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Article information
2019 , Volume 24, ¹ 3, p.33-43
Belov A.N., Turovtsev V.V., Orlov Y.D.
Errors in the numerical solution of the torsion Schrodinger equation with Mathieu functions basis set
The dependence for the Hamiltonian matrix elements of the Schrodinger torsion equation on the calculation errors of the Mathieu basis set is considered. The Mathieu functions are represented with continued fractions in this study. The analysis of the Mathieu function approximation algorithm using Fourier series expansion is carried out when the coefficients of the Fourier series are represented by convergent continued fractions. It is shown that the major contribution to the errors at the Fourier coefficient calculation is made by the error accumulating in the corresponding elements of the continued fraction. Recurrence relations for the absolute and relative errors of the kept elements of the continued fraction and the Fourier expansion coefficients are obtained. It is shown and illustrated by a numerical example that the absolute and relative errors of the Fourier expansion coefficients in the proposed algorithm are negligible. It is noted that the maximum relative errors of continued fraction are in the highest elements of the kept part. The results of our work are used to estimate the calculation error in the integrals containing Mathieu functions. These integrals constitute the Hamiltonian matrix elements of the Schr¨odinger torsion equation. We developed an algorithm to estimate of the calculation accuracy of the Hamiltonian matrix elements of the Schro¨dinger torsion equation in the basis set of Mathieu functions. We provide the example of this algorithm. The results of the work indicate the adequacy and effectiveness at the application of the Mathieu function basis set to the solution of the Schr?dinger torsion equation.
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Keywords: continued fraction, Mathieu function, Schrodinger equation, internal rotation, computational error
doi: 10.25743/ICT.2019.24.3.003
Author(s):
Belov Alexander Nikolaevich
Position: Senior Fellow
Office: Tver State University
Address: 170100, Russia, Tver, Sadovyi per, 35
E-mail: Belov.AN@tversu.ru
Turovtsev Vladimir Vladimirovich
Associate Professor
Position: Associate Professor
Office: Medical Informatics department
Address: 170100, Russia, Tver, Sovietskay str, 4
E-mail: turovtsev.vv@tversu.ru
Orlov Yuriy Dimitrievich
Professor
Position: Professor
Office: Tver State University
Address: 170100, Russia, Tver, Sadovyi per, 35
E-mail: orlov.yd@tversu.ru
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Bibliography link:
Belov A.N., Turovtsev V.V., Orlov Y.D. Errors in the numerical solution of the torsion Schrodinger equation with Mathieu functions basis set // Computational technologies. 2019. V. 24. ¹ 3. P. 33-43
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