The iterative methods for solving nonlinear matrix equation X+A⋆X−1A+B⋆X−1B=Q

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R E S E A R C H

Open Access

The iterative methods for solving nonlinear

matrix equation

X

+

A

X

–

A

+

B

X

–

B

=

Q

Sarah Vaezzadeh

1

_{, Seyyed Mansour Vaezpour}

1

_{, Reza Saadati}

2

_{and Choonkil Park}

3*

*_{Correspondence:}

[email protected]

3_{Research Institute for Natural}

Sciences, Hanyang University, Seoul, 133-791, Korea

Full list of author information is available at the end of the article

Abstract

In this paper, we study the matrix equationX+AX–1_A₊_B_X–1_B₌_Q_{, where}_A_and_B

are square matrices, andQis a positive definite matrix, and propose the iterative methods for finding positive definite solutions of the matrix equation. Also, general convergence results for the basic fixed point iteration for these equations are given. Some numerical examples are presented to show the usefulness of the iterations.

MSC: 65F10; 65F30; 65H10; 15A24

Keywords: nonlinear matrix equation; positive deﬁnite solution; inversion-free variant iterative method; convergence rate

1 Introduction

In this paper, we consider the matrix equation

X+AX–A+BX–B=Q, ()

whereAandBare square matrices,Qis a positive deﬁnite matrix. It is easy to see that matrix equation () can be reduced to

X+AX–A+BX–B=I, ()

whereIis the identity matrix. Trying to solve special linear systems [] leads to solving nonlinear matrix equations of the above types as follows.

For a linear systemMx=f with

M=

⎛ ⎜ ⎝

Q  A

 Q B

A _B _Q

⎞ ⎟ ⎠

positive deﬁnite, we rewriteM=M˜ +K, where

˜

M=

⎛ ⎜ ⎝

X  A

 X B

A _B _Q

⎞ ⎟

⎠, K=

⎛ ⎜ ⎝

Q–X  

 Q–X 

  

⎞ ⎟ ⎠.

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Moreover, we decomposeM˜ to theLUdecomposition

˜

M=

⎛ ⎜ ⎝

X  A

 X B

A _B _Q

⎞ ⎟ ⎠=

⎛ ⎜ ⎝

I  

 I 

A_X– _B_X– _I

⎞ ⎟ ⎠

⎛ ⎜ ⎝

X  A

 X B

  X

⎞ ⎟ ⎠.

A decomposition ofM˜ exists if and only ifXis a positive deﬁnite solution of matrix equation (). Solving the linear systemMy˜ =f is equivalent to solving two linear systems with a lower and upper block triangular system matrix. To compute the solution ofMx=f

fromy, the Woodbury formula [] can be applied.

The matrix equationX+A_X–_A₌_Q_{has been studied extensively by many authors [–} ]. Several conditions for the existence of positive definite solutions and some iterations to find maximal positive definite solutions for these equations were discussed. Apparently, matrix equation () generalizes the matrix equationX+A_X–_A₌_Q_.

Matrix equation () was studied in [], and based on some conditions, the authors proved that matrix equation () has positive definite solutions. They also proposed two iterative methods to find the Hermitian positive definite solutions of matrix equation (). They did not analyze the convergence rate of proposed algorithms.

In this paper, we propose two algorithms. We will show that Algorithm () is more ac-curate than Algorithm () pointed out in []. Also, Algorithm () needs less operation in comparison with Algorithm (). The following notations are used throughout the rest of the paper. The notationA≥ (A> ) means thatAis Hermitian positive semideﬁnite (positive deﬁnite). For Hermitian matricesAandB, we writeA≥B(A>B) ifA–B≥ (> ). Similarly, byλ(A) andλn we denote, respectively, the maximal and the minimal eigenvalues ofA. The norm used in this paper is the spectral norm of the matrixA,i.e.,

A= (λ(AA))  _.

2 Fixed point theorems

Lemma [] If C and P are Hermitian matrices of the same order with P> ,then CPC+

P–≥C.

In [] an algorithm that avoids matrix inversion for every iteration, called inversion-free variant of the basic fixed point iteration, and a theorem to find a Hermitian positive definite solution of matrix equation () were proposed as follows.

Algorithm [] Let

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

X=Y=I,

Xn+=I–AYnA–BYnB,

Yn+= Yn–YnXnYn, n= , , , . . . .

()

Theorem [] Assume that matrix equation()has a positive deﬁnite solution.Then Algorithm()deﬁnes a monotonically decreasing matrix sequence{Xn}converging to the

positive deﬁnite matrix X which is a solution of matrix equation().Also,the sequence{Yn}

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Although it is not mentioned in the previous theorem that the sequence{Xn}converges to the maximal Hermitian positive deﬁnite solution of equation (), during the proof of the theorem in [], it is obvious. So,X=X∞, whereX∞is the maximal positive deﬁnite solution of matrix equation () in Theorem .

The problem of convergence rate for Algorithm () was not considered in []. We now establish the following result.

Theorem  If matrix equation()has a positive deﬁnite solution,for Algorithm()and any> ,we have

for all n large enough.

Proof From Algorithm (), we have

Yn+= Yn–Yn equality () is true since

Xn+–X∞=A

This completes the proof.

The above proof shows that Algorithm () should be modiﬁed as follows to improve the preceding convergence properties.

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is the maximal Hermitian positive deﬁnite solution of equation().Also,the sequence{Yn}

deﬁned in Algorithm()deﬁnes a monotonically increasing sequence converging to X_∞–.

Proof LetX+be a positive deﬁnite solution of matrix equation (). It is clear that

X≥X≥ · · · ≥Xn≥X+, Y≤Y≤ · · · ≤Yn≤X+– ()

is true forn= . Assume () is true forn=k. By Lemma , we have that

Yk+= Yk–YkXkYk≤X–K ≤X+–.

Therefore,

Xk+=I–AYk+A≥I–AX+–A=X+.

SinceYk≤Xk–– ≤Xk–, we haveYk–≥Xk. Thus

Yk+–Yk=Yk

Y_k––Xk

Yk≥

and

Xk+–Xk= –A(Yk+–Yk)A≤.

We have now proved () forn=k+ . Therefore, () is true for alln, andlim_n_→∞Xnand

limn→∞Ynexist. So, we havelimXn=X∞andlimYn=X_∞–.

Similar to Theorem , we can state the following theorem.

Theorem  If matrix equation()has a positive deﬁnite solution for Algorithm()and any> ,then we have

Yn+–X–∞≤AX∞–+BX–∞+



Yn–X∞–

and

Xn–X∞ ≤

A+BYn–X∞– ()

Now, we can see that Algorithm () can be faster than Algorithm () from the estimates in Theorems  and .

Algorithm  Take

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

X=I, Y=I,

Yn+= (I–Xn)Yn+I,

Xn+=I–AYn+A–BYn+B, n= , , , . . . .

()

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Theorem  If matrix equation()has a positive deﬁnite solution and the two sequences

{Xn}and{Yn}are determined by Algorithm(),then{Xn}is monotone decreasing and

converges to the maximal Hermitian positive deﬁnite solution X∞.Also,the sequence{Yn}

deﬁned in Algorithm()is a monotonically increasing sequence converging to X–

∞.

Proof LetX+be a positive deﬁnite solution of equation (). We prove that

X≥X≥ · · · ≥Xn≥X+ ()

and

Y≤Y≤ · · · ≤Yn≤X–+ . ()

SinceX+is a solution of matrix equation (),X=I≥X+. Also, we haveY=Y=I, and so

X=I–AA–BB≥I–AX+–A–B

X–₊ B=X+,

i.e.,X≥X≥X+.

For the sequence{Yn}, sinceI≤X+–,Y=Y=I≤X+–.

Thus inequalities () and () are true forn= . Now, assume that inequalities () and () are true forn=k,i.e.,

X≥X≥ · · · ≥Xk≥X+

and

Y≤Y≤ · · · ≤Yk≤X+–.

We show that inequalities () and () are true forn=k+ . We have

Yk+= (I–Xk)Yk+I≥(I–Xk–)Yk–+I=Yk

and

Yk+= (I–Xk)Yk+I≤(I–X+)X+–+I=X+–,

i.e.,Yk≤Yk+≤X+–. Then

Xk–Xk+=A(Yk+–Yk)A+B(Yk+–Yk)B,

andXk+≤Xk, sinceYk≤Yk+. On the other hand,

Xk+ =I–AYk+A–BYk+B

≥I–A_X–

+ A–BX–+ B

=X+, ()

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Then the above inequalities are true for alln, alsolimn→∞Xnandlimn→∞Ynexist. By taking limit on Algorithm (), we havelimn→∞Xn=X∞andlimn→∞Yn=X∞–, whereX∞

is the maximal positive deﬁnite solution of matrix equation ().

By Algorithm (), we haveI–XnYn=Yn+–Yn. Then, for small> ,I–XnYncan be one stopping condition.

Theorem  If matrix equation()has a positive deﬁnite solution and after n iterative steps of Algorithm(),the inequalityI–XnYn<implies

Xn+AXn–A+B

X_n–B–I≤A+BX_∞–.

Proof Since

Xn+AX–n A+B

X_n–B–I=Xn–Xn++A

X_n––Yn+

A+BX–_n –Yn+

B

=AYn+–Xn–+Xn––Yn

A

+BYn+–Xn–+Xn––Yn

B

+AX_n––Yn+

A+BX_n––Yn+

B ()

=AX_n–(I–XnYn)A+BX–n (I–XnYn)B,

Xn+AXn–A+B

X_n–B–I≤A+BX_∞–I–XnYn

≤A+BX_∞–.

This completes the proof.

Theorem  If Xn> for every n,then matrix equation()has a Hermitian positive deﬁnite

solution.

Proof SinceXn>  for everyn, the proof of the monotonicity of{Yn} and{Xn}noted in Theorem  remains valid. Therefore, the sequence{Xn}is monotone decreasing and bounded from below by the zero matrix. Thenlim_n_→∞Xn=Xexists. We claim that the sequence{Yn}is bounded above. Suppose that it does not hold. Then, for everym> , there existsnmsuch thatmI<Ynm. Since eachXnis positive deﬁnite for everyn, we have

A_Y

nA+BYnB<I for everyn.

Furthermore, sinceAorBare nonsingular for everym> , we have

mAA+BB<AYnmA+B

YnmB<I.

By [, Lemma .],

mAA+BB<I for everym,

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in Algorithm () implies that

Y= (I–X)Y+I,

X=I–AYA–BYB.

SinceY≤I,X=Y–> , and henceX=I–A_X–_A_–_B_X–_B_{. Then matrix equation ()}

has a positive deﬁnite solution.

Theorem  If matrix equation()has a positive deﬁnite solution andA<_ andB<

Proof From Algorithm (), we have

Yn+= (I–Xn)Yn+I

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3 Numerical examples

In this section, we present some numerical examples to show the effectiveness of the new inversion-free variant of the basic fixed point iteration methods. Hermitian positive defi-nite solutions of matrix equation () for different matricesAandBare computed. We will compare the suggested algorithms, Algorithm () and Algorithm (), by Algorithm (). All programs were written in MATLAB.

Example  Consider equation () with

A= 

Algorithm () needs six iterations to get the solution

X=

Algorithm () needs six iterations to get the solution

X=

We can easily see that Algorithm () is more accurate than Algorithm (). Algorithm () needs six iterations to get the solution

X=

Example  Consider equation () with

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Algorithm () after  iterations gives the solution

Algorithm () after  iterations gives the solution

X=

Algorithm () after  iterations gives the solution

X=

We can see that Algorithm () needs to ﬁnd a Hermitian positive deﬁnite solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics and Computer Science, Amirkabir University of Technology, Hafez Ave., P.O. Box 15914,}

Tehran, Iran. 2_{Department of Mathematics, Iran University of Science and Technology, Tehran, Iran.}3_{Research Institute for}

Natural Sciences, Hanyang University, Seoul, 133-791, Korea.

Acknowledgements

The authors are grateful to the two anonymous reviewers for their valuable comments and suggestions.

Received: 13 June 2013 Accepted: 22 July 2013 Published: 6 August 2013 References

1. Buzbee, BL, Golub, GH, Nielson, CW: On direct methods for solving Poisson’s equations. SIAM J. Numer. Anal.7, 627-656 (1970)

2. Housholder, AS: The Theory of Matrices in Numerical Analysis. Blaisdell, New York (1964)

3. Engwerda, JC, Ran, AC, Rijkeboer, AL: Necessary and suﬃcient conditions for the existence of a positive deﬁnite solution of the matrix equationX+A_X–1_A₌_Q_{. Linear Algebra Appl.}₁₈₆_{, 255-275 (1993)}

4. Engwerda, JC: On the existence of a positive deﬁnite solution of the matrix equationX+AT_X–1_A₌_I_{. Linear Algebra}

Appl.194, 91-108 (1993)

5. Lancaster, P, Rodman, L: Algebraic Riccati Equations. Oxford Science Publishers, Oxford (1995) 6. Zhan, X, Xie, J: On the matrix equationX+AT_X–1_A₌_I_{. Linear Algebra Appl.}₂₄₇_{, 337-345 (1996)}

7. Anderson, WN, Morley, TD, Trapp, GE: Positive solutions of the matrix equationX=A–B_X–1_B_{. Linear Algebra Appl.} 134, 53-62 (1990)

(10)

9. Shufang, X: On the maximal solution for the matrix equationX+AT_X–1_A₌_I_{. Acta Sci. Nat. Univ. Pekin.}₃₆_{, 29-38 (2000)}

10. Guo, CH, Lancaster, P: Iterative solution of two matrix equations. Math. Comput.68, 1589-1603 (1999)

11. Meini, B: Eﬃcient computation of the extreme solutions ofX+A_X–1_A₌_Q_and_X_–_A_X–1_A₌_Q_{. Math. Comput.}₇₁_,

1189-1204 (2002)

12. El-Sayed, SM, Ran, ACM: On an iteration method for solving a class of nonlinear matrix equation. SIAM J. Matrix Anal. Appl.23, 632-645 (2001)

13. Ivanov, IG, Hasanov, VI, Uhlig, F: Improved methods and starting values to solve the matrix equationsX±A_X–1_A₌_I

iteratively. Math. Comput.74, 263-278 (2004)

14. El-Sayed, SM, Al-Dbiban, AM: A new inversion free iteration for solving the equationX+A_X–1_A₌_Q_{. J. Comput. Appl.}

Math.181, 148-156 (2005)

15. Long, JH, Hu, XY, Zhang, L: On the Hermitian positive deﬁnite solution of the matrix equationX+A_X–1_A₊_B_X–1_B₌_I_.

Bull. Braz. Math. Soc.39, 371-386 (2008)

16. Chen, G, Huang, X, Yang, X: Vector Optimization: Set-Valued and Variational Analysis. Springer, Berlin (2005)

doi:10.1186/1687-1847-2013-229

Cite this article as:Vaezzadeh et al.:The iterative methods for solving nonlinear matrix equation