order. Overloading multiplication on series objects is a combination (discrete convolution) of coefficients. This idea leads to algorithms for other operations and functions on series objects. A survey of more advanced topics in automatic differentiation includes an introduction to the reverse mode (our implementation is forward mode) and considerations in arbitrary-order multivariable series computation.
For any typical multivariable expression f , point a in the domain of f , and positive integer maxorder, this method produces the numerical values of all partial derivatives at a up through order maxorder. By the technique known as automatic differentiation, theoretically exact results are obtained using numerical (as opposed to symbolic) manipulation. The key ideas are a hyperpyramid data structure and a generalized Leibniz's rule. Any expression in n variables corresponds to a hyperpyramid array, in n -dimensional space, containing the numerical values of all unique partial derivatives (not wasting space on different permutations of derivatives). The arrays for simple expressions are combined by hyperpyramid operators to form the arrays for more complicated expressions. These operators are facilitated by a generalized Leibniz's rule which, given a product of multivariable functions, produces any partial derivative by forming the minimum number of products (between two lower partials) together with a product of binomial coefficients. The algorithms are described in abstract pseudo-code. A section on implementation shows how these ideas can be converted into practical and efficient programs in a typical computing environment. For any specific problem, only the expression itself would require recoding.
Automatic differentiation, manipulating numerical vectors of coefficients, is the efficient way to compute multivariable Taylor series. This does not require symbolic differentiation or numerical approximation but uses exact formulas applied to numerical arrays. Arrays of Taylor series coefficients of any elementary function can be built-up, as the array for each component (combination or function) is a combination of its argument arrays. The functions TIMES and EXP display the algorithmic ideas that enable all of the other standard functions. We study the interesting recursive formulas for these combinations, the resulting algorithms, and the implementation in APL. To handle all coefficients in n variables up to order m , the arrays are hyper-pyramid data structures, considered conceptually as n -dimensional but implemented as one-dimensional arrays. Unlike previous work, this implementation does not require huge arrays for binomial coefficients and indirect referencing. This APL*PLUS III implementation loops through one nested reference array and takes sub-arrays from another for a practical solution to this problem that can make tremendous demands on time and space.
Introduction. Let X and Y be Banach spaces and T: X -> 7 a given non-zero operator. (Throughout, "operator" means "bounded linear map".) Under what circumstances is it true that TK is closed for every closed bounded convex subset K of XΊ Evidently this is trivially true if X is reflexive, so suppose this is not the case. Here are some of the equivalences obtained in our main result of §2, Theorem 2.3. (For any Banach space Z, let B z = {z e Z: ||z|| < 1}.)
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