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
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机译:令f为实区间I的邻域V中的亚纯函数,使得f z; f(z)= 1gΩV nI。令W(x)是一个权函数,在I的端点处可能带有一些可积分的奇点。积分I W(f)= Z I f(x)W(x)dx的评估问题;对应用程序有自己的兴趣。从理论上讲,对于各种权重W(x),基于有理函数(GRQF)的高斯正交公式在几何上收敛到I W(f)。但是,所谓的困难极点,即靠近[a; b],产生数值不稳定性。 W. Gautschi(1999)开发了一些例程,可以在f的某些极点困难时计算GRQF的节点和系数。作者和U. Fidalgo(2006)发现了一种不同于Gautschi方法的方法,该方法已成功地用于计算同时比例正交公式(SRQF)。本文提出了适用于GRQF的SRQF方法的一种版本,即使在f的某些极点应视为困难的极点时,也可以有效地评估I W(f)。该过程包括使用[a; b]将实际极点从I移开,以便可以精确计算出度量dπ(x)= W(x)dx的修正矩。当某些极点非常困难时,该方法的微小变化会改进数值估计。一些数值测试显示与以前的结果进行了比较
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