Computation of Gauss-type quadrature formulas

Laurie, Dirk

doi:10.1016/s0377-0427(00)00506-9

Cited by 60 publications

(47 citation statements)

References 34 publications

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“…The number of terms in the infinite series is related to the order of the polynomial: a higher-order polynomial approximation will be more accurate. There are many papers that cover the detailed mathematics of orthogonal polynomials [26,44,4,33,58,28] and their use in trajectory optimization [29,62,18,30,19,53,3,57,23,24,15,21]. Here we will focus on the practical implementation details and on gaining a qualitative understanding of how orthogonal collocation works.…”

Section: B1 Full Solutionmentioning

confidence: 99%

An Introduction to Trajectory Optimization: How to Do Your Own Direct Collocation

Kelly¹

2017

SIAM Rev.

446

262

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Abstract. This paper is an introductory tutorial for numerical trajectory optimization with a focus on direct collocation methods. These methods are relatively simple to understand and effectively solve a wide variety of trajectory optimization problems. Throughout the paper we illustrate each new set of concepts by working through a sequence of four example problems. We start by using trapezoidal collocation to solve a simple one-dimensional toy problem and work up to using Hermite-Simpson collocation to compute the optimal gait for a bipedal walking robot. Along the way, we cover basic debugging strategies and guidelines for posing well-behaved optimization problems. The paper concludes with a short overview of other methods for trajectory optimization. We also provide an electronic supplement that contains well-documented MATLAB code for all examples and methods presented. Our primary goal is to provide the reader with the resources necessary to understand and successfully implement their own direct collocation methods.

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Section: B1 Full Solutionmentioning

confidence: 99%

An Introduction to Trajectory Optimization: How to Do Your Own Direct Collocation

Kelly¹

2017

SIAM Rev.

446

262

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“…Even for a good choice of the auxiliary polynomials (such as the Chebyshev polynomials) and modified moments the situation is not simple. There are distribution functions for which the condition numbers are small, but there are other distribution functions for which the condition numbers grow exponentially with the number of nodes k. Moreover, the assumption that the modified moments are known accurately is difficult to satisfy; see [38,Sect. 3.2].…”

Section: T1mentioning

confidence: 99%

“…This conclusion has been examined by Laurie [36], who pointed out that the negative statement is linked to the use of the max-norm for vectors. He suggested instead measuring the perturbation of the weights in the componentwise relative sense [36, p. 179], [38,Sect. 6].…”

Section: T1mentioning

confidence: 99%

“…2.7], [18]. The recent encyclopedic book by Gautschi [23], his surveys [18,24] and the survey by Laurie [38] describe the state-of-the-art of Gauss-Christoffel quadrature computation, and can be recommended as fundamental reading for anyone interested in related problems.…”

Section: Introductionmentioning

confidence: 99%

“…252 and Sect. 2], and for recent analysis to [1,23,38]. Despite the vast literature on related subjects, the problem of sensitivity of the Gauss-Christoffel quadrature has, to our knowledge, not been posed or examined in the literature.…”

Section: Introductionmentioning

confidence: 99%

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On sensitivity of Gauss–Christoffel quadrature

2007

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In numerical computations the question how much does a function change under perturbations of its arguments is of central importance. In this work, we investigate sensitivity of Gauss-Christoffel quadrature with respect to small perturbations of the distribution function. In numerical quadrature, a definite integral is approximated by a finite sum of functional values evaluated at given quadrature nodes and multiplied by given weights. Consider a sufficiently smooth integrated function uncorrelated with the perturbation of the distribution function. Then it seems natural that given the same number of function evaluations, the difference between the quadrature approximations is of the same order as the difference between the (original and perturbed) approximated integrals. 123 148 D. P. O'Leary et al.several other sensitivity problems, motivated, in particular, by computation of the quadrature nodes and weights from moments, have been thoroughly studied by many authors. We survey existing particular results and show that even a small perturbation of a distribution function can cause large differences in Gauss-Christoffel quadrature estimates. We then discuss conditions under which the Gauss-Christoffel quadrature is insensitive under perturbation of the distribution function, present illustrative examples, and relate our observations to known conjectures on some sensitivity problems.

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Numerical Quadrature Rules

2020

The Nyström Method in Electromagnetics

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Computation of Gauss-type quadrature formulas

Cited by 60 publications

References 34 publications

An Introduction to Trajectory Optimization: How to Do Your Own Direct Collocation

An Introduction to Trajectory Optimization: How to Do Your Own Direct Collocation

On sensitivity of Gauss–Christoffel quadrature

Numerical Quadrature Rules

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