ON THE PURE IMAGINARY QUATERNIONIC LEAST SQUARES SOLUTIONS OF MATRIX EQUATION†

Journal of Applied Mathematics & Informatics.
2016.
Jan,
34(1_2):
95-106

- Received : March 23, 2015
- Accepted : June 29, 2015
- Published : January 30, 2016

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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.
R
,
Q
=
R
+
R _{i}
+
R _{j}
+
R _{k}
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
=
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}
,
A^{T}
,
Ā
and
A^{H}
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}
,...,
a_{n}
) ∈
F
^{m×n}
, define vec(
A
) =
. The inverse mapping of vec(·) from
R^{mn}
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
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
,
Q
_{1}
=
Re
(
Q
),
Q
_{2}
=
Im
(
Q
),
E
_{1}
= (
Re
(
B
_{1}
)
Re
(
B
_{2}
)
Im
(
B
_{1}
)
Im
(
B
_{2}
))
^{T}
and
,
And the set of solution
J_{L}
is expressed as
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
where ║·║
_{F}
denotes the Frobenius norm. And the unique minimum-norm solution of the LS problem given by
where
A
^{†}
denotes the Moore-Penrose of
A
.
A
=
A
_{1}
+
A
_{2}
i
+
A
_{3}
j
+
A
_{4}
k
∈
Q
^{m×n}
and
A_{l}
∈
R
^{m×n}
(l = 1, 2, 3, 4), define
The real matrix
A^{R}
is known as real representation of the quaternion matrix
A
. The set of all matrices shaped as (3) is denoted by
. Obviously, the relation between
Q
^{m×n}
and
is one-to-one correspondence.
Let
Then
P_{t}
,
Q_{t}
,
R_{t}
,
S_{t}
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
Remark 2.1.
Form above property (a), we know that the mapping
Q
^{m×n}
→
is an isomorphism.
Theorem 2.2.
For any V
∈
R
^{4m×n}
, (
V
,
Q_{m}V
,
R_{m}V
,
S_{m}V
)
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:
Then we review the LSQR algorithm proposed in
[11]
for solving the following LS problem:
with given
M
∈
R
^{m×n}
and vector
f
∈
R^{m}
, whose normal equation is
The algorithm is summarized as follows.
Algorithm LSQR
We can choose
as convergence criteria, where
τ
> 0 is a small tolerance. Obviously, there is no storage requirement for all the vector
v_{i}
and
u_{i}
.
And we can easily obtain the following theorem that if linear equation (5) has a solution
x
^{∗}
∈
R
(
M^{T}M
) ∈
R
(
M^{T}
), 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.
with given matrices
A
∈
Q
^{m×n}
and
B
∈
Q
^{m×p}
. Then we can find problem (6) is equivalent to
which is a constrained LS problem with given matrices
Next, we will deduce the iterative method to find the pure imaginary quaternionic solution of the QLS problem (1). For any
define
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
Theorem 3.1.
Suppose Ƒ is a pure imaginary quaternionic constrained matrix, then
where
Proof
. First, we know
and
Hence, we have
Therefore, let
and from the above we have
Then because of
we can know that F is of full column rank and
that is
Because
where
M
⊗
N
denote the Kronecker product of matrices
M
and
N
, the QLS problem (6) is equivalent to
with
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
M_{v}
and
M^{T}_{u}
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
. Let
Then we have
where
Therefore, we can get the following algorithm.
Algorithm LSQR-P.
Algorithm LSQR-P can compute the minimum-norm solution
x
= vec
_{i}
(
X^{R}
) of (8), that is
Again,
so we have the following result.
Theorem 3.2.
The solution generated by Algorithm LSQR-P is the minimum- norm solution of problem (6).
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
ε_{k}
=
log
10(║
AX
−
B
║
_{(F)}
) and iteration number
K
.
The relation between error ε_{k} 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
η_{k}
=
log
10(║
M^{T}
(
M_{x}
−
f
)║
_{2}
) where
M, f
defined by (9). Then
Fig. 4.2
plots the relation between error
η_{k}
and iteration number
K
.
The relation between error η_{k} and iterative number K with different N
Notice that in the above case, the equation
AX
=
B
is not consistent and we use
η_{k}
= ║
M^{T}
(
f
−
M_{xk}
)║
_{2}
=
<
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
ε_{k}
=
log
10(║
AX
−
B
║
_{(F)}
),
η_{k}
=
log
10(║
X
−
X_{k}
║
_{F}
/║
X
║
_{F}
) and iteration number
K
.
The relation between error η_{k} , ε_{k} 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.
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

1. Introduction

Let
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2. Preliminary

For any
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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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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.
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BIO

Wu L.
,
Cain B.
(1996)
The re-nonnegative definite solutions to the matrix inverse problem AX = B
Linear Algebra Appl.
236
137 -
146
** DOI : 10.1016/0024-3795(94)00142-1**

Meng C.J.
,
Hu X.Y.
,
Zhang L.
(2005)
The skew symmetric orthogonal solution of the matrix equation AX = B
Linear Algebra Appl.
402
303 -
318
** DOI : 10.1016/j.laa.2005.01.022**

Jia Zhigang
,
Wei Musheng
,
Ling Sitao
(2013)
A new structure-preserving method for quater-nion Hermitian eigenvalue problems
Journal of Comput. and Appl. Math.
239
12 -
24
** DOI : 10.1016/j.cam.2012.09.018**

Jang T.
,
Chen L.
(2007)
Algebraic algorithm for least squares problem in quaternionic quantum theory
Comput. Phys. Comm.
176
481 -
485
** DOI : 10.1016/j.cpc.2006.12.005**

Yuan Shifang
,
Wang Qingwen
,
Duan Xuefeng
(2013)
On solutions of the quaternion matrix equation AX = B and their application in color image restoration
Appl. Math. Comput.
221
10 -
20
** DOI : 10.1016/j.amc.2013.05.069**

Au-Yeung Y.H.
,
Cheng C.M.
(2006)
On the pure imaginary quaternionic solutions of the Hurwitz matrix equations
Linear Algebra Appl.
419
630 -
642
** DOI : 10.1016/j.laa.2006.06.005**

Wang Q.W.
,
Li C.K.
(2009)
Ranks and the least-norm of the general solution to a system of quaternion matrix equation
Linear Algebra Appl.
430
1626 -
1640
** DOI : 10.1016/j.laa.2008.05.031**

Wang M.H.
,
Wei M.S.
,
Feng Y.
(2008)
An iterative algorithm for least squares problem in quaternionic quantum theory
Comput. Phys. Comm.
179
203 -
207
** DOI : 10.1016/j.cpc.2008.02.016**

Jang T.
(2005)
Algebraic mehtods for diagonalization of a quaternion matrix in quaternionic quantum theory
J. Math. Phys.
46
052106 -
** DOI : 10.1063/1.1896386**

Wang M.
(2010)
On positive definite solutions of quaternionic matrix equations
World Academy of Science Engineering and Technology
37
535 -
537

Paige C.C.
,
Saunders A.
(1982)
LSQR: An algorithm for sparse linear equations and sparse least squares
Appl. Math. Comput.
8
43 -
71

Toutounian F.
,
Karimi S.
(2006)
Global least squares method (Gl-LSQR) for solving general linear systems with several right-hand sides
Appl. Math. Comput.
178
452 -
460
** DOI : 10.1016/j.amc.2005.11.065**

Ling Sitao
(2010)
Hermitian tridiagonal solution with the least norm to quaternionic least squares problem
Comput. Phys. Comm.
181
481 -
488
** DOI : 10.1016/j.cpc.2009.10.019**

Peng Z.Y.
(2010)
A matrix LSQR iterative method to solve matrix equation AXB = C
International Journal of Computer Mathematics
87
1820 -
1830
** DOI : 10.1080/00207160802516875**

Citing 'ON THE PURE IMAGINARY QUATERNIONIC LEAST SQUARES SOLUTIONS OF MATRIX EQUATION†
'

@article{ E1MCA9_2016_v34n1_2_95}
,title={ON THE PURE IMAGINARY QUATERNIONIC LEAST SQUARES SOLUTIONS OF MATRIX EQUATION†}
,volume={1_2}
, url={http://dx.doi.org/10.14317/jami.2016.095}, DOI={10.14317/jami.2016.095}
, number= {1_2}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={WANG, MINGHUI
and
ZHANG, JUNTAO}
, year={2016}
, month={Jan}