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\title{Validated Function Evaluation using
         Polynomial Approximation
         from Truncated Chebyshev Series}

\author{Walter Kr\"amer} 

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\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ }

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\auffil{The author is with Universit\"at Karlsruhe,
  Institut f\"ur Wissenschaftliches Rechnen
  und Mathematische Modellbildung,
  D-76128 Karlsruhe, Germany,
  e-mail: ae36@rz.uni-karlsruhe.de.}

The
truncated Chebyshev series of a function can be used to get an efficient
direct polynomial approximation for the function under consideration.
Then the polynomial approximation can be evaluated by Horner's scheme.

To get a verified lower and upper bound for the exact function value,
we must consider the following error components:
\begin{itemize}
\item the truncation error;
\item the error of the coefficients of the truncated
Chebyshev series;
\item  the error of the coefficients of the direct polynomial,
as well as 
\item all rounding errors inherent in the floating-point
evaluation of Horner's method. 
\end{itemize}
A realistic upper
bound for these errors (it can be derived a priori as we will
demonstrate) can be used to formulate an interval routine which is able to
compute an enclosure of the exact function range over an interval.

As an example, we will discuss a routine for the natural logarithm.
The routine will be based on the rapidly converging
Chebyshev series for the function
  \[ f(x)= \frac{x+1}{x-1} \cdot\ln(x). \]
Let $T_k(x)$ denote the $k$-th Chebyshev polynomial.
If we denote
$$s= 3 - 2\sqrt{2}, \ \  y= \frac{x-1}{s \cdot (x+1)}\ \ \alpha=\arcsin(s),$$
then for all $$x \in            [\frac{1}{\sqrt{2}}, \sqrt{2}],$$ we
have 
$$f(x) =\sum_{k=0}^{\infty} c_{2k} T_{2k}(y),$$
where 
the coefficients $c_{2k}$ can be generated by the following recurrent scheme:
  \[c_0:= \frac{8 \alpha}{s}, \]
  \[c_{2k+2} := \frac{8 \tan^{2k+1} \alpha}{(2k+1) s} \; - \; c_{2k} \;,
             \;\;\; k = 0, 1, 2, \ldots \]

Using leading terms of this expansion, we compute
the coefficients of this Chebyshev sum
and the coefficients 
of the direct polynomial approximation by using multiple-precision
interval arithmetic.

Finally, a priori error estimations considering all kinds of errors
mentioned above will be shown, yielding an interval routine producing
validated results on a computing device for all admissible (floating-point)
interval arguments.

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