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Symbolic Computation of Divided Differences
, 1999
"... Divided differences are enormously useful in developing stable and accurate numerical formulas. For example, programs to compute f(x)  f(y) as might occur in integration, can be notoriously inaccurate. Such problems can be cured by approaching these computations through divided difference formulati ..."
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Cited by 3 (1 self)
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Divided differences are enormously useful in developing stable and accurate numerical formulas. For example, programs to compute f(x)  f(y) as might occur in integration, can be notoriously inaccurate. Such problems can be cured by approaching these computations through divided difference formulations. This paper provides a guide to divided difference theory and practice, with a special eye toward the needs of computer algebra systems that should be programmed to deal with these oftenmessy formulas.
Excerpts from a proposal to the National Science Foundation on Programming Environments and Tools for Advanced Scientific Computation
"... "by hand" mathematical models whose consequences can be simulated by running computer programs. These programs are typically written in Fortran, but are increasingly being written in C, C++, or other languages that have better tools for abstraction and data structures. In part, the computa ..."
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"by hand" mathematical models whose consequences can be simulated by running computer programs. These programs are typically written in Fortran, but are increasingly being written in C, C++, or other languages that have better tools for abstraction and data structures. In part, the computational approach we advocate involves the use of computers in that earlier "by hand" stage of model formulation, using symbolic mathematics. The blossoming of this area via commercial programs typified by Mathematica and Maple, as well as some less widely used but still viable competitors (Axiom, Macsyma, MuPAD) might suggests that this approach is (a) successful and (b) needs no more academic research. Actually, the relatively higher level of activity (and funding) in Europe has demonstrated that important results remain to be found in advancement of algorithms and building systems. Work at RISCLinz (Austria), ETH (Zurich), CAN (Netherlands), INRIA (France) and the multinational POSSO project are ju
Applications and Methods for Recognition of (Anti)Symmetric Functions
, 2008
"... One of the important advantages held by computer algebra systems (CAS) over purelynumerical computational frameworks is that the CAS can provide a higherlevel “symbolic ” viewpoint for problem solving. Sometimes this can convert apparently impossible problems to trivial ones. Sometimes the symboli ..."
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One of the important advantages held by computer algebra systems (CAS) over purelynumerical computational frameworks is that the CAS can provide a higherlevel “symbolic ” viewpoint for problem solving. Sometimes this can convert apparently impossible problems to trivial ones. Sometimes the symbolic perspective can provide information about questions which cannot be directly answered, or questions which might be hard to pose. For example, we might be able to analyze the asymptotic behavior of a solution to a differential equation even though we cannot solve the equation. One route to implicitly solving problems is the use of symmetry arguments. In this paper we suggest how, through symmetry, one can solve a large class of definite integration problems, including some that we found could not be solved by computer algebra systems. One case of symmetry provides for recognition of periodicity, and this solves additional problems, since removal of periodic components can be important in integration and in asymptotic expansions. 1
DRAFT: Comments on Sets in Computer Algebra Systems, especially including Infinite Indexed Sets
, 2012
"... Computing with “sets ” is wellexplored in the programminglanguage and datastructure literature. Many languages have one or more set representations as well as operations for these sets. Unfortunately, the notion of set in mathematics is far more powerful than the notional support offered by ordin ..."
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Computing with “sets ” is wellexplored in the programminglanguage and datastructure literature. Many languages have one or more set representations as well as operations for these sets. Unfortunately, the notion of set in mathematics is far more powerful than the notional support offered by ordinary programming languages. The programming languages ’ notations and operations work for explicit finite sets only, not (for