Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 301971, 7 pages
doi:10.1155/2012/301971
Research Article
A Fast Algorithm of Moore-Penrose Inverse for
the Loewner-Type Matrix
QiuJuan Tong,1, 2 SanYang Liu,1 and Quan Lu3
1
School of Science, Xidian University, Xi’an 710071, China
School of Science, Xi’an University of Post and Telecommunications, Xi’an 710061, China
3
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China
2
Correspondence should be addressed to QiuJuan Tong, xiaotong@xupt.edu.cn
Received 25 July 2011; Accepted 8 October 2011
Academic Editor: Tadeusz Kaczorek
Copyright q 2012 QiuJuan Tong et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
In this paper, we present a fast algorithm of Moore-Penrose inverse for m × n Loewner-type matrix
with full column rank by forming a special block matrix and studying its inverse. Its computation
−1
complexity is Omn On2 , but it is Omn2 On3 by using L LT L LT .
1. Introduction
Loewner matrix was first studied by Loewner in 1934 in 1. He studied the relations of
various Loewner matrices via the characteristic of the monotone matrix function and the
problem of rational interpolation at that time. Since then, more further studies had been
carried out by many scientists in 2–5, such as Belevitch, Donoghue Jr., Fiedler, Chen and
Zhang. From their mentioned works, we can find various properties of Loewner matrix and
its application in the rational interpolation. In 1984, Vavřı́n presented a fast algorithm for
the inverse of Loewner matrix in 6, then the fast algorithm for the Loewner system was
got accordingly. Rost and Vavřı́n presented a fast algorithm for the system whose coefficient
is a Loewner-Vandermonde matrix from 1995 to 1996 in 7, 8. Lu gave a fast triangular
factorization algorithm for the symmetrical Loewner-type matrix in 2003 in 9. Xu et al.
and so forth gave a fast triangular factorization algorithm for the inverse of symmetrical
Loewner-type matrix in 2003 in 10. In 2009, Tong et al. gave a fast algorithm of the MoorePenrose inverse for m × n symmetrical Loewner-type matrix 11. In this paper, we generalize
the fast algorithm of symmetrical Loewner-type matrix to the Loewner-type matrix. The
theory and computation of generalized inverse matrix arise in various applications such as
optimization theory, control theory, computation mathematics, and mathematical statistics.
2
Mathematical Problems in Engineering
Therefore, there are very important theoretical and practical significance when we study the
fast algorithm of Moore-Penrose inverse for the Loewner-type matrix.
This paper is organized as follows: in Section 1, we present some preliminaries results.
The fast algorithm of Moore-Penrose inverse for Loewner-type matrix is driven in Section 2.
We give some numerical examples to illustrate the fast algorithm obtained in Section 3.
2. Preliminaries
A Loewner matrix L1 is a matrix of the form L1 lij m×n ξi − ηj /αi − βj m,n
, where
i,j1
βj . A Loewnerαi , βj , ξi , and ηj i 1, 2, . . . , m; j 1, 2, . . . , n are given numbers and αi /
type matrix L is a matrix which satisfies
− LD DL
l
pj qjT ,
2.1
j1
j
j T
j
diagα1 , α2 , . . . , αm , D diagβ1 , β2 , . . . , βn , pj p , p , . . . , pm , and qj where D
2
1
j
j T
j
q1 , q2 , . . . , qn . Loewner matrix L1 is a special case of Loewner-type matrix, and it satisfies
1 − L1 D ξ1 , . . . , ξm T 1, . . . , 1 − 1, . . . , 1T η1 , . . . , ηn .
DL
Let n be the rank of m × n Loewner-type matrix L. Forming an m n matrix
−Im L
,
M
LT 0
2.2
we obtain
−1
M
⎞ ⎛
⎛ T
−1
−1
LL − Im
L
L LT L LT − Im L LT L
⎠⎝
⎝ −1
−1
L
LT L
LT L LT
LT L
−1
⎞
⎠.
2.3
It is obvious that we may obtain the Moore-Penrose inverse L of Loewner-type matrix
L by 2.3 if we can get M−1 .
Now let us begin to find M−1 .
The following result will be useful for getting M−1 .
Lemma 2.1 see 12. Let all the leading principal submatrices Si i 1, 2, . . . , n of matrix S sij n×n whose order is n be invertible. Linear system Sx b be given, where b b1 , b2 , . . . , bn T .
Note that bi b1 , b2 , . . . , bi T , and let xi xi1 , xi2 , . . . , xii T , ui u1i , u2i , . . . , uii T be solution
i
vectors of linear systems Si xi bi , Si ui ei differently, then
xi where σi bi −
i−1
j1 sij xi−1,j .
xi−1
0
σi ui ,
2.4
Mathematical Problems in Engineering
3
3. Fast Algorithm of the Moore-Penrose Inverse for
Loewner-Type Matrix
By using 2.1, we know that M satisfies
D
D
M−M
D
D
pj l
0Tm qjT −
0n
j1
0m
pjT 0Tn
qj
.
3.1
Let all the leading principal submatrices Mi i 1, 2, . . . , m n of M be invertible. If
i > m, by virtue of 3.1, we have
D
Mi − Mi
Di−m
j
Let gi
systems
D
Di−m
j
⎞
⎛
⎡
⎤
0m pj jT
⎣
0Tm qi−m − ⎝ j ⎠ pjT 0Ti−m ⎦.
0
qi−m
i−m
j1
l
j T
j
j
j
gi1 , gi2 , . . . , gii , hi
j
Mi gi
pj
⎛
j
Mi hi
,
j T
j
hi1 , hi2 , . . . , hii be solution vectors of linear
0i−m
3.2
⎞
0m
⎝ j ⎠
qi−m
i m, m 1, . . . , m n
j
3.3
j
differently, then using Mm −Im , we obtain gm −pj , hm 0m . By virtue of Lemma 2.1 we
have
⎛
j
gi ⎝
j
where σi
j
gi−1
0
⎞
⎠ σ j ui ,
⎞
⎛
j
hi−1
⎠ τ j ui ,
⎝
i
0
j
hi
i
3.4
j
j
j
j
T
m
qi−m − m
k1 lk,i−m gi−1,k , τi
k1 lk,i−m hi−1,k , and ui u1i , u2i , . . . , uii is a
i
of Mi ui ei . Multiplying 3.2 by M−1
i on the left and on the right differently,
−
solution vector
we obtain
M−1
i
D
Di−m
−
i
D
Di−m
Multiplying 3.5 by ei and noting that
Di−m
βi−m ui −
D
M−1
i D
Di−m
ui l j1
j jT
gi hi
i
j jT
− hi gi
.
3.5
i
ei βi−m ei , we have
l j1
j j
j j
hii gi − gii hi
.
3.6
4
Mathematical Problems in Engineering
j
j
j
j
Substituting 3.4 in 3.6 and noting that gii σi uii , hii τi uii , we have
βi−m ui −
⎛ ⎛
⎛
⎞
⎞⎞
j
j
l
g
h
j
j
ui uii ⎝τi ⎝ i−1 ⎠ − σi ⎝ i−1 ⎠⎠,
0
0
j1
D
Di−m
3.7
and hence,
uki ⎧
⎪
u
⎪
⎪ ii
⎪
⎪
⎨
l
j1
⎪
⎪
uii
⎪
⎪
⎪
⎩
l
j1
j j
j j
τi gi−1,k − σi hi−1,k
βi−m − αk
j j
j j
τi gi−1,k − σi hi−1,k
k 1, 2, . . . , m,
3.8
k m 1, m 2, . . . , i − 1.
βi−m − βk−m
i
Now, let us look for uii . Choosing the ith of Mi ui ei , namely
m
lk,i−m uki ,
3.9
k1
and using 3.8, we have
uii m
k1 lk,i−m /
βi−m − αk
1
l
j1
j j
j j
τi gi−1,k − σi hi−1,k
.
3.10
Note that M−1 v1 , v2 , . . . , vmn , where vi v1i , v2i , . . . , vmn,i T is the ith column of
M−1 , then
vmn umn .
mn
Letting i m n in 3.5 and multiplying it by ek
βk−m vk −
D
D
vk 3.11
k > m on the right, we have
l j1
j
j
j
j
hmn,k gmn − gmn,k hmn .
3.12
So, we obtain the fast algorithm of Moore-Penrose inverse L of Loewner-type matrix
L by 2.3, 3.4, and 3.8∼3.12.
Algorithm
Step 1.
j
gm −pj ,
j
hm 0m
j 1, 2, . . . , l ,
3.13
Mathematical Problems in Engineering
5
For i m 1, m 2, . . . , m n,
j
σi
−
m
k1
j
j
lk,i−m gi−1,k ,
tik l j1
τi
j j
j
qi−m −
j j
τi gi−1,k − σi hi−1,k
λi ⎛
j
gi
⎝
j
gi−1
0
⎞
k 1, 2, . . . , i − 1,
lk,i−m
tik ,
k1 β
i−m − αk
uii
tik
βi−m − βk−m
⎠
k1
j
σi ui ,
j 1, 2, . . . , l ,
j
lk,i−m hi−1,k
m
uii
tik
uki βi−m − αk
uki m
uii 1
,
λi
3.14
k 1, 2, . . . , m,
k m 1, m 2, . . . , i − 1,
⎛
j
hi
⎝
j
hi−1
0
⎞
⎠ τ j ui
i
j 1, 2, . . . , l .
Step 2.
vi,mn ui,mn
i 1, 2, . . . , m.
3.15
For k m 1, . . . m n − 1,
vik 1
l βk−m − αi j1
j
j
j
j
gmn,i hmn,k − hmn,i gmn,k
i 1, . . . , m,
3.16
then
⎛
⎜
⎜
L ⎜
⎝
v1,m1 · · · vm,m1
..
.
..
.
⎞
⎟
⎟
⎟.
⎠
3.17
v1,mn · · · vm,mn
The algorithm requires 7l − 3mn ln2 Om multiplication and division operations
and 5l − 3mn l/2n2 Om addition and subtraction operations, and the computation
−1
complexity is Omn On2 , but it is Omn2 On3 by using L LT L LT .
4. Numerical Examples
We get the Moore-Penrose inverse matrix of Loewner-type matrix with Fortran program in
computer, and the results of the partial numerical examples are given as follows in Table 1
the error is measured by the 2-norm of vector, and the time is measured by second.
6
Mathematical Problems in Engineering
Table 1
Orders of matrix
m
n
10000
20
20000
20
30000
20
40000
20
60000
20
−1
L LT L LT
Err.
Tim.
8.109e−14
11
2.788e−14
31
2.106e−14
89
1.001e−14
201
3.760e−14
705
Fast algorithm
Err.
Tim.
7.254e−16
2
2.398e−15
2
2.019e−15
5
9.108e−14
8
4.901e−14
36
Example 1. Consider
i − 1π
,
αi m−n1
p1 i q
1
j 1,
βj ξi
,
2
j 1 π
,
mn−1
p2 i 1,
q
2
j −ηj ,
ξi −1i i − mn,
p3 i q
3
ξi
,
2
j 1,
ηj j j−m .
p4 i 1;
q
4.1
4
j −ηj .
i 1, 2, . . . , m; j 1, 2, . . . , n.
From above examples and many more examples not given, we find that the stability
−1
of the fast algorithm is very good, and it needs shorter time than that of L LT L LT .
Acknowledgment
This work is being supported by the National Natural Science Foundation Grants of China
no. 60974082.
References
1 K. Loewner, “Uber mototone matrixfunktionen,” Math, vol. 38, pp. 177–216, 1934.
2 V. Belevitch, “Interpolation matrices,” Philips Research Report, vol. 25, no. 5, pp. 337–369, 1970.
3 W. F. Donoghue Jr., Monotone Matrix Functions and Analytic Continuation, Springer, New York, NY,
USA, 1974.
4 M. Fiedler, “Hankel and Loewner matrices,” Linear Algebra and Its Applications, vol. 58, pp. 75–95,
1984.
5 G. N. Chen and H. P. Zhang, “More on Loewner matrices,” Linear Algebra and Its Applications, vol.
203-204, pp. 265–300, 1994.
6 Z. Vavřı́n, “Inverses of Loewner matrices,” Linear Algebra and Its Applications, vol. 63, pp. 227–236,
1984.
7 K. Rost and Z. Vavřı́n, “Recursive solution of Loewner-Vandermonde systems of equations. II,” Linear
Algebra and Its Applications, vol. 223-224, pp. 597–617, 1995.
8 K. Rost and Z. Vavřı́n, “Recursive solution of Loewner-Vandermonde systems of equations. I,” Linear
Algebra and Its Applications, vol. 233, pp. 51–65, 1996.
9 Q. Lu, “A fast algorithm for the triangular factorization of the symmetric Loewner type matrix,”
Journal of Engineering Mathematics, vol. 20, no. 2, pp. 139–142, 2003 Chinese.
10 M. Xu, Z. Xu, and Z.-K. Shi, “A fast triangular factorization algorithm for the inverse of symmetric
loewner-type matrix,” Journal of China Three Gorges University, vol. 6, pp. 552–554, 2003 Chinese.
Mathematical Problems in Engineering
7
11 Q. Tong, Q. Lu, S. Liu, and J. Chai, “A fast algorithm of Moore-Penrose inverse for the symmetric
Loewner-type matrix,” in Proceedings of the International Conference on Information Engineering and
Computer Science (ICIECS ’09), pp. 2155–2158, December 2009.
12 Z. Xu, K.-Y. Zhang, and Q. Lu, Fast Algorithm of Toeplitz-Like Matrix(in Chinese), Northwest
Polytechnical University Press, Xi’an, China, 1999.
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