Numerical Methods Based on Whittaker Cardinal, or Sinc Functions

Stenger, Frank

doi:10.1137/1023037

Cited by 290 publications

(175 citation statements)

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“…During the last three decades the a e-mail: hsafouhi@ualberta.ca SCM has been used extensively to solve many problems in numerical analysis. The applications include numerical integration, linear and non-linear ordinary differential equations, partial differential equations, interpolation and approximations to functions [7,8]. The SCM applied to Sturm-Liouville problems consists of expanding the solution of a SL problem using a basis of Sinc functions.…”

Section: Introductionmentioning

confidence: 99%

Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

Gaudreau

Safouhi

2016

EPJ Web of Conferences

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Abstract. Recently, we used the Sinc collocation method with the double exponential transformation to compute eigenvalues for singular Sturm-Liouville problems. In this work, we show that the computation complexity of the eigenvalues of such a differential eigenvalue problem can be considerably reduced when its operator commutes with the parity operator. In this case, the matrices resulting from the Sinc collocation method are centrosymmetric. Utilizing well known properties of centrosymmetric matrices, we transform the problem of solving one large eigensystem into solving two smaller eigensystems. We show that only 1 N+1 of all components need to be computed and stored in order to obtain all eigenvalues, where 2N + 1 corresponds to the dimension of the eigensystem. We applied our result to the Schrödinger equation with the anharmonic potential and the numerical results section clearly illustrates the substantial gain in efficiency and accuracy when using the proposed algorithm.

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Section: Introductionmentioning

confidence: 99%

Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

Gaudreau

Safouhi

2016

EPJ Web of Conferences

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“…The properties of (2.2), when a finite number of terms is used, has been extensively studied. A comprehensive survey of these approximation properties is found in [6]. For approximations over an arbitrary interval the following definition is needed.…”

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confidence: 99%

Symmetrization of the Sinc-Galerkin Method for Boundary Value Problems

Lund¹

1986

Mathematics of Computation

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Abstract. The Sinc-Galerkin method developed in [5], when applied to the second-order selfadjoint boundary value problem, gives rise to a nonsymmetric coefficient matrix. The technique in [5] is based on weighting the Galerkin inner products in such a way that the method will handle boundary value problems with regular singular points. In particular, the method does an accurate job of handling problems with singular solutions (the first or a higher derivative of the solution is unbounded at one or both of the boundary points). Using n function evaluations, the method of [5] converges at the rate exp(-icjñ), where k is independent of n. In this paper it is shown that, by changing the weight function used in the Galerkin inner products, the coefficient matrix can be made symmetric. This symmetric method is applicable to a slightly more restrictive set of boundary value problems than the method of [5], The present method, however, still handles a wide class of singular problems and also has the same exp(-Ky7i ) convergence rate.

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“…(Such techniques appear in more detail in [4] and elsewhere.) In Section 3 we present and verify our integral approximation formulas.…”

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confidence: 99%

“…Underlying Properties and Assumptions. A general review of sine functions and their uses has recently been given by Stenger in [4]. We therefore only outline properties important to our present goals, and refer the reader to [4] for further references.…”

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confidence: 99%