Polynomials in Approximation and Numerical Integration
Gregory Jimenez and Vladimir Vanegas
SUNY at New Paltz
Preface
This is the report of two students, Gregory Jimenez and Vladimir Vanegas,
concerning their work during the Spring and Summer I semesters, 2004, on a project that
I mentored. Mr. Jimenez was a junior majoring in engineering and Mr. Vanegas was a
junior majoring in computer science. The students were selected in cooperation with Ms.
Deb Gould, Director of the SUNY New Paltz Alliance for Minority Participation (AMP).
AMP is a SUNY-wide project for increased minority participation in science and
engineering that is funded by a grant from The National Science Foundation. During
Spring, 2004, the students met with me and also worked independently to carry out the
experiments that are described in the sequel. During the Summer I, 2004 semester the
work continued full-time under the auspices of NSF Research Grant DMS-0201430.
My research under this grant concerns multivariable moment and cubature problems, so
the project is designed to give training and experience in this area. The students'
mathematics background included two semesters of calculus, so the project began at an
elementary level. The first goal was to experiment with how Taylor and Bernstein
polynomials could be used to approximate a function on an interval. Using Mathematica,
the students studied the convergence of Taylor and Bernstein polynomials with respect to
the regularity of the function, the size of the interval, and the location of the interval.
The next main topic concerns the use of polynomials in developing methods of numerical
integration. The emphasis is on an approach to Gaussian quadrature using "moment
matrices" that was developed in [1] and [4]. The usual approach to Gaussian quadrature
involves the recursive calculation of Legendre polynomials, whose roots are the Gaussian
nodes. By contrast, the moment matrix approach is non-recursive and dynamic, the chief
obstacle being the inversion of the poorly conditioned moment matrix. As the students
discovered in their work, this obstacle can be overcome by using Mathematica's highprecision arithmetic. The students' experiments concerned the rate of convergence of
Gaussian quadrature with respect to the regularity of the function being integrated, the
size and location of the interval of integration, and comparison with Simpson's Rule.
The students later carried out experiments using an iterated Gaussian quadrature rule for
rectangular regions and an analogous iterated rule for planar regions bounded by 2
parallel lines and 2 curves. It is well-known that Gaussian quadrature on an interval [a,b]
gives exact results for polynomials up to degree 2n+1 with the fewest nodes possible,
n+1. The iterated rule for the rectangle has (n+1)2 nodes and is exact for polynomials
p(x,y) whose degrees in x and y are each less than 2n+1, but such a rule is certainly not
"minimal". The search for minimal rules is a difficult unsolved problem (cf. [4]). In some
earlier student projects with seniors, the students were able to discover new 2dimesnional minimal quadrature rules using 2-dimensional moment matrices, but in this
project we did not have sufficient time to pursue 2-dimensional minimal rules.
I am grateful to AMP and NSF for supporting this project and to Gregory and
Vladimir for their hard work.
Lawrence Fialkow
Department of Computer Science
SUNY at New Paltz
Introduction
In this project we studied how polynomials can be used to approximate functions
and to develop rules to approximate integrals. We first studied how Taylor polynomials
and Bernstein polynomials can be used to approximate functions. We conducted many
experiments in Mathematica to measure how these polynomial approximations would
behave with a variety of distinct functions. The second main topic concerns numerical
integration.
We used a version of Gaussian quadrature developed in [1] to study
approximation of integrals. We did this for integrals over an interval [a, b], a rectangle
[a, b] x [c, d], and over more general non-rectangular regions. Throughout this, we
conducted experiments to test the theory and also to test the limits of Mathematica
software.
Acknowledgement
This project began in Spring 2004 under the auspices of SUNY-NSF alliance for
Minority Participation (AMP), mentored by Prof. L. Fialkow (Dept. of Computer
Science). The project continued during the Summer Section 1, 2004, mentored by Prof.
L. Fialkow, sponsored by a grant from the National Science Foundation (DMS-0201430)