Iteration with Stepsize Parameter and Condition Numbers for a Nonlinear Matrix Equation

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1 Electronic Journal of Linear Algebra Volume 34 Volume ) Article Iteration with Stesize Parameter and Condition Numbers for a Nonlinear Matrix Equation Syed M Raza Shah Naqvi Pusan National University raza@usanackr Jie Meng Pusan National University mengjiehw@163com Hyun-Min Kim Pusan National University hyunmin@usanackr Follow this and additional works at: Part of the Analysis Commons and the Numerical Analysis and Comutation Commons Recommended Citation M Raza Shah Naqvi Syed; Meng Jie; and Kim Hyun-Min 2018) "Iteration with Stesize Parameter and Condition Numbers for a Nonlinear Matrix Equation" Electronic Journal of Linear Algebra Volume DOI: htts://doiorg/ / This Article is brought to you for free and oen access by Wyoming Scholars Reository It has been acceted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Reository For more information lease contact scholcom@uwyoedu

2 ITERATION WITH STEPSIZE PARAMETER AND CONDITION NUMBERS FOR A NONLINEAR MATRIX EQUATION SYED M RAZA SHAH NAQVI JIE MENG AND HYUN-MIN KIM Abstract In this aer the nonlinear matrix equation X + A T XA = Q where is a ositive integer A is an arbitrary n n matrix and Q is a symmetric ositive definite matrix is considered A fixed-oint iteration with stesize arameter for obtaining the symmetric ositive definite solution of the matrix equation is roosed The exlicit exressions of the normwise mixed and comonentwise condition numbers are derived Several numerical examles are resented to show the efficiency of the roosed iterative method with roer stesize arameter and the sharness of the three kinds of condition numbers Key words Matrix equation Symmetric ositive definite Fixed-oint iteration Condition number Mixed and comonentwise AMS subject classifications 15A24 65F10 65H10 1 Introduction We consider the nonlinear matrix equations 11) X + A T XA = Q where is a ositive integer A R n n and Q is an n n symmetric ositive definite matrix This tye of nonlinear matrix equation has been studied recently by several authors see [6 9 15] For the case = 1 equation 11) reduces to a symmetric linear matrix equation [16] which aears in each ste of Newton s method for solving nonlinear matrix equations such as the one aears in Chater 7 of [19] For the case > 1 equation 11) is equivalent to Y + A T Y 1 A = Q with Y = X This is an examle of the equation studied by El-Sayed and Ran [5]; see also [17] Jia and Wei [10] studied the matrix equation X s +A T X t A = Q where s and t are both nonnegative integers and they roved that a symmetric ositive definite solution exists if λ max A T A) λ min Q)λ max Q)) t s They have also showed that the ositive definite solution could be unique under some certain condition Meng and Kim [13] studied equation 11) and roosed two elegant estimates of the ositive definite solution and three basic fixed-oint iterations for obtaining the solution For the erturbation analysis of equation 11) Jia and Wei [10] investigated the algebraic erturbation analysis of the unique symmetric ositive solution and they defined one normwise condition number In [22] Wang Yang and Li derived the exlicit exressions of the normwise mixed and comonentwise condition numbers for the nonlinear matrix equation X + A F X)A = Q Motivated by this we investigate the mixed and comonentwise condition numbers of equation 11) As a continuation of the revious work we first roose a fixed-oint iteration with stesize arameter Received by the editors on November Acceted for ublication on October Handling Editor: James G Nagy Corresonding Author: Hyun-Min Kim This research was suorted by Basic Science Research Program through the National Research Foundation of Korea NRF) funded by the Ministry of Education 2017R1D1A3B ) and the National Research Foundation of Korea NRF) Grant funded by the Korean Government MSIP) NRF-2017R1A5A ) Deartment of Mathematics Pusan National University Busan Korea raza@usanackr mengjiehw@163com hyunmin@usanackr)

3 Syed M Raza Shah Naqvi J Meng and H-M Kim 218 for obtaining the symmetric ositive definite solution of equation 11) Based on the couled fixed-oint theory we rove that the matrix sequences generated by the roosed iteration with stesize arameter are convergent We then investigate two kinds of normwise condition numbers and derive the exlicit exressions of mixed and comonentwise condition numbers for equation 11) This aer is organized as follows In Section 2 we roose a fixed-oint iteration with stesize arameter for obtaining the symmetric ositive definite solution of equation 11) and we rove the convergence of the matrix sequence In Section 3 we derive the exlicit exressions of the normwise mixed and comonentwise condition numbers In Section 4 we give some numerical examles to show the efficiency of the roosed iterative method and the sharness of the three kinds of condition numbers We begin with some notations used throughout this aer R n n stands for the set of n n matrices with elements on field R 2 F and are the sectral norm Frobenius norm and matrix row norm resectively For a matrix B = b ij ) R n n B max is the matrix norm given by B max = max ij b ij and B is the matrix whose elements are b ij The set of all n n ositive definite matrices is resented by P n) For a Hermitian matrix H λ min H) and λ max H) denote the minimal eigenvalue and the maximal eigenvalue resectively Similarly σ min H) and σ max H) denote the minimal and the maximal singular value resectively For Hermitian matrices X and Y X Y X > Y ) means that X Y is ositive semidefinite definite) and [αi βi] denotes the matrices set {X : X αi 0 and βi X 0} 2 Iteration with stesize arameter In this section we roose a fixed-oint iteration with stesize arameter for obtaining the symmetric ositive definite solution of equation 11) In [13] there was a basic fixed-oint iteration as follows: { X 0 = αm 1 22) X k+1 = Q A T X k A) 1 where α is a ositive solution of the following equations 1 α = λ min Q βa T N 1A) 1 λmin Q AT N 1A) 1 β = λ max Q αa T M 1A) 1 λmaxq) with M 1 = λ 1 min Q λ 1 max Q)A A)I and N 1 = λ 1 max Q)I Based on the above iteration 22) we roose a fixed-oint iteration with stesize arameter as follows: { X 0 = σi σ [a b] 23) X k+1 = 1 α)x k + αq A T X k A) 1 where a = α 0 1) λ min Q) λ max A T A)λ 1 max Q) b = λ max Q) ) 1 ) 1

4 219 Iteration With Stesize Parameter and Condition Numbers for a Nonlinear Matrix Equation To rove the convergence of iteration 23) we introduce the following well-known results: Lemma 21 Löwner-Heinz inequality [23 Theorem 11]) If A B 0 and 0 r 1 then A r B r Lemma 22 [3]) If 0 < θ 1 and P and Q are ositive definite matrices of the same order with P Q bi > 0 then P θ Q θ θb θ 1 P Q and P θ Q θ θb θ+1) P Q Here stands for one kind of matrix norm F : X X X is a mixed monotone maing if for any x y X x 1 x 2 X x 1 x 2 F x 1 y) F x 2 y) y 1 y 2 X y 1 y 2 F x y 1 ) F x y 2 ) Theorem 23 [1 Theorem 3]) Let X ) be a artially ordered set and suose there is a metric d on X such that X d) is a comlete metric sace Let F : X X X be a mixed monotone maing for which there exists a constant k [0 1) such that for each x u y v If there exist x 0 y 0 X such that then there exist x ȳ X such that df x y) F u v)) + df y x) F v u)) k[dx u) + dy v)] x 0 F x 0 y 0 ) and y 0 F y 0 x 0 ) x = F x ȳ) and ȳ = F ȳ x) Moreover the matrix sequences {x k } and {y k } generated by { xk+1 = F x k y k ) y k+1 = F y k x k ) converge resectively to x and ȳ And if every air x y) X X has a lower bound or an uer bound then x = ȳ Theorem 23 is a generalization of the couled fixed-oint theorem obtained by Bhaskar and Lakshmikantham [2] and it will be used in our roof of the convergence of iteration 23) Theorem 24 If λ min Q) > λ max A T A)λ 1 max Q) a1 A 2 2 < 1 then the matrix sequence {X k } generated by iteration 23) is convergent to a symmetric ositive definite solution of equation 11) Proof Let Ω = {X : X = X T and ai X bi} under Frobenius norm Ω F ) is a Banach sace Define F on Ω Ω by F X Y ) = 1 α)x + αq A T Y A) 1 Clearly F is a mixed monotone maing And F : Ω Ω Ω Indeed for any X Y Ω alying Lemma 21 yields F X Y ) = 1 α)x + αq A T Y A) 1 1 α)ai + αq ba T A) 1 1 α)ai + αai = ai

5 Syed M Raza Shah Naqvi J Meng and H-M Kim 220 and F X Y ) 1 α)bi + αq aa T A) 1 1 α)bi + αq 1 bi Let k = 1 α 1 A 2 2 a 1 )) and clearly k 0 1) For any X Y U V Ω such that X U Y V note that Q A XA a I and Q A Y A a I by Lemma 22 we have and similarly Then F X Y ) F U V ) F 1 α) X U F + α Q A T Y A) 1 Q A T V A) 1 F 1 α) X U F + αa1 A 2 2 V Y F F Y X) F V U) F 1 α) Y V F + αa1 A 2 2 X U F ) F X Y ) F U V ) F + F Y X) F V U) F k X U F + Y V F Let X 0 = ai Y 0 = bi since F Ω Ω) Ω it is trivial that X 0 F X 0 Y 0 ) and Y 0 F Y 0 X 0 ) By Theorem 23 there are X Ȳ Ω such that X = F X Ȳ ) and Ȳ = F Ȳ X) Since every X Y ) Ω Ω has a lower bound or an uer bound we can get X = Ȳ Define the matrix sequences {X k } and {Y k } by X 0 = ai Y 0 = bi X k+1 = F X k Y k ) = 1 α)x k + αq AT Y k A) 1 Y k+1 = F Y k X k ) = 1 α)y k + αq AT X k A) 1 According to Theorem 23 we have lim k X k = X and lim k Y k = Ȳ

6 221 Iteration With Stesize Parameter and Condition Numbers for a Nonlinear Matrix Equation To show the matrix sequence {X k } generated by 23) is convergent we first show that X k X k Y k is true for k = 0 1 Note that X k+1 = F X k X k ) and X 0 X 0 Y 0 by the mixed monotonicity of F we have F X 0 Y 0) F X 0 X 0 ) F Y 0 X 0) that is X 1 X 1 Y 1 By mathematical induction it is easy to get X k X k Y k for all k = 0 1 Since lim k X k = X = Ȳ = lim Y k k it follows that {X k } converges to X we can see that X satisfies equation 11) Theorem 25 Let X + be a ositive definite solution of equation 11) Consider the sequence {X k } generated by iteration 23) Assume that X k X + for some k then X k+1 X k Proof Assume that X k X + holds for one integer k then from Lemma 21 it follows that X k+1 = 1 α)x k + αq A T X k A) 1 1 α)x k + αq A T X + A) 1 = 1 α)x k + αx + X k Corollary 26 Let X + be a ositive definite solution of equation 11) Consider the sequence {X k } generated by 23) Assume that X k X + for some k then X k+1 X k 3 Normwise mixed and comonentwise condition numbers In this section we investigate the normwise mixed and comonentwise condition numbers of equation 11) Consider the erturbed equation 34) X + X) + A + A) T X + X)A + A) = Q + Q where A Q R n n Subtracting 11) from 34) yields 35) X + X) X = Q A T X A + A T XA + A T XA) E where E = A T X A + A T X A + A T X A + A T X A For notational simlicity we introduce a function φ : N N R n n R n n R n n defined in [20] as { φi 0)X Y ) = X i φ0 j)x Y ) = Y j i N φi j)x Y ) = Xφi 1 j) + Y φi j 1) ) X Y ) i j N + where N is the set of natural numbers and N + = N {0} We can easily get φ0 0)X Y ) = I n

7 Syed M Raza Shah Naqvi J Meng and H-M Kim 222 and φn 1)X Y ) = Using the notations given above we can also have Then 36) X + X) X = X + Y ) n = n X n i Y X i i=0 n φn i i)x Y ) i=0 φ i i)x X) X i=0 n N = φ 0)X X) + φ 1 1)X X) + G X) X 1 = X 1 j XX j + G X) j=0 where G X) = i=2 φ i i)x X) Combining 35) and 36) and alying the vec oerator yields 37) 1 ) X j X 1 j + A T A T vec X) j=0 = vec Q) ) ) I A T X) + A T X) I)Π vec A) + O Q A 2 F where Π R n2 n 2 is the vec ermutation satisfying ΠvecA) = veca T ) and here it is defined by Π = n i=1 j=1 n E ij n n) E ji n n) ) with E ij = e n) i e n) j ) T R n n and e n) i is the i-th column of the identity matrix I O Q A 2 F is the first order aroximation of X with resect to Q A) According to the imlicit function theorem we can get X 0 as Q A) 0 From Theorem 1632 in [8] we can get A T X) I)Π = ΠI A T X)) Then omitting the higher order terms in 37) yields 38) 1 ) X j X 1 j + A T A T vec X) j=0 = vec Q) ) I n 2 + Π)I A T X) vec A) Define a ma Φ : R 2n2 R n2 by 39) Φ : ω = vecq) T veca) T ) T vecx) where X is the unique symmetric ositive definite solution of equation 11)

8 223 Iteration With Stesize Parameter and Condition Numbers for a Nonlinear Matrix Equation 31 Normwise condition number According to Rice [18] and [21] we investigate two kinds of normwise condition numbers of ma Φ which are defined by 310) where Krel i X F = lim su i = 1 2 δ 0 i δ δ X F 1 = Q A) F Q A) F and 2 = Q Q F A A F ) F If Φ is Fréchet differentiable at ω = vecq) veca) T ) T from Theorem 4 in [18] we have 311) where Φ ω) is the Fréchet derivative of Φ at ω K 1 relx) = Φ ω) 2 Q A) F X F Let 1 S = X j X 1 j + A T A T T = j=0 I n 2 ) I n 2 + Π)I A T X)) v = vec Q) T vec A) T ) T Then 38) can be written as 312) Svec X) = T v Note that if a1 A 2 2 < 1 then 1 σ min S) λ min X j X 1 j) A 2 2 a 1 A 2 2 > 0 j=0 which shows that S is invertible Then it follows from 312) that Φ is Fréchet differentiable and 313) Φ ω) = S 1 T Therefore according to 311) the following theorem can be obtained Theorem 31 Let X be the ositive definite solution of matrix equation 11) If a1 A 2 2 the relative condition number krel 1 of equation 11) defined in 310) is given by < 1 then k 1 rel = S 1 T 2 Q A) F X F Theorem 32 Let X be the ositive definite solution of matrix equation 11) If a1 A 2 2 the relative condition number krel 2 of equation 11) defined in 310) is given by < 1 then 314) where k 2 rel = [ Q F Σ r A F Ω r ] 2 X F Σ r = S 1 and Ω r = S 1 I A T X) + XA) T I)Π)

9 Syed M Raza Shah Naqvi J Meng and H-M Kim 224 Proof Note that 312) can be rewritten as 315) vec X) = S 1 T 1 r 1 where It follows from 315) that vec Q) T 1 = T diag Q F A F ) and r 1 = veca) ) T Q F A F X F = S 1 T 1 r ) S 1 T 1 2 r 1 2 [ Q F S 1 A F S 1 I n 2 + Π)I A T X)) ] 2 r 1 2 Since r 1 2 = 2 according to 310) when i = 2) and inequality 316) we arrive at 314) 32 Mixed and comonentwise condition numbers We now consider the the mixed and comonentwise condition numbers of matrix equation 11) We first introduce some definitions and useful results about these two condition numbers For any a b R n we define a/b = [c 1 c 2 c n ] T with a i /b i if b i 0 c i = 0 if a i = b i = 0 otherwise Then we introduce one distance function da b) = a b)/b = max i=12n { } a i b i In the rest of this aer we assume da b) < for any air a b) And we extend the function d to matrices M and N by dm N) = dvecm) vecn)) For ɛ > 0 we denote B 0 a ɛ) = {x dx a) ɛ} Definition 33 [7]) Let F : R R q be a continuous maing defined on an oen set DomF ) R such that 0 / DomF ) and F a) 0 for a given a R 1) The mixed condition number of F at a is defined by mf a) = lim ɛ 0 su x B 0 aɛ) x a b i F x) F a) F a) 1 dx a) 2) Suose F a) = [ f 1 a) f 2 a) f q a) ] T such that fj a) 0 for j = 1 2 q The comonentwise condition number of F at a is defined by cf a) = lim ɛ 0 su x B 0 aɛ) x a df x) F a)) dx a) From Gohberg and Koltracht [7] or Cucker Diao and Wei [4] if F is Fréchet differentiable at a the exlicit exressions of the mixed and comonentwise condition numbers of F at a are given by the following lemma

10 225 Iteration With Stesize Parameter and Condition Numbers for a Nonlinear Matrix Equation Lemma 34 Suose F is Fréchet differentiable at a We have 1) if F a) 0 then mf a) = F a)diaga) F a) = F a) a F a) ; 2) if F a) = [ f 1 a) f 2 a) f q a) ] T such that fj a) 0 for j = 1 2 q then cf a) = diagf a))) 1 F a)diaga) = F a) a F a) Theorem 35 Let X be the symmetric ositive definite solution of equation 11) If a1 A 2 2 < 1 then the mixed and comonentwise condition numbers of matrix equation 11) are given by S 1 vec Q ) + S 1 I n 2 Π ) I A T X) ) vec A ) mφ) = and cφ) = S 1 vec Q )+ S 1 I n 2 +Π I A X)) T vec A ) vec X ) Furthermore we have two simle uer bounds for mφ) and cφ) as follows: and m U Φ) := S 1 Q + A T X A + A T AX max ) mϕ) c U Φ) := diag 1 vecx) ) S 1 = Q + A T X A + A T XA max cϕ) Proof From 313) we know Φ ω) = S 1 T where ω = vecq) T veca) T ) T according to 1) of Lemma 34 we get mφ) = S 1 T ω vecx) [ S 1 S 1 I n 2 + Π ) I A T X) ) ] ) vecq) veca) = S 1 vec Q ) + S 1 I n 2 + Π ) I A T X) ) vec A ) = Similarly from 2) of Lemma 34 we get cφ) = S 1 T ω vecx) = Note that ) S 1 vec Q )+ S 1 I n 2 +Π I A X)) T vec A ) vec X ) S 1 T ω S 1 T ω S 1 T ω S 1 vec Q ) + In 2 + Π)I A T X)) vec A ) S 1 Q + A T X A + A T AX max

11 Syed M Raza Shah Naqvi J Meng and H-M Kim 226 it follows that Similarly it holds that mφ) S 1 T ω = S 1 Q + A T X A + A T XA max The roof is comleted cφ) = diag 1 vecx)) S 1 T ω diag 1 vecx))s 1 T ω diag 1 vecx))s 1 Q + A T X A + A T XA max 4 Numerical examles In this section we give three examles to show the efficiency of the roosed iterative method with stesize arameter and the sharness of the three kinds of condition numbers The stesize arameter is different case by case Our exeriments were done in Matlab 7100 which has the unit roundoff µ and the iterations terminate if the relative residuals ρx k ) satisfies flx k ρx k ) = + AT X k A Q) F X k F + AT F X k F A F + Q F nµ In [13] we also roosed the following two iterative methods which work better than iteration 22) { X 0 = γi 41) X k+1 = Q A T X k A) 1 and 42) { X 0 = 0 X k+1 = Q A T Q A T X k A) 1 A) 1 where γ > 0 in 41) is a real number such that γ + γλ max A T A) = λ min Q) We will comare the roosed iteration 23) with iteration 22) and the above two iterations Examle 41 Examle 41 [13]) Let matrix A = rand10) 10 2 Q = eye10) and = We aly the basic fixed-oint iteration 22) 41) 42) and the iteration 23) with stesize arameter on equation 11) Figure 1 shows the iterations of 23) with different values of α when = 3 and 7 resectively The iterations before convergence the CPU time and the relative residuals are shown in 1 and Table 2 From Figure 1 we can see that the stesize arameter affects the erformance of iteration 23) significantly and when it is close to 1 in our examle it is close to 09) iteration 23) works more efficiently From Table 1 and Table 2 we can see that iteration 23) with roer stesize arameter converges much faster and uses less time for obtaining the symmetric ositive definite solution

12 227 Iteration With Stesize Parameter and Condition Numbers for a Nonlinear Matrix Equation Figure 1 Iterations with different arameter α Iteration 23) with stesize arameter Iteration 22) α Its) CPU time Relative res Its) CPU time relative res Table 1 Comarison of iteration 23) with iteration 22) Examle 42 Let A = Q = eye4) and = 3 The erturbations in coefficient matrices are given by A = rand4) 10 j ) A and Q = rand4) 10 j ) Q where j is a ositive integer and is the Hadamard roduct Using iteration 23) with X 0 = ai we can get the unique ositive definite solution X of equation 11) and X of the corresonding erturbed equation From Theorem 31 and Theorem 32 we get two local normwise erturbation bounds X F krel 1 1 and X F X F k 2 rel 2 Let ɛ 0 = min {ɛ : A ɛ A Q ɛ Q } We obtain the local mixed and comonentwise erturbation bounds X max / ɛ 0 m U ϕ) X F

13 Syed M Raza Shah Naqvi J Meng and H-M Kim 228 Iteration 23) with stesize arameter Iteration 41) Iteration 42) α Its) CPU time Its) CPU time Its) CPU time Table 2 Comarisons of iteration 23) with iteration 41) and iteration 42) j X F X F 28299e e e e-007 krel 1 ϕ) e e e e-006 krel 2 ϕ) e e e e-007 X max 32166e e e e-007 m U ϕ)ɛ e e e e-007 vec X) vecx) 11047e e e e-006 c U ϕ)ɛ e e e e-005 Table 3 Uer bounds given by normwise mixed and comonentwise condition numbers and vec X)/vecX) max ɛ 0 c U ϕ) Table 3 shows that the erturbation bounds given by the three kinds of condition numbers are very shar Examle 43 Let A = Q =

14 229 Iteration With Stesize Parameter and Condition Numbers for a Nonlinear Matrix Equation j X F X F 17180e e e e-15 krel 1 ϕ) e e e e-14 krel 2 ϕ) e e e e-15 X max 18107e e e e-15 m U ϕ)ɛ e e e e-14 vec X) vecx) 26329e e e e-12 c U ϕ)ɛ e e e e-11 Table 4 Uer bounds given by normwise mixed and comonentwise condition numbers and = 100 The erturbations in coefficient matrices are given by A = rand6) 10 j ) A and Q = rand6) 10 j ) Q where j is a ositive integer and is the Hadamard roduct This is another examle But in this examle we have a slightly greater order of matrix and also greater ower of The results are shown in Table 4 And it can be easily deduced from the results that our estimations are almost tight 5 Conclusion In this aer we roose an iteration with stesize arameter for obtaining the symmetric ositive definite solution 11) Different stesize arameter affects the erformance of the roosed iteration significantly but it works efficiently with a roer stesize arameter We also investigate the normwise mixed and comonentwise condition numbers of equation 11) Acknowledgment The authors thank the valuable comments and suggestions of an anonymous referee and of the handing editor Professor James Nagy that imroved the resentation of the aer REFERENCES [1] V Berinde Generalized couled fixed oint theorems for mixed monotone maings in artially ordered metric saces Nonlinear Analysis: Theory Methods Alications 74: [2] TG Bhaskar and V Lakshmikantham Fixed oint theorems in artially ordered metric saces and alications Nonlinear Analysis: Theory Methods Alications 65: [3] R Bhatia Matrix Analysis Graduate Texts in Mathematics Vol 169 Sringer-Verlag New York 1997 [4] F Cucker H Diao and Y Wei On mixed and comonentwise condition mumbers fro Moore-Penrose inverse and linear least squares roblems Mathematics of Comutation 76: [5] SM El-Sayed and ACM Ran On an iteration method for solving a class of nonlinear matrix equations SIAM Journal on Matrix Analysis and Alications 23: [6] A Ferrante and BC Levy Hermitian solutions of the eqaution X = Q + NX 1 N Linear Algebra and its Alications 246: [7] A Gohberg and I Koltracht Mixed comonentwise and structed condition numbers SIAM Journal on Matrix Analysis and Alications 14: [8] DA Harville Matrix Algebra from a Statistician s Persective Sringer-Verlag New York 1997 [9] IG Ivanov VI Hasanov and F Uhlig Imroved methods and starting values to solve the matrix equations X±A X 1 A = I iteratively Mathematics of Comutations 74: [10] Z Jia and M Wei Solvability and sensitivity theory of olynomial matrix equation X s +A T X t A = Q Alied Mathematics and Comutation 209:

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