## Algorithm 840

 swMATH ID: 4463 Software Authors: Boyd, John P. Description: Algorithm 840: Computation of grid points, quadrature weights and derivatives for spectral element methods using prolate spheroidal wave functions – prolate elements High order domain decomposition methods using a basis of Legendre polynomials, known variously as “spectral elements” or “$$p$$-type finite elements”, have become very popular. Recent studies suggest that accuracy and efficiency can be improved by replacing Legendre polynomials by prolate spheroidal wave functions of zeroth order. In this article, we explain the practicalities of computing all the numbers needed to switch bases: the grid points $$x_j$$, the quadrature weights $$w_j$$, and the values of the prolate functions and their derivatives at the grid points. The prolate functions themselves are computed by a Legendre-Galerkin discretization of the prolate differential equation; this yields a symmetric tridiagonal matrix. The prolate functions are then defined by Legendre series whose coefficients are the eigenfunctions of the matrix eigenproblem. The grid points and weights are found simultaneously through a Newton iteration. For large $$N$$ and $$c$$, the iteration diverges from a first guess of the Legendre-Lobatto points and weights. Fortunately, the variations of the $$x_j$$ and $$w_j$$ with $$c$$ are well-approximated by a symmetric parabola over the whole range of interest. This makes it possible to bypass the continuation procedures of earlier authors. Homepage: http://dl.acm.org/citation.cfm?id=1055538 Related Software: Matlab; DLMF; Differentiation Matrix Suite; mp toolbox; Regularization tools; algorithm 577; Chebfun Cited in: 27 Publications
all top 5

### Cited by 29 Authors

 6 Hogan, Jeffrey A. 6 Lakey, Joseph D. 6 Wang, Lilian 5 Zhang, Jing 4 Boyd, John Philip 3 Tian, Yan 2 Carin, Lawrence 2 Kovvali, Narayan 2 Lin, Wenbin 2 Zhang, Zhimin 1 Couchman, Luise S. 1 Feichtinger, Hans Georg 1 Ghaffari, Hamed Baghal 1 Hrycak, Tomasz 1 Huang, Chia-Chien 1 Kong, Wai Yip 1 Kroger, James 1 Li, Huiyuan 1 Rokhlin, Vladimir 1 Rong, Zhijian 1 Schmutzhard, Sebastian 1 Shen, Jie 1 Taylor, Mark Alan 1 Wang, Yingwei 1 Wingate, Beth A. 1 Xia, Jianlin 1 Xiang, Shuhuang 1 Zhao, Zhiqin 1 Žitňan, Peter
all top 5

### Cited in 17 Serials

 3 Applied and Computational Harmonic Analysis 3 The Journal of Fourier Analysis and Applications 2 Computer Physics Communications 2 Journal of Computational Physics 2 Journal of Engineering Mathematics 2 Mathematics of Computation 2 Numerical Algorithms 1 ACM Transactions on Mathematical Software 1 Applied Mathematics and Computation 1 Journal of Computational and Applied Mathematics 1 Applied Mathematics Letters 1 Journal of Scientific Computing 1 SIAM Journal on Scientific Computing 1 Advances in Applied Clifford Algebras 1 Engineering Analysis with Boundary Elements 1 Sampling Theory in Signal and Image Processing 1 Numerical Mathematics: Theory, Methods and Applications
all top 5

### Cited in 11 Fields

 19 Numerical analysis (65-XX) 9 Special functions (33-XX) 8 Harmonic analysis on Euclidean spaces (42-XX) 5 Partial differential equations (35-XX) 5 Information and communication theory, circuits (94-XX) 4 Ordinary differential equations (34-XX) 4 Approximations and expansions (41-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Fluid mechanics (76-XX) 1 Quantum theory (81-XX) 1 Statistical mechanics, structure of matter (82-XX)