# Optimization of Rational Approximations by Continued Fractions

Serdica Journal of Computing (2007)

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

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topBlomquist, 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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