Construction of gauss integration methods with respect to freud-type weight functions and application of these methods to highly oscillatory integrals

Highly oscillatory integrals arise in many applications in science, engineering, and medicine. Therefore, various methods have been proposed for numerically computing these integrals. One of these methods is the so called numerical steepest descent method, which has the highest asymptotic order. However when there is a stationary point in the interval of integration this method requires the construction of Gauss rules relative to a Freud weight function wr(x)=exp(-x^r), where r is a positive integer. Construction of these rules requires the calculation of coefficients in the three-term recurrance relations of orthogonal polynomials with respect to wr, because these coefficients can not be obtained analytically. In this thesis, the recurrance coefficients are obtained numerically by using the Chebyshev algorithm. These coefficients are used to form the Jacobi matrix, from which the nodes and weights of the desired Gauss rules are easily obtained. The Gauss rules obtained in this way were used to compute some selected highly oscillatory integrals. Numerical results show that the new Gauss rules are stable, very efficient, and give high accurate results. Key Words: Highly Oscillatory Integrals, Gauss Integration Methods, The Numerical Steepest Descent Method, Orthogonal Polynomials, Three-Term Recurrence Relation , Jacobi Matrix, Chebyshev Algorithm.