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

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Abstract

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.

Introduction

Quaternions and quaternion matrices play essential roles in quaternionic quantum mechanics and field theory [1], [2], [3], [4], [5], [6], [7], [8]. In the study of theory and numerical computations of quaternionic quantum theory, in order to well understand the perturbation theory [1], experimental proposals [3], [4], [5] and theoretical discussions [6], [7], [8] underlying the quaternionic formulations of energy eigenvalue equationH|ϕn=En|ϕn and so on, one often meets problems that finding approximate solutions of quaternion linear systems, such as the approximate solution of quaternion linear matrix equation AXBE that is approximate when error occurs in the observation matrix E, i.e. quaternionic least squares (QLS) problem, which has received lots of attention in mathematics and physics recently [9], [10], [11], [12], [13], [14].

As is well known, the main difficulty dealing with the QLS problem lies in the non-commutation of the multiplicity of quaternions, thus the general methods for least squares (LS) problem in real number field or complex number field can't be directly used. By means of complex representation, T. Jiang studied algebraic algorithms for QLS problem [10], QLS eigenproblem [11] and QLS problem with constraints [12], [13] in quaternionic quantum theory, and obtained some theoretical results. In [14], the QLS problem in quaternionic quantum theory was reconsidered by real representation, taking advantage of structures of the real representation matrices, the authors derived an operable iterative method called LSQR-Q algorithm for finding the minimum-norm solution of the QLS problem, which is more appropriate to large scale system. To our knowledge, Hermitian tridiagonal solution with the least norm to the QLS problem has not been investigated so far in the literature.

In real number filed or complex number field, many problems in physical applications, such as heat exchange, diffusing process and structure analyses, etc., give rise to a linear system whose coefficient matrix is symmetric tridiagonal or Hermitian tridiagonal. As fundamental tools, symmetric (Hermitian) tridiagonal matrices were widely studied and applied in distinct points of view [15], [16], [17], [18], [19]. In [15], analytical formula for the inversion of symmetrical tridiagonal matrices was presented which could be used to derive an exact analytical solution for the one-dimensional discrete Poisson equation with Dirichlet boundary conditions, paper [17] obtained exact energy level correlators for the Gaussian ensemble of finite tridiagonal symmetric matrices. By relaxing the diagonal constraint on the matrix representation of eigenvalue wave equation to tridiagonal form, a closed form solution of the three-dimensional time-independent Schrödinger equation for a charged particle in the field of a point electric dipole that could carry a nonzero net charge was obtained [19].

Motivated by the work of [14], and keeping the applications and interests of tridiagonal matrix in physical applications mentioned above, in this paper we consider the Hermitian tridiagonal solution with the least norm to the QLS problem in quaternionic quantum theory.

Let R be the real number field, C the complex number field, Q=RRiRjRk the quaternion field, where ij=ji=k, i2=j2=k2=ijk=1. Fm×n denotes the set of m×n matrices over field F, F3n×n denotes the set of tridiagonal matrices of order n. For any quaternion x=x0+x1i+x2j+x3k where xiR, i=0,1,2,3, the conjugate of quaternion x is x¯=x0x1ix2jx3k and |x|=xx¯=x02+x12+x22+x32. For any matrix AFm×n, AT, A, A denote the transpose, conjugate transpose and Moore–Penrose inverse of A, respectively. A(i:j,k:l) represents the submatrix of A containing the intersection of rows i to j and columns k to l. As a special case, A(:,i:j) is the ith to jth columns of A and A(i:j,:) the ith to jth rows of A. AB=(aijB) denotes the Kronecker product of two matrices A and B, ei the ith column of identity matrix with appropriate size.

Given a tridiagonal matrix AF3n×n with the formA=(a1c1b1a2c2bn2an1cn1bn1an), if AR3n×n and AT=A, then A is a symmetric tridiagonal matrix, if AR3n×n and AT=A, then A is a skew-symmetric tridiagonal matrix, if AQ3n×n and A=A, then A is a Hermitian tridiagonal matrix, we denote the set of symmetric tridiagonal matrix, skew-symmetric tridiagonal matrix and Hermitian tridiagonal quaternion matrix by SR3n×n, ASR3n×n, HQ3n×n, respectively. It is not difficult to verify that if A=A1+A2i+A3j+A4kHQ3n×n, then A1SR3n×n,A2,A3,A4ASR3n×n.

For any quaternion matrix A=A1+A2i+A3j+A4kQm×n, the Frobenius norm of quaternion matrix is defined in [14] to beA(F)=A1F2+A2F2+A3F2+A4F2=12ARF, where AR is the real representation matrix of A with the formAR=(A1A2A3A4A2A1A4A3A3A4A1A2A4A3A2A1)R4m×4n, the set of all matrices shaped like (1.4) is denoted by QRm×n and F is the Frobenius norm.

We consider the following quaternionic least squares (QLS) problem. For given data matrices AQm×n, BQn×s, an observation matrix EQm×s, writeHL={X|XHQ3n×n,AXBE(F)=minYHQ3n×nAYBE(F)},

we are asked to find a matrix XˆHL such thatXˆ(F)=minXHLX(F).

Section snippets

Basis matrices and subvectors

In view of the special structure of Hermitian tridiagonal quaternion matrices, we construct basis matrices and subvectors, study the relationship between Hermitian tridiagonal matrices and basis matrices and subvectors in this section. For more discusses we refer to Appendix A at the end of the paper.

For any matrix X=(x1,x2,,xn)Fn×n, definevec(X)=(x1x2xn)Fn2,vecj(X)=(X(2,1)X(3,2)X(4,3)X(n,n1))Fn1,veci(X)=(X(1,1)X(2,1)X(2,2)X(3,2)X(n1,n1)X(n,n1)X(n,n))F2n1. veci(X) and vecj(X) are

Algorithm for the QLS problem (1.5)–(1.6)

In this section, we study the QLS problem mentioned in the first section by the LSQR-Q algorithm. In view of the special structure of Hermitian tridiagonal quaternion matrix, our main task lies in constructing constrained matrices in the algorithm.

LetPt=(It0000It0000It0000It),Qt=(0It00It000000It00It0),Rt=(00It0000ItIt0000It00),St=(000It00It00It00It000).

Lemma 3.1

(See [14].) For any VR4m×n, (V,QmV,RmV,SmV) is a real representation matrix of some quaternion matrix.

For any matrix XQRn×p,X=(X1X2X3

Preconditioning

Wiegmann [21] gave the following singular value decomposition (SVD) of a quaternion matrix.

Theorem 4.1

(See [21].) Let AQm×n with rank(A)=r. Then there exist unitary quaternion matrices UQm×m and VQn×n such thatUAV=(Σr000), where Σr=diag(σ1,σ2,,σr), σ1σ2σr>0, and σ1,σ2,,σr are the nonzero singular values of quaternion matrix A.

Taking the real representation operation in both sides of (4.1), we get a SVD of real representation matrix ARURTARVR=(Σr000)R. It follows that the singular values of AR

Numerical experiments

In this section, we first verify the effectiveness of our Algorithm 3.1, then we illustrate that our preconditioning strategy is efficient by two examples. The experiments are performed by MATLAB 7.3.0 on PC machine (Intel Celeron 2.66 GHz, Memory 512 MB), all functions are defined by MATLAB 7.3.0.

Example 5.1

Given m=n=s=N, A=A1+A2i+A3j+A4k, B=B1+B2i+B3j+B4k, E=E1+E2i+E3j+E4k. Take A1=A2=hilb(m), A3=A4=rand(m,n), B1=B2=ones(n), B3=B4=eye(n), Et=rand(m,s), t=1,2,3,4. Letηk=log10(MTrk2), where rk=fMxk, M,

Acknowledgements

The authors are indebted to the anonymous referee and the editor for their helpful comments and suggestions, which improved the presentation of this paper.

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    This paper is supported by the National Natural Science Foundations of China (Nos. 10771073, 10671086) and Shanghai Leading Academic Discipline Project (No. S30405).

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