A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree N polynomial in O(N(logN)2/ log logN) 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, which can then be evaluated by using the di...

In this note, we provide simple convergence analysis for the algebraic multilevel iteration methods [37, 51]. We consider two examples of AMLI methods with different polynomial acceleration. The first one is based on shifted and scaled Chebyshev polynomial and the other on the polynomial of best approximation to x−1 on a finite interval [λmin , λmax ], 0 < λmin < λmax in the ‖ ·‖∞ norm. The con...

In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multiorder fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is ve...

We introduce the Shifted Legendre Symbol Problem and some variants along with efficient quantum algorithms to solve them. The problems and their algorithms are different from previous work on quantum computation in that they do not appear to fit into the framework of the Hidden Subgroup Problem. The classical complexity of the problem is unknown, despite the various results on the irregularity ...

Exact and approximate analytical solutions of linear and nonlinear singular two-point boundary value problems (BVPs) are obtained for the first time by the Legendre operational matrix of differentiation. Different from other numerical techniques, shifted Legendre polynomials and their properties are employed for deriving a general procedure for forming this matrix. The accuracy of the technique...

In this paper, two collocation methods based on the shifted Legendre polynomials are proposed for solving system of nonlinear Fredholm-Volterra integro-differential equations. The equation considered in paper involves derivative unknown functions integral term, which makes its numerical solution more complicated. We first introduce a single-step method interval [0, 1]. Next, multi-step version ...

The Legendre polynomial expansion method (LPEM), which has been successfully applied to homogenous and longitudinally inhomogeneous gratings [J. Opt. Soc. Am. B24, 2676 (2007)], is now generalized for the efficient analysis of arbitrary-shaped surface relief gratings. The modulated region is cut into a few sufficiently thin arbitrary-shaped subgratings of equal spatial period, where electromagn...