# Condition Numbers of the Nonlinear Matrix EquationXp-A⁎eXA=I

Authors: Chacha Stephen Chacha; Syed Muhammad Raza Shah Naqvi
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We explore the condition numbers of the nonlinear matrix equationXp-A⁎eXA=I. Explicit expressions for the normwise, mixed, and componentwise condition numbers are derived. The upper bounds for the mixed and componentwise condition numbers are obtained. The numerical result favors the fact that our estimations are fairly sharp. Also, the relative upper perturbation bounds give satisfactory results for small perturbations in the input data.


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## Body

## 1. Introduction

We consider the nonlinear matrix equation(1)Xp-A∗eXA=Ip>1,where A is a real or complex square matrix, I is an identity matrix, eX is the matrix exponential function, and p is a positive integer. The basic general form of (1) is X+A∗FXA=Q(Q>0), and it occurs in the analysis of ladder networks, the dynamic programming, control theory, stochastic filtering, and statistics [1]. Ran and Reurings in [2] studied the solutions and perturbation theory for a general matrix equation X+A∗FXA=Q, where F represent a map from the set of all positive semidefinite matrices into a space of complex matrices and satisfy some monotonicity properties. Recently, Gao in [3] studied the Hermitian positive definite solution (HPDS) of the nonlinear matrix equation X-A∗eXA=I which is (1) for p=1 and derived some necessary and sufficient conditions for the existence of the HPDS. In [4], authors derived the explicit expressions for the normwise, mixed, and componentwise condition numbers and their upper bounds for the nonlinear matrix equation A0+∑i=1kσiAi∗XpiAi=0, σi=±1,k≥2. Authors in [5] presented a perturbation analysis of the matrix equation C+∑i=1rσiAiXBi+DXsE=0, for positive integers r,s≥2, and employed Lyapunov majorant and fixed point-point principle to derive both local and nonlocal bounds. For more details about condition numbers, see ([6, 7]) and the references therein.To the best of our knowledge, no one has studied the condition numbers of (1). Thus, the objective of this study is to derive the explicit expressions for the normwise, mixed, and componentwise condition numbers as well as the local upper bounds for mixed and componentwise condition numbers of (1). Finally, we give a comparative analysis for the computed condition numbers.The following notations will be used throughout this paper:Krelj stands for normwise condition number; “≔’’ means equal by definition; Bϵ0a stands for a ball with center a and radius ϵ; domF stands for domain of F; Cn×n denotes the set of all complex n×n Hermitian matrices; if A,B∈Cn×n, then A≥0(B>0) means that A is positive semidefinite (positive definite) and A≥BA>B means that A-B≥0A-B>0; the notation ρ(■) stands for spectral radius; A∗,AT signify the transpose conjugate and transpose of matrix A, respectively; ■F and ■2 denote the Frobenius norm and usual spectral norm, respectively; given A=aij∈Cm×n and B∈Cp×q, the Kronecker product is A⨂B=aijB∈Cmp×nq; the operator vec:Cm×n→Cmn is defined by vecA=a11,…,am1,a12,…,am2,…,a1n,…,amnT; cond■ means the ratio of the largest singular value to the smallest; A=aij∈R+n×n stands for the absolute value of A.


## 2. Preliminaries

In this section, we provide useful definitions and lemmas that will be applied in our proofs in the next sections.Definition 1.
The condition number of a matrix functionf:Cn×n→Cn×n at a point X∈Cn×n is defined as (2)condf,X≔limϵ→0⁡supE≤ϵX⁡fX+E-fXϵfX,for any matrix norm. The computed condition number measures the stability or sensitivity of a problem. In this case, the problem is said to be well-structured or well-posed or well-conditioned if the condition number is small and ill-conditioned if the condition number is large, where the definition of large or small condition number is problem dependent.Definition 2.
The Fréchet derivative of a matrix functionf:Cn×n→Cn×n at a point X∈Cn×n is a linear operator(3)Cn×n⟶LfX,ECn×nE⟼LfX,Esuch that fX+E-fX-LfX,E=oE for all E∈Cn×n. The operator LfX,E denotes the Fréchet derivative of f at X in the direction E. If such an operator exists, f is said to be Fréchet differentiable and(4)LfX,E≔maxE≠0⁡LfX,EE.Lemma 3 (see [8], Lemma 4.3.1).
Suppose thatA∈Cm×n, B∈Cp×q, C∈Cp×q, and X∈Cn×p. Then,(5)vecAXB=BT⊗AvecX.Lemma 4 (see [9], pp. 178, Theorem 3.3.16(a, b)).
LetA,B∈Mm,n be given and let q=minm,n. Then, following inequalities hold for the decreasingly ordered singular values of A,B,A+B and AB∗:(I)
σi+j-1A+B≤σiA+σjB,(II)
σi+j-1AB∗≤σiA+σjB.Lemma 5 (see [10], Theorem 15).
LetA∈Mn and B∈Mm. If λ is an eigenvalue of A and μ is an eigenvalue of B, then λμ is an eigenvalue of A⊗B.For easy expansion and simplification of matrix polynomials, we need Lemma6.Lemma 6 (see [11]).
LetZ+ denote a set of positive integers including zero and Z++=Z+∖0 such that Φ:Z+×Z+×Cn×n×Cn×n→Cn×n. Then,(i)
{Φi,0X,Y=Xi,Φ0,jX,Y=Xj,i∈Z+, Φi,jX,Y=XΦi-1,jX,Y+YΦi,j-1X,Y, i,j∈Z++}.(ii)
Φ0,0X,Y=I, and for Φk,1X,Y=∑i=1kXk-iYXi.(iii)
X+Yn=∑i=0nΦn-i,iX,Y,n∈Z+.


## 3. Normwise, Mixed, and Componentwise Condition Numbers

In this section, we concentrate on the derivation of the explicit expressions for the normwise, mixed, and componentwise condition numbers. In order to derive the explicit expressions for the normwise, mixed, and componentwise condition numbers of (1), we consider the perturbed equation (6).(6)X~p-A∗~eX~A~=I~.Replacing eX by ∑k=0∞1/k!Xk in (6) yields(7)X~p-A∗~∑k=0∞1k!X~kA~=I~.Now, let us make small perturbations in the matrices A,X, and I as shown in (8)X+ΔXp-A+ΔA∗∑k=0∞1k!X+ΔXkA+ΔA=I+ΔI.Subtracting (8) by (1) yields(9)X+ΔXp-Xp=ΔI+A+ΔA∗∑k=0∞1k!X+ΔXkA+ΔA-A∑k=0∞1k!Xk.Using Lemma 6, we have(10)X+ΔXp-Xp=∑j=0pΦp-1,jX,ΔX-Xp=Φp,0X,X+Φp-1,1X,ΔX+HΔX=∑j=0p-1Xp-1-jΔXXj+HΔX,where HΔX=∑j=2pΦp-j,jX,ΔX. Because HΔX have higher orders of ΔX, we omit it and consider only the first term of (10). Then, replacing X+ΔXp-Xp in (9) by ∑j=0p-1Xp-1-jΔXXj gives (11)∑j=0p-1Xp-1-jΔXXj=ΔI+A∗∑k=0∞1k!∑i=0k-1Xk-1-iΔXXiA+A∗eXΔA+ΔA∗eXA+OH2.Using the fact that X is symmetric and A is real and applying the vec operator in (11), we get(12)∑j=0p-1Xj⨂Xp-1-jvecΔX=vecΔI+∑k=0∞∑i=0k-1ATXi⨂A∗Xp-1-ivecΔX+I⨂A∗eXvecΔA+ATeX⨂IvecΔA∗+OH2.Combining the terms with vecΔX in (12) yields(13)∑j=0p-1Xj⨂Xp-1-jvecΔX-∑k=0∞∑i=0k-1ATXi⨂A∗Xp-1-ivecΔX=vecΔII⨂A∗eXvecΔA+ATeX⨂IvecΔA∗+OH2.Then, we have(14)∑j=0p-1Xj⨂Xp-1-j-∑k=0∞∑i=0k-1ATXi⨂A∗Xp-1-ivecΔX=vecΔI+I⨂A∗eX+ATeX⨂IΠvecΔA+OH2.In (14), the term OH2=ΔI,ΔAF2 refers to higher order approximation of ΔX with respect to ΔI,ΔA and Π∈Rn2×n2 is the vec permutation operator satisfying ΠvecA=vecAT and it is defined as(15)Π=∑i=1n∑j=1nEijn×n⨂Ejin×n,where Eij=einejnT∈Rn×n and ein is the ith column of the identity matrix I.Let us define a mapΨ:L2n2⟼Ln2, where, L represents a real or complex space. We have Ψ:Γ=vecIT,vecATT→vecX, where the matrices I,A∈Ln×n. According to the implicit function theorem, it is apparent that ΔX→0 as ΔI,ΔA→0, because ΔX is a function of ΔI,ΔA.


### 3.1. Normwise Condition Numbers

In this subsection, we define the two kinds of normwise condition numbers. According to Rice [12], the two kinds of normwise condition numbers of map Ψ are defined by(16)Krelj=limδ→0⁡supΔj≤δ⁡ΔXFδXF,j=1,2,whereΔ1=ΔI,ΔAFI,AFandΔ2=ΔIIF,ΔAAFF.Now, suppose that Ψ is differentiable at Γ=vecIT,vecATT, then using Theorem 4 in [12], we have (17)Krel1=Ψ′Γ2I,AFXF,where Ψ′Γ is a Fréchet derivative of Ψ at Γ. It follows that(18)BvecΔX=vecΔI+I⨂A∗eX+ATeX⨂IΠvecΔA,where(19)B=∑j=0p-1Xj⨂Xp-1-j-∑k=0∞∑i=0k-1ATXi⨂A∗Xp-1-i.Denoting(20)Λ=In2,I⨂A∗eX+ATeX⨂IΠ,we have(21)BvecΔX=ΛΓ.Now, we prove that matrixB is nonsingular in Theorem 7.Theorem 7.
Suppose thatX is the Hermitian positive definite solution of (1) and A is a real matrix with ρA<1/e such that (p-1)λpX>K-1)λ2AeλX. Then,(22)B=∑j=0p-1Xj⨂Xp-1-j-∑k=0∞∑i=0k-1ATXi⨂A∗Xk-1-iis nonsingular.Proof.
Denote(23)D≔∑j=0p-1Xj⨂Xp-1-j,andC≔∑k=0∞∑i=0k-1ATXi⨂A∗Xk-1-i.Then, we have (24)λB+C=λD.Using Lemmas 4 and 5, it follows that(25)∑j=0p-1λjXλp-1-jX≤λB+∑k=0∞1k!∑i=0kλiXλ2Aλk-1-iX,and(26)∑j=0p-1λjXλp-1-jX-∑k=0∞1k!∑i=0kλiXλ2Aλk-1-iX≤λB.From X=I+A∗eXA1/p, we know that A∗eXA>0; this implies that I<X, which means that 1< λX. Therefore, λ-1Xp-1λpX-k-1λ2AeλX≤λB.
Sincep-1λpX>k-1λ2AeλX, then we have 0<λB. Therefore, we conclude that B is invertible.
From Theorem7 above, since B is invertible, then we have vecΔX=B-1ΛΓ, and the Kronecker Fréchet derivative(27)Ψ′I,A=B-1Λ.Explicit expressions for the two kinds of normwise condition numbers are derived in Theorem8.Theorem 8.
Suppose thatX is the HPD solution of (1) and B is nonsingular. Then, ①
Krel1=B-1Λ2I,AF/XF,②
Krel2=IFB-1,BFB-1Ξ2/XF,
where(28)B=∑j=0p-1Xj⨂Xp-1-j-∑k=0∞∑i=0k-1ATXi⨂A∗Xp-1-i,Λ=In2,I⨂A∗eX+ATeX⨂IΠ,and (29)Ξ=I⨂A∗eX+ATeX⨂IΠ.Proof.
①
Using the fact thatΨ′I,A=B-1Λ and (17), we can easily get (30)Krel1=B-1Λ2I,AFXF. ②
In this case, we rewritevecΔX=B-1ΛΓ as vecΔX=B-1Λ1κ, where Λ1=ΛdiagIF,AF and κ=ΔI/IF,ΔA/AFF, then we have vecΔX2=ΔXF. It follows that(31)ΔXF=B-1Λκ2≤B-1Λ2κ2=IFB-1,B-1I⨂A∗eX+ATeX⨂IΠ2κ2.Finally, using (16) and (31) and Δ2=κ2, we see that (32)Krel2=IFB-1,AFB-1Ξ2XF.


### 3.2. Mixed and Componentwise Condition Numbers

In this subsection, we derive the explicit expressions for mixed and componentwise condition numbers of (1). The following distance function is introduced before defining mixed and componentwise condition numbers. For any vectors(33)u=u1,u2,⋯,umT,v=v1,v2,⋯,vmT,we  defineu.v=w1,w2,⋯,wmT.wi=uiviifvi≠0,0ifui=vi=0,∞otherwise,  for  alli=1,2,⋯,m.Now, let us define a distance function (34)du,v=u-v.v∞=maxi⁡ui-vi.vi.In the rest of this paper, we assume du,v is finite for any pair u,v, and we extend the function d to matrices. That is for any matrices X,Y, we have, dX,Y=dvecX,vecY. For ϵ>0, we denote Bϵ0a=x∣da,x≤ϵ.Based on the work by Gohberg and Koltracht [13] on mixed, componentwise, and structured condition numbers, we have Definition 9.Definition 9.
LetF:Rp→Rq be a continuous mapping defined on an open interval set domF⊂Rp such that 0∉domF and Fa≠0 for a given a∈Rp.(I)
The mixed condition number ofF at a is defined by(35)mF,a=limϵ→0⁡supx∈Bϵ0ax≠a⁡Fx-Fa∞Fa∞1dx,a.(II)
The componentwise condition number ofF at a is defined by(36)cF,a=limϵ→0⁡supx∈Bϵ0ax≠a⁡dFx,Fadx,a.If F is Fréchet differentiable at point a. Then, the explicit expressions of the mixed and componentwise condition numbers of F at a are given by Lemma 10.Lemma 10 (see [13, 14]).
AssumeF is Fréchet differentiable at point a, we have (1)
ifFa≠0, then(37)mF,a=F′adiaga∞Fa∞=F′aa∞Fa∞,(2)
ifFa=f1a,f2a⋯,fqaT such that fja≠0, for j=1,2,⋯,q, then (38)cF,a=diagFa-1F′adiaga∞=F′aa.Fa∞.Now, we derive the explicit expressions for the mixed and componentwise condition numbers and their upper bounds in Theorem 11.Theorem 11.
LetX be the Hermitian positive definite solution of (1). Define the mapping(39)Ψ:A,I⟼vecX=ΨA,I.(I)
LetmΨ,X denote the mixed condition number defined by (35). Then mΨ,X has the explicit expression (40)mΨ,X=Θ∞Xmax,where(41)Θ=B-1vecI+B-1I⨂A∗eX+ATeX⨂IΠvecA,(II)
LetcΨ,X denote the componentwise condition number defined by (36). Then, cΨ,X has the explicit expression (42)cΨ,X=Θ.vecX∞,where Θ is the same as in item (I).
Moreover, we define two simple upper bounds formΨ,X and cΨ,X given by(43)mupΨ,X=Xmax-1B-1∞I+2A∗eXAmax≥mΨ,Xand(44)cupΨ,X=diag-1vecXB-1∞I+2A∗eXAmax≥cΨ,X,respectively.Proof.
We first prove (I) using(1) of Lemma 10. In this case, for a=vec([I,A]), we obtain that the mixed condition number is mΨ,X=Ψ′aa∞/Ψa∞. It follows from (27) that(45)mΨ,X=B-1ΛΓ∞vecX∞=Θ∞Xmax,where (46)Θ=B-1ΛΓ=B-1In2,B-1I⨂A∗eX+ATeX⨂IΠvecIvecA=B-1vecI+B-1I⨂A∗eX+ATeX⨂IΠvecA.It also holds that (47)Θ∞=B-1ΛΓ∞≤B-1ΛΓ∞=B-1∞ΛΓ∞=B-1∞I+2A∗eXAmax.So it follows from (45) that the upper bound of mΨ,X is given by(48)mΨ,X≤Xmax-1B-1∞I+2A∗eXAmax=mupΨ,X.Now we prove item (II) for the componentwise condition number cΨ,X of (1). Using (27) together with item (2) of Lemma 10 yields (49)cΨ,X=Ψ′aa.Ψa∞=B-1ΛΓ.vecX∞=Θ.vecX∞,where Θ is the same as in item (I).
Likewise, to estimate the upper bound ofcΨ,X, we have (50)cupΨ,X=diag-1vecXB-1∞I+2A∗eXAmax≥cΨ,X.This completes our proof.


## 3.1. Normwise Condition Numbers

In this subsection, we define the two kinds of normwise condition numbers. According to Rice [12], the two kinds of normwise condition numbers of map Ψ are defined by(16)Krelj=limδ→0⁡supΔj≤δ⁡ΔXFδXF,j=1,2,whereΔ1=ΔI,ΔAFI,AFandΔ2=ΔIIF,ΔAAFF.Now, suppose that Ψ is differentiable at Γ=vecIT,vecATT, then using Theorem 4 in [12], we have (17)Krel1=Ψ′Γ2I,AFXF,where Ψ′Γ is a Fréchet derivative of Ψ at Γ. It follows that(18)BvecΔX=vecΔI+I⨂A∗eX+ATeX⨂IΠvecΔA,where(19)B=∑j=0p-1Xj⨂Xp-1-j-∑k=0∞∑i=0k-1ATXi⨂A∗Xp-1-i.Denoting(20)Λ=In2,I⨂A∗eX+ATeX⨂IΠ,we have(21)BvecΔX=ΛΓ.Now, we prove that matrixB is nonsingular in Theorem 7.Theorem 7.
Suppose thatX is the Hermitian positive definite solution of (1) and A is a real matrix with ρA<1/e such that (p-1)λpX>K-1)λ2AeλX. Then,(22)B=∑j=0p-1Xj⨂Xp-1-j-∑k=0∞∑i=0k-1ATXi⨂A∗Xk-1-iis nonsingular.Proof.
Denote(23)D≔∑j=0p-1Xj⨂Xp-1-j,andC≔∑k=0∞∑i=0k-1ATXi⨂A∗Xk-1-i.Then, we have (24)λB+C=λD.Using Lemmas 4 and 5, it follows that(25)∑j=0p-1λjXλp-1-jX≤λB+∑k=0∞1k!∑i=0kλiXλ2Aλk-1-iX,and(26)∑j=0p-1λjXλp-1-jX-∑k=0∞1k!∑i=0kλiXλ2Aλk-1-iX≤λB.From X=I+A∗eXA1/p, we know that A∗eXA>0; this implies that I<X, which means that 1< λX. Therefore, λ-1Xp-1λpX-k-1λ2AeλX≤λB.
Sincep-1λpX>k-1λ2AeλX, then we have 0<λB. Therefore, we conclude that B is invertible.
From Theorem7 above, since B is invertible, then we have vecΔX=B-1ΛΓ, and the Kronecker Fréchet derivative(27)Ψ′I,A=B-1Λ.Explicit expressions for the two kinds of normwise condition numbers are derived in Theorem8.Theorem 8.
Suppose thatX is the HPD solution of (1) and B is nonsingular. Then, ①
Krel1=B-1Λ2I,AF/XF,②
Krel2=IFB-1,BFB-1Ξ2/XF,
where(28)B=∑j=0p-1Xj⨂Xp-1-j-∑k=0∞∑i=0k-1ATXi⨂A∗Xp-1-i,Λ=In2,I⨂A∗eX+ATeX⨂IΠ,and (29)Ξ=I⨂A∗eX+ATeX⨂IΠ.Proof.
①
Using the fact thatΨ′I,A=B-1Λ and (17), we can easily get (30)Krel1=B-1Λ2I,AFXF. ②
In this case, we rewritevecΔX=B-1ΛΓ as vecΔX=B-1Λ1κ, where Λ1=ΛdiagIF,AF and κ=ΔI/IF,ΔA/AFF, then we have vecΔX2=ΔXF. It follows that(31)ΔXF=B-1Λκ2≤B-1Λ2κ2=IFB-1,B-1I⨂A∗eX+ATeX⨂IΠ2κ2.Finally, using (16) and (31) and Δ2=κ2, we see that (32)Krel2=IFB-1,AFB-1Ξ2XF.


## 3.2. Mixed and Componentwise Condition Numbers

In this subsection, we derive the explicit expressions for mixed and componentwise condition numbers of (1). The following distance function is introduced before defining mixed and componentwise condition numbers. For any vectors(33)u=u1,u2,⋯,umT,v=v1,v2,⋯,vmT,we  defineu.v=w1,w2,⋯,wmT.wi=uiviifvi≠0,0ifui=vi=0,∞otherwise,  for  alli=1,2,⋯,m.Now, let us define a distance function (34)du,v=u-v.v∞=maxi⁡ui-vi.vi.In the rest of this paper, we assume du,v is finite for any pair u,v, and we extend the function d to matrices. That is for any matrices X,Y, we have, dX,Y=dvecX,vecY. For ϵ>0, we denote Bϵ0a=x∣da,x≤ϵ.Based on the work by Gohberg and Koltracht [13] on mixed, componentwise, and structured condition numbers, we have Definition 9.Definition 9.
LetF:Rp→Rq be a continuous mapping defined on an open interval set domF⊂Rp such that 0∉domF and Fa≠0 for a given a∈Rp.(I)
The mixed condition number ofF at a is defined by(35)mF,a=limϵ→0⁡supx∈Bϵ0ax≠a⁡Fx-Fa∞Fa∞1dx,a.(II)
The componentwise condition number ofF at a is defined by(36)cF,a=limϵ→0⁡supx∈Bϵ0ax≠a⁡dFx,Fadx,a.If F is Fréchet differentiable at point a. Then, the explicit expressions of the mixed and componentwise condition numbers of F at a are given by Lemma 10.Lemma 10 (see [13, 14]).
AssumeF is Fréchet differentiable at point a, we have (1)
ifFa≠0, then(37)mF,a=F′adiaga∞Fa∞=F′aa∞Fa∞,(2)
ifFa=f1a,f2a⋯,fqaT such that fja≠0, for j=1,2,⋯,q, then (38)cF,a=diagFa-1F′adiaga∞=F′aa.Fa∞.Now, we derive the explicit expressions for the mixed and componentwise condition numbers and their upper bounds in Theorem 11.Theorem 11.
LetX be the Hermitian positive definite solution of (1). Define the mapping(39)Ψ:A,I⟼vecX=ΨA,I.(I)
LetmΨ,X denote the mixed condition number defined by (35). Then mΨ,X has the explicit expression (40)mΨ,X=Θ∞Xmax,where(41)Θ=B-1vecI+B-1I⨂A∗eX+ATeX⨂IΠvecA,(II)
LetcΨ,X denote the componentwise condition number defined by (36). Then, cΨ,X has the explicit expression (42)cΨ,X=Θ.vecX∞,where Θ is the same as in item (I).
Moreover, we define two simple upper bounds formΨ,X and cΨ,X given by(43)mupΨ,X=Xmax-1B-1∞I+2A∗eXAmax≥mΨ,Xand(44)cupΨ,X=diag-1vecXB-1∞I+2A∗eXAmax≥cΨ,X,respectively.Proof.
We first prove (I) using(1) of Lemma 10. In this case, for a=vec([I,A]), we obtain that the mixed condition number is mΨ,X=Ψ′aa∞/Ψa∞. It follows from (27) that(45)mΨ,X=B-1ΛΓ∞vecX∞=Θ∞Xmax,where (46)Θ=B-1ΛΓ=B-1In2,B-1I⨂A∗eX+ATeX⨂IΠvecIvecA=B-1vecI+B-1I⨂A∗eX+ATeX⨂IΠvecA.It also holds that (47)Θ∞=B-1ΛΓ∞≤B-1ΛΓ∞=B-1∞ΛΓ∞=B-1∞I+2A∗eXAmax.So it follows from (45) that the upper bound of mΨ,X is given by(48)mΨ,X≤Xmax-1B-1∞I+2A∗eXAmax=mupΨ,X.Now we prove item (II) for the componentwise condition number cΨ,X of (1). Using (27) together with item (2) of Lemma 10 yields (49)cΨ,X=Ψ′aa.Ψa∞=B-1ΛΓ.vecX∞=Θ.vecX∞,where Θ is the same as in item (I).
Likewise, to estimate the upper bound ofcΨ,X, we have (50)cupΨ,X=diag-1vecXB-1∞I+2A∗eXAmax≥cΨ,X.This completes our proof.


## 4. Numerical Experiments

In this section, we provide some numerical examples and results. Our tests were carried out in MATLAB mark 22.0 on an Intel(R) Core(TM)i3-4005u CPU@1.7GHz 1.70GHz with 64-bit operating system. Four examples are considered. In Example1, we evaluate normwise, mixed, and componentwise condition numbers for different tridiagonal matrix sizes. In Example 2, two cases are considered, in the first case, a badly scaled input matrix A is considered and the three kinds of condition numbers are computed. In the second case, a well-known doubly stochastic matrix of different sizes is considered and the computed condition numbers are compared. In Example 3, we make some small random perturbation in matrices I and A and evaluate the local upper perturbation bounds for the computed condition numbers. In Example 4, we consider a symmetric matrix A with some specified perturbations in the matrices I and A and evaluate condition numbers and their local upper perturbation bounds.In each example a comparison table for the computed condition numbers is provided and a general remark is provided for all results.Forϵ>0, we define ϵ0=maxϵ:ΔA≤ϵA,ΔI≤ϵI, A~=A+ΔA, I~=I+ΔI, and X, X~ are solutions of (1) and (6), respectively. The solutions X and X~ are computed by a fixed point algorithm. We obtain the local normwise, mixed, and componentwise condition numbers as follows. ζN=ΔXF/XF, ζM=ΔXmax/Xmax, and ζC=vecΔX./vecXmax. Then, we also define the relative mixed and componentwise local upper perturbation bounds as ΔXmax/Xmax≤ϵ0mupΨ,X and vecΔX./vecXmax≤ϵ0cupΨ,X, respectively.Here, we propose a fixed point algorithm to compute the solutionsX and X~.Fixed Point Algorithm (I)
Input ann×n matrix with ρA<1/e, tolerance error tol=n×eps, and an initial guess X0≥I, where n is the size of matrix A and eps ≃1.1×10-16 is the standard machine precision. (II)
Fori=0,1,2,⋯, compute Xi+1=I+A∗eXA1/p and relative residual(51)res=Xp-A∗eXA+IFXpF+A∗eXA+IF. (III)
Exit the loop if res≤tol. Otherwise, go to step (II). (IV)
Display the solutionX.Example 1.
We consider the tridiagonal matrixA=Wn/25, where Wn=diag(1,2,1).The matrixWn is generated by a MATLAB function “full (gallery (‘tridiag’, n, 1,2,1))’’, where n denotes the size of matrix A. For p=2 and using fixed point algorithm we evaluate the solution of (1) and compute relative normwise, mixed, and componentwise condition numbers. The summary of results is recorded in Table 1.Table 1
Condition numbers for Example1 evaluated using different tridiagonal matrices with size n.
n K r e l 1 K r e l 2 m ( Ψ , X ) c ( Ψ , X ) c o n d (X) 4 0.1991 0.5114 0.5066 11.4838 1.0071 6 0.2096 0.5124 0.5066 23.1583 1.0079 8 0.2140 0.5128 0.5066 52.7581 1.0082 10 0.2162 0.5130 0.5066 134.5177 1.0084Example 2.
(I)
We consider (1) with A=2/50010-17203110-140.05. Then, we evaluate the normwise mixed and condition numbers for different p-values. We employ our proposed fixed point method with X0=eye(n). A summary of results is displayed in Table 2. (II)
We consider (1) with a well-known doubly stochastic matrix which has applications in communication theory and graph theory [15]. We generate a doubly stochastic matrix A=aij=H/N such that ∑iaij=∑jaij=1, where H=magicn, N=sum(H(1,:)), and n is the size of A. Then, we apply our proposed fixed point method with X0=eye(n). A summary of results is recorded in Table 3Table 2
Normwise and mixed condition numbers for differentp-values using an ill-conditioned input matrix A.
p K r e l 1 K r e l 2 m ( Ψ , X ) 2 34.6169 22.7843 1.6223 4 6.5804 4.4013 0.8919 6 3.5024 2.3663 0.6400 9 2.0393 1.3890 0.4498Table 3
Normwise and mixed condition numbers evaluated at different doubly stochastic input matrixA.
( p , n) K r e l 1 K r e l 2 m ( Ψ , X ) c ( Ψ , X ) (3,3) 8.7952 5.2523 3.4629 12.6425 (4,4) 3.4727 2.0392 1.4709 9.7038 (5,5) 2.2008 1.2491 0.8735 8.1702Remark. In Table 2, the numerical result indicates that the mixed condition number is much smaller than the normwise condition number.Equation (1) converges to a Hermitian positive definite solution if ρA<1/e as considered in Theorem 7 and in our proposed fixed point method. In our several trials such matrices did not satisfy componentwise condition numbers to be less than the normwise condition number. Only mixed condition number yielded the best result.From Table3, all the computed condition numbers decrease as the size of doubly stochastic matrix and p-values are increased.Example 3.
In this example, we use the same matrixA as in Example 1 for n=3 and p=2. We also set ΔA=10-h×rand(size(A))/4 and ΔI=2ΔA as the perturbations in the matrices A and I, respectively. Δ1 and Δ2 are used as they were previously defined and h is a positive integer.The proposed fixed point algorithm is used to obtain the solutions of (1) and (6), then a summary of results is recorded in Table 4.Table 4
Condition numbers and relative upper perturbation bounds for Example3 evaluated at h=6,8,10,12 and n=3.
h 6 8 10 12 K r e l 1 Δ 1 1.2816×10-7 1.3364×10-9 1.2573×10-11 1.2935×10-13 ζ N 3.0635×10-7 2.8668×10-9 2.9382×10-11 2.7542×10-13 ζ M 5.6886×10-7 5.0152×10-9 5.7033×10-11 5.9011×10-13 ζ C 5.6991×10-7 5.0237×10-9 5.7129×10-11 5.9093×10-13 K r e l 2 Δ 2 4.8743×10-6 5.6170×10-8 4.8345×10-10 5.2817×10-12 ϵ 0 m u p Ψ , X 5.5744×10-6 5.9432×10-8 5.2362×10-10 5.7641×10-12 ϵ 0 c u p Ψ , X 5.5987×10-6 5.9691×10-8 5.2590×10-10 5.7893×10-12 ϵ 0 c o n d ( X) 1.0861×10-5 1.1579×10-7 1.0202×10-9 1.1230×10-11Example 4.
In this example, we consider (1) and (6) in which p=2 and (52)A=0.03820.01570.03950.015700.04780.033950.04780.1065.We also suppose that the perturbations in A and I are(53)ΔA=-0.2-0.30.10.1-0.10.1-0.10.10.2×10-handΔI=-0.30.20.10.1-0.20.3-0.10.1-0.3×10h,where h is a positive integer. Then, using the proposed fixed point algorithm we evaluate solutions X and X~ for (1) and (6), respectively. A summary of results is recorded in Table 5.Table 5
Condition numbers and relative upper perturbation bounds for Example4 evaluated at h=6,8,10,12.
h 6 8 10 12 ζ N 1.6657×10-7 1.6657×10-9 1.6656×10-11 1.6601×10-13 K r e l 1 Δ 1 1.8385×10-7 1.8385×10-9 1.8385×10-11 1.8385×10-13 ζ M 2.9229×10-7 2.9229×10-9 2.9228×10-11 2.9087×10-13 ζ C 3.0312×10-7 3.0312×10-9 3.0310×10-11 3.0164×10-13 K r e l 2 Δ 2 1.8081×10-6 1.8081×10-8 1.8083×10-10 1.8081×10-12 ϵ 0 m u p Ψ , X 1.9060×10-6 1.9060×10-8 1.9060×10-10 1.9060×10-12 ϵ 0 c u p Ψ , X 1.9376×10-6 1.9376×10-8 1.9376×10-10 1.9376×10-12 ϵ 0 c o n d ( X) 3.3083×10-6 3.3083×10-8 3.3083×10-10 3.3083×10-12Remarks (I)
In Table1, both mixed and normwise condition numbers indicate that our nonlinear matrix equation is well-conditioned since the computed condition number is very small for different matrix sizes used in the experiment. However, the componentwise condition number shows that our equation is ill-conditioned. Moreover, the componentwise and normwise condition numbers increase as the matrix size increases whereas the results for the mixed condition number remains constant. (II)
In Tables4 and 5, all the computed condition numbers are fairly sharp and the local upper perturbation bounds for the mixed and componentwise condition numbers exist as it was expected. (III)
In Tables4 and 5, all condition numbers decrease as the perturbation in the matrices I and A are decreased.


## 5. Conclusion

In this paper, we studied the normwise, mixed, and componentwise condition numbers of (1). Also, we derived the explicit expressions normwise mixed and componentwise condition numbers. Local upper bounds for the mixed and componentwise condition numbers exists as it was expected. A comparative analysis for the studied condition numbers is carried out. Componentwise condition number showed the worst results among the computed condition numbers as shown in Tables 1 and 3. In general, (1) seems to be well-conditioned since the computed condition numbers are relatively small. Results in Tables 1, 2, 3, 4, and 5 indicate that the mixed condition number can reveal the true sensitivity of the problem when its input data are badly scaled or sparse.



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