Calculation of Gauss quadrature rules

By Gene H. Golub, John H. Welsch. 1967

Subfields: Natural Language Processing, Numerical Methods & Scientific Computing

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Natural Language Processing

In Numerical Methods & Scientific Computing

Citations in the corpus

An Investigation into Neural Net Optimization via Hessian Eigenvalue Density (2019, methodology): “The algorithm we use is due to Golub and Welsch [10].”
Discrete-Time Nonlinear Filtering Algorithms Using Gauss–Hermite Quadrature (2007, methodology): “Instead of finding quadrature points using root-finding methods, which may be mathematically unstable, a computationally better approach is presented in [ 19 ] to find the quadrature points and weights.”
An Introduction to Trajectory Optimization: How to Do Your Own Direct Collocation (2017, methodology): “methods for computing these points are given by [24, 27], although various sub-routines can be found with a quick internet search.”
Use of the Probability Integral Transformation to Fit Nonlinear Mixed-Effects Models With Nonnormal Random Effects (2006, methodology): “q , and the standard Gauss-Hermite weights η q and abscissas zq can be obtained from tables (Abramowitz and Stegun 1972, table 25.10) and an algorithm proposed by Golub (1973) and Golub and Welsch (1969) .”
Optical wireless links with spatial diversity over strong atmospheric turbulence channels (2009, methodology): “From the algorithm proposed in [25], if W (x) is the joint pdf of the rvs Im, the K-point GQR can be computed using the first 2K−1 moments of Im, which are derived in closed-form in (5).”
Bayesian Filtering and Smoothing (2013, background): “Note that we do not need to form the polynomial and then compute its roots, but instead it is numerically more stable to compute the roots as eigenvalues of a suitable tridiagonal matrix (Golub and Welsch, 1969).”
ROBUST HYPERPARAMETER ESTIMATION PROTECTS AGAINST HYPERVARIABLE GENES AND IMPROVES POWER TO DETECT DIFFERENTIAL EXPRESSION (2016, methodology): “This function computes Gauss quadrature weights and nodes using an adaption of the algorithm and Fortran code published by Golub and Welsch (1969).”
Fast Estimation of tr(f(A)) via Stochastic Lanczos Quadrature (2017, methodology): “Next, the integral can be estimated using the Gauss quadrature rule [23]”
Gaussian quadrature for multiple orthogonal polynomials (2005, background): “It is known that multiple orthogonal polynomials satisfy a recurrence relation of order r + 1; see, e.g., [14, Section 24; 6, 12 ,16,18].”
PyHessian: Neural Networks Through the Lens of the Hessian (2019, background): “(12) This integral can be estimated by the Gauss quadrature rule [24], φσ(t) ≈ q ∑ i=1 ωif(ti; t, σ), (13) where (ωi, ti) is the weight-node pair to estimate the integral.”
Special Classes of Orthogonal Polynomials and Corresponding Quadratures of Gaussian Type (2012, methodology): “The Golub and Welsch procedure [8] was implemented in several programming packages including the most known ORTHPOL developed by Gautschi [4] in 1994.”