A numerical approach for Gaussian rational formulas to handle difficult poles

Abstract

Let f be a meromorphic function in a neighborhood V of the real interval I , such that f z ; f ( z ) = 1g Ω V n I . Let W ( x ) be a weight function with possibly some integrable singularities at the end points of I . The problem of evaluating the integral I W ( f ) = Z I f ( x ) W ( x ) dx; has its own interest in applications. It is a theoretical fact that for a variety of weights W ( x ) , Gaussian quadrature formulas based on rational functions (GRQF) converge geometrically to I W ( f ) . However, the so-called difficult poles, that is, those poles which are close to [ a; b ] , produce numerical instability. W. Gautschi (1999) has de- veloped routines to calculate nodes and coefficients for a GRQF when some poles of f are difficult. The authors and U. Fidalgo (2006) have found a method different from Gautschi’s which has been succesfully applied to compute simultaneous ratio- nal quadrature formulas (SRQF). This paper presents a version of the SRQF approach adapted to GRQF for evaluating I W ( f ) efficiently even when some poles of f should be considered as difficult ones. The procedure consists in the use of smoothing trans- formations of [ a; b ] to move real poles away from I , so that the modified moments of the measure dπ ( x ) = W ( x ) dx can be computed with accuracy. A slight variant of the method improves the numerical estimates when some poles are very difficult. Some numerical tests are shown to be compared with previous results