Abstract
We explore the condition numbers of the nonlinear matrix equation . 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.
1. Introduction
We consider the nonlinear matrix equationwhere is a real or complex square matrix, is an identity matrix, is the matrix exponential function, and is a positive integer. The basic general form of (1) is , 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 , where 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 which is (1) for 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 , . Authors in [5] presented a perturbation analysis of the matrix equation , for positive integers , 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: stands for normwise condition number; “’’ means equal by definition; stands for a ball with center and radius ; stands for domain of ; denotes the set of all complex Hermitian matrices; if , then means that is positive semidefinite (positive definite) and means that ; the notation (■) stands for spectral radius; signify the transpose conjugate and transpose of matrix , respectively; and denote the Frobenius norm and usual spectral norm, respectively; given and , the Kronecker product is ; the operator is defined by ; means the ratio of the largest singular value to the smallest; stands for the absolute value of .
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 function at a point is defined as 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 function at a point is a linear operatorsuch that for all . The operator denotes the Fréchet derivative of at in the direction . If such an operator exists, is said to be Fréchet differentiable and
Lemma 3 (see [8], Lemma 4.3.1). Suppose that , , , and . Then,
Lemma 4 (see [9], pp. 178, Theorem 3.3.16(a, b)). Let be given and let . Then, following inequalities hold for the decreasingly ordered singular values of and :(I),(II)
Lemma 5 (see [10], Theorem 15). Let and . If is an eigenvalue of and is an eigenvalue of , then is an eigenvalue of .
For easy expansion and simplification of matrix polynomials, we need Lemma 6.
Lemma 6 (see [11]). Let denote a set of positive integers including zero and such that . Then,(i), , .(ii), and for .(iii)
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).Replacing by in (6) yieldsNow, let us make small perturbations in the matrices , and as shown in Subtracting (8) by (1) yieldsUsing Lemma 6, we havewhere . Because have higher orders of , we omit it and consider only the first term of (10). Then, replacing in (9) by gives Using the fact that is symmetric and is real and applying the operator in (11), we getCombining the terms with vec in (12) yieldsThen, we haveIn (14), the term refers to higher order approximation of with respect to and is the vec permutation operator satisfying and it is defined aswhere and is the column of the identity matrix .
Let us define a map , where, represents a real or complex space. We have , where the matrices . According to the implicit function theorem, it is apparent that as , because is a function of .
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 are defined byNow, suppose that is differentiable at , then using Theorem 4 in [12], we have where is a Fréchet derivative of at . It follows thatwhereDenotingwe have
Now, we prove that matrix is nonsingular in Theorem 7.
Theorem 7. Suppose that is the Hermitian positive definite solution of (1) and is a real matrix with such that . Then,is nonsingular.
Proof. DenoteThen, we have Using Lemmas 4 and 5, it follows thatandFrom , we know that ; this implies that , which means that 1< . Therefore, .
Since , then we have . Therefore, we conclude that is invertible.
From Theorem 7 above, since is invertible, then we have , and the Kronecker Fréchet derivative
Explicit expressions for the two kinds of normwise condition numbers are derived in Theorem 8.
Theorem 8. Suppose that is the HPD solution of (1) and is nonsingular. Then, ①,②, whereand
Proof. ①Using the fact that and (17), we can easily get ②In this case, we rewrite as , where and , then we have . It follows thatFinally, using (16) and (31) and , we see that
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 vectorsNow, let us define a distance function In the rest of this paper, we assume is finite for any pair , and we extend the function to matrices. That is for any matrices , we have, . For , we denote .
Based on the work by Gohberg and Koltracht [13] on mixed, componentwise, and structured condition numbers, we have Definition 9.
Definition 9. Let be a continuous mapping defined on an open interval set such that and for a given .(I)The mixed condition number of at is defined by (II)The componentwise condition number of at is defined byIf is Fréchet differentiable at point . Then, the explicit expressions of the mixed and componentwise condition numbers of at are given by Lemma 10.
Lemma 10 (see [13, 14]). Assume is Fréchet differentiable at point , we have (1)if , then(2)if such that , for , then Now, we derive the explicit expressions for the mixed and componentwise condition numbers and their upper bounds in Theorem 11.
Theorem 11. Let be the Hermitian positive definite solution of (1). Define the mapping(I)Let denote the mixed condition number defined by (35). Then has the explicit expression where(II)Let denote the componentwise condition number defined by (36). Then, has the explicit expression where is the same as in item (I).
Moreover, we define two simple upper bounds for and given byandrespectively.
Proof. We first prove (I) using of Lemma 10. In this case, for , we obtain that the mixed condition number is . It follows from (27) thatwhere It also holds that So it follows from (45) that the upper bound of is given byNow we prove item (II) for the componentwise condition number of (1). Using (27) together with item of Lemma 10 yields where is the same as in item (I).
Likewise, to estimate the upper bound of , we have 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 [email protected] 1.70GHz with 64-bit operating system. Four examples are considered. In Example 1, 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 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 and and evaluate the local upper perturbation bounds for the computed condition numbers. In Example 4, we consider a symmetric matrix with some specified perturbations in the matrices and 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.
For , we define , , , and , are solutions of (1) and (6), respectively. The solutions and are computed by a fixed point algorithm. We obtain the local normwise, mixed, and componentwise condition numbers as follows. , , and . Then, we also define the relative mixed and componentwise local upper perturbation bounds as and , respectively.
Here, we propose a fixed point algorithm to compute the solutions and .
Fixed Point Algorithm(I)Input an matrix with , tolerance error tol=eps, and an initial guess , where is the size of matrix and eps is the standard machine precision.(II)For , compute and relative residual(III)Exit the loop if res . Otherwise, go to step (II).(IV)Display the solution .
Example 1. We consider the tridiagonal matrix , where .
The matrix is generated by a MATLAB function “full (gallery (‘tridiag’, , 1,2,1))’’, where denotes the size of matrix . For 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.
Example 2. (I)We consider (1) with . Then, we evaluate the normwise mixed and condition numbers for different -values. We employ our proposed fixed point method with =eye(). 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 such that , where magic, sum, and is the size of . Then, we apply our proposed fixed point method with =eye(). A summary of results is recorded in Table 3
Remark. 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 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 Table 3, all the computed condition numbers decrease as the size of doubly stochastic matrix and -values are increased.
Example 3. In this example, we use the same matrix as in Example 1 for and . We also set and as the perturbations in the matrices and , respectively. and are used as they were previously defined and 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.
Example 4. In this example, we consider (1) and (6) in which and We also suppose that the perturbations in and arewhere is a positive integer. Then, using the proposed fixed point algorithm we evaluate solutions and for (1) and (6), respectively. A summary of results is recorded in Table 5.
Remarks(I)In Table 1, 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 Tables 4 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 Tables 4 and 5, all condition numbers decrease as the perturbation in the matrices and 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.
Data Availability
Due to the nature of our subject, all the necessary steps are included in the submitted manuscript. However, if more details will be required we will provide them immediately.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors would like to thank Professor Hyun-Min Kim for his remarkable guidance during this research work. This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) (NRF-2017R1A5A1015722).