The Double Exponential Sinc Collocation Method for Singular Sturm-Liouville Problems

Gaudreau, Philippe; Slevinsky, Richard M.; Safouhi, Hassan

doi:10.48550/arxiv.1409.7471

Cited by 3 publications

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“…In [34], we apply the DESCM to obtain an approximation to the eigenvalues λ of equation ( 5). We initially applied Eggert et al's transformation to equation ( 5) since it was shown that the proposed change of variable results in a symmetric discretized system when using the Sinc collocation method [35].…”

Section: Definitions and Basic Propertiesmentioning

confidence: 99%

“…and µ are approximations of the eigenvalues λ of equation (7). For more details on the application of the SCM, we refer the readers to [9].…”

Section: Definitions and Basic Propertiesmentioning

confidence: 99%

“…By evaluating the resulting approximation at the Sinc collocation points separated by a fixed mesh size h, one obtains a matrix eigenvalue problem or generalized matrix eigenvalue problem for which the eigenvalues are approximations to the eigenvalues of the SL operator. In [9], we used the double exponential Sinc collocation method (DESCM) to compute the eigenvalues of singular Sturm-Liouville boundary value problems. The DESCM leads to a generalized eigenvalue problem where the matrices are symmetric and positive-definite.…”

Section: Introductionmentioning

confidence: 99%

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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: Definitions and Basic Propertiesmentioning

confidence: 99%

“…and µ are approximations of the eigenvalues λ of equation (7). For more details on the application of the SCM, we refer the readers to [9].…”

Section: Definitions and Basic Propertiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Gaudreau

Safouhi

2016

EPJ Web of Conferences

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“…The Sinc function and Sinc collocation method have been used extensively since their introduction to solve a variety of numerical problems [10][11][12]. The applications include numerical integration, linear and non-linear ordinary differential equations as well as partial differential equations [13][14][15][16][17][18][19][20][21][22]. The single exponential Sinc collocation method (SESCM) has been shown to offer an exponential convergence rate and works well in the presence of singularities.…”

Section: Introductionmentioning

confidence: 99%

“…The combination of SCM with the DE transformation was used to compute eigenvalues of the anharmonic oscillator V (x) = n i=1 a i x 2i [13] and to Sturm-Liouville boundary value problems [14]. This method which is referred to as DESCM is shown to be highly accurate, efficient and stable for computing the energy eigenvalues of the Schrödinger equation.…”

Section: Introductionmentioning

confidence: 99%

On the Computation of Eigenvalues of the Anharmonic Coulombic Potential

2017

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In this work, we propose a method combining the Sinc collocation method with the double exponential transformation for computing the eigenvalues of the anharmonic Coulombic potential. We introduce a scaling factor that improves the convergence speed and the stability of the method. Further, we apply this method to Coulombic potentials leading to a highly efficient and accurate computation of the eigenvalues.

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Double exponential sinc-collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function

Gaudreau

Safouhi

2017

Journal of Mathematical Physics

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In the present contribution, we apply the double exponential Sinc-collocation method (DESCM) to the one-dimensional time independent Schrödinger equation for a class of rational potentials of the form V (x) = p(x)/q(x). This algorithm is based on the discretization of the Hamiltonian of the Schrödinger equation using Sinc expansions. This discretization results in a generalized eigenvalue problem where the eigenvalues correspond to approximations of the energy values of the corresponding Hamiltonian. A systematic numerical study is conducted, beginning with test potentials with known eigenvalues and moving to rational potentials of increasing degree.

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The Double Exponential Sinc Collocation Method for Singular Sturm-Liouville Problems

Cited by 3 publications

References 0 publications

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

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

On the Computation of Eigenvalues of the Anharmonic Coulombic Potential

Double exponential sinc-collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function

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