Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2011, Article ID 609863, 10 pages
doi:10.1155/2011/609863
Research Article
A Note on the Inversion of Sylvester Matrices in
Control Systems
Hongkui Li and Ranran Li
College of Science, Shandong University of Technology, Shandong 255049, China
Correspondence should be addressed to Hongkui Li, lhk8068@163.com
Received 21 November 2010; Revised 28 February 2011; Accepted 31 March 2011
Academic Editor: Gradimir V. Milovanović
Copyright q 2011 H. Li and R. Li. 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.
We give a sufficient condition the solvability of two standard equations of Sylvester matrix by
using the displacement structure of the Sylvester matrix, and, according to the sufficient condition,
we derive a new fast algorithm for the inversion of a Sylvester matrix, which can be denoted as
a sum of products of two triangular Toeplitz matrices. The stability of the inversion formula for
a Sylvester matrix is also considered. The Sylvester matrix is numerically forward stable if it is
nonsingular and well conditioned.
1. Introduction
Let Rx be the space of polynomials over the real numbers. Given univariate polynomials
0, where
fx, gx ∈ Rx, a1 /
fx a1 xn a2 xn−1 · · · an1 ,
a1 /
0,
gx b1 xm b2 xm−1 · · · bm1 ,
Let S denote the Sylvester matrix of fx and gx :
⎫
⎛
⎞⎪
⎪
⎪
a1 a2 · · · · · · an1
⎪
⎪
⎜
⎟⎬
a1 a2 · · · · · · an1
⎜
⎟
⎜
⎟ ⎪ m row
⎜
⎟⎪
..
.. ..
⎜
⎟⎪
⎪
.
.
.
⎜
⎟⎪
⎭
⎜
⎟
a
a
·
·
·
·
·
·
a
1
2
n1 ⎟
⎜
S⎜
⎟ ⎫
⎜ b1 b2 · · · bm bm1
⎟
⎜
⎟ ⎪
⎪
⎜
⎟ ⎪
b
b
·
·
·
b
b
1
2
m
m1
⎪
⎜
⎟ ⎪
⎜
⎟ ⎬
.
.
.
.
⎜
⎟
.
.
.
.
.
.
.
.
⎝
⎠ ⎪ n row
⎪
⎪
⎪
b1 b 2
· · · bm bm1
⎪
⎭
b1 /
0.
1.1
1.2
2
Mathematical Problems in Engineering
Sylvester matrix is applied in many science and technology fields. The solutions of
Sylvester matrix equations and matrix inequations play an important role in the analysis and
design of control systems. In determining the greatest common divisor of two polynomials,
the Sylvester matrix plays a vital role, and the magnitude of the inverse of the Sylvester matrix
is important in determining the distance to the closest polynomials which have a common
root. Assuming that all principal submatrices of the matrix are nonsingular, in 1, Jing Yang
et al. have given a fast algorithm for the inverse of Sylvester matrix by using the displacement
structure of m n-order Sylvester matrix.
By using the displacement structure of the Sylvester matrix, in this paper, we give
a sufficient condition the solvability of two standard equations of Toeplitz matrix, and,
according to the sufficient condition, we derive a new fast algorithm for the inversion
of a Sylvester matrix, which can be denoted as a sum of products of two triangular
Toeplitz matrices. At last, the stability of the inversion formula for a Sylvester matrix is also
considered. The Sylvester matrix is numerically forward stable if it is nonsingular and well
conditioned.
In this paper, · 2 always denotes the Euclidean or spectral norm and · F the
Frobenius norm.
2. Preliminary Notes
In this section, we present a lemma that is important to our main results.
Lemma 2.1 see 2, Section 2.4.8. Let A, B ∈
arithmetic with machine precision ε, one has that
flαA αA E,
flA B A B E,
flAB AB E,
n,n
and α ∈
. Then for any floating-point
√
EF ≤ ε|α|AF ≤ ε n|α|A2 ,
√
EF ≤ εA BF ≤ ε nA B2 ,
2.1
EF ≤ εnAF BF .
ε2 , and Oε
ε.
As usual, one neglects the errors of Oε2 , O
3. Sylvester Inversion Formula
In this section, we present our main results.
Theorem 3.1. Let matrix S be a Sylvester matrix; then it satisfies the formula
KS − SK em f T − emn g T ,
3.1
where K 0 e1 e2 · · · emn−1 is a m n × m n shift matrix,
T
f b1 · · · bm bm1 − a1 · · · −an ,
T
g 0 · · · 0 b1 · · · bm .
3.2
Mathematical Problems in Engineering
3
Proof. We have that
⎛
0 a1
···
an1
0
··· 0
⎛
⎞
0 a1 · · · an1
0 ··· 0
⎜
⎟ ⎜
..
⎜
⎜
a1
· · · an1 · · · 0⎟
⎜
⎟ ⎜
.
⎜
⎟ ⎜
..
⎜
⎟ ⎜
⎜
⎟ ⎜
.
⎜
⎟ ⎜
⎜
⎟ ⎜
⎟ − ⎜0 · · · a1
KS − SK ⎜
·
·
·
b
0
·
·
·
0
···
b
m1
⎜ 1
⎟ ⎜
⎜
⎟ ⎜
⎜
⎜
b1 · · · bm1 0 · · · 0⎟
⎜
⎟ ⎜ b1 · · · bm1
⎜
⎟ ⎜
⎜
⎟ ⎜
..
..
⎜
⎟ ⎜
.
.
⎝
⎠ ⎝
0
⎛
···
0 ··· 0
0
0
···
0
0 · · · −b1
···
−bm
0
···
0
0 · · · −b1
···
−bm
b1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
an ⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
· · · bm
⎞
⎜
⎟
⎜0
···
0 ⎟
⎜
⎟
⎜
⎟
..
⎜
⎟
⎜
⎟
.
⎜
⎟
⎜
⎟
⎜
⎜b1 · · · bm bm1 − a1 −a2 · · · −an ⎟
⎟
⎜
⎟
⎜0
⎟
·
·
·
0
⎜
⎟
⎜
⎟
⎜
⎟
.
⎜0
..
0 ⎟
⎝
⎠
⎛
⎞
⎟
⎜
⎜0
···
0 ⎟
⎟
⎜
⎟
⎜
..
⎟
⎜
⎟
⎜
.
⎟
⎜
⎟
⎜
T
T
⎜
em f − emn g ⎜b1 · · · bm bm1 − a1 −a2 · · · −an ⎟
⎟.
⎟
⎜
⎜0
···
0 ⎟
⎟
⎜
⎟
⎜
⎟
⎜
.
⎜0
..
0 ⎟
⎠
⎝
3.3
So KS − SK em f T − emn g T .
Theorem 3.2. Let matrix S be a Sylvester matrix and x x1 x2 · · · xmn T , y y1 y2 · · · ymn T , μ μ1 μ2 · · · μmn T , and V V1 V2 · · · Vmn T the solutions of the
systems of equations Sx em , Sy emn , ST μ f, and ST V g, respectively, where em and emn
are both m n × 1 vectors; then
a S is invertible, and the column vector wj j 1, 2, . . . , mn of S−1 satisfies the recurrence
relation
wmn y,
wi−1 Kwi μi x − Vi y,
i m n, . . . , 3, 2,
3.4
4
Mathematical Problems in Engineering
b
S−1 S1 U1 S2 U2
⎛
ymn ymn−1 · · ·
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
ymn
y2
..
.
y3
..
.
..
.
ymn
⎛
xmn xmn−1 · · ·
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
xmn
⎞⎛
⎞
1
⎟⎜
⎟
⎟⎜−V
⎟
1
mn
y2 ⎟
⎜
⎟
⎟⎜
⎟
⎟⎜ .
⎟
.
.
.. ⎟
.
.
.
⎜
⎟
.
.
.
⎟
. ⎟
⎟⎜
⎜
⎟
⎟⎜
⎟
⎟
−V
−V
·
·
·
1
3
4
⎠
ymn−1 ⎠⎝
−V2 −V3 · · · −Vmn 1
ymn
y1
x2
..
.
x3
..
.
..
.
xmn
3.5
⎞⎛
⎞
0
⎟⎜
⎟
⎟⎜μ
⎟
mn 0
x2 ⎟
⎜
⎟
⎟⎜
⎟
⎟⎜ .
⎟
.
.
.. ⎟
.
.
.
⎜
⎟.
.
.
.
⎟
. ⎟
⎟⎜
⎜
⎟
⎟⎜
⎟
⎟
μ
μ
·
·
·
0
4
⎠
xmn−1 ⎠⎝ 3
μ2 μ3 · · · μmn 0
xmn
x1
Proof. By Theorem 3.1 Sx em and Sy emn , we have that
KS SK em f T − emn g T SK Sxf T − Syg T S K xf T − yg T ,
3.6
so
K i S K i−1 S K xf T − yg T
K i−2 KS K xf T − yg T
2
K i−2 S K xf T − yg T
3.7
..
.
i
S K xf T − yg T .
Hence, we have that
i
K i emn K i Sy S K xf T − yg T y,
i 0, 1, . . . , m n − 1.
3.8
Let
i
wmn−i K xf T − yg T y,
i 0, 1, . . . , m n − 1.
3.9
Mathematical Problems in Engineering
5
It is easy to see that K i emn emn−i , and by 3.8
Swmn−i K i emn emn−i ,
i 0, 1, . . . , m n − 1.
3.10
From Sy emn , we have that wmn y. Let X w1 w2 · · · wmn ; then
SX Sw1 w2 · · · wmn Sw1 Sw2 · · · Swmn Imn ,
3.11
so the matrix S is invertible and the inverse of S is the matrix X.
From 3.1, we have that
S−1 KS − SKS−1 S−1 em f T − emn g T S−1 ,
3.12
S−1 K − KS−1 xμT − yV T .
3.13
and thus
So
S−1 K − KS−1 ei xμT − yV T ei .
3.14
Since Kei ei−1 , we have that
wi−1 Ksi μi x − Vi y,
i m n, . . . , 3, 2,
3.15
and hence
wmn y,
wi−1 Kwi μi x − Vi y,
i m n, . . . , 3, 2.
3.16
For b, by3.4
wmn y,
wmn−1 Ky μmn x − Vmn y,
wmn−2 K 2 y Kμmn x − KVmn y μmn−1 x − Vmn−1 y,
···
w1 K mn−1 y K mn−2 μmn x − K mn−2 Vmn y · · · μ2 x − V2 y.
3.17
6
Mathematical Problems in Engineering
So
S−1 w1 w2 · · · wmn
⎛
⎞
1
⎜
⎜−Vmn 1
⎜
⎜
mn−1
⎜
.. . .
mn−2
y K
y · · · Ky y ⎜ ...
K
.
.
⎜
⎜
⎜ −V3 −V4 · · ·
⎝
1
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
−V2 −V3 · · · −Vmn 1
⎞
⎛
0
⎟
⎜
⎟
⎜μmn 0
⎟
⎜
⎟
⎜
mn−1
⎟
⎜
.
.
.. . . .
x K mn−2 x · · · Kx x ⎜ ..
K
⎟
⎟
⎜
⎟
⎜
⎟
⎜ μ3 μ4 · · · 0
⎠
⎝
μ2
⎛
ymn ymn−1 · · ·
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
ymn
y2
..
.
y3
..
.
..
.
ymn
y1
μ3 · · · μmn 0
⎞⎛
⎞
3.18
1
⎟⎜
⎟
⎟⎜−V
⎟
1
mn
y2 ⎟
⎟
⎟⎜
⎜
⎟
⎟⎜ .
⎟
.
.
.. ⎟
.
.
.
⎜
⎟
.
.
.
⎜
⎟
. ⎟
⎟⎜
⎟
⎟⎜
⎟
−V3 −V4 · · ·
1
⎠
ymn−1 ⎟
⎠⎝
−V
−V
·
·
·
−V
1
2
3
mn
ymn
⎞⎛
⎛
⎞
x1
xmn xmn−1 · · · x2
0
⎟⎜
⎜
⎟
⎟⎜μ
⎜
.
⎟
⎜
mn 0
xmn . . x3
x2 ⎟
⎟
⎟⎜
⎜
⎜
⎟
⎟⎜ .
⎜
⎟
.
.
⎟
⎜
.
.
.
.
..
..
⎜
⎜
⎟.
.
.
⎟⎜ .
.
.
.
⎟
.
⎟⎜
⎜
⎟
⎟⎜
⎜
⎟
⎜
μ
μ4 · · · 0
⎠
xmn xmn−1 ⎟
⎠⎝ 3
⎝
μ
μ
·
·
·
μ
0
2
3
mn
xmn
This completes the proof.
4. Stability Analysis
In this section, we will show that the Sylvester inversion formula presented in this paper is
evaluation forward stable.
If, for all well conditioned problems, the computed solution x is close to the true
solution x, in the sense that the relative error x − x
2 /x2 is small, then we call the
algorithm forward stable the author called this weakly in 3. Round-off errors will occur
in the matrix computation.
Mathematical Problems in Engineering
7
Theorem 4.1. Let matrix S be a nonsingular Sylvester matrix and well conditioned; then the formula
in Theorem 3.2 is forward stable.
Proof. Assume that we have computed the solutions x,
y,
μ
, and V of Sx em , Sy emn ,
T
T
S μ f, and S V g in Theorem 3.2 which are perturbed by the normwise relative errors
bounded by ε,
2 ≤ x2 1 ε,
x
y ≤ y 1 ε,
2
2
μ ≤ μ 1 ε,
2
2
V ≤ V 2 1 ε.
2
4.1
Thus, we have that
√
m n y 2 ,
√
U1 F ≤ m n 1 V 22 ,
S1 F ≤
S2 F ≤
√
m nx2 ,
U2 F ≤
√
m nμ2 .
4.2
The inversion formula in Theorem 3.2, using the perturbed solutions x,
y,
μ
, and V ,
can be expressed as
1 S2 U
2
S−1 fl S1 U
flS1 ΔS1 U1 ΔU1 S2 ΔS2 U2 ΔU2 4.3
S−1 ΔS1 U1 S1 ΔU1 ΔS2 U2 S2 ΔU2 E F.
Here, and in the sequel, E is the matrix containing the error which results from computing the
matrix products and F contains the error from subtracting the matrices. For the error matrices
ΔS1 , ΔU1 , ΔS2 , and ΔU2 , we have that
√
ΔS1 F ≤ εS1 F ≤ ε m ny2 ,
√
ΔU1 F ≤ εU1 F ≤ ε m n 1 V 22 ,
√
ΔS2 F ≤ εS2 F ≤ ε m nx2 ,
√
ΔU2 F ≤ εU2 F ≤ ε m nμ2 .
4.4
By Lemma 2.1, we have the following bounds on E and F:
E2 ≤ EF
≤ εm nS1 F U1 F S2 F U2 F 2 2
≤ εm n
y 2 1 V 2 x2 μ 2
≤ εm n2 y2 1 V 22 x2 μ2
F2 ≤
√
m nεS−1 .
2
4.5
8
Mathematical Problems in Engineering
Consequently, adding all these error bounds, by 4.3, we have that
√
−1
ε m nε y2 1 V 22 x2 μ2 m nεS−1 .
S − S−1 ≤ m n2
2
2
4.6
From the equations Sx em , Sy emn ST μ f, and ST V g in Theorem 3.2, we have
that
y ≤ S−1 ,
2
2
V 2 ≤ S−1 g 2 ,
2
x2 ≤ S−1 ,
2
μ ≤ S−1 f 2 .
2
2
4.7
Thus, the relative error is
−1
ε m nε y2 1 V 22 x2 μ2
m n2
S − S−1 √
2
2 ≤
ε mn
S−1 S−1 2
2
√
−1 ≤ m n2
ε m nε 1 S g 2 f 2 ε m n.
4.8
2
Since S is well conditioned, S−1 2 is finite. It is easy to see that g2 , f2 are finite.
Therefore, the formula presented in Theorem 3.2 is forward stable.
This completes the proof.
5. Numerical Example
This section gives an example to illustrate our results. All the following tests are performed
by MATLAB 7.0.
Example 5.1. Given fx x 1, gx x2 x 1, that
is,a1 1, a2 1, b1 1, b2 1, b3 1, n 1, m 2, f 1, 1, 0T , g 0, 1, 1T , and S So
110
011
111
⎞⎛
1 1 0
0 1
⎟⎜
⎜
⎟⎜
⎟ ⎜
⎜
⎟
⎜
⎜
⎟
⎜
⎟
KS − SK ⎝0 0 1⎠⎝0 1 1⎠ − ⎝0 1 1⎠⎝0 0
1 1 1
0 0
0 0 0
1 1 1
⎛
⎞ ⎛
⎞ ⎛
⎞
0 1 1
0 1 1
0 0 0
⎜
⎟ ⎜
⎟ ⎜
⎟
⎟ ⎜
⎟ ⎜
⎟
⎜
⎝1 1 1⎠ − ⎝0 0 1⎠ ⎝1 1 0 ⎠,
0 0 0
0 1 1
0 −1 −1
⎛ ⎞
⎛ ⎞
⎛
0
0
0 0
⎜ ⎟
⎜ ⎟
⎜
T
T
⎜
⎟
⎜
⎟
⎜
em f − emn g ⎝1⎠ 1 1 0 − ⎝0⎠ 0 1 1 ⎝1 1
0
1
0 −1
⎛
0 1 0
Therefore, KS − SK em f T − emn g T .
⎞⎛
1 1 0
⎞
⎛
.
⎞
0
⎟
1⎟
⎠
0
0
5.1
⎞
⎟
0⎟
⎠.
−1
5.2
Mathematical Problems in Engineering
9
By the condition of Theorem 3.2, we can get
x −1, 1, 0T ,
y 1, −1, 1T ,
Obviously, S is invertible and S−1 ⎛
0
0 −1 1
1 1 −1
−1 0 1
μ 1, 0, 0T ,
V 0, 1, 0T .
5.3
. And it is easy to see that w3 y,
⎞
⎜ ⎟
⎟
w1 ⎜
⎝1⎠
−1
⎛
⎞⎛ ⎞
⎛ ⎞ ⎛ ⎞
0 1 0
−1
−1
1
⎜
⎟⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎝0 0 1⎠⎝ 1 ⎠ 0⎝ 1 ⎠ − ⎝−1⎟
⎠
0 0 0
0
0
1
Sw2 μ2 x − V2 y,
⎛ ⎞
−1
⎜ ⎟
⎟
w2 ⎜
⎝1⎠
0
⎛
⎞⎛ ⎞
⎛ ⎞
⎛ ⎞
0 1 0
1
−1
1
⎜
⎟⎜ ⎟
⎜ ⎟
⎜ ⎟
⎟⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜
⎝0 0 1⎠⎝−1⎠ 0⎝ 1 ⎠ − 0⎝−1⎠
0 0 0
1
0
1
S−1
5.4
Sw3 μ3 x − V3 y,
⎛
⎞
0 −1 1
⎜
⎟
⎟
⎜
⎝ 1 1 −1⎠
−1 0 1
⎛
⎞⎛
⎞ ⎛
⎞⎛
⎞
1 −1 1
1
0 1 −1
0
⎜
⎟⎜
⎟ ⎜
⎟⎜
⎟
⎟⎜
⎟ ⎜
⎟⎜
⎟
⎜
⎝ 1 −1⎠⎝ 0 1 ⎠ ⎝ 0 1 ⎠⎝0 0 ⎠
1
−1 0 1
0
0 0 0
S1 U1 S2 U2 .
Acknowledgments
This work was supported financially by the Youth Foundation of Shandong Province,
China ZR2010EQ014, Shandong Province Natural Science Foundation ZR2010AQ026,
and National Natural Science Foundation of China 11026047.
10
Mathematical Problems in Engineering
References
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Methods and Computer Applications, vol. 31, no. 2, pp. 92–98, 2010.
2 G. H. Golub and C. F. Van Loan, Matrix Computations, vol. 3 of Johns Hopkins Series in the Mathematical
Sciences, Johns Hopkins University Press, Baltimore, Md, USA, 2nd edition, 1989.
3 J. R. Bunch, “The weak and strong stability of algorithms in numerical linear algebra,” Linear Algebra
and Its Applications, vol. 88/89, pp. 49–66, 1987.
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International Journal of
Advances in
Combinatorics
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Mathematical Physics
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Volume 2014
Journal of
Complex Analysis
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Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
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Stochastic Analysis
Abstract and
Applied Analysis
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International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
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Volume 2014
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Volume 2014
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Volume 2014
Optimization
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Volume 2014
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Volume 2014