Computing orthogonal rational functions with poles near the boundary
Computers & Mathematics with Applications
Eigendecomposition of images correlated on S1, S2, and SO(3) using spectral theory
IEEE Transactions on Image Processing
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Three methods are compared for computing n points xi and weights w(xi) for Gauss--Legendre quadrature. The eigensystem method computes the points as the eigenvalues of a symmetric tridiagonal matrix, and the weights are computed from its eigenvectors. The resulting error is O(n) for the points and $O(n^2)$ for the weights. Using an inverse Taylor series computation for the points, Yakimiw [J. Comput. Phys., 129 (1996), pp. 406--430] reduced the point error growth to O(1). He also reduced the error in the weights to O(n) and obtained uniform pointwise relative accuracy. However, the points xi cluster quadratically near the end points $x = \pm 1$, where for some n = nmax they coalesce or even exchange to within machine accuracy, and computation fails. Although this has not been a problem in the past, it could soon become so because weather models are now approaching resolutions with n 1000. Here we compare these methods to the Fourier--Newton method, which consists of Newton's method for computing the transformed points $\theta_i = \arccos x_i$ as the zeros of the Fourier representations of the Legendre polynomials $P_n(\theta)$. The $\theta_i$ are near equally spaced, particularly in the neighborhood of x = 0. For a fixed arithmetic precision, this extends the range of computable points and weights to $n=n_{max}^2$, where nmax is the limit for methods based on xi. Both the accuracy and speed of the three methods are tabulated.