Interval Mathematical Library Based on
Chebyshev and Taylor Series Expansion
ALEXEI G. ERSHOV
Ledas Ltd., 6, Lavrentiev ave., 630090 Novosibirsk, Russia, e-mail: firstname.lastname@example.org
TAMARA P. KASHEVAROVA
Institute of Informatics Systems of SB RAS, 6, Lavrentiev ave., 630090 Novosibirsk, Russia,
(Received: 30 October 2004; accepted: 1 March 2005)
Abstract. In this paper, we present a mathematical library designed for use in interval solvers of
nonlinear systems of equations. The library computes the validated upper and lower bounds of ranges
of values of elementary mathematical functions on an interval, which are optimal in most cases.
Computation of elementary functions is based on their expansion in Chebyshev and Taylor series and
uses the rounded directions setting mechanism. Some original techniques developed by the authors
are applied in order to provide high speed and accuracy of the computation.
Interval analysis actively developed during the last three decades brings out the
computational mathematics to a qualitatively new level. Interval methods enable us
to compute validated interval estimates of solutions sets in the form of multidimen-
sional boxes. Moreover, the use of intervals makes it possible to correctly handle
the errors caused by run-time roundings.
The interval data representation necessitates the development of the correspond-
ing mathematical software library. There are several interval mathematical libraries
available nowadays, but almost all of them have certain shortcomings. For example,
the integer-valued arithmetic is used in implementation of Pascal–XSC library of
mathematical functions (see ), which gives precise results of the ﬂoating-point
computation, but increases the run-time signiﬁcantly.
For this reason, the authors have developed and implemented the algorithms for
validated computation of the upper and lower bounds of elementary mathematical
functions by means of directed roundings. These algorithms are based on expansion
in Chebyshev and Taylor series. They provide high accuracy of computation of the
lower and upper bounds, as well as performance comparable with that of the stan-
dard mathematical functions of the C library. The developed interval mathematical
library is used in interval solvers of nonlinear systems of equations SibCalc (see ),
LEDAS Math Solver (see http://www.ledas.com/solver.php).