Many problems in scientific computing require the evaluation of Gauss quadrature rules. It is important to be able to estimate the quadrature error in these rules. Error estimates or error bounds often can be computed...
详细信息
Many problems in scientific computing require the evaluation of Gauss quadrature rules. It is important to be able to estimate the quadrature error in these rules. Error estimates or error bounds often can be computed by evaluating an additional related Gauss-type formula such as a Gauss-Radau, Gauss-Lobatto, anti-Gauss, averaged Gauss, or optimal averaged Gauss rule. This paper presents software for both the evaluation of a single Gauss quadrature rule and the calculation of a pair of a Gauss rule and a related Gauss-type rule. The software is based on a divide-and-conquer method. This method is compared to both an available and a new implementation of the golub-welsch algorithm, which is the classical approach to evaluate a single Gauss quadrature rule. Timings on a laptop computer show the divide-and-conquer method to be competitive except for the computation of a single quadrature rule with very few nodes.
We explore the computational issues concerning a new algorithm for the classical least-squares approximation of N samples by an algebraic polynomial of degree at most n when the number N of the samples is very large. ...
详细信息
We explore the computational issues concerning a new algorithm for the classical least-squares approximation of N samples by an algebraic polynomial of degree at most n when the number N of the samples is very large. The algorithm is based on a recent idea about accurate numerical approximations of sums with large numbers of terms. For a fixed n, the complexity of our algorithm in double precision accuracy is O(1). It is faster and more precise than the standard algorithm in MATLAB.
Three methods are reviewed for computing optimal weights and abscissas which can be used in the quadrature method of moments (QMOM): the product-difference algorithm (PDA), the long quotient-modified difference algori...
详细信息
Three methods are reviewed for computing optimal weights and abscissas which can be used in the quadrature method of moments (QMOM): the product-difference algorithm (PDA), the long quotient-modified difference algorithm (LQMDA, variants are also called Wheeler algorithm or Chebyshev algorithm), and the golub-welsch algorithm (GWA). The PDA is traditionally used in applications. It is discussed that the PDA fails in certain situations whereas the LQMDA and the GWA are successful. Numerical studies reveal that the LQMDA is also more efficient than the PDA. (C) 2012 Elsevier Ltd. All rights reserved.
暂无评论