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ON THE PURE IMAGINARY QUATERNIONIC LEAST SQUARES SOLUTIONS OF MATRIX EQUATION†
ON THE PURE IMAGINARY QUATERNIONIC LEAST SQUARES SOLUTIONS OF MATRIX EQUATION†
Journal of Applied Mathematics & Informatics. 2016. Jan, 34(1_2): 95-106
Copyright © 2016, Korean Society of Computational and Applied Mathematics
  • Received : March 23, 2015
  • Accepted : June 29, 2015
  • Published : January 30, 2016
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MINGHUI WANG
JUNTAO ZHANG

Abstract
In this paper, according to the classical LSQR algorithm forsolving least squares (LS) problem, an iterative method is proposed for finding the minimum-norm pure imaginary solution of the quaternionic least squares (QLS) problem. By means of real representation of quaternion matrix, the QLS's correspongding vector algorithm is rewrited back to the matrix-form algorthm without Kronecker product and long vectors. Finally, numerical examples are reported that show the favorable numerical properties of the method. AMS Mathematics Subject Classification : 65H05, 65F10.
Keywords
1. Introduction
Let R , Q = R + Ri + Rj + Rk and IQ m×n denote the real number field, the quaternion field and the set of all m × n pure imaginary quaternion matrices, respectively, where i 2 = j 2 = k 2 = −1, ij = − ji = k . For any x = x 1 + x 2 i + x 3 j + x 4 k Q , the conjugate of quaternion x is
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= x 1 x 2 i x 3 j x 4 k .
Let F m×n denotes the set of m × n matrices on F . For any A F m×n , AT , Ā and AH present the transpose, conjugate and conjugate transpose 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 .
For any A = ( a 1 ,..., an ) ∈ F m×n , define vec( A ) =
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. The inverse mapping of vec(·) from Rmn to R m×n which is denoted by mat(·) satisfies mat(vec( A )) = A .
Quaternions and quaternion matrices have many applications in quaternionic quantum mechanics and field theory. Based on the study of [5] , we also discuss the quaternion matrix equation
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where A and B are given matrices of suitable size defined over the quaternion field. In this paper, we will derive an operable iterative method for finding the minimum-norm pure imaginary solution of the QLS problem, which is more appropriate to large scale system.
Many people have studied the matrix equation (1) and others constrained matrix equation, see [1 , 2 , 12 , 13 , 14 , etc.] . For the real, complex and quaternion matrix equations, there are many results, see [3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , etc.].
In [5] , the least squares pure imaginary solution with the least norm was given of the quaternion matrix equation (1) by using the complex representation of quaternion matrix and the Moore-Penrose. For A = A 1 + A 2 j Q s×m , B = B 1 + B 2 j Q s×n , let
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, Q 1 = Re ( Q ), Q 2 = Im ( Q ), E 1 = ( Re ( B 1 ) Re ( B 2 ) Im ( B 1 ) Im ( B 2 )) T and
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,
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And the set of solution JL is expressed as
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where Y is an arbitrary matrix of appropriate size. However, the method is not easy to realize in large scale system which motivated us to find an operable iterative method. Also Au-Yeung and Cheng in [6] studied the pure imaginary quaternionic solutions of the Hurwitz matrix equations.
Firstly, let us review the real least squares problem. In the LS problem, given A R m×n and B R n×p for finding a real matrix X so that
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where ║·║ F denotes the Frobenius norm. And the unique minimum-norm solution of the LS problem given by
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where A denotes the Moore-Penrose of A .
2. Preliminary
For any A = A 1 + A 2 i + A 3 j + A 4 k Q m×n and Al R m×n (l = 1, 2, 3, 4), define
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The real matrix AR is known as real representation of the quaternion matrix A . The set of all matrices shaped as (3) is denoted by
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. Obviously, the relation between Q m×n and
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is one-to-one correspondence.
Let
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Then Pt , Qt , Rt , St are unitary matrices, and by the definition of real representation, we can obtain the following results which given by T. Jang [4] and M. Wang [8] .
Proposition 2.1. Let A,B Q m×n , C Q n×s , α R . Then
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Remark 2.1. Form above property (a), we know that the mapping Q m×n
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is an isomorphism.
Theorem 2.2. For any V R 4m×n , ( V , QmV , RmV , SmV ) is a real representation matrix of some quaternion matrix .
Based on the definition of quaternion matrix norm in [8] , which denoted by ║·║ (F) can be proved a natural generality of Frobenius norm for complex matrices, it has the following properties:
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Then we review the LSQR algorithm proposed in [11] for solving the following LS problem:
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with given M R m×n and vector f Rm , whose normal equation is
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The algorithm is summarized as follows.
Algorithm LSQR
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We can choose
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as convergence criteria, where τ > 0 is a small tolerance. Obviously, there is no storage requirement for all the vector vi and ui .
And we can easily obtain the following theorem that if linear equation (5) has a solution x R ( MTM ) ∈ R ( MT ), then x generated by Algorithm LSQR is the minimum norm solution of (4). So we can have the solution generated by Algorithm LSQR is the minimum-norm solution of problem (4). Specifically, it was shown in [11] that this method is numerically more reliable even if M is ill-conditioned.
3. The matrix-form LSQR method for QLS problem
In this section, we give the definition of quaternionic least squares (QLS) problem on the basis of quaternion matrix norm which is shown in section 2, for
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with given matrices A Q m×n and B Q m×p . Then we can find problem (6) is equivalent to
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which is a constrained LS problem with given matrices
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Next, we will deduce the iterative method to find the pure imaginary quaternionic solution of the QLS problem (1). For any
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define
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Obviously, there is an one to one linear mapping from the long-vector space vec( R 4n×4p ) to the independent parameter space vec i ( R 3n×p ). Let Ƒ denote the pure imaginary quaternionic constrained matrix which defines linear mapping from vec i ( R 3n×p ) to vec( R 4n×4p ), that is
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Theorem 3.1. Suppose Ƒ is a pure imaginary quaternionic constrained matrix, then
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where
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Proof . First, we know
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and
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Hence, we have
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Therefore, let
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and from the above we have
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Then because of
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we can know that F is of full column rank and
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that is
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Because
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where M N denote the Kronecker product of matrices M and N , the QLS problem (6) is equivalent to
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with
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Now, we will apply Algorithm LSQR to problem (8) and the vector iteration of it will be transformed into matrix form so that the Kronecker product and Ƒ can be released. Then we transform the matrix-vector product of Mv and MTu back to a matrix-matrix form so as to let vector v and u be matrix V and U respectively, which required in Algorithm LSQR.
Let mat( α ) represent the matrix form of a vector α , For any v R 3np and u = vec( U ) ∈ R 16mp , where
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. Let
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Then we have
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where
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Therefore, we can get the following algorithm.
Algorithm LSQR-P.
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Algorithm LSQR-P can compute the minimum-norm solution x = vec i ( XR ) of (8), that is
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Again,
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so we have the following result.
Theorem 3.2. The solution generated by Algorithm LSQR-P is the minimum- norm solution of problem (6).
4. Numerical examples
In this section, we give three examples to illustrate the efficiency and investigate the performance of Algorithm LSQR-P which shown to be numerically reliable in various circumstances. All functions are defined by Matlab 7.0.
Example 4.1. Given [ m, n, p ] = N , A = A 1 + A 2 i + A 3 j + A 4 k , X = X 1 + X 2 i + X 3 j + X 4 k , B = AX , with A 1 , A 2 , A 3 , A 4 defined by rand ( m, n ) respectively. Given X 1 = zeros ( n, p ) and X 2 , X 3 , X 4 defined by rand ( n, p ) respectively. Then Fig. 4.1 plots the relation between error εk = log 10(║ AX B (F) ) and iteration number K .
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The relation between error εk and iterative number K with different N
Notice that in the above case, the equation AX = B is consistent and has a unique solution. From Fig. 4.1 we find our algorithm is effective.
Example 4.2. Given [ m, n, p ] = N , A = A 1 + A 2 i + A 3 j + A 4 k , B = B 1 + B 2i + B 3j + B 4 k , with A 1 , A 2 , A 3 , A 4 , B 1 , B 2 , B 3 , B 4 defined by rand ( m, n ) respectively. Let ηk = log 10(║ MT ( Mx f )║ 2 ) where M, f defined by (9). Then Fig. 4.2 plots the relation between error ηk and iteration number K .
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The relation between error ηk and iterative number K with different N
Notice that in the above case, the equation AX = B is not consistent and we use ηk = ║ MT ( f Mxk )║ 2 =
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< τ = 10 −12 as convergence criteria. From Fig. 4.2 , we also find our algorithm work well.
Example 4.3. Given m = n = p = 10, A = A 1 + A 2 i + A 3 j + A 4 k , X = X 1 + X 2 i + X 3 j + X 4 k , B = AX , with A 1 = hilb ( m ), A 2 = pascal ( m ), A 3 = ones ( m, n ), A 4 = pascal ( m ). Given X 1 = zeros ( n, p ) and X 2 , X 3 , X 4 defined by rand ( n, p ) respectively. In this case, the condition number of M is 3.9927 × 10 9 , therefore this system is ill-conditioned. Then Fig. 4.3 plots the relation between error εk = log 10(║ AX B (F) ), ηk = log 10(║ X Xk F /║ X F ) and iteration number K .
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The relation between error ηk, εk and iterative number K
Notice that the equation (1) is consistent and has a unique solution. The algorithm performance is not very well when the system very ill-conditioned. From Fig. 4.3 we find our algorithm is also effective.
BIO
Minghui Wang Ph.D
Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, P.R. China.
e-mail: mhwang@yeah.net
Juntao Zhang M.D.
Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, P.R. China.
e-mail: qustzhangjuntao@163.com
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