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Hermitian tridiagonal solution with the least norm to quaternionic least squares problem. (English) Zbl 1205.81086
Summary: Quaternionic least squares (QLS) is an efficient method for solving approximate problems in quaternionic quantum theory. In view of the extensive applications of Hermitian tridiagonal matrices in physics, in this paper we list some properties of basis matrices and subvectors related to tridiagonal matrices, and give an iterative algorithm for finding Hermitian tridiagonal solution with the least norm to the quaternionic least squares problem by making the best use of structure of real representation matrices, we also propose a preconditioning strategy for the Algorithm LSQR-Q in {\it M. Wang, M. Wei} and {\it Y. Feng} [ibid. 179, No. 4, 203--207 (2008; Zbl 1197.81138)] and our algorithm. Numerical experiments are provided to verify the effectiveness of our method.

MSC:
81Q10Selfadjoint operator theory in quantum theory, including spectral analysis
15B33Matrices over special rings (quaternions, finite fields, etc.)
81-08Computational methods (quantum theory)
65D99Numerical approximation
93E24Least squares and related methods in stochastic control
Software:
LSQR
WorldCat.org
Full Text: DOI
References:
[1] Adler, S. L.: Quaternionic quantum mechanics and quantum fields, (1994) · Zbl 0885.00019
[2] Adler, S. L.: Quaternionic quantum field theory, Commun. math. Phys. 104, 611 (1986) · Zbl 0594.58059 · doi:10.1007/BF01211069
[3] Kaiser, H.; George, E. A.; Werner, S. A.: Neutron interferometrix search for quaternions in quantum mechanics, Phys. rev. A 29, 2276-2279 (1984)
[4] Klein, A. G.: Schrödinger inviolate: neutron optical searches for violations of quantum mechanics, Physica B 151, 44-49 (1988)
[5] Peres, A.: Proposed test for complex versus quaternion quantum theory, Phys. rev. Lett. 42, 683-686 (1979)
[6] Davies, A. J.: Quaternionic Dirac equation, Phys. rev. D 41, 2628-2630 (1990)
[7] Davies, A. J.; Mckellar, B. H.: Non-relativistic quaternionic quantum mechanics, Phys. rev. A 40, 4209-4214 (1989)
[8] Davies, A. J.; Mckellar, B. H.: Observability of quaternionic quantum mechanics, Phys. rev. A 46, 3671-3675 (1992)
[9] Jiang, T.: Algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory, J. math. Phys. 46, 052106 (2005) · Zbl 1110.81096 · doi:10.1063/1.1896386
[10] Jiang, T.; Chen, L.: Algebraic algorithms for least squares problem in quaternionic quantum theory, Comput. phys. Commun. 176, 481-485 (2007) · Zbl 1196.81026 · doi:10.1016/j.cpc.2006.12.005
[11] Jiang, T.; Chen, L.: An algebraic method for Schrödinger equations in quaternionic quantum mechanics, Comput. phys. Commun. 178, 795-799 (2008) · Zbl 1196.81027 · doi:10.1016/j.cpc.2008.01.038
[12] Jiang, T.; Zhao, J.; Wei, M.: A new technique of quaternion equality constrained least squares problem, J. comput. Appl. math. 216, 509-513 (2008) · Zbl 1146.65032 · doi:10.1016/j.cam.2007.06.005
[13] Jiang, T.; Wei, M.: Equality constrained least squares least problem over quaternion filed, Appl. math. Lett. 16, 883-888 (2003) · Zbl 1048.65036 · doi:10.1016/S0893-9659(03)90012-7
[14] Wang, M.; Wei, M.; Feng, Y.: An iterative algorithm for least squares problem in quaternionic quantum theory, Comput. phys. Commun. 179, 203-207 (2008) · Zbl 1197.81138 · doi:10.1016/j.cpc.2008.02.016
[15] Hu, G. Y.; O’connell, R. F.: Analytical inversion of symmetric tridiagonal matrices, J. phys. A math. Gen. 29, 1511-1513 (1996) · Zbl 0914.15002 · doi:10.1088/0305-4470/29/7/020
[16] Yamani, H. A.; Abdelmonem, M. S.: The analytic inversion of any finite symmetric tridiagonal matrix, J. phys. A math. Gen. 30, 2889-2893 (1997) · Zbl 0915.15002 · doi:10.1088/0305-4470/30/8/029
[17] Vagov, A. V.; Vorov, O. K.: Gaussian ensemble of tridiagonal symmetric random matrices, Phys. lett. A 232, 91-98 (1997) · Zbl 1053.82508 · doi:10.1016/S0375-9601(97)00342-3
[18] Godunov, S. K.; Malyshev, A. N.: On a special basis of approximate eigenvectors with local supports for an isolated narrow cluster of eigenvalues of a symmetric tridiagonal matrix, Comput. math. Math. phys. 48, 1089-1099 (2008)
[19] Alhaidari, A. D.: Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment, Ann. phys. 323, 1709-1728 (2008) · Zbl 1192.81408 · doi:10.1016/j.aop.2007.12.005
[20] Paige, C. C.; Saunders, M. A.: LSQR: an algorithm for sparse linear equations and sparse least squares, ACM trans. Math. software 8, No. 1, 43-71 (1982) · Zbl 0478.65016 · doi:10.1145/355984.355989
[21] Wiegmann, N. A.: Some theorems on matrices with real quaternion elements, Canad. J. Math. 7, 191-201 (1955) · Zbl 0064.01604 · doi:10.4153/CJM-1955-024-x
[22] Van Der Sluis, A.: Condition numbers and equilibration of matrices, Numer. math. 14, 14-23 (1969) · Zbl 0182.48906 · doi:10.1007/BF02165096