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Gauss and Clenshaw–Curtis quadrature, like Legendre and Chebyshev spectral methods, make use of grids strongly clustered at boundaries. From the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal. Nevertheless such methods may " waste " a factor of π/2 with respect to each space dimension. We propose new… (More)

An efficient algorithm for the accurate computation of Gauss–Legendre and Gauss– Jacobi quadrature nodes and weights is presented. The algorithm is based on Newton's root-finding method with initial guesses and function evaluations computed via asymptotic formulae. The n-point quadrature rule is computed in O(n) operations to an accuracy of essentially… (More)

- J Shi, J B Potash, J A Knowles, M M Weissman, W Coryell, W A Scheftner +20 others
- Molecular psychiatry
- 2011

A genome-wide association study was carried out in 1020 case subjects with recurrent early-onset major depressive disorder (MDD) (onset before age 31) and 1636 control subjects screened to exclude lifetime MDD. Subjects were genotyped with the Affymetrix 6.0 platform. After extensive quality control procedures, 671 424 autosomal single nucleotide… (More)

A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree N polynomial in O(N (log N) 2 / log log N) operations is derived. The basis of the algorithm is to rewrite a well-known asymptotic formula for Legendre polynomials of large degree as a weighted linear combination of Chebyshev polynomials,… (More)

By exploiting conformal maps to vertically slit regions in the complex plane, a recently developed rational spectral method [27] is able to solve PDEs with interior layer-like behaviour using significantly fewer collocation points than traditional spectral methods. The conformal maps are chosen to 'enlarge the region of analyticity' in the solution: an idea… (More)

An O(N 2) algorithm for the convolution of compactly supported Legendre series is described. The algorithm is derived from the convolution theorem for Legendre polynomials and the recurrence relation satisfied by spherical Bessel functions. Combining with previous work yields an O(N 2) algorithm for the convolution of Chebyshev series. Numerical results are… (More)

The function of prostaglandin-F2 alpha (PGF2 alpha) in reproductive behavior of adult female paradise fish (Macropodus opercularis) was evaluated. In experiment 1, females were allowed to spawn normally with a male and were left with that male until the following day when testing occurred. Subjects were then randomly assigned to one of three groups: PGF2… (More)

- N Hale, A Townsend, Nicholas Hale, Alex Townsend
- 2012

An efficient algorithm for the accurate computation of Gauss–Legendre and Gauss– Jacobi quadrature nodes and weights is presented. The algorithm is based on Newton's root-finding method with initial guesses and function evaluations computed via asymptotic formulae. The n-point quadrature rule is computed in O(n) operations to an accuracy of essentially… (More)