Research Article Solvability Theory and Iteration Method...
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Research Article Solvability Theory and Iteration Method for One Self-Adjoint Polynomial Matrix Equation Zhigang Jia, 1 Meixiang Zhao, 1 Minghui Wang, 2 and Sitao Ling 3 1 School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China 2 Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, China 3 College of Science, China University of Mining and Technology, Jiangsu 221116, China Correspondence should be addressed to Minghui Wang; [email protected] Received 19 October 2013; Revised 30 March 2014; Accepted 13 April 2014; Published 7 May 2014 Academic Editor: Zhi-Hong Guan Copyright © 2014 Zhigang Jia et al. is 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. e solvability theory of an important self-adjoint polynomial matrix equation is presented, including the boundary of its Hermitian positive definite (HPD) solution and some sufficient conditions under which the (unique or maximal) HPD solution exists. e algebraic perturbation analysis is also given with respect to the perturbation of coefficient matrices. An efficient general iterative algorithm for the maximal or unique HPD solution is designed and tested by numerical experiments. 1. Introduction In this paper, we consider the following self-adjoint polyno- mial matrix equation: − ∗ = , (1) where , are positive integers, , ∈ C × , and >0. As far as we know, the solvability of (1) is not completely solved untill now. In many fields of applied mathematics, engineering, and economic sciences, (1) plays an important role. e famous discrete-time algebraic Lyapunov equation (DALE) is exactly (1) with = = 1. Undoubtedly, DALE is one of the most important mathematical problems in signal processing, system, and control theory and many others (e.g., see the monographs [1, 2]). If is stable (with respect to the unit circle), DALE has a unique Hermitian positive definite (HPD) solution. Such strong relation between the spectral property of and the solvability theory is fortunately owned by (1), which can be considered as a nonlinear DALE if ̸ =1 or ̸ =1. What about the following algebraic Riccati equation: 2 + ∗ + − ∗ − = 0, (2) where , , ∈ C × , ∗ =≥0, and ∗ =>0? Defining := + and := + 2 − ∗ , we can immediately get (1) with = 2 and = 1 as an equivalent form of (2). As we all know, solving algebraic Riccati equations is an important task in the linear-quadratic regulator problem, Kalman filtering, ∞ -control, model reduction problems, and so forth. See [1, 3–5] and the references therein. Many numerical methods have been proposed, such as invariant subspace methods [6], Schur method [7], doubling algorithm [8], and structure-preserving doubling algorithm [9, 10]. At the same time the perturbation theory was developed in [11–15], as well as the unified methods for the discrete-time and continuous-time algebraic Riccati equations [16, 17]. A general iteration method for (1) given in this paper can be seen as a new algorithm for the algebraic Riccati equation (2), setting =2 and =1. Apart from the above applications, (1) is appealing from the mathematical viewpoint since it unifies a large class of systems of polynomial matrix equations. Many nonlinear matrix equations are special cases of (1). For example, nonlinear matrix equations, − ∗ = (see, e.g., [18, 19]), are equivalence models of − ∗ = and = 1/ , where , are positive integers and = /. In a rather general form, Ran and Reurings [18] investigated + ∗ F() = (>0) for its positive semidefinite solutions under the assumption that the function F(⋅) is monotone and − ∗ F() is positively definite. Besides, Lee and Lim [20] Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 681605, 7 pages http://dx.doi.org/10.1155/2014/681605
Transcript of Research Article Solvability Theory and Iteration Method...
Research ArticleSolvability Theory and Iteration Method for One Self-AdjointPolynomial Matrix Equation
Zhigang Jia1 Meixiang Zhao1 Minghui Wang2 and Sitao Ling3
1 School of Mathematics and Statistics Jiangsu Normal University Xuzhou 221116 China2Department of Mathematics Qingdao University of Science and Technology Qingdao 266061 China3 College of Science China University of Mining and Technology Jiangsu 221116 China
Correspondence should be addressed to Minghui Wang mhwangqusteducn
Received 19 October 2013 Revised 30 March 2014 Accepted 13 April 2014 Published 7 May 2014
Academic Editor Zhi-Hong Guan
Copyright copy 2014 Zhigang Jia et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The solvability theory of an important self-adjoint polynomialmatrix equation is presented including the boundary of itsHermitianpositive definite (HPD) solution and some sufficient conditions under which the (unique or maximal) HPD solution exists Thealgebraic perturbation analysis is also given with respect to the perturbation of coefficient matrices An efficient general iterativealgorithm for the maximal or unique HPD solution is designed and tested by numerical experiments
1 Introduction
In this paper we consider the following self-adjoint polyno-mial matrix equation
119883119904
minus 119860lowast
119883119905
119860 = 119876 (1)
where 119904 119905 are positive integers 119860119876 isin C119899times119899 and 119876 gt 0 Asfar as we know the solvability of (1) is not completely solveduntill now
In many fields of applied mathematics engineering andeconomic sciences (1) plays an important role The famousdiscrete-time algebraic Lyapunov equation (DALE) is exactly(1) with 119904 = 119905 = 1 Undoubtedly DALE is one of themost important mathematical problems in signal processingsystem and control theory and many others (eg see themonographs [1 2]) If 119860 is stable (with respect to the unitcircle) DALEhas a uniqueHermitian positive definite (HPD)solution Such strong relation between the spectral propertyof 119860 and the solvability theory is fortunately owned by (1)which can be considered as a nonlinear DALE if 119904 = 1 or 119905 = 1What about the following algebraic Riccati equation
1198842
+ 119861lowast
119884 + 119884119861 minus 119860lowast
119884119860 minus 119877 = 0 (2)
where 119860 119861 119877 isin C119899times119899 119861lowast = 119861 ge 0 and 119877lowast = 119877 gt 0 Defining119883 = 119884 + 119861 and 119876 = 119877 + 119861
2
minus 119860lowast
119861119860 we can immediately
get (1) with 119904 = 2 and 119905 = 1 as an equivalent form of(2) As we all know solving algebraic Riccati equations isan important task in the linear-quadratic regulator problemKalman filtering 119867
infin-control model reduction problems
and so forth See [1 3ndash5] and the references therein Manynumerical methods have been proposed such as invariantsubspacemethods [6] Schurmethod [7] doubling algorithm[8] and structure-preserving doubling algorithm [9 10] Atthe same time the perturbation theory was developed in[11ndash15] as well as the unified methods for the discrete-timeand continuous-time algebraic Riccati equations [16 17] Ageneral iteration method for (1) given in this paper can beseen as a new algorithm for the algebraic Riccati equation (2)setting 119904 = 2 and 119905 = 1
Apart from the above applications (1) is appealing fromthe mathematical viewpoint since it unifies a large class ofsystems of polynomial matrix equations Many nonlinearmatrix equations are special cases of (1) For examplenonlinear matrix equations 119883 minus 119860
lowast
119883119902
119860 = 119876 (see eg[18 19]) are equivalence models of 119884119904 minus 119860lowast119884119905119860 = 119876 and119884 = 119883
1119904 where 119904 119905 are positive integers and 119902 = 119905119904 In arather general form Ran and Reurings [18] investigated 119883 +
119860lowastF(119883)119860 = 119876 (119876 gt 0) for its positive semidefinite solutions
under the assumption that the functionF(sdot) ismonotone and119876minus119860lowastF(119876)119860 is positively definite Besides Lee and Lim [20]
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 681605 7 pageshttpdxdoiorg1011552014681605
2 Journal of Applied Mathematics
proved that (1) has a unique HPD solution when |119904| ge 1 ge |119905|and |119905119904| lt 1 See [21ndash25] formore recent results on nonlinearmatrix equations To the best of our knowledge (1) with 119904 lt 119905(without monotony in hand) has not been discussed Thesefacts motivate us to study polynomial matrix equation (1)
This paper is organized as follows In Section 2 we deducethe existence and uniqueness conditions of HPD solutionsof (1) in Section 3 we derive the algebraic perturbationtheory for the unique or maximal solution of (1) finallyin Section 4 we provide an iterative algorithm and twonumerical experiments
We begin with some notations used throughout thispaper F119898times119899 stands for the set of119898times119899matrices with elementson field F (F is R or C) If 119867 is a Hermitian matrix onF119899times119899 120582min(119867) and 120582max(119867) stand for the minimal and themaximal eigenvalues respectively Denote the singular valuesof a matrix 119860 isin F119898times119899 by 120590
1(119860) ge sdot sdot sdot ge 120590
119897(119860) ge 0
where 119897 = min119898 119899 Suppose that 119883 and 119884 are Hermitianmatrices we write 119883 ge 119884(119883 gt 119884) if 119883 minus 119884 is positivelysemidefinite (definite) and denote the matrices set 119883 | 119883 minus
120572119868 ge 0 and 120573119868 minus 119883 ge 0 by [120572119868 120573119868]
2 Solvability of Self-Adjoint PolynomialMatrix Equation
In this section we study the solvability theory of (1) assumingthat 119860 is nonsingular that is 120582min(119860
lowast
119860) gt 0 To do this weneed two simple but useful functions defined on the positiveabscissa axis
1198921(119909) = 119909
119904
minus 120582max (119860lowast
119860) 119909119905
minus 120582max (119876)
1198922(119909) = 119909
119904
minus 120582min (119860lowast
119860) 119909119905
minus 120582min (119876) (3)
The following two famous inequalities will be usedfrequently in the remaining of this paper
Lemma 1 (Lowner-Heinz inequality [26 Theorem 11]) If119860 ge 119861 ge 0 and 0 le 119903 le 1 then 119860119903 ge 119861119903
Lemma 2 (see [27 Theorem 21]) Let 119860 and 119861 be positiveoperators on a Hilbert space 119867 such that119872
1119868 ge 119860 ge 119898
1119868 gt
01198722119868 ge 119861 ge 119898
2119868 gt 0 and 0 lt 119860 le 119861 Then
119860119905
le (1198721
1198981
)
119905minus1
119861119905
119860119905
le (1198722
1198982
)
119905minus1
119861119905 (4)
hold for any 119905 ge 1
21 Maximal Solution of (1) with 119904 lt 119905 Now we derive anecessary condition and a sufficient condition for existenceof HPD solutions of (1) with 119904 lt 119905 With 119892
1(119909) and 119892
2(119909) in
hand we can easily get the distribution of eigenvalues of theHPD solution119883 of (1)
Theorem 3 Suppose that 120582max(119860lowast
119860) le (119904119905)((119905 minus 119904)
120582max(119876)119905)(119905minus119904)119904 and 119883 isin C119899times119899 is an HPD solution of (1) then
for any eigenvalue 120582(119883) of119883
1205731le 120582 (119883) le 120572
1or 1205722le 120582 (119883) le 120573
2 (5)
where 1205721 1205722are two positive roots of 119892
1(119909) and 120573
1 1205732are two
positive roots of 1198922(119909)
Proof From Theorem 3316(d) in Horn and Johnson [28]one can see that
120590119894(119860lowast
119883119905
119860) le 120590119894(119883119905
) 1205902
1(119860) that is
120582119894(119860lowast
119883119905
119860) le 120582119894(119883119905
) 120582max (119860lowast
119860)
119894 = 1 119899
(6)
If 119860 is nonsingular
120590119894(119883119905
) = 120590119894((119860minus1
)lowast
119860lowast
119883119905
119860119860minus1
) le 120590minus2
119899(119860) 120590119894(119860lowast
119883119905
119860)
(7)
That means
120590119894(119860lowast
119883119905
119860) ge 120590119894(119883119905
) 1205902
119899(119860) that is
120582119894(119860lowast
119883119905
119860) ge 120582119894(119883119905
) 120582min (119860lowast
119860)
119894 = 1 119899
(8)
The above equations still hold if119860 is singular since120590119899(119860) = 0
that is 120582min(119860lowast
119860) = 0 in this case Applying Weyl theoremin Horn and Johnson [29]119883119904 = 119876 + 119860
lowast
119883119905
119860 implies
120582(119883)119904
minus 120582max (119860lowast
119860) 120582(119883)119905
120582max (119876) le 0
120582(119883)119904
minus 120582min (119860lowast
119860) 120582(119883)119905
minus 120582min (119876) ge 0(9)
Define a function 119891(119909) = 119909119904
minus 1198862
119909119905
minus 119902 119886 gt 0 119902 gt
0 Then the only positive stationary point of 119891(119909) is 1199090=
((119905119904)1198862
)1(119904minus119905) If 1198862 le (119904119905)((119905 minus 119904)119902119905)
(119905minus119904)119904 119891(119909) has twopositive roots 119909
1and 119909
2 with 1199021119904 lt 119909
1le 1199090le 1199092lt 1198862(119904minus119905)
So 120582max(119860lowast
119860) le (119904119905)(119905minus119904)120582max(119876)119905(119905minus119904)119904 implies that 119892
1(119909)
has two roots 1205721 1205722gt 0 and 119892
2(119909) has two roots 120573
1 1205732gt 0
Since 1198922(119909) ge 119892
1(119909) (120582min(119876))
1119904
le 1205731le 1205721le 1205722le 1205732le
(120582min(119860lowast
119860))1(119904minus119905) Then from (9) we obtain (5)
If (1) has an HPD solution its eigenvalues may skipbetween [120573
1 1205721] and [120572
2 1205732] Next what we take more atten-
tion on is the HPD solution with its eigenvalues distributedonly on one interval
Theorem 4 Suppose that 120582max(119860lowast
119860) le (119904119905)((119905 minus 119904)
120582max(119876)119905)(119905minus119904)119904
(1) Equation (1) has an HPD solution119883 isin [1205731119868 1205721119868] and
if 120582min(119860lowast
119860) gt 119904120572119904minus1
1(119905120573119905minus1
1)minus1 such119883 exists uniquely
(2) Equation (1) has an HPD solution 119885 isin [1205722119868 1205732119868] and
if 120582min(119860lowast
119860) gt 119904120573119904minus1
2(119905120572119905minus1
2)minus1 such 119885 exists uniquely
Journal of Applied Mathematics 3
Proof (1) Let ℎ1(119883) = (119876 + 119860
lowast
119883119905
119860)1119904 where 119883 isin
[(120582min(119876))1119904
119868 (119904(120582max(119860lowast
119860)119905))1(119905minus119904)
119868] Lemmas 1 and 2and 119905 minus 119904 gt 0 imply
(120582min (119876))1119904
119868 le ℎ1(119883)
le 120582max (119876) + 120582max (119860lowast
119860)[119904
(120582max (119860lowast119860) 119905)
]
119905(119905minus119904)
1119904
119868
le [119904
120582max (119860lowast119860) 119905
]
119904(119905minus119904)times1119904
119868 = [119904
120582max (119860lowast119860) 119905
]
1(119905minus119904)
119868
(10)
Applying Brouwerrsquos fixed-point theorem ℎ1(119883) has a fixed
point 119883 isin [(120582min(119876))1119904
119868 (119904(120582max(119860lowast
119860)119905))1(119905minus119904)
119868] ThenfromTheorem 3119883 isin [120573
1119868 1205721119868]
We now prove the uniqueness of 119883 under the addi-tional condition that 120582min(119860
lowast
119860) gt 119904120572119904minus1
1(119905120573119905minus1
1)minus1 Suppose
119884 isin [(120582min(119876))1119904
119868 (119904(120582max(119860lowast
119860)119905))1(119905minus119904)
119868] is another HPDsolution of (1) and 119884 =119883 It has been known that
10038171003817100381710038171003817119883119905
minus 11988411990510038171003817100381710038171003817119865=100381710038171003817100381710038171003817(119860minus1
)lowast
(119883119904
minus 119884119904
)119860minus1100381710038171003817100381710038171003817119865
le (120582min (119860lowast
119860))minus11003817100381710038171003817119883119904
minus 1198841199041003817100381710038171003817119865
(11)
Then from 119883119904
minus 119884119904
119865le 119904120572119904minus1
1119883 minus 119884
119865and 119883119905 minus 119884119905
119865ge
119905120573119905minus1
1119883 minus 119884
119865
119883 minus 119884119865
le 119904120572119904minus1
1[119905120573119905minus1
1120582min (119860
lowast
119860)]minus1
119883 minus 119884119865lt 119883 minus 119884
119865
(12)
which is impossible Hence119883 = 119884(2) Let ℎ
2(119885) = [(119860
minus1
)lowast
(119885119904
minus 119876)119860minus1
]1119905 where 119885 isin
[1205722119868 1205732119868] ℎ2(119885) is continuous and
ℎ2(1205722119868) le ℎ
2(119885) le ℎ
2(1205732119868) (13)
because (119860minus1
)lowast
(120572119904
2119868 minus 119876)119860
minus1
le (119860minus1
)lowast
(119885119904
minus 119876)119860minus1
le
(119860minus1
)lowast
(120573119904
2119868minus119876)119860
minus1 By Lemmas 1 and 2 and Brouwerrsquos fixed-point theorem it is sufficient to prove ℎ
2(1205722119868) ge 120572
2119868 and
ℎ2(1205732119868) le 120573
2119868 in order for an HPD solution 119885 isin [120572
2119868 1205732119868]
to exist The existence of such 119885 follows from inequalities
ℎ2(1205722119868)
= [(119860minus1
)lowast
(120572119904
2119868 minus 119876)119860
minus1
]
1119905
ge [(119860minus1
)lowast
(120572119904
2119868 minus 120582max (119876) 119868) 119860
minus1
]
1119905
ge [(120582max (119860lowast
119860))minus1
(120572119904
2119868 minus 120582max (119876) 119868)]
1119905
= 1205722119868
ℎ2(1205732119868)
= [(119860minus1
)lowast
(120573119904
2119868 minus 119876)119860
minus1
]
1119905
le [(119860minus1
)lowast
(120573119904
2119868 minus 120582min (119876) 119868) 119860
minus1
]
1119905
le [(120582min (119860lowast
119860))minus1
(120573119904
2119868 minus 120582min (119876) 119868)]
1119905
= 1205732119868
(14)
Next we prove the uniqueness of 119885 under the additionalcondition that 120582min(119860
lowast
119860) gt 119904120573119904minus1
2(119905120572119905minus1
2)minus1 Suppose (1) has
two different HPD solutions 119885 and 119884 on [1205722119868 1205732119868] Then
10038171003817100381710038171003817119885119905
minus 11988411990510038171003817100381710038171003817119865=100381710038171003817100381710038171003817(119860minus1
)lowast
(119885119904
minus 119884119904
)119860minus1100381710038171003817100381710038171003817119865
le (120582min (119860lowast
119860))minus11003817100381710038171003817119885119904
minus 1198841199041003817100381710038171003817119865
le (120582min (119860lowast
119860))minus1
119904120573119904minus1
2119885 minus 119884
119865
(15)
Moreover if 120582min(119860lowast
119860) gt 119904120573119904minus1
2(119905120572119905minus1
2)minus1 applying the
inequality 119885119905 minus 119884119905119865ge 119905120572119905minus1
2119885 minus 119884
119865 we have
119885 minus 119884119865le (119905120572119905minus1
2120582min (119860
lowast
119860))minus1
119904120573119904minus1
2119885 minus 119884
119865lt 119885 minus 119884
119865
(16)
which is impossible Hence 119884 = 119885
The maximal solution (see eg [30 31]) of (1) is definedas follows
Definition 5 An HPD solution 119883119872
isin C119899times119899 of (1) is themaximal solution if for any HPD solution 119884 isin C119899times119899 of (1)there is119883
119872ge 119884
So the second term of Theorem 4 implies that the maxi-mal solution of (1) is on [120572
2119868 1205732119868]
Theorem 6 Suppose that 120582max(119860lowast
119860) le (119904119905)(119905(119905 minus
119904)120582max(119876))(119904minus119905)119904 and120582min(119860
lowast
119860) gt 119904120573119904minus1
2(119905120572119905minus1
2)minus1 then (1) has
a maximal solution 119883max isin [1205722119868 1205732119868] which can be computedby
119883119894= [(119860
minus1
)lowast
(119883119904
119894minus1minus 119876)119860
minus1
]
1119905
119894 = 1 2 (17)
with the initial value1198830= 1205732119868
Proof Let 120585 = (119905120572119905minus1
2120582min(119860
lowast
119860))minus1
119904120573119904minus1
2 then 120585 lt 1 From
the proof of Theorem 4 (2)
119905120572119905minus1
2
1003817100381710038171003817119883119894+1 minus 1198831198941003817100381710038171003817119865le10038171003817100381710038171003817119883119905
119894+1minus 119883119905
119894
10038171003817100381710038171003817119865
=100381710038171003817100381710038171003817(119860minus1
)lowast
(119883119904
119894minus 119883119904
119894minus1)119860minus1100381710038171003817100381710038171003817119865
le (120582min (119860lowast
119860))minus11003817100381710038171003817119883119904
119894minus 119883119904
119894minus1
1003817100381710038171003817119865
le (120582min (119860lowast
119860))minus1
119904120573119904minus1
2
1003817100381710038171003817119883119894 minus 119883119894minus11003817100381710038171003817119865
(18)
4 Journal of Applied Mathematics
Then1003817100381710038171003817119883119894+1 minus 119883119894
1003817100381710038171003817119865le 120585
1003817100381710038171003817119883119894 minus 119883119894minus11003817100381710038171003817119865le 12058511989410038171003817100381710038171198831 minus 1198830
1003817100381710038171003817119865 (19)
which indicates the convergence of matrix series 1198830 1198831
1198832 generated by (17)Set 119883
0= 1205732119868 Assuming 119883
119894isin [1205722119868 1205732119868] then from
inequalities (14) we have
1205722119868 le ℎ (120572
2119868) le 119883
119894+1
= [(119860minus1
)lowast
(119883119904
119894minus 119876)119860
minus1
]
1119905
le ℎ (1205732119868) le 120573
2119868
(20)
That means for any 119894 = 0 1 2 119883119894isin [1205722119868 1205732119868] By
Theorem 4 (2) we can see that 119883max = lim119894rarr+infin
119883119894is the
unique HPD solution of (1) on [1205722119868 1205732119868]
Now we prove the maximality of 119883max Suppose that 119883is an arbitrary HPD solution of (1) then 119883
0ge 119883 and
Theorem 3 implies1198831199050ge 119883119905 (since119883
0= 1205732119868) Assuming that
119883119905
119894ge 119883119905 Lemma 1 with 119904119905 lt 1 implies
119883119905
119894+1= (119860minus1
)lowast
[(119883119905
119894)119904119905
minus 119876]119860minus1
ge (119860minus1
)lowast
[(119883119905
)119904119905
minus 119876]119860minus1
= 119883119905
(21)
Then119883119905max = lim119894rarr+infin
119883119905
119894ge 119883119905 which implies that119883max ge 119883
by the Lowner-Heinz inequality
Note that similar iteration formula ever appeared in somepapers such as [20 21] for other nonlinear matrix equationsHere we firstly proved that the iteration form (17) preservesthe maximality of119883
119894over all HPD solutions of (1)
22 Unique Solution of (1) with 119904 ge 119905 If 119904 gt 119905 Lee and Lim[20 Theorem 94] show that (1) always has a unique HPDsolution denoted by 119883
119906 Now we give an upper bound and
a lower bound of 119883119906and suggest an iteration method for
computing119883119906
As defined in (3) 1198921(119909) and 119892
2(119909) with 119904 gt 119905 have unique
positive roots denoted by 1205741and 1205742 respectively
Since 1198921(120582(119883119906)) le 0 and 119892
2(120582(119883119906)) le 0 120574
2le 120582(119883
119906) le 1205741
Theorem 7 If 119904 gt 119905 (1) has a unique HPD solution119883119906isin [1205742119868 1205741119868] Let 119883
0= 1205741119868 or 120574
2119868 then matrix series
1198830 1198831 1198832 generated by
119883119894= (119876 + 119860
lowast
119883119905
119894minus1119860)1119904
119894 = 0 1 2 (22)
will converge to119883119906
Proof Weonly need to prove the convergence ofmatrix series1198830 1198831 1198832 Set119883
0= 1205741119868 From (22) we have
1198831= (119876 + 120574
119905
1119860lowast
119860)1119904
le (120582max (119876) + 120574119905
1120582max (119860
lowast
119860))1119904
119868 = 1205741119868
(23)
and then1198831199041le 119883119904
0 Assuming that119883119904
119894le 119883119904
119894minus1
119883119904
119894+1= 119876 + 119860
lowast
119883119905
119894119860 = 119876 + 119860
lowast
(119883119904
119894)119905119904
119860
le 119876 + 119860lowast
(119883119904
119894minus1)119905119904
119860 = 119883119904
119894
(24)
Then for any 119894 = 0 1 2 we have 119883119904119894+1
le 119883119904
119894and then
119883119894+1
le 119883119894by Lowner-Heinz inequality On the other hand
1198830ge 1205742119868 implies 119883
119894ge 1205742119868 for any 119894 = 0 1 2 because if
119883119894minus1
ge 1205742119868 then
119883119894= (119876 + 119860
lowast
119883119905
119894minus1119860)1119904
ge (119876 + 120574119905
2119860lowast
119860)1119904
ge (120582min (119876) + 120574119905
2120582min (119860
lowast
119860))1119904
119868 = 1205742119868
(25)
Then 1198830 1198831 1198832 with 119883
0= 1205741119868 is a decreasingly
monotone matrix series with a lower bound 1205742119868 Similarly we
can prove that 1198830 1198831 1198832 generated by (22) with 119883
0=
1205742119868 is an increasingly monotone matrix series with an upper
bound 1205741119868Therefore the convergence of 119883
0 1198831 1198832 has
been proved
From the above proof we can see that the iteration form(22) preserves the minimality (119883
0= 1205741119868) or maximality
(1198830= 1205742119868) of119883
119894in process
If 119904 = 119905 (1) can be reduced to a linear matrix equation119884minus119860lowast
119884119860 = 119876 which is the discrete-time algebraic Lyapunovequation (DALE) or Hermitian Stein equation [1 Page 5]assuming that 119884 = 119883
119904 It is well known that if 119860 is d-stable(see [1]) 119884 minus 119860
lowast
119884119860 = 119876 has a unique solution and matrixseries 119884
0 1198841 1198842 generated by 119884
119894+1= 119876+119860
lowast
119884119894119860 with an
initial value 1198840 will converge to the unique solution Besides
it is not difficult to get an expression of the unique solution119883119906= (suminfin
119895=0(119860lowast
)119895
119876119860119895
)1119904 applying [32 Theorem 1 Section
132] [1 Theorem 1118] and the results in Section 64 [28]Now we have presented the solvability theory of the self-
adjoint polynomial matrix equation (1) in three cases Ageneral iterative algorithm for its maximal solution (119904 lt 119905)or unique solution (119904 ge 119905) will be given in Section 4 Before itwe study the algebraic perturbation of themaximal or uniquesolution of (1)
3 Algebraic Perturbation Analysis
In this section we present the algebraic perturbation analysisof the HPD solution of (1) with respect to the perturbationof its coefficient matrices Similar to [30] we define theperturbed matrix equation of (1) as
119883119904
minus 119860lowast
119883119905
119860 = 119876 (26)
where 119860 = 119860 + Δ119860 isin C119899times119899 and 119876 = 119876 + Δ119876 isin C119899times119899 Wealways suppose that (1) has a maximal (or unique) solutiondenoted by 119883
119872isin [1205722119868 1205732119868] and (26) has a maximal (or
unique) solution denoted by119883119872isin [2119868 1205732119868]
Now we present the perturbation bound for 119883119872
when119904 = 119905 Define a function 120591
120591 (120572 120573) = 119904120572119904minus1
minus 119905120573119905minus1
1198602
2 (120572 120573) isin R
2
(27)
Journal of Applied Mathematics 5
Theorem 8 Let 120576 gt 0 be an arbitrary real number and 120591(2
1205732) ge 0 If
Δ119860119865lt (119860
2
2+2120576
3120591 (2 1205732)10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
minus119905
2
)
12
minus 1198602
Δ119876119865lt1
3120591 (2 1205732) 120576
(28)
then10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865lt 120576 (29)
Proof It is easy to induce that
10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865ge (
119904minus1
sum
119896=0
119904minus1minus119896
2120572119896
2)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
ge 119904119904minus1
2
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
10038171003817100381710038171003817119883119905
119872minus 119883119905
119872
10038171003817100381710038171003817119865le (
119905minus1
sum
119896=0
120573119905minus1minus119896
2120573119896
2)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le 119905120573119905minus1
2
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
(30)
Then from (1) and (26) we have
120591 (2 1205732)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le 21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ119860119865+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ1198602
119865+ Δ119876
119865
(31)
Since 120591(2 1205732) gt 0
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le (120591 (2 1205732))minus1
(21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ119860119865
+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ1198602
119865+ Δ119876
119865)
(32)
Then for an arbitrary 120576 gt 0 if Δ119860119865
lt (1198602
2+
(21205763)120591(2 1205732)119883119872minus119905
2)12
minus 1198602
and Δ119876119865
lt
(13)120591(2 1205732)120576 we have (29)
If 119904 = 119905 for an arbitrary 120576 gt 0 define
984858 (120576) = 1198602+ (119860
2
2+2120576
3120588)
12
(33)
where
120588 =10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
[119904119904minus1
2(1 minus 119860
2
2)]minus1
(34)
Theorem9 Let 120576 gt 0 be an arbitrary real number and 1198602lt
1 If
Δ119860119865lt2120576
3(120588984858 (120576))
minus1
Δ119876119865lt
120576
3120588
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
(35)
then10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865lt 120576 (36)
Table 1 Iteration CPU time (seconds) and residue for solving (1)with 119904 = 119905
(119904 119905)Algorithm 1 MONO
Ite CPU Res Ite CPU Res(2 1) 9 00541 45275119890 minus 13 200 21031 00016(1 2) 200 10275 20297119890 minus 07 mdash mdash mdash(8 5) 10 00716 59909119890 minus 13 200 22284 00034(5 8) 200 11048 31059119890 minus 05 mdash mdash mdash(30 15) 9 00743 52317119890 minus 13 200 23051 00029(15 30) 200 12865 20838119890 minus 08 mdash mdash mdash(300 150) 10 00886 79960119890 minus 13 200 22683 00031(150 300) 200 14187 28384119890 minus 07 mdash mdash mdash
Proof Similar to the proof of Theorem 8 we can induce that
(1 minus 1198602
2)10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865
le 21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ119860119865+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ1198602
119865+ Δ119876
119865
(37)
Then10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865
le (1 minus 1198602
2)minus1
(21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ119860119865
+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ1198602
119865+ Δ119876
119865)
(38)
With the help of (30) and (34) (38) implies10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865le 120588 (Δ119860
119865+ 2119860
2Δ119860119865+ Δ119876
119865) (39)
Then if Δ119860119865lt (21205763)(120588984858(120576))
minus1 and Δ119876119865lt (1205763120588)119883
119872119904
2
we have (36)
Theorems 8 and 9 make sure that the perturbation of119883119872
can be controlled if Δ119860 and Δ119876 have a proper upper bound
4 Algorithm and Numerical Experiments
In this section we give a general iterative algorithm for themaximal or unique solutions of (1) and two numerical exper-iments All reported results were obtained using MATLAB-R2012b on a personal computer with 24GHz Intel Core i7and 8GB 1600MHz DDR3
Example 10 Let matrices 119860 = rand(100) times 10minus2 and 119876 =
eye(100) With tol = 10minus12 and not more than 200 iterationswe apply Algorithm 1 to compute the maximal or uniqueHPD solutions of (1) with 119904 = 119905 and compare the results withthose by the iteration method from [33] (denoted by MONOin Table 1)
Table 1 shows iterations CPU times before convergenceand the residues of the computed HPD solution 119883 definedby
119890 (119904 119905) =
1003817100381710038171003817119883119904
minus 119860lowast
119883119905
119860 minus 1198761003817100381710038171003817119865
[119860 119876]119865
(40)
6 Journal of Applied Mathematics
Table 2 Iteration CPU time (seconds) and residue for solving (1) with 119904 = 119905 and different initial solutions
(119904 119905 1198830)
Algorithm 1 MONOIte CPU Res Ite CPU Res
(1 1 1205751119868119899) 20 00223 34947119890 minus 13 20 00202 34947119890 minus 13
(1 1 1205752119868119899) 31 00258 73027119890 minus 13 31 00305 73027119890 minus 13
(2 2 1205751119868119899) 20 01170 89978119890 minus 13 200 26421 00037
(2 2 1205752119868119899) 43 05475 96421119890 minus 13 200 27224 00037
(10 10 1205751119868119899) 29 01890 29296119890 minus 13 200 29717 00059
(10 10 1205752119868119899) 157 30859 71154119890 minus 13 200 29788 00059
Step 1 Compute 120582max(119860lowast
119860) 120582min(119860lowast
119860) 120582max(119876) 120582min(119876)Step 2 Input (3)Step 3 If 119904 lt 119905 run Steps 4-5 if 119905 lt 119904 run Steps 6-7 otherwise run Steps 8-9Step 4 Compute the roots 120572
1 1205722of 1198921(119909) and 120573
1 1205732of 1198922(119909) respectively
Step 5 Let 1198830= 1205732119868 run (17)
Step 6 Compute the root 1205741of 1198921(119909) and the root 120574
2of 1198922(119909) respectively
Step 7 Let 1198850= 1205741119868 run (22)
Step 8 Compute the root 1205751of 1198921(119909) and the root 120575
2of 1198922(119909) respectively
Step 9 If 120582max(119860lowast
119860) lt 1 and 1205751ge 1205752 then let 119883
0= 1205751119868 and run (22)
Algorithm 1 Given matrices 119860119876 isin C119899times119899 and positive integers 119904 119905
From Table 1 we can see that it takes more iterations andCPU times to solve themaximal solution of (1) with 119904 lt 119905 thanto solve the unique solution of (1) with 119904 gt 119905 At the same timethe accuracy of the latter is better than the formerMONOcannot be used to solve (1) with 119904 lt 119905 and it costs more iterationsand CPU times than Algorithm 1 when solving (1) with 119904 gt 119905
Now we use Example 41 of [33] to test our method
Example 11 Let 119860 = 05119861119861infin
with 119861 = [119861119894119895]119899times119899 119887119894119895= 119894 +
119895 + 1 and let 119876 = eye(119899) with 119899 = 100 We solve (1) with119904 = 119905 and with two different initial solutions The iterationsCPU times and the residues of the computation are reportedin Table 2
Table 2 shows that for Algorithm 1 the choice 1198830= 1205751119868119899
is better than 1198830= 1205752119868119899 When 119904 and 119905 rise MONO might
lose its efficiency It seems not proper to apply the iterationmethod designed for 119884 minus 119860lowast119884119905119904119860 = 119876 with 119884 = 119883
119904 to solve119883119904
minus 119860lowast
119883119905
119860 = 119876 although they are equivalent to each otherin theory
5 Conclusion
In this paper we considered the solvability of the self-adjointpolynomial matrix equation (1) Sufficient conditions weregiven to guarantee the existence of the maximal or uniqueHPD solutions of (1) The algebraic perturbation analysisincluding perturbation bounds was also developed for (1)under the perturbation of given coefficient matrices At lasta general iterative algorithm with maximality preserved inprocess was presented for the maximal or unique solutionwith two numerical experiments reported
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Zhigang Jiarsquos research was supported in part by NationalNatural Science Foundation of China under Grants 11201193and 11171289 and a project funded by the Priority AcademicProgram Development of Jiangsu Higher EducationInstitutions Minghui Wangrsquos research was supported inpart by the National Natural Science Foundation of China(Grant no 11001144) the Science and Technology Programof Shandong Universities of China (J11LA04) and theResearch Award Fund for Outstanding Young Scientists ofShandong Province in China (BS2012DX009) Sitao Lingrsquosresearch was supported in part by National Natural ScienceFoundations of China under Grant 11301529 PostdoctoralScience Foundation of China under Grant 2013M540472 andJiangsu Planned Projects for Postdoctoral Research Funds1302036CThe authors would like to thank three anonymousreferees for giving valuable comments and suggestions
References
[1] H Abou-Kandil G Freiling V Ionescu and G Jank MatrixRiccati Equations in Control and Systems Theory BirkhauserBasel Switzerland 2003
[2] I Gohberg P Lancaster and L Rodman Matrix PolynomialsAcademic Press New York NY USA 1982
[3] P Benner A J Laub and V Mehrmann ldquoBenchmarks for thenumerical solution of algebraic Riccati equationsrdquo IEEE ControlSystems Magazine vol 17 no 5 pp 18ndash28 1997
Journal of Applied Mathematics 7
[4] S Bittanti A Laub and J CWillems EdsTheRiccati EquationCommunications and Control Engineering Series SpringerBerlin Germany 1991
[5] P Lancaster andL RodmanAlgebraic Riccati Equations OxfordScience Publications Oxford University Press New York NYUSA 1995
[6] A J Laub ldquoInvariant subspace methods for the numerical solu-tion of Riccati equationsrdquo inThe Riccati Equation S Bittanti AJ Laub and J C Willems Eds Communications and ControlEngineering pp 163ndash196 Springer Berlin Germany 1991
[7] A J Laub ldquoA Schur method for solving algebraic Riccatiequationsrdquo IEEE Transactions on Automatic Control vol 24 no6 pp 913ndash921 1979
[8] M Kimura ldquoDoubling algorithm for continuous-time algebraicRiccati equationrdquo International Journal of Systems Science vol20 no 2 pp 191ndash202 1989
[9] E K-W Chu H-Y FanW-W Lin and C-SWang ldquoStructure-preserving algorithms for periodic discrete-time algebraic Ric-cati equationsrdquo International Journal of Control vol 77 no 8pp 767ndash788 2004
[10] E K-W Chu H-Y Fan andW-W Lin ldquoA structure-preservingdoubling algorithm for continuous-time algebraic Riccati equa-tionsrdquo Linear Algebra and Its Applications vol 396 pp 55ndash802005
[11] N J Higham ldquoPerturbation theory and backward error for119860119883 minus 119883119861 = 119862rdquo BIT Numerical Mathematics vol 33 no 1 pp124ndash136 1993
[12] M Konstantinov and P Petkov ldquoNote on perturbation theoryfor algebraic Riccati equationsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 21 no 1 pp 327ndash354 2000
[13] J-G Sun ldquoResidual bounds of approximate solutions of thealgebraic Riccati equationrdquoNumerischeMathematik vol 76 no2 pp 249ndash263 1997
[14] J-G Sun ldquoBackward error for the discrete-time algebraicRiccati equationrdquo Linear Algebra and Its Applications vol 259pp 183ndash208 1997
[15] J-G Sun ldquoBackward perturbation analysis of the periodicdiscrete-time algebraic Riccati equationrdquo SIAM Journal onMatrix Analysis and Applications vol 26 no 1 pp 1ndash19 2004
[16] V Mehrmann ldquoA step towards a unified treatment of continu-ous and discrete time control problemsrdquo Linear Algebra and ItsApplications vol 241ndash243 pp 749ndash779 1996
[17] H-G Xu ldquoTransformations between discrete-time andcontinuous-time algebraic Riccati equationsrdquo Linear Algebraand Its Applications vol 425 no 1 pp 77ndash101 2007
[18] A C M Ran and M C B Reurings ldquoOn the nonlinear matrixequation 119883 + 119860
lowastF (119883)119860 = 119876 solutions and perturbationtheoryrdquo Linear Algebra and Its Applications vol 346 pp 15ndash262002
[19] A C M Ran M C B Reurings and L Rodman ldquoA pertur-bation analysis for nonlinear selfadjoint operator equationsrdquoSIAM Journal on Matrix Analysis and Applications vol 28 no1 pp 89ndash104 2006
[20] H Lee and Y Lim ldquoInvariant metrics contractions and nonlin-ear matrix equationsrdquo Nonlinearity vol 21 no 4 pp 857ndash8782008
[21] J Cai and G-L Chen ldquoOn the Hermitian positive definitesolutions of nonlinear matrix equation 119883
119904
+ 119860119883minus119905
119860 = 119876rdquoApplied Mathematics and Computation vol 217 no 1 pp 117ndash123 2010
[22] X-F Duan Q-W Wang and A-P Liao ldquoOn the matrix equa-tion arising in an interpolation problemrdquo Linear andMultilinearAlgebra vol 61 no 9 pp 1192ndash1205 2013
[23] Z-G Jia and M-S Wei ldquoSolvability and sensitivity analysisof polynomial matrix equation 119883
119904
+ 119860119879
119883119905
119860 = 119876rdquo AppliedMathematics and Computation vol 209 no 2 pp 230ndash2372009
[24] M-H Wang M-S Wei and S Hu ldquoThe extremal solution ofthe matrix equation 119883119904 + 119860lowast119883minus119902119860 = 119868rdquo Applied Mathematicsand Computation vol 220 pp 193ndash199 2013
[25] B Zhou G-B Cai and J Lam ldquoPositive definite solutions ofthe nonlinear matrix equation 119883 + 119860
119867
119883minus1
119860 = 119868rdquo AppliedMathematics and Computation vol 219 no 14 pp 7377ndash73912013
[26] X-Z Zhan Matrix Inequalities vol 1790 of Lecture Notes inMathematics Springer Berlin Germany 2002
[27] T Furuta ldquoOperator inequalities associated with Holder-McCarthy and Kantorovich inequalitiesrdquo Journal of Inequalitiesand Applications vol 1998 Article ID 234521 1998
[28] R A Horn and C R Johnson Topics in Matrix AnalysisCambridge University Press Cambridge UK 1991
[29] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990
[30] X-G Liu and H Gao ldquoOn the positive definite solutions of thematrix equations 119883119904 minus 119860119879119883minus119905119860 = 119868
119899rdquo Linear Algebra and Its
Applications vol 368 pp 83ndash97 2003[31] S-F Xu ldquoPerturbation analysis of the maximal solution of
the matrix equation 119860lowast
119883minus1
119860 = 119875rdquo Linear Algebra and ItsApplications vol 336 pp 61ndash70 2001
[32] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985
[33] S M El-Sayed and A C M Ran ldquoOn an iteration method forsolving a class of nonlinear matrix equationsrdquo SIAM Journal onMatrix Analysis and Applications vol 23 no 3 pp 632ndash6452001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
proved that (1) has a unique HPD solution when |119904| ge 1 ge |119905|and |119905119904| lt 1 See [21ndash25] formore recent results on nonlinearmatrix equations To the best of our knowledge (1) with 119904 lt 119905(without monotony in hand) has not been discussed Thesefacts motivate us to study polynomial matrix equation (1)
This paper is organized as follows In Section 2 we deducethe existence and uniqueness conditions of HPD solutionsof (1) in Section 3 we derive the algebraic perturbationtheory for the unique or maximal solution of (1) finallyin Section 4 we provide an iterative algorithm and twonumerical experiments
We begin with some notations used throughout thispaper F119898times119899 stands for the set of119898times119899matrices with elementson field F (F is R or C) If 119867 is a Hermitian matrix onF119899times119899 120582min(119867) and 120582max(119867) stand for the minimal and themaximal eigenvalues respectively Denote the singular valuesof a matrix 119860 isin F119898times119899 by 120590
1(119860) ge sdot sdot sdot ge 120590
119897(119860) ge 0
where 119897 = min119898 119899 Suppose that 119883 and 119884 are Hermitianmatrices we write 119883 ge 119884(119883 gt 119884) if 119883 minus 119884 is positivelysemidefinite (definite) and denote the matrices set 119883 | 119883 minus
120572119868 ge 0 and 120573119868 minus 119883 ge 0 by [120572119868 120573119868]
2 Solvability of Self-Adjoint PolynomialMatrix Equation
In this section we study the solvability theory of (1) assumingthat 119860 is nonsingular that is 120582min(119860
lowast
119860) gt 0 To do this weneed two simple but useful functions defined on the positiveabscissa axis
1198921(119909) = 119909
119904
minus 120582max (119860lowast
119860) 119909119905
minus 120582max (119876)
1198922(119909) = 119909
119904
minus 120582min (119860lowast
119860) 119909119905
minus 120582min (119876) (3)
The following two famous inequalities will be usedfrequently in the remaining of this paper
Lemma 1 (Lowner-Heinz inequality [26 Theorem 11]) If119860 ge 119861 ge 0 and 0 le 119903 le 1 then 119860119903 ge 119861119903
Lemma 2 (see [27 Theorem 21]) Let 119860 and 119861 be positiveoperators on a Hilbert space 119867 such that119872
1119868 ge 119860 ge 119898
1119868 gt
01198722119868 ge 119861 ge 119898
2119868 gt 0 and 0 lt 119860 le 119861 Then
119860119905
le (1198721
1198981
)
119905minus1
119861119905
119860119905
le (1198722
1198982
)
119905minus1
119861119905 (4)
hold for any 119905 ge 1
21 Maximal Solution of (1) with 119904 lt 119905 Now we derive anecessary condition and a sufficient condition for existenceof HPD solutions of (1) with 119904 lt 119905 With 119892
1(119909) and 119892
2(119909) in
hand we can easily get the distribution of eigenvalues of theHPD solution119883 of (1)
Theorem 3 Suppose that 120582max(119860lowast
119860) le (119904119905)((119905 minus 119904)
120582max(119876)119905)(119905minus119904)119904 and 119883 isin C119899times119899 is an HPD solution of (1) then
for any eigenvalue 120582(119883) of119883
1205731le 120582 (119883) le 120572
1or 1205722le 120582 (119883) le 120573
2 (5)
where 1205721 1205722are two positive roots of 119892
1(119909) and 120573
1 1205732are two
positive roots of 1198922(119909)
Proof From Theorem 3316(d) in Horn and Johnson [28]one can see that
120590119894(119860lowast
119883119905
119860) le 120590119894(119883119905
) 1205902
1(119860) that is
120582119894(119860lowast
119883119905
119860) le 120582119894(119883119905
) 120582max (119860lowast
119860)
119894 = 1 119899
(6)
If 119860 is nonsingular
120590119894(119883119905
) = 120590119894((119860minus1
)lowast
119860lowast
119883119905
119860119860minus1
) le 120590minus2
119899(119860) 120590119894(119860lowast
119883119905
119860)
(7)
That means
120590119894(119860lowast
119883119905
119860) ge 120590119894(119883119905
) 1205902
119899(119860) that is
120582119894(119860lowast
119883119905
119860) ge 120582119894(119883119905
) 120582min (119860lowast
119860)
119894 = 1 119899
(8)
The above equations still hold if119860 is singular since120590119899(119860) = 0
that is 120582min(119860lowast
119860) = 0 in this case Applying Weyl theoremin Horn and Johnson [29]119883119904 = 119876 + 119860
lowast
119883119905
119860 implies
120582(119883)119904
minus 120582max (119860lowast
119860) 120582(119883)119905
120582max (119876) le 0
120582(119883)119904
minus 120582min (119860lowast
119860) 120582(119883)119905
minus 120582min (119876) ge 0(9)
Define a function 119891(119909) = 119909119904
minus 1198862
119909119905
minus 119902 119886 gt 0 119902 gt
0 Then the only positive stationary point of 119891(119909) is 1199090=
((119905119904)1198862
)1(119904minus119905) If 1198862 le (119904119905)((119905 minus 119904)119902119905)
(119905minus119904)119904 119891(119909) has twopositive roots 119909
1and 119909
2 with 1199021119904 lt 119909
1le 1199090le 1199092lt 1198862(119904minus119905)
So 120582max(119860lowast
119860) le (119904119905)(119905minus119904)120582max(119876)119905(119905minus119904)119904 implies that 119892
1(119909)
has two roots 1205721 1205722gt 0 and 119892
2(119909) has two roots 120573
1 1205732gt 0
Since 1198922(119909) ge 119892
1(119909) (120582min(119876))
1119904
le 1205731le 1205721le 1205722le 1205732le
(120582min(119860lowast
119860))1(119904minus119905) Then from (9) we obtain (5)
If (1) has an HPD solution its eigenvalues may skipbetween [120573
1 1205721] and [120572
2 1205732] Next what we take more atten-
tion on is the HPD solution with its eigenvalues distributedonly on one interval
Theorem 4 Suppose that 120582max(119860lowast
119860) le (119904119905)((119905 minus 119904)
120582max(119876)119905)(119905minus119904)119904
(1) Equation (1) has an HPD solution119883 isin [1205731119868 1205721119868] and
if 120582min(119860lowast
119860) gt 119904120572119904minus1
1(119905120573119905minus1
1)minus1 such119883 exists uniquely
(2) Equation (1) has an HPD solution 119885 isin [1205722119868 1205732119868] and
if 120582min(119860lowast
119860) gt 119904120573119904minus1
2(119905120572119905minus1
2)minus1 such 119885 exists uniquely
Journal of Applied Mathematics 3
Proof (1) Let ℎ1(119883) = (119876 + 119860
lowast
119883119905
119860)1119904 where 119883 isin
[(120582min(119876))1119904
119868 (119904(120582max(119860lowast
119860)119905))1(119905minus119904)
119868] Lemmas 1 and 2and 119905 minus 119904 gt 0 imply
(120582min (119876))1119904
119868 le ℎ1(119883)
le 120582max (119876) + 120582max (119860lowast
119860)[119904
(120582max (119860lowast119860) 119905)
]
119905(119905minus119904)
1119904
119868
le [119904
120582max (119860lowast119860) 119905
]
119904(119905minus119904)times1119904
119868 = [119904
120582max (119860lowast119860) 119905
]
1(119905minus119904)
119868
(10)
Applying Brouwerrsquos fixed-point theorem ℎ1(119883) has a fixed
point 119883 isin [(120582min(119876))1119904
119868 (119904(120582max(119860lowast
119860)119905))1(119905minus119904)
119868] ThenfromTheorem 3119883 isin [120573
1119868 1205721119868]
We now prove the uniqueness of 119883 under the addi-tional condition that 120582min(119860
lowast
119860) gt 119904120572119904minus1
1(119905120573119905minus1
1)minus1 Suppose
119884 isin [(120582min(119876))1119904
119868 (119904(120582max(119860lowast
119860)119905))1(119905minus119904)
119868] is another HPDsolution of (1) and 119884 =119883 It has been known that
10038171003817100381710038171003817119883119905
minus 11988411990510038171003817100381710038171003817119865=100381710038171003817100381710038171003817(119860minus1
)lowast
(119883119904
minus 119884119904
)119860minus1100381710038171003817100381710038171003817119865
le (120582min (119860lowast
119860))minus11003817100381710038171003817119883119904
minus 1198841199041003817100381710038171003817119865
(11)
Then from 119883119904
minus 119884119904
119865le 119904120572119904minus1
1119883 minus 119884
119865and 119883119905 minus 119884119905
119865ge
119905120573119905minus1
1119883 minus 119884
119865
119883 minus 119884119865
le 119904120572119904minus1
1[119905120573119905minus1
1120582min (119860
lowast
119860)]minus1
119883 minus 119884119865lt 119883 minus 119884
119865
(12)
which is impossible Hence119883 = 119884(2) Let ℎ
2(119885) = [(119860
minus1
)lowast
(119885119904
minus 119876)119860minus1
]1119905 where 119885 isin
[1205722119868 1205732119868] ℎ2(119885) is continuous and
ℎ2(1205722119868) le ℎ
2(119885) le ℎ
2(1205732119868) (13)
because (119860minus1
)lowast
(120572119904
2119868 minus 119876)119860
minus1
le (119860minus1
)lowast
(119885119904
minus 119876)119860minus1
le
(119860minus1
)lowast
(120573119904
2119868minus119876)119860
minus1 By Lemmas 1 and 2 and Brouwerrsquos fixed-point theorem it is sufficient to prove ℎ
2(1205722119868) ge 120572
2119868 and
ℎ2(1205732119868) le 120573
2119868 in order for an HPD solution 119885 isin [120572
2119868 1205732119868]
to exist The existence of such 119885 follows from inequalities
ℎ2(1205722119868)
= [(119860minus1
)lowast
(120572119904
2119868 minus 119876)119860
minus1
]
1119905
ge [(119860minus1
)lowast
(120572119904
2119868 minus 120582max (119876) 119868) 119860
minus1
]
1119905
ge [(120582max (119860lowast
119860))minus1
(120572119904
2119868 minus 120582max (119876) 119868)]
1119905
= 1205722119868
ℎ2(1205732119868)
= [(119860minus1
)lowast
(120573119904
2119868 minus 119876)119860
minus1
]
1119905
le [(119860minus1
)lowast
(120573119904
2119868 minus 120582min (119876) 119868) 119860
minus1
]
1119905
le [(120582min (119860lowast
119860))minus1
(120573119904
2119868 minus 120582min (119876) 119868)]
1119905
= 1205732119868
(14)
Next we prove the uniqueness of 119885 under the additionalcondition that 120582min(119860
lowast
119860) gt 119904120573119904minus1
2(119905120572119905minus1
2)minus1 Suppose (1) has
two different HPD solutions 119885 and 119884 on [1205722119868 1205732119868] Then
10038171003817100381710038171003817119885119905
minus 11988411990510038171003817100381710038171003817119865=100381710038171003817100381710038171003817(119860minus1
)lowast
(119885119904
minus 119884119904
)119860minus1100381710038171003817100381710038171003817119865
le (120582min (119860lowast
119860))minus11003817100381710038171003817119885119904
minus 1198841199041003817100381710038171003817119865
le (120582min (119860lowast
119860))minus1
119904120573119904minus1
2119885 minus 119884
119865
(15)
Moreover if 120582min(119860lowast
119860) gt 119904120573119904minus1
2(119905120572119905minus1
2)minus1 applying the
inequality 119885119905 minus 119884119905119865ge 119905120572119905minus1
2119885 minus 119884
119865 we have
119885 minus 119884119865le (119905120572119905minus1
2120582min (119860
lowast
119860))minus1
119904120573119904minus1
2119885 minus 119884
119865lt 119885 minus 119884
119865
(16)
which is impossible Hence 119884 = 119885
The maximal solution (see eg [30 31]) of (1) is definedas follows
Definition 5 An HPD solution 119883119872
isin C119899times119899 of (1) is themaximal solution if for any HPD solution 119884 isin C119899times119899 of (1)there is119883
119872ge 119884
So the second term of Theorem 4 implies that the maxi-mal solution of (1) is on [120572
2119868 1205732119868]
Theorem 6 Suppose that 120582max(119860lowast
119860) le (119904119905)(119905(119905 minus
119904)120582max(119876))(119904minus119905)119904 and120582min(119860
lowast
119860) gt 119904120573119904minus1
2(119905120572119905minus1
2)minus1 then (1) has
a maximal solution 119883max isin [1205722119868 1205732119868] which can be computedby
119883119894= [(119860
minus1
)lowast
(119883119904
119894minus1minus 119876)119860
minus1
]
1119905
119894 = 1 2 (17)
with the initial value1198830= 1205732119868
Proof Let 120585 = (119905120572119905minus1
2120582min(119860
lowast
119860))minus1
119904120573119904minus1
2 then 120585 lt 1 From
the proof of Theorem 4 (2)
119905120572119905minus1
2
1003817100381710038171003817119883119894+1 minus 1198831198941003817100381710038171003817119865le10038171003817100381710038171003817119883119905
119894+1minus 119883119905
119894
10038171003817100381710038171003817119865
=100381710038171003817100381710038171003817(119860minus1
)lowast
(119883119904
119894minus 119883119904
119894minus1)119860minus1100381710038171003817100381710038171003817119865
le (120582min (119860lowast
119860))minus11003817100381710038171003817119883119904
119894minus 119883119904
119894minus1
1003817100381710038171003817119865
le (120582min (119860lowast
119860))minus1
119904120573119904minus1
2
1003817100381710038171003817119883119894 minus 119883119894minus11003817100381710038171003817119865
(18)
4 Journal of Applied Mathematics
Then1003817100381710038171003817119883119894+1 minus 119883119894
1003817100381710038171003817119865le 120585
1003817100381710038171003817119883119894 minus 119883119894minus11003817100381710038171003817119865le 12058511989410038171003817100381710038171198831 minus 1198830
1003817100381710038171003817119865 (19)
which indicates the convergence of matrix series 1198830 1198831
1198832 generated by (17)Set 119883
0= 1205732119868 Assuming 119883
119894isin [1205722119868 1205732119868] then from
inequalities (14) we have
1205722119868 le ℎ (120572
2119868) le 119883
119894+1
= [(119860minus1
)lowast
(119883119904
119894minus 119876)119860
minus1
]
1119905
le ℎ (1205732119868) le 120573
2119868
(20)
That means for any 119894 = 0 1 2 119883119894isin [1205722119868 1205732119868] By
Theorem 4 (2) we can see that 119883max = lim119894rarr+infin
119883119894is the
unique HPD solution of (1) on [1205722119868 1205732119868]
Now we prove the maximality of 119883max Suppose that 119883is an arbitrary HPD solution of (1) then 119883
0ge 119883 and
Theorem 3 implies1198831199050ge 119883119905 (since119883
0= 1205732119868) Assuming that
119883119905
119894ge 119883119905 Lemma 1 with 119904119905 lt 1 implies
119883119905
119894+1= (119860minus1
)lowast
[(119883119905
119894)119904119905
minus 119876]119860minus1
ge (119860minus1
)lowast
[(119883119905
)119904119905
minus 119876]119860minus1
= 119883119905
(21)
Then119883119905max = lim119894rarr+infin
119883119905
119894ge 119883119905 which implies that119883max ge 119883
by the Lowner-Heinz inequality
Note that similar iteration formula ever appeared in somepapers such as [20 21] for other nonlinear matrix equationsHere we firstly proved that the iteration form (17) preservesthe maximality of119883
119894over all HPD solutions of (1)
22 Unique Solution of (1) with 119904 ge 119905 If 119904 gt 119905 Lee and Lim[20 Theorem 94] show that (1) always has a unique HPDsolution denoted by 119883
119906 Now we give an upper bound and
a lower bound of 119883119906and suggest an iteration method for
computing119883119906
As defined in (3) 1198921(119909) and 119892
2(119909) with 119904 gt 119905 have unique
positive roots denoted by 1205741and 1205742 respectively
Since 1198921(120582(119883119906)) le 0 and 119892
2(120582(119883119906)) le 0 120574
2le 120582(119883
119906) le 1205741
Theorem 7 If 119904 gt 119905 (1) has a unique HPD solution119883119906isin [1205742119868 1205741119868] Let 119883
0= 1205741119868 or 120574
2119868 then matrix series
1198830 1198831 1198832 generated by
119883119894= (119876 + 119860
lowast
119883119905
119894minus1119860)1119904
119894 = 0 1 2 (22)
will converge to119883119906
Proof Weonly need to prove the convergence ofmatrix series1198830 1198831 1198832 Set119883
0= 1205741119868 From (22) we have
1198831= (119876 + 120574
119905
1119860lowast
119860)1119904
le (120582max (119876) + 120574119905
1120582max (119860
lowast
119860))1119904
119868 = 1205741119868
(23)
and then1198831199041le 119883119904
0 Assuming that119883119904
119894le 119883119904
119894minus1
119883119904
119894+1= 119876 + 119860
lowast
119883119905
119894119860 = 119876 + 119860
lowast
(119883119904
119894)119905119904
119860
le 119876 + 119860lowast
(119883119904
119894minus1)119905119904
119860 = 119883119904
119894
(24)
Then for any 119894 = 0 1 2 we have 119883119904119894+1
le 119883119904
119894and then
119883119894+1
le 119883119894by Lowner-Heinz inequality On the other hand
1198830ge 1205742119868 implies 119883
119894ge 1205742119868 for any 119894 = 0 1 2 because if
119883119894minus1
ge 1205742119868 then
119883119894= (119876 + 119860
lowast
119883119905
119894minus1119860)1119904
ge (119876 + 120574119905
2119860lowast
119860)1119904
ge (120582min (119876) + 120574119905
2120582min (119860
lowast
119860))1119904
119868 = 1205742119868
(25)
Then 1198830 1198831 1198832 with 119883
0= 1205741119868 is a decreasingly
monotone matrix series with a lower bound 1205742119868 Similarly we
can prove that 1198830 1198831 1198832 generated by (22) with 119883
0=
1205742119868 is an increasingly monotone matrix series with an upper
bound 1205741119868Therefore the convergence of 119883
0 1198831 1198832 has
been proved
From the above proof we can see that the iteration form(22) preserves the minimality (119883
0= 1205741119868) or maximality
(1198830= 1205742119868) of119883
119894in process
If 119904 = 119905 (1) can be reduced to a linear matrix equation119884minus119860lowast
119884119860 = 119876 which is the discrete-time algebraic Lyapunovequation (DALE) or Hermitian Stein equation [1 Page 5]assuming that 119884 = 119883
119904 It is well known that if 119860 is d-stable(see [1]) 119884 minus 119860
lowast
119884119860 = 119876 has a unique solution and matrixseries 119884
0 1198841 1198842 generated by 119884
119894+1= 119876+119860
lowast
119884119894119860 with an
initial value 1198840 will converge to the unique solution Besides
it is not difficult to get an expression of the unique solution119883119906= (suminfin
119895=0(119860lowast
)119895
119876119860119895
)1119904 applying [32 Theorem 1 Section
132] [1 Theorem 1118] and the results in Section 64 [28]Now we have presented the solvability theory of the self-
adjoint polynomial matrix equation (1) in three cases Ageneral iterative algorithm for its maximal solution (119904 lt 119905)or unique solution (119904 ge 119905) will be given in Section 4 Before itwe study the algebraic perturbation of themaximal or uniquesolution of (1)
3 Algebraic Perturbation Analysis
In this section we present the algebraic perturbation analysisof the HPD solution of (1) with respect to the perturbationof its coefficient matrices Similar to [30] we define theperturbed matrix equation of (1) as
119883119904
minus 119860lowast
119883119905
119860 = 119876 (26)
where 119860 = 119860 + Δ119860 isin C119899times119899 and 119876 = 119876 + Δ119876 isin C119899times119899 Wealways suppose that (1) has a maximal (or unique) solutiondenoted by 119883
119872isin [1205722119868 1205732119868] and (26) has a maximal (or
unique) solution denoted by119883119872isin [2119868 1205732119868]
Now we present the perturbation bound for 119883119872
when119904 = 119905 Define a function 120591
120591 (120572 120573) = 119904120572119904minus1
minus 119905120573119905minus1
1198602
2 (120572 120573) isin R
2
(27)
Journal of Applied Mathematics 5
Theorem 8 Let 120576 gt 0 be an arbitrary real number and 120591(2
1205732) ge 0 If
Δ119860119865lt (119860
2
2+2120576
3120591 (2 1205732)10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
minus119905
2
)
12
minus 1198602
Δ119876119865lt1
3120591 (2 1205732) 120576
(28)
then10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865lt 120576 (29)
Proof It is easy to induce that
10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865ge (
119904minus1
sum
119896=0
119904minus1minus119896
2120572119896
2)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
ge 119904119904minus1
2
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
10038171003817100381710038171003817119883119905
119872minus 119883119905
119872
10038171003817100381710038171003817119865le (
119905minus1
sum
119896=0
120573119905minus1minus119896
2120573119896
2)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le 119905120573119905minus1
2
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
(30)
Then from (1) and (26) we have
120591 (2 1205732)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le 21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ119860119865+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ1198602
119865+ Δ119876
119865
(31)
Since 120591(2 1205732) gt 0
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le (120591 (2 1205732))minus1
(21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ119860119865
+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ1198602
119865+ Δ119876
119865)
(32)
Then for an arbitrary 120576 gt 0 if Δ119860119865
lt (1198602
2+
(21205763)120591(2 1205732)119883119872minus119905
2)12
minus 1198602
and Δ119876119865
lt
(13)120591(2 1205732)120576 we have (29)
If 119904 = 119905 for an arbitrary 120576 gt 0 define
984858 (120576) = 1198602+ (119860
2
2+2120576
3120588)
12
(33)
where
120588 =10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
[119904119904minus1
2(1 minus 119860
2
2)]minus1
(34)
Theorem9 Let 120576 gt 0 be an arbitrary real number and 1198602lt
1 If
Δ119860119865lt2120576
3(120588984858 (120576))
minus1
Δ119876119865lt
120576
3120588
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
(35)
then10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865lt 120576 (36)
Table 1 Iteration CPU time (seconds) and residue for solving (1)with 119904 = 119905
(119904 119905)Algorithm 1 MONO
Ite CPU Res Ite CPU Res(2 1) 9 00541 45275119890 minus 13 200 21031 00016(1 2) 200 10275 20297119890 minus 07 mdash mdash mdash(8 5) 10 00716 59909119890 minus 13 200 22284 00034(5 8) 200 11048 31059119890 minus 05 mdash mdash mdash(30 15) 9 00743 52317119890 minus 13 200 23051 00029(15 30) 200 12865 20838119890 minus 08 mdash mdash mdash(300 150) 10 00886 79960119890 minus 13 200 22683 00031(150 300) 200 14187 28384119890 minus 07 mdash mdash mdash
Proof Similar to the proof of Theorem 8 we can induce that
(1 minus 1198602
2)10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865
le 21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ119860119865+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ1198602
119865+ Δ119876
119865
(37)
Then10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865
le (1 minus 1198602
2)minus1
(21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ119860119865
+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ1198602
119865+ Δ119876
119865)
(38)
With the help of (30) and (34) (38) implies10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865le 120588 (Δ119860
119865+ 2119860
2Δ119860119865+ Δ119876
119865) (39)
Then if Δ119860119865lt (21205763)(120588984858(120576))
minus1 and Δ119876119865lt (1205763120588)119883
119872119904
2
we have (36)
Theorems 8 and 9 make sure that the perturbation of119883119872
can be controlled if Δ119860 and Δ119876 have a proper upper bound
4 Algorithm and Numerical Experiments
In this section we give a general iterative algorithm for themaximal or unique solutions of (1) and two numerical exper-iments All reported results were obtained using MATLAB-R2012b on a personal computer with 24GHz Intel Core i7and 8GB 1600MHz DDR3
Example 10 Let matrices 119860 = rand(100) times 10minus2 and 119876 =
eye(100) With tol = 10minus12 and not more than 200 iterationswe apply Algorithm 1 to compute the maximal or uniqueHPD solutions of (1) with 119904 = 119905 and compare the results withthose by the iteration method from [33] (denoted by MONOin Table 1)
Table 1 shows iterations CPU times before convergenceand the residues of the computed HPD solution 119883 definedby
119890 (119904 119905) =
1003817100381710038171003817119883119904
minus 119860lowast
119883119905
119860 minus 1198761003817100381710038171003817119865
[119860 119876]119865
(40)
6 Journal of Applied Mathematics
Table 2 Iteration CPU time (seconds) and residue for solving (1) with 119904 = 119905 and different initial solutions
(119904 119905 1198830)
Algorithm 1 MONOIte CPU Res Ite CPU Res
(1 1 1205751119868119899) 20 00223 34947119890 minus 13 20 00202 34947119890 minus 13
(1 1 1205752119868119899) 31 00258 73027119890 minus 13 31 00305 73027119890 minus 13
(2 2 1205751119868119899) 20 01170 89978119890 minus 13 200 26421 00037
(2 2 1205752119868119899) 43 05475 96421119890 minus 13 200 27224 00037
(10 10 1205751119868119899) 29 01890 29296119890 minus 13 200 29717 00059
(10 10 1205752119868119899) 157 30859 71154119890 minus 13 200 29788 00059
Step 1 Compute 120582max(119860lowast
119860) 120582min(119860lowast
119860) 120582max(119876) 120582min(119876)Step 2 Input (3)Step 3 If 119904 lt 119905 run Steps 4-5 if 119905 lt 119904 run Steps 6-7 otherwise run Steps 8-9Step 4 Compute the roots 120572
1 1205722of 1198921(119909) and 120573
1 1205732of 1198922(119909) respectively
Step 5 Let 1198830= 1205732119868 run (17)
Step 6 Compute the root 1205741of 1198921(119909) and the root 120574
2of 1198922(119909) respectively
Step 7 Let 1198850= 1205741119868 run (22)
Step 8 Compute the root 1205751of 1198921(119909) and the root 120575
2of 1198922(119909) respectively
Step 9 If 120582max(119860lowast
119860) lt 1 and 1205751ge 1205752 then let 119883
0= 1205751119868 and run (22)
Algorithm 1 Given matrices 119860119876 isin C119899times119899 and positive integers 119904 119905
From Table 1 we can see that it takes more iterations andCPU times to solve themaximal solution of (1) with 119904 lt 119905 thanto solve the unique solution of (1) with 119904 gt 119905 At the same timethe accuracy of the latter is better than the formerMONOcannot be used to solve (1) with 119904 lt 119905 and it costs more iterationsand CPU times than Algorithm 1 when solving (1) with 119904 gt 119905
Now we use Example 41 of [33] to test our method
Example 11 Let 119860 = 05119861119861infin
with 119861 = [119861119894119895]119899times119899 119887119894119895= 119894 +
119895 + 1 and let 119876 = eye(119899) with 119899 = 100 We solve (1) with119904 = 119905 and with two different initial solutions The iterationsCPU times and the residues of the computation are reportedin Table 2
Table 2 shows that for Algorithm 1 the choice 1198830= 1205751119868119899
is better than 1198830= 1205752119868119899 When 119904 and 119905 rise MONO might
lose its efficiency It seems not proper to apply the iterationmethod designed for 119884 minus 119860lowast119884119905119904119860 = 119876 with 119884 = 119883
119904 to solve119883119904
minus 119860lowast
119883119905
119860 = 119876 although they are equivalent to each otherin theory
5 Conclusion
In this paper we considered the solvability of the self-adjointpolynomial matrix equation (1) Sufficient conditions weregiven to guarantee the existence of the maximal or uniqueHPD solutions of (1) The algebraic perturbation analysisincluding perturbation bounds was also developed for (1)under the perturbation of given coefficient matrices At lasta general iterative algorithm with maximality preserved inprocess was presented for the maximal or unique solutionwith two numerical experiments reported
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Zhigang Jiarsquos research was supported in part by NationalNatural Science Foundation of China under Grants 11201193and 11171289 and a project funded by the Priority AcademicProgram Development of Jiangsu Higher EducationInstitutions Minghui Wangrsquos research was supported inpart by the National Natural Science Foundation of China(Grant no 11001144) the Science and Technology Programof Shandong Universities of China (J11LA04) and theResearch Award Fund for Outstanding Young Scientists ofShandong Province in China (BS2012DX009) Sitao Lingrsquosresearch was supported in part by National Natural ScienceFoundations of China under Grant 11301529 PostdoctoralScience Foundation of China under Grant 2013M540472 andJiangsu Planned Projects for Postdoctoral Research Funds1302036CThe authors would like to thank three anonymousreferees for giving valuable comments and suggestions
References
[1] H Abou-Kandil G Freiling V Ionescu and G Jank MatrixRiccati Equations in Control and Systems Theory BirkhauserBasel Switzerland 2003
[2] I Gohberg P Lancaster and L Rodman Matrix PolynomialsAcademic Press New York NY USA 1982
[3] P Benner A J Laub and V Mehrmann ldquoBenchmarks for thenumerical solution of algebraic Riccati equationsrdquo IEEE ControlSystems Magazine vol 17 no 5 pp 18ndash28 1997
Journal of Applied Mathematics 7
[4] S Bittanti A Laub and J CWillems EdsTheRiccati EquationCommunications and Control Engineering Series SpringerBerlin Germany 1991
[5] P Lancaster andL RodmanAlgebraic Riccati Equations OxfordScience Publications Oxford University Press New York NYUSA 1995
[6] A J Laub ldquoInvariant subspace methods for the numerical solu-tion of Riccati equationsrdquo inThe Riccati Equation S Bittanti AJ Laub and J C Willems Eds Communications and ControlEngineering pp 163ndash196 Springer Berlin Germany 1991
[7] A J Laub ldquoA Schur method for solving algebraic Riccatiequationsrdquo IEEE Transactions on Automatic Control vol 24 no6 pp 913ndash921 1979
[8] M Kimura ldquoDoubling algorithm for continuous-time algebraicRiccati equationrdquo International Journal of Systems Science vol20 no 2 pp 191ndash202 1989
[9] E K-W Chu H-Y FanW-W Lin and C-SWang ldquoStructure-preserving algorithms for periodic discrete-time algebraic Ric-cati equationsrdquo International Journal of Control vol 77 no 8pp 767ndash788 2004
[10] E K-W Chu H-Y Fan andW-W Lin ldquoA structure-preservingdoubling algorithm for continuous-time algebraic Riccati equa-tionsrdquo Linear Algebra and Its Applications vol 396 pp 55ndash802005
[11] N J Higham ldquoPerturbation theory and backward error for119860119883 minus 119883119861 = 119862rdquo BIT Numerical Mathematics vol 33 no 1 pp124ndash136 1993
[12] M Konstantinov and P Petkov ldquoNote on perturbation theoryfor algebraic Riccati equationsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 21 no 1 pp 327ndash354 2000
[13] J-G Sun ldquoResidual bounds of approximate solutions of thealgebraic Riccati equationrdquoNumerischeMathematik vol 76 no2 pp 249ndash263 1997
[14] J-G Sun ldquoBackward error for the discrete-time algebraicRiccati equationrdquo Linear Algebra and Its Applications vol 259pp 183ndash208 1997
[15] J-G Sun ldquoBackward perturbation analysis of the periodicdiscrete-time algebraic Riccati equationrdquo SIAM Journal onMatrix Analysis and Applications vol 26 no 1 pp 1ndash19 2004
[16] V Mehrmann ldquoA step towards a unified treatment of continu-ous and discrete time control problemsrdquo Linear Algebra and ItsApplications vol 241ndash243 pp 749ndash779 1996
[17] H-G Xu ldquoTransformations between discrete-time andcontinuous-time algebraic Riccati equationsrdquo Linear Algebraand Its Applications vol 425 no 1 pp 77ndash101 2007
[18] A C M Ran and M C B Reurings ldquoOn the nonlinear matrixequation 119883 + 119860
lowastF (119883)119860 = 119876 solutions and perturbationtheoryrdquo Linear Algebra and Its Applications vol 346 pp 15ndash262002
[19] A C M Ran M C B Reurings and L Rodman ldquoA pertur-bation analysis for nonlinear selfadjoint operator equationsrdquoSIAM Journal on Matrix Analysis and Applications vol 28 no1 pp 89ndash104 2006
[20] H Lee and Y Lim ldquoInvariant metrics contractions and nonlin-ear matrix equationsrdquo Nonlinearity vol 21 no 4 pp 857ndash8782008
[21] J Cai and G-L Chen ldquoOn the Hermitian positive definitesolutions of nonlinear matrix equation 119883
119904
+ 119860119883minus119905
119860 = 119876rdquoApplied Mathematics and Computation vol 217 no 1 pp 117ndash123 2010
[22] X-F Duan Q-W Wang and A-P Liao ldquoOn the matrix equa-tion arising in an interpolation problemrdquo Linear andMultilinearAlgebra vol 61 no 9 pp 1192ndash1205 2013
[23] Z-G Jia and M-S Wei ldquoSolvability and sensitivity analysisof polynomial matrix equation 119883
119904
+ 119860119879
119883119905
119860 = 119876rdquo AppliedMathematics and Computation vol 209 no 2 pp 230ndash2372009
[24] M-H Wang M-S Wei and S Hu ldquoThe extremal solution ofthe matrix equation 119883119904 + 119860lowast119883minus119902119860 = 119868rdquo Applied Mathematicsand Computation vol 220 pp 193ndash199 2013
[25] B Zhou G-B Cai and J Lam ldquoPositive definite solutions ofthe nonlinear matrix equation 119883 + 119860
119867
119883minus1
119860 = 119868rdquo AppliedMathematics and Computation vol 219 no 14 pp 7377ndash73912013
[26] X-Z Zhan Matrix Inequalities vol 1790 of Lecture Notes inMathematics Springer Berlin Germany 2002
[27] T Furuta ldquoOperator inequalities associated with Holder-McCarthy and Kantorovich inequalitiesrdquo Journal of Inequalitiesand Applications vol 1998 Article ID 234521 1998
[28] R A Horn and C R Johnson Topics in Matrix AnalysisCambridge University Press Cambridge UK 1991
[29] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990
[30] X-G Liu and H Gao ldquoOn the positive definite solutions of thematrix equations 119883119904 minus 119860119879119883minus119905119860 = 119868
119899rdquo Linear Algebra and Its
Applications vol 368 pp 83ndash97 2003[31] S-F Xu ldquoPerturbation analysis of the maximal solution of
the matrix equation 119860lowast
119883minus1
119860 = 119875rdquo Linear Algebra and ItsApplications vol 336 pp 61ndash70 2001
[32] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985
[33] S M El-Sayed and A C M Ran ldquoOn an iteration method forsolving a class of nonlinear matrix equationsrdquo SIAM Journal onMatrix Analysis and Applications vol 23 no 3 pp 632ndash6452001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
Proof (1) Let ℎ1(119883) = (119876 + 119860
lowast
119883119905
119860)1119904 where 119883 isin
[(120582min(119876))1119904
119868 (119904(120582max(119860lowast
119860)119905))1(119905minus119904)
119868] Lemmas 1 and 2and 119905 minus 119904 gt 0 imply
(120582min (119876))1119904
119868 le ℎ1(119883)
le 120582max (119876) + 120582max (119860lowast
119860)[119904
(120582max (119860lowast119860) 119905)
]
119905(119905minus119904)
1119904
119868
le [119904
120582max (119860lowast119860) 119905
]
119904(119905minus119904)times1119904
119868 = [119904
120582max (119860lowast119860) 119905
]
1(119905minus119904)
119868
(10)
Applying Brouwerrsquos fixed-point theorem ℎ1(119883) has a fixed
point 119883 isin [(120582min(119876))1119904
119868 (119904(120582max(119860lowast
119860)119905))1(119905minus119904)
119868] ThenfromTheorem 3119883 isin [120573
1119868 1205721119868]
We now prove the uniqueness of 119883 under the addi-tional condition that 120582min(119860
lowast
119860) gt 119904120572119904minus1
1(119905120573119905minus1
1)minus1 Suppose
119884 isin [(120582min(119876))1119904
119868 (119904(120582max(119860lowast
119860)119905))1(119905minus119904)
119868] is another HPDsolution of (1) and 119884 =119883 It has been known that
10038171003817100381710038171003817119883119905
minus 11988411990510038171003817100381710038171003817119865=100381710038171003817100381710038171003817(119860minus1
)lowast
(119883119904
minus 119884119904
)119860minus1100381710038171003817100381710038171003817119865
le (120582min (119860lowast
119860))minus11003817100381710038171003817119883119904
minus 1198841199041003817100381710038171003817119865
(11)
Then from 119883119904
minus 119884119904
119865le 119904120572119904minus1
1119883 minus 119884
119865and 119883119905 minus 119884119905
119865ge
119905120573119905minus1
1119883 minus 119884
119865
119883 minus 119884119865
le 119904120572119904minus1
1[119905120573119905minus1
1120582min (119860
lowast
119860)]minus1
119883 minus 119884119865lt 119883 minus 119884
119865
(12)
which is impossible Hence119883 = 119884(2) Let ℎ
2(119885) = [(119860
minus1
)lowast
(119885119904
minus 119876)119860minus1
]1119905 where 119885 isin
[1205722119868 1205732119868] ℎ2(119885) is continuous and
ℎ2(1205722119868) le ℎ
2(119885) le ℎ
2(1205732119868) (13)
because (119860minus1
)lowast
(120572119904
2119868 minus 119876)119860
minus1
le (119860minus1
)lowast
(119885119904
minus 119876)119860minus1
le
(119860minus1
)lowast
(120573119904
2119868minus119876)119860
minus1 By Lemmas 1 and 2 and Brouwerrsquos fixed-point theorem it is sufficient to prove ℎ
2(1205722119868) ge 120572
2119868 and
ℎ2(1205732119868) le 120573
2119868 in order for an HPD solution 119885 isin [120572
2119868 1205732119868]
to exist The existence of such 119885 follows from inequalities
ℎ2(1205722119868)
= [(119860minus1
)lowast
(120572119904
2119868 minus 119876)119860
minus1
]
1119905
ge [(119860minus1
)lowast
(120572119904
2119868 minus 120582max (119876) 119868) 119860
minus1
]
1119905
ge [(120582max (119860lowast
119860))minus1
(120572119904
2119868 minus 120582max (119876) 119868)]
1119905
= 1205722119868
ℎ2(1205732119868)
= [(119860minus1
)lowast
(120573119904
2119868 minus 119876)119860
minus1
]
1119905
le [(119860minus1
)lowast
(120573119904
2119868 minus 120582min (119876) 119868) 119860
minus1
]
1119905
le [(120582min (119860lowast
119860))minus1
(120573119904
2119868 minus 120582min (119876) 119868)]
1119905
= 1205732119868
(14)
Next we prove the uniqueness of 119885 under the additionalcondition that 120582min(119860
lowast
119860) gt 119904120573119904minus1
2(119905120572119905minus1
2)minus1 Suppose (1) has
two different HPD solutions 119885 and 119884 on [1205722119868 1205732119868] Then
10038171003817100381710038171003817119885119905
minus 11988411990510038171003817100381710038171003817119865=100381710038171003817100381710038171003817(119860minus1
)lowast
(119885119904
minus 119884119904
)119860minus1100381710038171003817100381710038171003817119865
le (120582min (119860lowast
119860))minus11003817100381710038171003817119885119904
minus 1198841199041003817100381710038171003817119865
le (120582min (119860lowast
119860))minus1
119904120573119904minus1
2119885 minus 119884
119865
(15)
Moreover if 120582min(119860lowast
119860) gt 119904120573119904minus1
2(119905120572119905minus1
2)minus1 applying the
inequality 119885119905 minus 119884119905119865ge 119905120572119905minus1
2119885 minus 119884
119865 we have
119885 minus 119884119865le (119905120572119905minus1
2120582min (119860
lowast
119860))minus1
119904120573119904minus1
2119885 minus 119884
119865lt 119885 minus 119884
119865
(16)
which is impossible Hence 119884 = 119885
The maximal solution (see eg [30 31]) of (1) is definedas follows
Definition 5 An HPD solution 119883119872
isin C119899times119899 of (1) is themaximal solution if for any HPD solution 119884 isin C119899times119899 of (1)there is119883
119872ge 119884
So the second term of Theorem 4 implies that the maxi-mal solution of (1) is on [120572
2119868 1205732119868]
Theorem 6 Suppose that 120582max(119860lowast
119860) le (119904119905)(119905(119905 minus
119904)120582max(119876))(119904minus119905)119904 and120582min(119860
lowast
119860) gt 119904120573119904minus1
2(119905120572119905minus1
2)minus1 then (1) has
a maximal solution 119883max isin [1205722119868 1205732119868] which can be computedby
119883119894= [(119860
minus1
)lowast
(119883119904
119894minus1minus 119876)119860
minus1
]
1119905
119894 = 1 2 (17)
with the initial value1198830= 1205732119868
Proof Let 120585 = (119905120572119905minus1
2120582min(119860
lowast
119860))minus1
119904120573119904minus1
2 then 120585 lt 1 From
the proof of Theorem 4 (2)
119905120572119905minus1
2
1003817100381710038171003817119883119894+1 minus 1198831198941003817100381710038171003817119865le10038171003817100381710038171003817119883119905
119894+1minus 119883119905
119894
10038171003817100381710038171003817119865
=100381710038171003817100381710038171003817(119860minus1
)lowast
(119883119904
119894minus 119883119904
119894minus1)119860minus1100381710038171003817100381710038171003817119865
le (120582min (119860lowast
119860))minus11003817100381710038171003817119883119904
119894minus 119883119904
119894minus1
1003817100381710038171003817119865
le (120582min (119860lowast
119860))minus1
119904120573119904minus1
2
1003817100381710038171003817119883119894 minus 119883119894minus11003817100381710038171003817119865
(18)
4 Journal of Applied Mathematics
Then1003817100381710038171003817119883119894+1 minus 119883119894
1003817100381710038171003817119865le 120585
1003817100381710038171003817119883119894 minus 119883119894minus11003817100381710038171003817119865le 12058511989410038171003817100381710038171198831 minus 1198830
1003817100381710038171003817119865 (19)
which indicates the convergence of matrix series 1198830 1198831
1198832 generated by (17)Set 119883
0= 1205732119868 Assuming 119883
119894isin [1205722119868 1205732119868] then from
inequalities (14) we have
1205722119868 le ℎ (120572
2119868) le 119883
119894+1
= [(119860minus1
)lowast
(119883119904
119894minus 119876)119860
minus1
]
1119905
le ℎ (1205732119868) le 120573
2119868
(20)
That means for any 119894 = 0 1 2 119883119894isin [1205722119868 1205732119868] By
Theorem 4 (2) we can see that 119883max = lim119894rarr+infin
119883119894is the
unique HPD solution of (1) on [1205722119868 1205732119868]
Now we prove the maximality of 119883max Suppose that 119883is an arbitrary HPD solution of (1) then 119883
0ge 119883 and
Theorem 3 implies1198831199050ge 119883119905 (since119883
0= 1205732119868) Assuming that
119883119905
119894ge 119883119905 Lemma 1 with 119904119905 lt 1 implies
119883119905
119894+1= (119860minus1
)lowast
[(119883119905
119894)119904119905
minus 119876]119860minus1
ge (119860minus1
)lowast
[(119883119905
)119904119905
minus 119876]119860minus1
= 119883119905
(21)
Then119883119905max = lim119894rarr+infin
119883119905
119894ge 119883119905 which implies that119883max ge 119883
by the Lowner-Heinz inequality
Note that similar iteration formula ever appeared in somepapers such as [20 21] for other nonlinear matrix equationsHere we firstly proved that the iteration form (17) preservesthe maximality of119883
119894over all HPD solutions of (1)
22 Unique Solution of (1) with 119904 ge 119905 If 119904 gt 119905 Lee and Lim[20 Theorem 94] show that (1) always has a unique HPDsolution denoted by 119883
119906 Now we give an upper bound and
a lower bound of 119883119906and suggest an iteration method for
computing119883119906
As defined in (3) 1198921(119909) and 119892
2(119909) with 119904 gt 119905 have unique
positive roots denoted by 1205741and 1205742 respectively
Since 1198921(120582(119883119906)) le 0 and 119892
2(120582(119883119906)) le 0 120574
2le 120582(119883
119906) le 1205741
Theorem 7 If 119904 gt 119905 (1) has a unique HPD solution119883119906isin [1205742119868 1205741119868] Let 119883
0= 1205741119868 or 120574
2119868 then matrix series
1198830 1198831 1198832 generated by
119883119894= (119876 + 119860
lowast
119883119905
119894minus1119860)1119904
119894 = 0 1 2 (22)
will converge to119883119906
Proof Weonly need to prove the convergence ofmatrix series1198830 1198831 1198832 Set119883
0= 1205741119868 From (22) we have
1198831= (119876 + 120574
119905
1119860lowast
119860)1119904
le (120582max (119876) + 120574119905
1120582max (119860
lowast
119860))1119904
119868 = 1205741119868
(23)
and then1198831199041le 119883119904
0 Assuming that119883119904
119894le 119883119904
119894minus1
119883119904
119894+1= 119876 + 119860
lowast
119883119905
119894119860 = 119876 + 119860
lowast
(119883119904
119894)119905119904
119860
le 119876 + 119860lowast
(119883119904
119894minus1)119905119904
119860 = 119883119904
119894
(24)
Then for any 119894 = 0 1 2 we have 119883119904119894+1
le 119883119904
119894and then
119883119894+1
le 119883119894by Lowner-Heinz inequality On the other hand
1198830ge 1205742119868 implies 119883
119894ge 1205742119868 for any 119894 = 0 1 2 because if
119883119894minus1
ge 1205742119868 then
119883119894= (119876 + 119860
lowast
119883119905
119894minus1119860)1119904
ge (119876 + 120574119905
2119860lowast
119860)1119904
ge (120582min (119876) + 120574119905
2120582min (119860
lowast
119860))1119904
119868 = 1205742119868
(25)
Then 1198830 1198831 1198832 with 119883
0= 1205741119868 is a decreasingly
monotone matrix series with a lower bound 1205742119868 Similarly we
can prove that 1198830 1198831 1198832 generated by (22) with 119883
0=
1205742119868 is an increasingly monotone matrix series with an upper
bound 1205741119868Therefore the convergence of 119883
0 1198831 1198832 has
been proved
From the above proof we can see that the iteration form(22) preserves the minimality (119883
0= 1205741119868) or maximality
(1198830= 1205742119868) of119883
119894in process
If 119904 = 119905 (1) can be reduced to a linear matrix equation119884minus119860lowast
119884119860 = 119876 which is the discrete-time algebraic Lyapunovequation (DALE) or Hermitian Stein equation [1 Page 5]assuming that 119884 = 119883
119904 It is well known that if 119860 is d-stable(see [1]) 119884 minus 119860
lowast
119884119860 = 119876 has a unique solution and matrixseries 119884
0 1198841 1198842 generated by 119884
119894+1= 119876+119860
lowast
119884119894119860 with an
initial value 1198840 will converge to the unique solution Besides
it is not difficult to get an expression of the unique solution119883119906= (suminfin
119895=0(119860lowast
)119895
119876119860119895
)1119904 applying [32 Theorem 1 Section
132] [1 Theorem 1118] and the results in Section 64 [28]Now we have presented the solvability theory of the self-
adjoint polynomial matrix equation (1) in three cases Ageneral iterative algorithm for its maximal solution (119904 lt 119905)or unique solution (119904 ge 119905) will be given in Section 4 Before itwe study the algebraic perturbation of themaximal or uniquesolution of (1)
3 Algebraic Perturbation Analysis
In this section we present the algebraic perturbation analysisof the HPD solution of (1) with respect to the perturbationof its coefficient matrices Similar to [30] we define theperturbed matrix equation of (1) as
119883119904
minus 119860lowast
119883119905
119860 = 119876 (26)
where 119860 = 119860 + Δ119860 isin C119899times119899 and 119876 = 119876 + Δ119876 isin C119899times119899 Wealways suppose that (1) has a maximal (or unique) solutiondenoted by 119883
119872isin [1205722119868 1205732119868] and (26) has a maximal (or
unique) solution denoted by119883119872isin [2119868 1205732119868]
Now we present the perturbation bound for 119883119872
when119904 = 119905 Define a function 120591
120591 (120572 120573) = 119904120572119904minus1
minus 119905120573119905minus1
1198602
2 (120572 120573) isin R
2
(27)
Journal of Applied Mathematics 5
Theorem 8 Let 120576 gt 0 be an arbitrary real number and 120591(2
1205732) ge 0 If
Δ119860119865lt (119860
2
2+2120576
3120591 (2 1205732)10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
minus119905
2
)
12
minus 1198602
Δ119876119865lt1
3120591 (2 1205732) 120576
(28)
then10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865lt 120576 (29)
Proof It is easy to induce that
10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865ge (
119904minus1
sum
119896=0
119904minus1minus119896
2120572119896
2)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
ge 119904119904minus1
2
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
10038171003817100381710038171003817119883119905
119872minus 119883119905
119872
10038171003817100381710038171003817119865le (
119905minus1
sum
119896=0
120573119905minus1minus119896
2120573119896
2)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le 119905120573119905minus1
2
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
(30)
Then from (1) and (26) we have
120591 (2 1205732)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le 21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ119860119865+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ1198602
119865+ Δ119876
119865
(31)
Since 120591(2 1205732) gt 0
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le (120591 (2 1205732))minus1
(21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ119860119865
+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ1198602
119865+ Δ119876
119865)
(32)
Then for an arbitrary 120576 gt 0 if Δ119860119865
lt (1198602
2+
(21205763)120591(2 1205732)119883119872minus119905
2)12
minus 1198602
and Δ119876119865
lt
(13)120591(2 1205732)120576 we have (29)
If 119904 = 119905 for an arbitrary 120576 gt 0 define
984858 (120576) = 1198602+ (119860
2
2+2120576
3120588)
12
(33)
where
120588 =10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
[119904119904minus1
2(1 minus 119860
2
2)]minus1
(34)
Theorem9 Let 120576 gt 0 be an arbitrary real number and 1198602lt
1 If
Δ119860119865lt2120576
3(120588984858 (120576))
minus1
Δ119876119865lt
120576
3120588
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
(35)
then10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865lt 120576 (36)
Table 1 Iteration CPU time (seconds) and residue for solving (1)with 119904 = 119905
(119904 119905)Algorithm 1 MONO
Ite CPU Res Ite CPU Res(2 1) 9 00541 45275119890 minus 13 200 21031 00016(1 2) 200 10275 20297119890 minus 07 mdash mdash mdash(8 5) 10 00716 59909119890 minus 13 200 22284 00034(5 8) 200 11048 31059119890 minus 05 mdash mdash mdash(30 15) 9 00743 52317119890 minus 13 200 23051 00029(15 30) 200 12865 20838119890 minus 08 mdash mdash mdash(300 150) 10 00886 79960119890 minus 13 200 22683 00031(150 300) 200 14187 28384119890 minus 07 mdash mdash mdash
Proof Similar to the proof of Theorem 8 we can induce that
(1 minus 1198602
2)10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865
le 21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ119860119865+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ1198602
119865+ Δ119876
119865
(37)
Then10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865
le (1 minus 1198602
2)minus1
(21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ119860119865
+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ1198602
119865+ Δ119876
119865)
(38)
With the help of (30) and (34) (38) implies10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865le 120588 (Δ119860
119865+ 2119860
2Δ119860119865+ Δ119876
119865) (39)
Then if Δ119860119865lt (21205763)(120588984858(120576))
minus1 and Δ119876119865lt (1205763120588)119883
119872119904
2
we have (36)
Theorems 8 and 9 make sure that the perturbation of119883119872
can be controlled if Δ119860 and Δ119876 have a proper upper bound
4 Algorithm and Numerical Experiments
In this section we give a general iterative algorithm for themaximal or unique solutions of (1) and two numerical exper-iments All reported results were obtained using MATLAB-R2012b on a personal computer with 24GHz Intel Core i7and 8GB 1600MHz DDR3
Example 10 Let matrices 119860 = rand(100) times 10minus2 and 119876 =
eye(100) With tol = 10minus12 and not more than 200 iterationswe apply Algorithm 1 to compute the maximal or uniqueHPD solutions of (1) with 119904 = 119905 and compare the results withthose by the iteration method from [33] (denoted by MONOin Table 1)
Table 1 shows iterations CPU times before convergenceand the residues of the computed HPD solution 119883 definedby
119890 (119904 119905) =
1003817100381710038171003817119883119904
minus 119860lowast
119883119905
119860 minus 1198761003817100381710038171003817119865
[119860 119876]119865
(40)
6 Journal of Applied Mathematics
Table 2 Iteration CPU time (seconds) and residue for solving (1) with 119904 = 119905 and different initial solutions
(119904 119905 1198830)
Algorithm 1 MONOIte CPU Res Ite CPU Res
(1 1 1205751119868119899) 20 00223 34947119890 minus 13 20 00202 34947119890 minus 13
(1 1 1205752119868119899) 31 00258 73027119890 minus 13 31 00305 73027119890 minus 13
(2 2 1205751119868119899) 20 01170 89978119890 minus 13 200 26421 00037
(2 2 1205752119868119899) 43 05475 96421119890 minus 13 200 27224 00037
(10 10 1205751119868119899) 29 01890 29296119890 minus 13 200 29717 00059
(10 10 1205752119868119899) 157 30859 71154119890 minus 13 200 29788 00059
Step 1 Compute 120582max(119860lowast
119860) 120582min(119860lowast
119860) 120582max(119876) 120582min(119876)Step 2 Input (3)Step 3 If 119904 lt 119905 run Steps 4-5 if 119905 lt 119904 run Steps 6-7 otherwise run Steps 8-9Step 4 Compute the roots 120572
1 1205722of 1198921(119909) and 120573
1 1205732of 1198922(119909) respectively
Step 5 Let 1198830= 1205732119868 run (17)
Step 6 Compute the root 1205741of 1198921(119909) and the root 120574
2of 1198922(119909) respectively
Step 7 Let 1198850= 1205741119868 run (22)
Step 8 Compute the root 1205751of 1198921(119909) and the root 120575
2of 1198922(119909) respectively
Step 9 If 120582max(119860lowast
119860) lt 1 and 1205751ge 1205752 then let 119883
0= 1205751119868 and run (22)
Algorithm 1 Given matrices 119860119876 isin C119899times119899 and positive integers 119904 119905
From Table 1 we can see that it takes more iterations andCPU times to solve themaximal solution of (1) with 119904 lt 119905 thanto solve the unique solution of (1) with 119904 gt 119905 At the same timethe accuracy of the latter is better than the formerMONOcannot be used to solve (1) with 119904 lt 119905 and it costs more iterationsand CPU times than Algorithm 1 when solving (1) with 119904 gt 119905
Now we use Example 41 of [33] to test our method
Example 11 Let 119860 = 05119861119861infin
with 119861 = [119861119894119895]119899times119899 119887119894119895= 119894 +
119895 + 1 and let 119876 = eye(119899) with 119899 = 100 We solve (1) with119904 = 119905 and with two different initial solutions The iterationsCPU times and the residues of the computation are reportedin Table 2
Table 2 shows that for Algorithm 1 the choice 1198830= 1205751119868119899
is better than 1198830= 1205752119868119899 When 119904 and 119905 rise MONO might
lose its efficiency It seems not proper to apply the iterationmethod designed for 119884 minus 119860lowast119884119905119904119860 = 119876 with 119884 = 119883
119904 to solve119883119904
minus 119860lowast
119883119905
119860 = 119876 although they are equivalent to each otherin theory
5 Conclusion
In this paper we considered the solvability of the self-adjointpolynomial matrix equation (1) Sufficient conditions weregiven to guarantee the existence of the maximal or uniqueHPD solutions of (1) The algebraic perturbation analysisincluding perturbation bounds was also developed for (1)under the perturbation of given coefficient matrices At lasta general iterative algorithm with maximality preserved inprocess was presented for the maximal or unique solutionwith two numerical experiments reported
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Zhigang Jiarsquos research was supported in part by NationalNatural Science Foundation of China under Grants 11201193and 11171289 and a project funded by the Priority AcademicProgram Development of Jiangsu Higher EducationInstitutions Minghui Wangrsquos research was supported inpart by the National Natural Science Foundation of China(Grant no 11001144) the Science and Technology Programof Shandong Universities of China (J11LA04) and theResearch Award Fund for Outstanding Young Scientists ofShandong Province in China (BS2012DX009) Sitao Lingrsquosresearch was supported in part by National Natural ScienceFoundations of China under Grant 11301529 PostdoctoralScience Foundation of China under Grant 2013M540472 andJiangsu Planned Projects for Postdoctoral Research Funds1302036CThe authors would like to thank three anonymousreferees for giving valuable comments and suggestions
References
[1] H Abou-Kandil G Freiling V Ionescu and G Jank MatrixRiccati Equations in Control and Systems Theory BirkhauserBasel Switzerland 2003
[2] I Gohberg P Lancaster and L Rodman Matrix PolynomialsAcademic Press New York NY USA 1982
[3] P Benner A J Laub and V Mehrmann ldquoBenchmarks for thenumerical solution of algebraic Riccati equationsrdquo IEEE ControlSystems Magazine vol 17 no 5 pp 18ndash28 1997
Journal of Applied Mathematics 7
[4] S Bittanti A Laub and J CWillems EdsTheRiccati EquationCommunications and Control Engineering Series SpringerBerlin Germany 1991
[5] P Lancaster andL RodmanAlgebraic Riccati Equations OxfordScience Publications Oxford University Press New York NYUSA 1995
[6] A J Laub ldquoInvariant subspace methods for the numerical solu-tion of Riccati equationsrdquo inThe Riccati Equation S Bittanti AJ Laub and J C Willems Eds Communications and ControlEngineering pp 163ndash196 Springer Berlin Germany 1991
[7] A J Laub ldquoA Schur method for solving algebraic Riccatiequationsrdquo IEEE Transactions on Automatic Control vol 24 no6 pp 913ndash921 1979
[8] M Kimura ldquoDoubling algorithm for continuous-time algebraicRiccati equationrdquo International Journal of Systems Science vol20 no 2 pp 191ndash202 1989
[9] E K-W Chu H-Y FanW-W Lin and C-SWang ldquoStructure-preserving algorithms for periodic discrete-time algebraic Ric-cati equationsrdquo International Journal of Control vol 77 no 8pp 767ndash788 2004
[10] E K-W Chu H-Y Fan andW-W Lin ldquoA structure-preservingdoubling algorithm for continuous-time algebraic Riccati equa-tionsrdquo Linear Algebra and Its Applications vol 396 pp 55ndash802005
[11] N J Higham ldquoPerturbation theory and backward error for119860119883 minus 119883119861 = 119862rdquo BIT Numerical Mathematics vol 33 no 1 pp124ndash136 1993
[12] M Konstantinov and P Petkov ldquoNote on perturbation theoryfor algebraic Riccati equationsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 21 no 1 pp 327ndash354 2000
[13] J-G Sun ldquoResidual bounds of approximate solutions of thealgebraic Riccati equationrdquoNumerischeMathematik vol 76 no2 pp 249ndash263 1997
[14] J-G Sun ldquoBackward error for the discrete-time algebraicRiccati equationrdquo Linear Algebra and Its Applications vol 259pp 183ndash208 1997
[15] J-G Sun ldquoBackward perturbation analysis of the periodicdiscrete-time algebraic Riccati equationrdquo SIAM Journal onMatrix Analysis and Applications vol 26 no 1 pp 1ndash19 2004
[16] V Mehrmann ldquoA step towards a unified treatment of continu-ous and discrete time control problemsrdquo Linear Algebra and ItsApplications vol 241ndash243 pp 749ndash779 1996
[17] H-G Xu ldquoTransformations between discrete-time andcontinuous-time algebraic Riccati equationsrdquo Linear Algebraand Its Applications vol 425 no 1 pp 77ndash101 2007
[18] A C M Ran and M C B Reurings ldquoOn the nonlinear matrixequation 119883 + 119860
lowastF (119883)119860 = 119876 solutions and perturbationtheoryrdquo Linear Algebra and Its Applications vol 346 pp 15ndash262002
[19] A C M Ran M C B Reurings and L Rodman ldquoA pertur-bation analysis for nonlinear selfadjoint operator equationsrdquoSIAM Journal on Matrix Analysis and Applications vol 28 no1 pp 89ndash104 2006
[20] H Lee and Y Lim ldquoInvariant metrics contractions and nonlin-ear matrix equationsrdquo Nonlinearity vol 21 no 4 pp 857ndash8782008
[21] J Cai and G-L Chen ldquoOn the Hermitian positive definitesolutions of nonlinear matrix equation 119883
119904
+ 119860119883minus119905
119860 = 119876rdquoApplied Mathematics and Computation vol 217 no 1 pp 117ndash123 2010
[22] X-F Duan Q-W Wang and A-P Liao ldquoOn the matrix equa-tion arising in an interpolation problemrdquo Linear andMultilinearAlgebra vol 61 no 9 pp 1192ndash1205 2013
[23] Z-G Jia and M-S Wei ldquoSolvability and sensitivity analysisof polynomial matrix equation 119883
119904
+ 119860119879
119883119905
119860 = 119876rdquo AppliedMathematics and Computation vol 209 no 2 pp 230ndash2372009
[24] M-H Wang M-S Wei and S Hu ldquoThe extremal solution ofthe matrix equation 119883119904 + 119860lowast119883minus119902119860 = 119868rdquo Applied Mathematicsand Computation vol 220 pp 193ndash199 2013
[25] B Zhou G-B Cai and J Lam ldquoPositive definite solutions ofthe nonlinear matrix equation 119883 + 119860
119867
119883minus1
119860 = 119868rdquo AppliedMathematics and Computation vol 219 no 14 pp 7377ndash73912013
[26] X-Z Zhan Matrix Inequalities vol 1790 of Lecture Notes inMathematics Springer Berlin Germany 2002
[27] T Furuta ldquoOperator inequalities associated with Holder-McCarthy and Kantorovich inequalitiesrdquo Journal of Inequalitiesand Applications vol 1998 Article ID 234521 1998
[28] R A Horn and C R Johnson Topics in Matrix AnalysisCambridge University Press Cambridge UK 1991
[29] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990
[30] X-G Liu and H Gao ldquoOn the positive definite solutions of thematrix equations 119883119904 minus 119860119879119883minus119905119860 = 119868
119899rdquo Linear Algebra and Its
Applications vol 368 pp 83ndash97 2003[31] S-F Xu ldquoPerturbation analysis of the maximal solution of
the matrix equation 119860lowast
119883minus1
119860 = 119875rdquo Linear Algebra and ItsApplications vol 336 pp 61ndash70 2001
[32] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985
[33] S M El-Sayed and A C M Ran ldquoOn an iteration method forsolving a class of nonlinear matrix equationsrdquo SIAM Journal onMatrix Analysis and Applications vol 23 no 3 pp 632ndash6452001
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
Then1003817100381710038171003817119883119894+1 minus 119883119894
1003817100381710038171003817119865le 120585
1003817100381710038171003817119883119894 minus 119883119894minus11003817100381710038171003817119865le 12058511989410038171003817100381710038171198831 minus 1198830
1003817100381710038171003817119865 (19)
which indicates the convergence of matrix series 1198830 1198831
1198832 generated by (17)Set 119883
0= 1205732119868 Assuming 119883
119894isin [1205722119868 1205732119868] then from
inequalities (14) we have
1205722119868 le ℎ (120572
2119868) le 119883
119894+1
= [(119860minus1
)lowast
(119883119904
119894minus 119876)119860
minus1
]
1119905
le ℎ (1205732119868) le 120573
2119868
(20)
That means for any 119894 = 0 1 2 119883119894isin [1205722119868 1205732119868] By
Theorem 4 (2) we can see that 119883max = lim119894rarr+infin
119883119894is the
unique HPD solution of (1) on [1205722119868 1205732119868]
Now we prove the maximality of 119883max Suppose that 119883is an arbitrary HPD solution of (1) then 119883
0ge 119883 and
Theorem 3 implies1198831199050ge 119883119905 (since119883
0= 1205732119868) Assuming that
119883119905
119894ge 119883119905 Lemma 1 with 119904119905 lt 1 implies
119883119905
119894+1= (119860minus1
)lowast
[(119883119905
119894)119904119905
minus 119876]119860minus1
ge (119860minus1
)lowast
[(119883119905
)119904119905
minus 119876]119860minus1
= 119883119905
(21)
Then119883119905max = lim119894rarr+infin
119883119905
119894ge 119883119905 which implies that119883max ge 119883
by the Lowner-Heinz inequality
Note that similar iteration formula ever appeared in somepapers such as [20 21] for other nonlinear matrix equationsHere we firstly proved that the iteration form (17) preservesthe maximality of119883
119894over all HPD solutions of (1)
22 Unique Solution of (1) with 119904 ge 119905 If 119904 gt 119905 Lee and Lim[20 Theorem 94] show that (1) always has a unique HPDsolution denoted by 119883
119906 Now we give an upper bound and
a lower bound of 119883119906and suggest an iteration method for
computing119883119906
As defined in (3) 1198921(119909) and 119892
2(119909) with 119904 gt 119905 have unique
positive roots denoted by 1205741and 1205742 respectively
Since 1198921(120582(119883119906)) le 0 and 119892
2(120582(119883119906)) le 0 120574
2le 120582(119883
119906) le 1205741
Theorem 7 If 119904 gt 119905 (1) has a unique HPD solution119883119906isin [1205742119868 1205741119868] Let 119883
0= 1205741119868 or 120574
2119868 then matrix series
1198830 1198831 1198832 generated by
119883119894= (119876 + 119860
lowast
119883119905
119894minus1119860)1119904
119894 = 0 1 2 (22)
will converge to119883119906
Proof Weonly need to prove the convergence ofmatrix series1198830 1198831 1198832 Set119883
0= 1205741119868 From (22) we have
1198831= (119876 + 120574
119905
1119860lowast
119860)1119904
le (120582max (119876) + 120574119905
1120582max (119860
lowast
119860))1119904
119868 = 1205741119868
(23)
and then1198831199041le 119883119904
0 Assuming that119883119904
119894le 119883119904
119894minus1
119883119904
119894+1= 119876 + 119860
lowast
119883119905
119894119860 = 119876 + 119860
lowast
(119883119904
119894)119905119904
119860
le 119876 + 119860lowast
(119883119904
119894minus1)119905119904
119860 = 119883119904
119894
(24)
Then for any 119894 = 0 1 2 we have 119883119904119894+1
le 119883119904
119894and then
119883119894+1
le 119883119894by Lowner-Heinz inequality On the other hand
1198830ge 1205742119868 implies 119883
119894ge 1205742119868 for any 119894 = 0 1 2 because if
119883119894minus1
ge 1205742119868 then
119883119894= (119876 + 119860
lowast
119883119905
119894minus1119860)1119904
ge (119876 + 120574119905
2119860lowast
119860)1119904
ge (120582min (119876) + 120574119905
2120582min (119860
lowast
119860))1119904
119868 = 1205742119868
(25)
Then 1198830 1198831 1198832 with 119883
0= 1205741119868 is a decreasingly
monotone matrix series with a lower bound 1205742119868 Similarly we
can prove that 1198830 1198831 1198832 generated by (22) with 119883
0=
1205742119868 is an increasingly monotone matrix series with an upper
bound 1205741119868Therefore the convergence of 119883
0 1198831 1198832 has
been proved
From the above proof we can see that the iteration form(22) preserves the minimality (119883
0= 1205741119868) or maximality
(1198830= 1205742119868) of119883
119894in process
If 119904 = 119905 (1) can be reduced to a linear matrix equation119884minus119860lowast
119884119860 = 119876 which is the discrete-time algebraic Lyapunovequation (DALE) or Hermitian Stein equation [1 Page 5]assuming that 119884 = 119883
119904 It is well known that if 119860 is d-stable(see [1]) 119884 minus 119860
lowast
119884119860 = 119876 has a unique solution and matrixseries 119884
0 1198841 1198842 generated by 119884
119894+1= 119876+119860
lowast
119884119894119860 with an
initial value 1198840 will converge to the unique solution Besides
it is not difficult to get an expression of the unique solution119883119906= (suminfin
119895=0(119860lowast
)119895
119876119860119895
)1119904 applying [32 Theorem 1 Section
132] [1 Theorem 1118] and the results in Section 64 [28]Now we have presented the solvability theory of the self-
adjoint polynomial matrix equation (1) in three cases Ageneral iterative algorithm for its maximal solution (119904 lt 119905)or unique solution (119904 ge 119905) will be given in Section 4 Before itwe study the algebraic perturbation of themaximal or uniquesolution of (1)
3 Algebraic Perturbation Analysis
In this section we present the algebraic perturbation analysisof the HPD solution of (1) with respect to the perturbationof its coefficient matrices Similar to [30] we define theperturbed matrix equation of (1) as
119883119904
minus 119860lowast
119883119905
119860 = 119876 (26)
where 119860 = 119860 + Δ119860 isin C119899times119899 and 119876 = 119876 + Δ119876 isin C119899times119899 Wealways suppose that (1) has a maximal (or unique) solutiondenoted by 119883
119872isin [1205722119868 1205732119868] and (26) has a maximal (or
unique) solution denoted by119883119872isin [2119868 1205732119868]
Now we present the perturbation bound for 119883119872
when119904 = 119905 Define a function 120591
120591 (120572 120573) = 119904120572119904minus1
minus 119905120573119905minus1
1198602
2 (120572 120573) isin R
2
(27)
Journal of Applied Mathematics 5
Theorem 8 Let 120576 gt 0 be an arbitrary real number and 120591(2
1205732) ge 0 If
Δ119860119865lt (119860
2
2+2120576
3120591 (2 1205732)10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
minus119905
2
)
12
minus 1198602
Δ119876119865lt1
3120591 (2 1205732) 120576
(28)
then10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865lt 120576 (29)
Proof It is easy to induce that
10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865ge (
119904minus1
sum
119896=0
119904minus1minus119896
2120572119896
2)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
ge 119904119904minus1
2
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
10038171003817100381710038171003817119883119905
119872minus 119883119905
119872
10038171003817100381710038171003817119865le (
119905minus1
sum
119896=0
120573119905minus1minus119896
2120573119896
2)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le 119905120573119905minus1
2
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
(30)
Then from (1) and (26) we have
120591 (2 1205732)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le 21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ119860119865+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ1198602
119865+ Δ119876
119865
(31)
Since 120591(2 1205732) gt 0
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le (120591 (2 1205732))minus1
(21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ119860119865
+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ1198602
119865+ Δ119876
119865)
(32)
Then for an arbitrary 120576 gt 0 if Δ119860119865
lt (1198602
2+
(21205763)120591(2 1205732)119883119872minus119905
2)12
minus 1198602
and Δ119876119865
lt
(13)120591(2 1205732)120576 we have (29)
If 119904 = 119905 for an arbitrary 120576 gt 0 define
984858 (120576) = 1198602+ (119860
2
2+2120576
3120588)
12
(33)
where
120588 =10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
[119904119904minus1
2(1 minus 119860
2
2)]minus1
(34)
Theorem9 Let 120576 gt 0 be an arbitrary real number and 1198602lt
1 If
Δ119860119865lt2120576
3(120588984858 (120576))
minus1
Δ119876119865lt
120576
3120588
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
(35)
then10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865lt 120576 (36)
Table 1 Iteration CPU time (seconds) and residue for solving (1)with 119904 = 119905
(119904 119905)Algorithm 1 MONO
Ite CPU Res Ite CPU Res(2 1) 9 00541 45275119890 minus 13 200 21031 00016(1 2) 200 10275 20297119890 minus 07 mdash mdash mdash(8 5) 10 00716 59909119890 minus 13 200 22284 00034(5 8) 200 11048 31059119890 minus 05 mdash mdash mdash(30 15) 9 00743 52317119890 minus 13 200 23051 00029(15 30) 200 12865 20838119890 minus 08 mdash mdash mdash(300 150) 10 00886 79960119890 minus 13 200 22683 00031(150 300) 200 14187 28384119890 minus 07 mdash mdash mdash
Proof Similar to the proof of Theorem 8 we can induce that
(1 minus 1198602
2)10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865
le 21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ119860119865+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ1198602
119865+ Δ119876
119865
(37)
Then10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865
le (1 minus 1198602
2)minus1
(21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ119860119865
+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ1198602
119865+ Δ119876
119865)
(38)
With the help of (30) and (34) (38) implies10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865le 120588 (Δ119860
119865+ 2119860
2Δ119860119865+ Δ119876
119865) (39)
Then if Δ119860119865lt (21205763)(120588984858(120576))
minus1 and Δ119876119865lt (1205763120588)119883
119872119904
2
we have (36)
Theorems 8 and 9 make sure that the perturbation of119883119872
can be controlled if Δ119860 and Δ119876 have a proper upper bound
4 Algorithm and Numerical Experiments
In this section we give a general iterative algorithm for themaximal or unique solutions of (1) and two numerical exper-iments All reported results were obtained using MATLAB-R2012b on a personal computer with 24GHz Intel Core i7and 8GB 1600MHz DDR3
Example 10 Let matrices 119860 = rand(100) times 10minus2 and 119876 =
eye(100) With tol = 10minus12 and not more than 200 iterationswe apply Algorithm 1 to compute the maximal or uniqueHPD solutions of (1) with 119904 = 119905 and compare the results withthose by the iteration method from [33] (denoted by MONOin Table 1)
Table 1 shows iterations CPU times before convergenceand the residues of the computed HPD solution 119883 definedby
119890 (119904 119905) =
1003817100381710038171003817119883119904
minus 119860lowast
119883119905
119860 minus 1198761003817100381710038171003817119865
[119860 119876]119865
(40)
6 Journal of Applied Mathematics
Table 2 Iteration CPU time (seconds) and residue for solving (1) with 119904 = 119905 and different initial solutions
(119904 119905 1198830)
Algorithm 1 MONOIte CPU Res Ite CPU Res
(1 1 1205751119868119899) 20 00223 34947119890 minus 13 20 00202 34947119890 minus 13
(1 1 1205752119868119899) 31 00258 73027119890 minus 13 31 00305 73027119890 minus 13
(2 2 1205751119868119899) 20 01170 89978119890 minus 13 200 26421 00037
(2 2 1205752119868119899) 43 05475 96421119890 minus 13 200 27224 00037
(10 10 1205751119868119899) 29 01890 29296119890 minus 13 200 29717 00059
(10 10 1205752119868119899) 157 30859 71154119890 minus 13 200 29788 00059
Step 1 Compute 120582max(119860lowast
119860) 120582min(119860lowast
119860) 120582max(119876) 120582min(119876)Step 2 Input (3)Step 3 If 119904 lt 119905 run Steps 4-5 if 119905 lt 119904 run Steps 6-7 otherwise run Steps 8-9Step 4 Compute the roots 120572
1 1205722of 1198921(119909) and 120573
1 1205732of 1198922(119909) respectively
Step 5 Let 1198830= 1205732119868 run (17)
Step 6 Compute the root 1205741of 1198921(119909) and the root 120574
2of 1198922(119909) respectively
Step 7 Let 1198850= 1205741119868 run (22)
Step 8 Compute the root 1205751of 1198921(119909) and the root 120575
2of 1198922(119909) respectively
Step 9 If 120582max(119860lowast
119860) lt 1 and 1205751ge 1205752 then let 119883
0= 1205751119868 and run (22)
Algorithm 1 Given matrices 119860119876 isin C119899times119899 and positive integers 119904 119905
From Table 1 we can see that it takes more iterations andCPU times to solve themaximal solution of (1) with 119904 lt 119905 thanto solve the unique solution of (1) with 119904 gt 119905 At the same timethe accuracy of the latter is better than the formerMONOcannot be used to solve (1) with 119904 lt 119905 and it costs more iterationsand CPU times than Algorithm 1 when solving (1) with 119904 gt 119905
Now we use Example 41 of [33] to test our method
Example 11 Let 119860 = 05119861119861infin
with 119861 = [119861119894119895]119899times119899 119887119894119895= 119894 +
119895 + 1 and let 119876 = eye(119899) with 119899 = 100 We solve (1) with119904 = 119905 and with two different initial solutions The iterationsCPU times and the residues of the computation are reportedin Table 2
Table 2 shows that for Algorithm 1 the choice 1198830= 1205751119868119899
is better than 1198830= 1205752119868119899 When 119904 and 119905 rise MONO might
lose its efficiency It seems not proper to apply the iterationmethod designed for 119884 minus 119860lowast119884119905119904119860 = 119876 with 119884 = 119883
119904 to solve119883119904
minus 119860lowast
119883119905
119860 = 119876 although they are equivalent to each otherin theory
5 Conclusion
In this paper we considered the solvability of the self-adjointpolynomial matrix equation (1) Sufficient conditions weregiven to guarantee the existence of the maximal or uniqueHPD solutions of (1) The algebraic perturbation analysisincluding perturbation bounds was also developed for (1)under the perturbation of given coefficient matrices At lasta general iterative algorithm with maximality preserved inprocess was presented for the maximal or unique solutionwith two numerical experiments reported
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Zhigang Jiarsquos research was supported in part by NationalNatural Science Foundation of China under Grants 11201193and 11171289 and a project funded by the Priority AcademicProgram Development of Jiangsu Higher EducationInstitutions Minghui Wangrsquos research was supported inpart by the National Natural Science Foundation of China(Grant no 11001144) the Science and Technology Programof Shandong Universities of China (J11LA04) and theResearch Award Fund for Outstanding Young Scientists ofShandong Province in China (BS2012DX009) Sitao Lingrsquosresearch was supported in part by National Natural ScienceFoundations of China under Grant 11301529 PostdoctoralScience Foundation of China under Grant 2013M540472 andJiangsu Planned Projects for Postdoctoral Research Funds1302036CThe authors would like to thank three anonymousreferees for giving valuable comments and suggestions
References
[1] H Abou-Kandil G Freiling V Ionescu and G Jank MatrixRiccati Equations in Control and Systems Theory BirkhauserBasel Switzerland 2003
[2] I Gohberg P Lancaster and L Rodman Matrix PolynomialsAcademic Press New York NY USA 1982
[3] P Benner A J Laub and V Mehrmann ldquoBenchmarks for thenumerical solution of algebraic Riccati equationsrdquo IEEE ControlSystems Magazine vol 17 no 5 pp 18ndash28 1997
Journal of Applied Mathematics 7
[4] S Bittanti A Laub and J CWillems EdsTheRiccati EquationCommunications and Control Engineering Series SpringerBerlin Germany 1991
[5] P Lancaster andL RodmanAlgebraic Riccati Equations OxfordScience Publications Oxford University Press New York NYUSA 1995
[6] A J Laub ldquoInvariant subspace methods for the numerical solu-tion of Riccati equationsrdquo inThe Riccati Equation S Bittanti AJ Laub and J C Willems Eds Communications and ControlEngineering pp 163ndash196 Springer Berlin Germany 1991
[7] A J Laub ldquoA Schur method for solving algebraic Riccatiequationsrdquo IEEE Transactions on Automatic Control vol 24 no6 pp 913ndash921 1979
[8] M Kimura ldquoDoubling algorithm for continuous-time algebraicRiccati equationrdquo International Journal of Systems Science vol20 no 2 pp 191ndash202 1989
[9] E K-W Chu H-Y FanW-W Lin and C-SWang ldquoStructure-preserving algorithms for periodic discrete-time algebraic Ric-cati equationsrdquo International Journal of Control vol 77 no 8pp 767ndash788 2004
[10] E K-W Chu H-Y Fan andW-W Lin ldquoA structure-preservingdoubling algorithm for continuous-time algebraic Riccati equa-tionsrdquo Linear Algebra and Its Applications vol 396 pp 55ndash802005
[11] N J Higham ldquoPerturbation theory and backward error for119860119883 minus 119883119861 = 119862rdquo BIT Numerical Mathematics vol 33 no 1 pp124ndash136 1993
[12] M Konstantinov and P Petkov ldquoNote on perturbation theoryfor algebraic Riccati equationsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 21 no 1 pp 327ndash354 2000
[13] J-G Sun ldquoResidual bounds of approximate solutions of thealgebraic Riccati equationrdquoNumerischeMathematik vol 76 no2 pp 249ndash263 1997
[14] J-G Sun ldquoBackward error for the discrete-time algebraicRiccati equationrdquo Linear Algebra and Its Applications vol 259pp 183ndash208 1997
[15] J-G Sun ldquoBackward perturbation analysis of the periodicdiscrete-time algebraic Riccati equationrdquo SIAM Journal onMatrix Analysis and Applications vol 26 no 1 pp 1ndash19 2004
[16] V Mehrmann ldquoA step towards a unified treatment of continu-ous and discrete time control problemsrdquo Linear Algebra and ItsApplications vol 241ndash243 pp 749ndash779 1996
[17] H-G Xu ldquoTransformations between discrete-time andcontinuous-time algebraic Riccati equationsrdquo Linear Algebraand Its Applications vol 425 no 1 pp 77ndash101 2007
[18] A C M Ran and M C B Reurings ldquoOn the nonlinear matrixequation 119883 + 119860
lowastF (119883)119860 = 119876 solutions and perturbationtheoryrdquo Linear Algebra and Its Applications vol 346 pp 15ndash262002
[19] A C M Ran M C B Reurings and L Rodman ldquoA pertur-bation analysis for nonlinear selfadjoint operator equationsrdquoSIAM Journal on Matrix Analysis and Applications vol 28 no1 pp 89ndash104 2006
[20] H Lee and Y Lim ldquoInvariant metrics contractions and nonlin-ear matrix equationsrdquo Nonlinearity vol 21 no 4 pp 857ndash8782008
[21] J Cai and G-L Chen ldquoOn the Hermitian positive definitesolutions of nonlinear matrix equation 119883
119904
+ 119860119883minus119905
119860 = 119876rdquoApplied Mathematics and Computation vol 217 no 1 pp 117ndash123 2010
[22] X-F Duan Q-W Wang and A-P Liao ldquoOn the matrix equa-tion arising in an interpolation problemrdquo Linear andMultilinearAlgebra vol 61 no 9 pp 1192ndash1205 2013
[23] Z-G Jia and M-S Wei ldquoSolvability and sensitivity analysisof polynomial matrix equation 119883
119904
+ 119860119879
119883119905
119860 = 119876rdquo AppliedMathematics and Computation vol 209 no 2 pp 230ndash2372009
[24] M-H Wang M-S Wei and S Hu ldquoThe extremal solution ofthe matrix equation 119883119904 + 119860lowast119883minus119902119860 = 119868rdquo Applied Mathematicsand Computation vol 220 pp 193ndash199 2013
[25] B Zhou G-B Cai and J Lam ldquoPositive definite solutions ofthe nonlinear matrix equation 119883 + 119860
119867
119883minus1
119860 = 119868rdquo AppliedMathematics and Computation vol 219 no 14 pp 7377ndash73912013
[26] X-Z Zhan Matrix Inequalities vol 1790 of Lecture Notes inMathematics Springer Berlin Germany 2002
[27] T Furuta ldquoOperator inequalities associated with Holder-McCarthy and Kantorovich inequalitiesrdquo Journal of Inequalitiesand Applications vol 1998 Article ID 234521 1998
[28] R A Horn and C R Johnson Topics in Matrix AnalysisCambridge University Press Cambridge UK 1991
[29] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990
[30] X-G Liu and H Gao ldquoOn the positive definite solutions of thematrix equations 119883119904 minus 119860119879119883minus119905119860 = 119868
119899rdquo Linear Algebra and Its
Applications vol 368 pp 83ndash97 2003[31] S-F Xu ldquoPerturbation analysis of the maximal solution of
the matrix equation 119860lowast
119883minus1
119860 = 119875rdquo Linear Algebra and ItsApplications vol 336 pp 61ndash70 2001
[32] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985
[33] S M El-Sayed and A C M Ran ldquoOn an iteration method forsolving a class of nonlinear matrix equationsrdquo SIAM Journal onMatrix Analysis and Applications vol 23 no 3 pp 632ndash6452001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
Theorem 8 Let 120576 gt 0 be an arbitrary real number and 120591(2
1205732) ge 0 If
Δ119860119865lt (119860
2
2+2120576
3120591 (2 1205732)10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
minus119905
2
)
12
minus 1198602
Δ119876119865lt1
3120591 (2 1205732) 120576
(28)
then10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865lt 120576 (29)
Proof It is easy to induce that
10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865ge (
119904minus1
sum
119896=0
119904minus1minus119896
2120572119896
2)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
ge 119904119904minus1
2
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
10038171003817100381710038171003817119883119905
119872minus 119883119905
119872
10038171003817100381710038171003817119865le (
119905minus1
sum
119896=0
120573119905minus1minus119896
2120573119896
2)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le 119905120573119905minus1
2
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
(30)
Then from (1) and (26) we have
120591 (2 1205732)10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le 21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ119860119865+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ1198602
119865+ Δ119876
119865
(31)
Since 120591(2 1205732) gt 0
10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865
le (120591 (2 1205732))minus1
(21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ119860119865
+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119905
2
Δ1198602
119865+ Δ119876
119865)
(32)
Then for an arbitrary 120576 gt 0 if Δ119860119865
lt (1198602
2+
(21205763)120591(2 1205732)119883119872minus119905
2)12
minus 1198602
and Δ119876119865
lt
(13)120591(2 1205732)120576 we have (29)
If 119904 = 119905 for an arbitrary 120576 gt 0 define
984858 (120576) = 1198602+ (119860
2
2+2120576
3120588)
12
(33)
where
120588 =10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
[119904119904minus1
2(1 minus 119860
2
2)]minus1
(34)
Theorem9 Let 120576 gt 0 be an arbitrary real number and 1198602lt
1 If
Δ119860119865lt2120576
3(120588984858 (120576))
minus1
Δ119876119865lt
120576
3120588
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
(35)
then10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865lt 120576 (36)
Table 1 Iteration CPU time (seconds) and residue for solving (1)with 119904 = 119905
(119904 119905)Algorithm 1 MONO
Ite CPU Res Ite CPU Res(2 1) 9 00541 45275119890 minus 13 200 21031 00016(1 2) 200 10275 20297119890 minus 07 mdash mdash mdash(8 5) 10 00716 59909119890 minus 13 200 22284 00034(5 8) 200 11048 31059119890 minus 05 mdash mdash mdash(30 15) 9 00743 52317119890 minus 13 200 23051 00029(15 30) 200 12865 20838119890 minus 08 mdash mdash mdash(300 150) 10 00886 79960119890 minus 13 200 22683 00031(150 300) 200 14187 28384119890 minus 07 mdash mdash mdash
Proof Similar to the proof of Theorem 8 we can induce that
(1 minus 1198602
2)10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865
le 21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ119860119865+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ1198602
119865+ Δ119876
119865
(37)
Then10038171003817100381710038171003817119883119904
119872minus 119883119904
119872
10038171003817100381710038171003817119865
le (1 minus 1198602
2)minus1
(21198602
10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ119860119865
+10038171003817100381710038171003817119883119872
10038171003817100381710038171003817
119904
2
Δ1198602
119865+ Δ119876
119865)
(38)
With the help of (30) and (34) (38) implies10038171003817100381710038171003817119883119872minus 119883119872
10038171003817100381710038171003817119865le 120588 (Δ119860
119865+ 2119860
2Δ119860119865+ Δ119876
119865) (39)
Then if Δ119860119865lt (21205763)(120588984858(120576))
minus1 and Δ119876119865lt (1205763120588)119883
119872119904
2
we have (36)
Theorems 8 and 9 make sure that the perturbation of119883119872
can be controlled if Δ119860 and Δ119876 have a proper upper bound
4 Algorithm and Numerical Experiments
In this section we give a general iterative algorithm for themaximal or unique solutions of (1) and two numerical exper-iments All reported results were obtained using MATLAB-R2012b on a personal computer with 24GHz Intel Core i7and 8GB 1600MHz DDR3
Example 10 Let matrices 119860 = rand(100) times 10minus2 and 119876 =
eye(100) With tol = 10minus12 and not more than 200 iterationswe apply Algorithm 1 to compute the maximal or uniqueHPD solutions of (1) with 119904 = 119905 and compare the results withthose by the iteration method from [33] (denoted by MONOin Table 1)
Table 1 shows iterations CPU times before convergenceand the residues of the computed HPD solution 119883 definedby
119890 (119904 119905) =
1003817100381710038171003817119883119904
minus 119860lowast
119883119905
119860 minus 1198761003817100381710038171003817119865
[119860 119876]119865
(40)
6 Journal of Applied Mathematics
Table 2 Iteration CPU time (seconds) and residue for solving (1) with 119904 = 119905 and different initial solutions
(119904 119905 1198830)
Algorithm 1 MONOIte CPU Res Ite CPU Res
(1 1 1205751119868119899) 20 00223 34947119890 minus 13 20 00202 34947119890 minus 13
(1 1 1205752119868119899) 31 00258 73027119890 minus 13 31 00305 73027119890 minus 13
(2 2 1205751119868119899) 20 01170 89978119890 minus 13 200 26421 00037
(2 2 1205752119868119899) 43 05475 96421119890 minus 13 200 27224 00037
(10 10 1205751119868119899) 29 01890 29296119890 minus 13 200 29717 00059
(10 10 1205752119868119899) 157 30859 71154119890 minus 13 200 29788 00059
Step 1 Compute 120582max(119860lowast
119860) 120582min(119860lowast
119860) 120582max(119876) 120582min(119876)Step 2 Input (3)Step 3 If 119904 lt 119905 run Steps 4-5 if 119905 lt 119904 run Steps 6-7 otherwise run Steps 8-9Step 4 Compute the roots 120572
1 1205722of 1198921(119909) and 120573
1 1205732of 1198922(119909) respectively
Step 5 Let 1198830= 1205732119868 run (17)
Step 6 Compute the root 1205741of 1198921(119909) and the root 120574
2of 1198922(119909) respectively
Step 7 Let 1198850= 1205741119868 run (22)
Step 8 Compute the root 1205751of 1198921(119909) and the root 120575
2of 1198922(119909) respectively
Step 9 If 120582max(119860lowast
119860) lt 1 and 1205751ge 1205752 then let 119883
0= 1205751119868 and run (22)
Algorithm 1 Given matrices 119860119876 isin C119899times119899 and positive integers 119904 119905
From Table 1 we can see that it takes more iterations andCPU times to solve themaximal solution of (1) with 119904 lt 119905 thanto solve the unique solution of (1) with 119904 gt 119905 At the same timethe accuracy of the latter is better than the formerMONOcannot be used to solve (1) with 119904 lt 119905 and it costs more iterationsand CPU times than Algorithm 1 when solving (1) with 119904 gt 119905
Now we use Example 41 of [33] to test our method
Example 11 Let 119860 = 05119861119861infin
with 119861 = [119861119894119895]119899times119899 119887119894119895= 119894 +
119895 + 1 and let 119876 = eye(119899) with 119899 = 100 We solve (1) with119904 = 119905 and with two different initial solutions The iterationsCPU times and the residues of the computation are reportedin Table 2
Table 2 shows that for Algorithm 1 the choice 1198830= 1205751119868119899
is better than 1198830= 1205752119868119899 When 119904 and 119905 rise MONO might
lose its efficiency It seems not proper to apply the iterationmethod designed for 119884 minus 119860lowast119884119905119904119860 = 119876 with 119884 = 119883
119904 to solve119883119904
minus 119860lowast
119883119905
119860 = 119876 although they are equivalent to each otherin theory
5 Conclusion
In this paper we considered the solvability of the self-adjointpolynomial matrix equation (1) Sufficient conditions weregiven to guarantee the existence of the maximal or uniqueHPD solutions of (1) The algebraic perturbation analysisincluding perturbation bounds was also developed for (1)under the perturbation of given coefficient matrices At lasta general iterative algorithm with maximality preserved inprocess was presented for the maximal or unique solutionwith two numerical experiments reported
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Zhigang Jiarsquos research was supported in part by NationalNatural Science Foundation of China under Grants 11201193and 11171289 and a project funded by the Priority AcademicProgram Development of Jiangsu Higher EducationInstitutions Minghui Wangrsquos research was supported inpart by the National Natural Science Foundation of China(Grant no 11001144) the Science and Technology Programof Shandong Universities of China (J11LA04) and theResearch Award Fund for Outstanding Young Scientists ofShandong Province in China (BS2012DX009) Sitao Lingrsquosresearch was supported in part by National Natural ScienceFoundations of China under Grant 11301529 PostdoctoralScience Foundation of China under Grant 2013M540472 andJiangsu Planned Projects for Postdoctoral Research Funds1302036CThe authors would like to thank three anonymousreferees for giving valuable comments and suggestions
References
[1] H Abou-Kandil G Freiling V Ionescu and G Jank MatrixRiccati Equations in Control and Systems Theory BirkhauserBasel Switzerland 2003
[2] I Gohberg P Lancaster and L Rodman Matrix PolynomialsAcademic Press New York NY USA 1982
[3] P Benner A J Laub and V Mehrmann ldquoBenchmarks for thenumerical solution of algebraic Riccati equationsrdquo IEEE ControlSystems Magazine vol 17 no 5 pp 18ndash28 1997
Journal of Applied Mathematics 7
[4] S Bittanti A Laub and J CWillems EdsTheRiccati EquationCommunications and Control Engineering Series SpringerBerlin Germany 1991
[5] P Lancaster andL RodmanAlgebraic Riccati Equations OxfordScience Publications Oxford University Press New York NYUSA 1995
[6] A J Laub ldquoInvariant subspace methods for the numerical solu-tion of Riccati equationsrdquo inThe Riccati Equation S Bittanti AJ Laub and J C Willems Eds Communications and ControlEngineering pp 163ndash196 Springer Berlin Germany 1991
[7] A J Laub ldquoA Schur method for solving algebraic Riccatiequationsrdquo IEEE Transactions on Automatic Control vol 24 no6 pp 913ndash921 1979
[8] M Kimura ldquoDoubling algorithm for continuous-time algebraicRiccati equationrdquo International Journal of Systems Science vol20 no 2 pp 191ndash202 1989
[9] E K-W Chu H-Y FanW-W Lin and C-SWang ldquoStructure-preserving algorithms for periodic discrete-time algebraic Ric-cati equationsrdquo International Journal of Control vol 77 no 8pp 767ndash788 2004
[10] E K-W Chu H-Y Fan andW-W Lin ldquoA structure-preservingdoubling algorithm for continuous-time algebraic Riccati equa-tionsrdquo Linear Algebra and Its Applications vol 396 pp 55ndash802005
[11] N J Higham ldquoPerturbation theory and backward error for119860119883 minus 119883119861 = 119862rdquo BIT Numerical Mathematics vol 33 no 1 pp124ndash136 1993
[12] M Konstantinov and P Petkov ldquoNote on perturbation theoryfor algebraic Riccati equationsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 21 no 1 pp 327ndash354 2000
[13] J-G Sun ldquoResidual bounds of approximate solutions of thealgebraic Riccati equationrdquoNumerischeMathematik vol 76 no2 pp 249ndash263 1997
[14] J-G Sun ldquoBackward error for the discrete-time algebraicRiccati equationrdquo Linear Algebra and Its Applications vol 259pp 183ndash208 1997
[15] J-G Sun ldquoBackward perturbation analysis of the periodicdiscrete-time algebraic Riccati equationrdquo SIAM Journal onMatrix Analysis and Applications vol 26 no 1 pp 1ndash19 2004
[16] V Mehrmann ldquoA step towards a unified treatment of continu-ous and discrete time control problemsrdquo Linear Algebra and ItsApplications vol 241ndash243 pp 749ndash779 1996
[17] H-G Xu ldquoTransformations between discrete-time andcontinuous-time algebraic Riccati equationsrdquo Linear Algebraand Its Applications vol 425 no 1 pp 77ndash101 2007
[18] A C M Ran and M C B Reurings ldquoOn the nonlinear matrixequation 119883 + 119860
lowastF (119883)119860 = 119876 solutions and perturbationtheoryrdquo Linear Algebra and Its Applications vol 346 pp 15ndash262002
[19] A C M Ran M C B Reurings and L Rodman ldquoA pertur-bation analysis for nonlinear selfadjoint operator equationsrdquoSIAM Journal on Matrix Analysis and Applications vol 28 no1 pp 89ndash104 2006
[20] H Lee and Y Lim ldquoInvariant metrics contractions and nonlin-ear matrix equationsrdquo Nonlinearity vol 21 no 4 pp 857ndash8782008
[21] J Cai and G-L Chen ldquoOn the Hermitian positive definitesolutions of nonlinear matrix equation 119883
119904
+ 119860119883minus119905
119860 = 119876rdquoApplied Mathematics and Computation vol 217 no 1 pp 117ndash123 2010
[22] X-F Duan Q-W Wang and A-P Liao ldquoOn the matrix equa-tion arising in an interpolation problemrdquo Linear andMultilinearAlgebra vol 61 no 9 pp 1192ndash1205 2013
[23] Z-G Jia and M-S Wei ldquoSolvability and sensitivity analysisof polynomial matrix equation 119883
119904
+ 119860119879
119883119905
119860 = 119876rdquo AppliedMathematics and Computation vol 209 no 2 pp 230ndash2372009
[24] M-H Wang M-S Wei and S Hu ldquoThe extremal solution ofthe matrix equation 119883119904 + 119860lowast119883minus119902119860 = 119868rdquo Applied Mathematicsand Computation vol 220 pp 193ndash199 2013
[25] B Zhou G-B Cai and J Lam ldquoPositive definite solutions ofthe nonlinear matrix equation 119883 + 119860
119867
119883minus1
119860 = 119868rdquo AppliedMathematics and Computation vol 219 no 14 pp 7377ndash73912013
[26] X-Z Zhan Matrix Inequalities vol 1790 of Lecture Notes inMathematics Springer Berlin Germany 2002
[27] T Furuta ldquoOperator inequalities associated with Holder-McCarthy and Kantorovich inequalitiesrdquo Journal of Inequalitiesand Applications vol 1998 Article ID 234521 1998
[28] R A Horn and C R Johnson Topics in Matrix AnalysisCambridge University Press Cambridge UK 1991
[29] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990
[30] X-G Liu and H Gao ldquoOn the positive definite solutions of thematrix equations 119883119904 minus 119860119879119883minus119905119860 = 119868
119899rdquo Linear Algebra and Its
Applications vol 368 pp 83ndash97 2003[31] S-F Xu ldquoPerturbation analysis of the maximal solution of
the matrix equation 119860lowast
119883minus1
119860 = 119875rdquo Linear Algebra and ItsApplications vol 336 pp 61ndash70 2001
[32] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985
[33] S M El-Sayed and A C M Ran ldquoOn an iteration method forsolving a class of nonlinear matrix equationsrdquo SIAM Journal onMatrix Analysis and Applications vol 23 no 3 pp 632ndash6452001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
Table 2 Iteration CPU time (seconds) and residue for solving (1) with 119904 = 119905 and different initial solutions
(119904 119905 1198830)
Algorithm 1 MONOIte CPU Res Ite CPU Res
(1 1 1205751119868119899) 20 00223 34947119890 minus 13 20 00202 34947119890 minus 13
(1 1 1205752119868119899) 31 00258 73027119890 minus 13 31 00305 73027119890 minus 13
(2 2 1205751119868119899) 20 01170 89978119890 minus 13 200 26421 00037
(2 2 1205752119868119899) 43 05475 96421119890 minus 13 200 27224 00037
(10 10 1205751119868119899) 29 01890 29296119890 minus 13 200 29717 00059
(10 10 1205752119868119899) 157 30859 71154119890 minus 13 200 29788 00059
Step 1 Compute 120582max(119860lowast
119860) 120582min(119860lowast
119860) 120582max(119876) 120582min(119876)Step 2 Input (3)Step 3 If 119904 lt 119905 run Steps 4-5 if 119905 lt 119904 run Steps 6-7 otherwise run Steps 8-9Step 4 Compute the roots 120572
1 1205722of 1198921(119909) and 120573
1 1205732of 1198922(119909) respectively
Step 5 Let 1198830= 1205732119868 run (17)
Step 6 Compute the root 1205741of 1198921(119909) and the root 120574
2of 1198922(119909) respectively
Step 7 Let 1198850= 1205741119868 run (22)
Step 8 Compute the root 1205751of 1198921(119909) and the root 120575
2of 1198922(119909) respectively
Step 9 If 120582max(119860lowast
119860) lt 1 and 1205751ge 1205752 then let 119883
0= 1205751119868 and run (22)
Algorithm 1 Given matrices 119860119876 isin C119899times119899 and positive integers 119904 119905
From Table 1 we can see that it takes more iterations andCPU times to solve themaximal solution of (1) with 119904 lt 119905 thanto solve the unique solution of (1) with 119904 gt 119905 At the same timethe accuracy of the latter is better than the formerMONOcannot be used to solve (1) with 119904 lt 119905 and it costs more iterationsand CPU times than Algorithm 1 when solving (1) with 119904 gt 119905
Now we use Example 41 of [33] to test our method
Example 11 Let 119860 = 05119861119861infin
with 119861 = [119861119894119895]119899times119899 119887119894119895= 119894 +
119895 + 1 and let 119876 = eye(119899) with 119899 = 100 We solve (1) with119904 = 119905 and with two different initial solutions The iterationsCPU times and the residues of the computation are reportedin Table 2
Table 2 shows that for Algorithm 1 the choice 1198830= 1205751119868119899
is better than 1198830= 1205752119868119899 When 119904 and 119905 rise MONO might
lose its efficiency It seems not proper to apply the iterationmethod designed for 119884 minus 119860lowast119884119905119904119860 = 119876 with 119884 = 119883
119904 to solve119883119904
minus 119860lowast
119883119905
119860 = 119876 although they are equivalent to each otherin theory
5 Conclusion
In this paper we considered the solvability of the self-adjointpolynomial matrix equation (1) Sufficient conditions weregiven to guarantee the existence of the maximal or uniqueHPD solutions of (1) The algebraic perturbation analysisincluding perturbation bounds was also developed for (1)under the perturbation of given coefficient matrices At lasta general iterative algorithm with maximality preserved inprocess was presented for the maximal or unique solutionwith two numerical experiments reported
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Zhigang Jiarsquos research was supported in part by NationalNatural Science Foundation of China under Grants 11201193and 11171289 and a project funded by the Priority AcademicProgram Development of Jiangsu Higher EducationInstitutions Minghui Wangrsquos research was supported inpart by the National Natural Science Foundation of China(Grant no 11001144) the Science and Technology Programof Shandong Universities of China (J11LA04) and theResearch Award Fund for Outstanding Young Scientists ofShandong Province in China (BS2012DX009) Sitao Lingrsquosresearch was supported in part by National Natural ScienceFoundations of China under Grant 11301529 PostdoctoralScience Foundation of China under Grant 2013M540472 andJiangsu Planned Projects for Postdoctoral Research Funds1302036CThe authors would like to thank three anonymousreferees for giving valuable comments and suggestions
References
[1] H Abou-Kandil G Freiling V Ionescu and G Jank MatrixRiccati Equations in Control and Systems Theory BirkhauserBasel Switzerland 2003
[2] I Gohberg P Lancaster and L Rodman Matrix PolynomialsAcademic Press New York NY USA 1982
[3] P Benner A J Laub and V Mehrmann ldquoBenchmarks for thenumerical solution of algebraic Riccati equationsrdquo IEEE ControlSystems Magazine vol 17 no 5 pp 18ndash28 1997
Journal of Applied Mathematics 7
[4] S Bittanti A Laub and J CWillems EdsTheRiccati EquationCommunications and Control Engineering Series SpringerBerlin Germany 1991
[5] P Lancaster andL RodmanAlgebraic Riccati Equations OxfordScience Publications Oxford University Press New York NYUSA 1995
[6] A J Laub ldquoInvariant subspace methods for the numerical solu-tion of Riccati equationsrdquo inThe Riccati Equation S Bittanti AJ Laub and J C Willems Eds Communications and ControlEngineering pp 163ndash196 Springer Berlin Germany 1991
[7] A J Laub ldquoA Schur method for solving algebraic Riccatiequationsrdquo IEEE Transactions on Automatic Control vol 24 no6 pp 913ndash921 1979
[8] M Kimura ldquoDoubling algorithm for continuous-time algebraicRiccati equationrdquo International Journal of Systems Science vol20 no 2 pp 191ndash202 1989
[9] E K-W Chu H-Y FanW-W Lin and C-SWang ldquoStructure-preserving algorithms for periodic discrete-time algebraic Ric-cati equationsrdquo International Journal of Control vol 77 no 8pp 767ndash788 2004
[10] E K-W Chu H-Y Fan andW-W Lin ldquoA structure-preservingdoubling algorithm for continuous-time algebraic Riccati equa-tionsrdquo Linear Algebra and Its Applications vol 396 pp 55ndash802005
[11] N J Higham ldquoPerturbation theory and backward error for119860119883 minus 119883119861 = 119862rdquo BIT Numerical Mathematics vol 33 no 1 pp124ndash136 1993
[12] M Konstantinov and P Petkov ldquoNote on perturbation theoryfor algebraic Riccati equationsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 21 no 1 pp 327ndash354 2000
[13] J-G Sun ldquoResidual bounds of approximate solutions of thealgebraic Riccati equationrdquoNumerischeMathematik vol 76 no2 pp 249ndash263 1997
[14] J-G Sun ldquoBackward error for the discrete-time algebraicRiccati equationrdquo Linear Algebra and Its Applications vol 259pp 183ndash208 1997
[15] J-G Sun ldquoBackward perturbation analysis of the periodicdiscrete-time algebraic Riccati equationrdquo SIAM Journal onMatrix Analysis and Applications vol 26 no 1 pp 1ndash19 2004
[16] V Mehrmann ldquoA step towards a unified treatment of continu-ous and discrete time control problemsrdquo Linear Algebra and ItsApplications vol 241ndash243 pp 749ndash779 1996
[17] H-G Xu ldquoTransformations between discrete-time andcontinuous-time algebraic Riccati equationsrdquo Linear Algebraand Its Applications vol 425 no 1 pp 77ndash101 2007
[18] A C M Ran and M C B Reurings ldquoOn the nonlinear matrixequation 119883 + 119860
lowastF (119883)119860 = 119876 solutions and perturbationtheoryrdquo Linear Algebra and Its Applications vol 346 pp 15ndash262002
[19] A C M Ran M C B Reurings and L Rodman ldquoA pertur-bation analysis for nonlinear selfadjoint operator equationsrdquoSIAM Journal on Matrix Analysis and Applications vol 28 no1 pp 89ndash104 2006
[20] H Lee and Y Lim ldquoInvariant metrics contractions and nonlin-ear matrix equationsrdquo Nonlinearity vol 21 no 4 pp 857ndash8782008
[21] J Cai and G-L Chen ldquoOn the Hermitian positive definitesolutions of nonlinear matrix equation 119883
119904
+ 119860119883minus119905
119860 = 119876rdquoApplied Mathematics and Computation vol 217 no 1 pp 117ndash123 2010
[22] X-F Duan Q-W Wang and A-P Liao ldquoOn the matrix equa-tion arising in an interpolation problemrdquo Linear andMultilinearAlgebra vol 61 no 9 pp 1192ndash1205 2013
[23] Z-G Jia and M-S Wei ldquoSolvability and sensitivity analysisof polynomial matrix equation 119883
119904
+ 119860119879
119883119905
119860 = 119876rdquo AppliedMathematics and Computation vol 209 no 2 pp 230ndash2372009
[24] M-H Wang M-S Wei and S Hu ldquoThe extremal solution ofthe matrix equation 119883119904 + 119860lowast119883minus119902119860 = 119868rdquo Applied Mathematicsand Computation vol 220 pp 193ndash199 2013
[25] B Zhou G-B Cai and J Lam ldquoPositive definite solutions ofthe nonlinear matrix equation 119883 + 119860
119867
119883minus1
119860 = 119868rdquo AppliedMathematics and Computation vol 219 no 14 pp 7377ndash73912013
[26] X-Z Zhan Matrix Inequalities vol 1790 of Lecture Notes inMathematics Springer Berlin Germany 2002
[27] T Furuta ldquoOperator inequalities associated with Holder-McCarthy and Kantorovich inequalitiesrdquo Journal of Inequalitiesand Applications vol 1998 Article ID 234521 1998
[28] R A Horn and C R Johnson Topics in Matrix AnalysisCambridge University Press Cambridge UK 1991
[29] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990
[30] X-G Liu and H Gao ldquoOn the positive definite solutions of thematrix equations 119883119904 minus 119860119879119883minus119905119860 = 119868
119899rdquo Linear Algebra and Its
Applications vol 368 pp 83ndash97 2003[31] S-F Xu ldquoPerturbation analysis of the maximal solution of
the matrix equation 119860lowast
119883minus1
119860 = 119875rdquo Linear Algebra and ItsApplications vol 336 pp 61ndash70 2001
[32] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985
[33] S M El-Sayed and A C M Ran ldquoOn an iteration method forsolving a class of nonlinear matrix equationsrdquo SIAM Journal onMatrix Analysis and Applications vol 23 no 3 pp 632ndash6452001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
[4] S Bittanti A Laub and J CWillems EdsTheRiccati EquationCommunications and Control Engineering Series SpringerBerlin Germany 1991
[5] P Lancaster andL RodmanAlgebraic Riccati Equations OxfordScience Publications Oxford University Press New York NYUSA 1995
[6] A J Laub ldquoInvariant subspace methods for the numerical solu-tion of Riccati equationsrdquo inThe Riccati Equation S Bittanti AJ Laub and J C Willems Eds Communications and ControlEngineering pp 163ndash196 Springer Berlin Germany 1991
[7] A J Laub ldquoA Schur method for solving algebraic Riccatiequationsrdquo IEEE Transactions on Automatic Control vol 24 no6 pp 913ndash921 1979
[8] M Kimura ldquoDoubling algorithm for continuous-time algebraicRiccati equationrdquo International Journal of Systems Science vol20 no 2 pp 191ndash202 1989
[9] E K-W Chu H-Y FanW-W Lin and C-SWang ldquoStructure-preserving algorithms for periodic discrete-time algebraic Ric-cati equationsrdquo International Journal of Control vol 77 no 8pp 767ndash788 2004
[10] E K-W Chu H-Y Fan andW-W Lin ldquoA structure-preservingdoubling algorithm for continuous-time algebraic Riccati equa-tionsrdquo Linear Algebra and Its Applications vol 396 pp 55ndash802005
[11] N J Higham ldquoPerturbation theory and backward error for119860119883 minus 119883119861 = 119862rdquo BIT Numerical Mathematics vol 33 no 1 pp124ndash136 1993
[12] M Konstantinov and P Petkov ldquoNote on perturbation theoryfor algebraic Riccati equationsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 21 no 1 pp 327ndash354 2000
[13] J-G Sun ldquoResidual bounds of approximate solutions of thealgebraic Riccati equationrdquoNumerischeMathematik vol 76 no2 pp 249ndash263 1997
[14] J-G Sun ldquoBackward error for the discrete-time algebraicRiccati equationrdquo Linear Algebra and Its Applications vol 259pp 183ndash208 1997
[15] J-G Sun ldquoBackward perturbation analysis of the periodicdiscrete-time algebraic Riccati equationrdquo SIAM Journal onMatrix Analysis and Applications vol 26 no 1 pp 1ndash19 2004
[16] V Mehrmann ldquoA step towards a unified treatment of continu-ous and discrete time control problemsrdquo Linear Algebra and ItsApplications vol 241ndash243 pp 749ndash779 1996
[17] H-G Xu ldquoTransformations between discrete-time andcontinuous-time algebraic Riccati equationsrdquo Linear Algebraand Its Applications vol 425 no 1 pp 77ndash101 2007
[18] A C M Ran and M C B Reurings ldquoOn the nonlinear matrixequation 119883 + 119860
lowastF (119883)119860 = 119876 solutions and perturbationtheoryrdquo Linear Algebra and Its Applications vol 346 pp 15ndash262002
[19] A C M Ran M C B Reurings and L Rodman ldquoA pertur-bation analysis for nonlinear selfadjoint operator equationsrdquoSIAM Journal on Matrix Analysis and Applications vol 28 no1 pp 89ndash104 2006
[20] H Lee and Y Lim ldquoInvariant metrics contractions and nonlin-ear matrix equationsrdquo Nonlinearity vol 21 no 4 pp 857ndash8782008
[21] J Cai and G-L Chen ldquoOn the Hermitian positive definitesolutions of nonlinear matrix equation 119883
119904
+ 119860119883minus119905
119860 = 119876rdquoApplied Mathematics and Computation vol 217 no 1 pp 117ndash123 2010
[22] X-F Duan Q-W Wang and A-P Liao ldquoOn the matrix equa-tion arising in an interpolation problemrdquo Linear andMultilinearAlgebra vol 61 no 9 pp 1192ndash1205 2013
[23] Z-G Jia and M-S Wei ldquoSolvability and sensitivity analysisof polynomial matrix equation 119883
119904
+ 119860119879
119883119905
119860 = 119876rdquo AppliedMathematics and Computation vol 209 no 2 pp 230ndash2372009
[24] M-H Wang M-S Wei and S Hu ldquoThe extremal solution ofthe matrix equation 119883119904 + 119860lowast119883minus119902119860 = 119868rdquo Applied Mathematicsand Computation vol 220 pp 193ndash199 2013
[25] B Zhou G-B Cai and J Lam ldquoPositive definite solutions ofthe nonlinear matrix equation 119883 + 119860
119867
119883minus1
119860 = 119868rdquo AppliedMathematics and Computation vol 219 no 14 pp 7377ndash73912013
[26] X-Z Zhan Matrix Inequalities vol 1790 of Lecture Notes inMathematics Springer Berlin Germany 2002
[27] T Furuta ldquoOperator inequalities associated with Holder-McCarthy and Kantorovich inequalitiesrdquo Journal of Inequalitiesand Applications vol 1998 Article ID 234521 1998
[28] R A Horn and C R Johnson Topics in Matrix AnalysisCambridge University Press Cambridge UK 1991
[29] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press Cambridge UK 1990
[30] X-G Liu and H Gao ldquoOn the positive definite solutions of thematrix equations 119883119904 minus 119860119879119883minus119905119860 = 119868
119899rdquo Linear Algebra and Its
Applications vol 368 pp 83ndash97 2003[31] S-F Xu ldquoPerturbation analysis of the maximal solution of
the matrix equation 119860lowast
119883minus1
119860 = 119875rdquo Linear Algebra and ItsApplications vol 336 pp 61ndash70 2001
[32] P Lancaster and M Tismenetsky The Theory of MatricesComputer Science and Applied Mathematics Academic PressOrlando Fla USA 2nd edition 1985
[33] S M El-Sayed and A C M Ran ldquoOn an iteration method forsolving a class of nonlinear matrix equationsrdquo SIAM Journal onMatrix Analysis and Applications vol 23 no 3 pp 632ndash6452001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
