# Optimization of Rational Approximations by Continued Fractions

• Volume: 1, Issue: 4, page 433-442
• ISSN: 1312-6555

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## Abstract

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The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006.To get guaranteed machine enclosures of a special function f(x), an upper bound ε(f) of the relative error is needed, where ε(f) itself depends on the error bounds ε(app); ε(eval) of the approximation and evaluation error respectively. The approximation function g(x) ≈ f(x) is a rational function (Remez algorithm), and with sufficiently high polynomial degrees ε(app) becomes sufficiently small. Evaluating g(x) on the machine produces a rather great ε(eval) because of the division of the two erroneous polynomials. However, ε(eval) can distinctly be decreased, if the rational function g(x) is substituted by an appropriate continued fraction c(x) which in general needs less elementary operations than the original rational function g(x). Numerical examples will illustrate this advantage.

## How to cite

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Blomquist, Frithjof. "Optimization of Rational Approximations by Continued Fractions." Serdica Journal of Computing 1.4 (2007): 433-442. <http://eudml.org/doc/11434>.

@article{Blomquist2007,
abstract = {The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006.To get guaranteed machine enclosures of a special function f(x), an upper bound ε(f) of the relative error is needed, where ε(f) itself depends on the error bounds ε(app); ε(eval) of the approximation and evaluation error respectively. The approximation function g(x) ≈ f(x) is a rational function (Remez algorithm), and with sufficiently high polynomial degrees ε(app) becomes sufficiently small. Evaluating g(x) on the machine produces a rather great ε(eval) because of the division of the two erroneous polynomials. However, ε(eval) can distinctly be decreased, if the rational function g(x) is substituted by an appropriate continued fraction c(x) which in general needs less elementary operations than the original rational function g(x). Numerical examples will illustrate this advantage.},
author = {Blomquist, Frithjof},
journal = {Serdica Journal of Computing},
keywords = {C-XSC; Continued Fractions; Error Bounds; Special Functions; error function; special function; rational function; Remez algorithm},
language = {eng},
number = {4},
pages = {433-442},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Optimization of Rational Approximations by Continued Fractions},
url = {http://eudml.org/doc/11434},
volume = {1},
year = {2007},
}

TY - JOUR
AU - Blomquist, Frithjof
TI - Optimization of Rational Approximations by Continued Fractions
JO - Serdica Journal of Computing
PY - 2007
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 1
IS - 4
SP - 433
EP - 442
AB - The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006.To get guaranteed machine enclosures of a special function f(x), an upper bound ε(f) of the relative error is needed, where ε(f) itself depends on the error bounds ε(app); ε(eval) of the approximation and evaluation error respectively. The approximation function g(x) ≈ f(x) is a rational function (Remez algorithm), and with sufficiently high polynomial degrees ε(app) becomes sufficiently small. Evaluating g(x) on the machine produces a rather great ε(eval) because of the division of the two erroneous polynomials. However, ε(eval) can distinctly be decreased, if the rational function g(x) is substituted by an appropriate continued fraction c(x) which in general needs less elementary operations than the original rational function g(x). Numerical examples will illustrate this advantage.
LA - eng
KW - C-XSC; Continued Fractions; Error Bounds; Special Functions; error function; special function; rational function; Remez algorithm
UR - http://eudml.org/doc/11434
ER -

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