Hermitian tridiagonal solution with the least norm to quaternionic least squares problem

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 Wang, Wei and Feng (2008) [14] and our algorithm. Numerical experiments are provided to verify the effectiveness of our method.

[1]  Anthony G. Klein,et al.  Schrödinger inviolate: Neutron optical searches for violations of quantum mechanics , 1988 .

[2]  R. F. O’Connell,et al.  Analytical inversion of symmetric tridiagonal matrices , 1996 .

[3]  A. V. Vagov,et al.  Gaussian ensemble of tridiagonal symmetric random matrices , 1997 .

[4]  S. Adler,et al.  Quaternionic quantum mechanics and quantum fields , 1995 .

[5]  Davies.  Quaternionic Dirac equation. , 1990, Physical review. D, Particles and fields.

[6]  A. Peres.  Proposed test for complex versus quaternion quantum theory , 1979 .

[7]  Tongsong Jiang,et al.  Algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory , 2005 .

[8]  M. S. Abdelmonem,et al.  CORRIGENDUM: The analytic inversion of any finite symmetric tridiagonal matrix , 1997 .

[9]  Davies,et al.  Nonrelativistic quaternionic quantum mechanics in one dimension. , 1989, Physical review. A, General physics.

[10]  Musheng Wei,et al.  Equality constrained least squares problem over quaternion field , 2003, Appl. Math. Lett..

[11]  Michael A. Saunders,et al.  LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares , 1982, TOMS.

[12]  N. Wiegmann,et al.  Some Theorems On Matrices With Real Quaternion Elements , 1955, Canadian Journal of Mathematics.

[13]  Davies,et al.  Observability of quaternionic quantum mechanics. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[14]  A. Sluis.  Condition numbers and equilibration of matrices , 1969 .

[15]  Sidney D. Drell,et al.  Relativistic Quantum Mechanics , 1965 .

[16]  Jianli Zhao,et al.  A new technique of quaternion equality constrained least squares problem , 2008 .

[17]  Yan Feng,et al.  An iterative algorithm for least squares problem in quaternionic quantum theory , 2008, Comput. Phys. Commun..

[18]  Alexander Malyshev,et al.  On a special basis of approximate eigenvectors with local supports for an isolated narrow cluster of eigenvalues of a symmetric tridiagonal matrix , 2008 .

[19]  Li Chen,et al.  Algebraic algorithms for least squares problem in quaternionic quantum theory , 2007, Comput. Phys. Commun..

[20]  Li Chen,et al.  An algebraic method for Schrödinger equations in quaternionic quantum mechanics , 2008, Comput. Phys. Commun..

[21]  Samuel A. Werner,et al.  Neutron interferometric search for quaternions in quantum mechanics , 1984 .

[22]  Stephen L. Adler,et al.  Quaternionic quantum field theory. , 1985, Physical review letters.

[23]  A. D. Alhaidari.  Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment , 2008 .