Symbolic-numeric interface: A review

Ng, Edward W.

doi:10.1007/3-540-09519-5_85

Cited by 13 publications

(1 citation statement)

References 38 publications

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“…Approximate expressions were also investigated for use in algebraic computation. Most of the earlier works aimed at combining numeric and algebraic computation, as surveyed by [BH79] and [Ng79]. Following these works, some authors began to construct algorithms using approximate expressions.…”

Section: Introduetionmentioning

confidence: 99%

A study of approximate polynomials, I —Representation and arithmetic—

Sasaki

1995

Japan J. Indust. Appl. Math.

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By approximate polynomials in this paper, we mean polynomials with uumerical coefficients represented approximately. Algebraic operations on approximate polynomials are becoming important rapidly not only in application areas but also in algebraic algorithm construction; however, study of approximate polynomials in the context of algebraic computation is very poor so far. In studying approximate polynomials, we must consider two points, 1) the "error term" may be quite large in approximate algebra, and 2) approximate polynomials show various pathological features occasionally. For example, as for 1), the error term may be of relative magnitude of 10 -3 or even 10 -2 in approximate factorization, andas for 2), conventional division algorithm leads to very inaccurate results ir the leading coefficient of the divisor is small. This paper proposes a simple representation, presents algorithms for basic arithmetic operations, and clarifies some pathologicat properties, of approximate polynomials. In particular, presented is a division algorithm which maintains accuracy as highly as possible. IntroduetionIn this paper, by approximate expressions we mean mathematical expressions with numerical coefficients represented approximately. Hence, an approximate expression may contain small errors or uncertainties in its coefficients. For example, considera polynomial with floating number coefficients, a typical example of approximate expression, where by floating number we mean fixed-precision floatingpoint number. If M bits are used to represent the mantissa of a floating number, an error of relative magnitude 2 -M enters into the number. The error may be caused by other reasons. For example, an expression determined experimentally contains experimental errors in its coefficients. Approximate expressions appear frequently in science and technology, and numerical evaluation and approximation of mathematical expressions have been studied quite thorough]y in numerical analysis. Approximate expressions were also investigated for use in algebraic computation. Most of the earlier works aimed at combining numeric and algebraic computation, as surveyed by [BH79] and [Ng79].Following these works, some authors began to construct algorithms using approximate expressions. Consider, for example, solving bivariate algebraic equations in *

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Section: Introduetionmentioning

confidence: 99%