Alex Townsend

A Fast and Well-Conditioned Spectral Method

A spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients and general boundary conditions. The method leads to matrices that are al...

FAST AND ACCURATE COMPUTATION OF GAUSS-LEGENDRE AND GAUSS-JACOBI QUADRATURE NODES AND WEIGHTS ∗

An efficient algorithm for the accurate computation of Gauss--Legendre and Gauss--Jacobi quadrature nodes and weights is presented. The algorithm is based on Newtons root-finding method with initial guesses and function evaluations computed via asymptotic formulae. The

An extension of Chebfun to two dimensions

n

Continuous analogues of matrix factorizations

-point quadrature rule is computed in

Vector Spaces of Linearizations for Matrix Polynomials: A Bivariate Polynomial Approach

\mathcal{O}(n)

A Fast, Simple, and Stable Chebyshev--Legendre Transform Using an Asymptotic Formula

operations to an accuracy of essentially double precision for any

Computing with Functions in Spherical and Polar Geometries I. The Sphere

n\geq 100

The automatic solution of partial differential equations using a global spectral method

.

Computing the common zeros of two bivariate functions via Bézout resultants

An object-oriented MATLAB system is described that extends the capabilities of Chebfun to smooth functions of two variables defined on rectangles. Functions are approximated to essentially machine precision by using iterative Gaussian elimination with complete pivoting to form “chebfun2” objects representing low rank approximations. Operations such as integration, differentiation, function evaluation, and transforms are particularly efficient. Global optimization, the singular value decomposition, and rootfinding are also extended to chebfun2 objects. Numerical applications are presented.

AN ALGORITHM FOR THE CONVOLUTION OF LEGENDRE SERIES

Analogues of singular value decomposition (SVD), QR, LU and Cholesky factorizations are presented for problems in which the usual discrete matrix is replaced by a ‘quasimatrix’, continuous in one dimension, or a ‘cmatrix’, continuous in both dimensions. Two challenges arise: the generalization of the notions of triangular structure and row and column pivoting to continuous variables (required in all cases except the SVD, and far from obvious), and the convergence of the infinite series that define the cmatrix factorizations. Our generalizations of triangularity and pivoting are based on a new notion of a ‘triangular quasimatrix’. Concerning convergence of the series, we prove theorems asserting convergence provided the functions involved are sufficiently smooth.