C-XSC Continued Fractions Error Bounds Special Functions
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Journal of Computing, Vol. 1, No 4, (2007), 433p-442p
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.
The paper has been presented at the 12th International Conference on Applications of
Computer Algebra, Varna, Bulgaria, June, 2006.