Abstract
This paper mainly investigates the verification of real eigenvalues of the real symmetric and persymmetric matrices. For a real symmetric or persymmetric matrix, we use eig code in Matlab to obtain its real eigenvalues on the basis of numerical computation and provide an algorithm to compute verified error bound such that there exists a perturbation matrix of the same type within the computed error bound whose exact real eigenvalues are the computed real eigenvalues.
1. Introduction
The eigenvalues of a matrix are one of the important tools to solve the complex mathematical problems in the fields of image processing, quantum mechanics, chemistry, and so on [1–3]. The matrices involved in practical problems often have algebraic structure. The preservation of matrix algebraic structure is helpful to keep the physical background of the matrix eigenvalue problem [4, 5]. For a matrix with special algebraic structure, the computation of its eigenvalues is very important in practical problems.
Many scholars have carried out a lot of work to compute the eigenvalues of matrices with special algebraic structure. Using QR decomposition, Bunse-Gerstner et al. [6] designed a stable algorithm to compute the eigenvalues of structured matrices. Higham and Higham [7, 8] estimated the backward error bound and the condition number for computing the generalized eigenvalue and generalized eigenvector of structured matrix. Based on interval calculation, Rump [9] provided the sensitivity analysis of eigenvalues of structured matrix under structured perturbation and proved that, for circulant matrix, Toeplitz matrix, symmetric Toeplitz matrix, and symmetric matrix, the structured condition number is equal to the unstructured condition number under normwise perturbations. Byers and Kressner [10] used the condition number of invariant subspace of the structured matrix to compute the eigenvalues of the structured matrix. Shiozaki [11] proposed an effective algorithm to compute the structured eigenvalues in the field of quantum chemistry. Alon et al. [12] studied the concentration of the maximum eigenvalue of random symmetric matrix whose diagonal and upper diagonal entries are independent real random variables. By Gaussian probability density function with the same mean and variance, Edwards and Jones [13] proposed a straightforward method to analyse the characteristic spectrum of the large symmetric matrix. Given a symmetric matrix whose entries depend on a parameter, Hiriart-Urruty and Ye [14] investigated the first-order sensitivity of all the eigenvalues. Hladík et al [15] considered the eigenvalue problem about the symmetric matrix with perturbed interval entries. Hernandez et al. [16] proposed a greedy algorithm to compute the selected eigenpairs of a large sparse symmetric matrix by exploiting localization features of the eigenvector. Reid [17] showed some useful eigenvalue and eigenvector properties of symmetric and persymmetric matrices.
In this paper, we use Rump interval method and Kantorovich theorem to compute the verified error bounds of real eigenvalues of the given real symmetric and persymmetric matrices, such that there exists a perturbation matrix of the same type within the computed error bound whose exact real eigenvalues are the computed real eigenvalues. To be precise, we transform the verification of real eigenvalues of real symmetric and persymmetric matrices into the verification of a root of a nonlinear system. We utilize the Rump interval method to compute the constants appearing in Kantorovich theorem about the nonlinear system and then use Kantorovich theorem to compute verified error bound of the zero vector as an approximate solution.
The paper is organized as follows. Section 2 is devoted as a preparation of this paper. The main theory and algorithm are, respectively, given in Section 3. Section 4 gives some examples to demonstrate the performance of our algorithm.
2. Notation and Preliminaries
Let and denote the set of natural numbers and real numbers, respectively. For a matrix , is a vector obtained by reshaping all elements of into a single column vector, designates a submatrix of by selecting from the th to th rows, and represents a submatrix of by selecting from the th to th columns. Let denote an zero matrix and be the identity matrix of order . For an matrix , let denote the nullspace of .
Definition 1 (see [18]). Given a matrix with , the corank of is defined by . For a threshold , if the singular values of matrix satisfy thatthen we say that has numerical -corank , denoted by .
Let represent the set of all intervals. A matrix and vector with interval entries are, respectively, called interval vector and interval matrix. Given an interval matrix if an arbitrary real matrix satisfying is nonsingular, then we call the interval matrix a nonsingular interval matrix. Rump [19] developed INTLAB toolbox in Matlab for interval arithmetic. For a nonlinear system if the Jacobian matrix of the system is Lipschitz continuous on some domain, Kantorovich [20] established Kantorovich theorem, which gives a sufficient condition to judge whether the Newton iterative method converges or not by the information of the initial approximate value on some domain.
Lemma 1 (see [21]). Given if the spectral radius of the matrix is less than 1, then is nonsingular.
Theorem 1 (see [22]). If function in INTLAB runs successfully for a given interval matrix and interval right-hand side vector , then the computed interval satisfies the following condition:
Theorem 2 (see [20]). Let with , where are continuously differentiable functions and denotes the Jacobian matrix of the system . Suppose is an approximate solution satisfying the condition that is invertible. Let be a constant such that and a Lipschitz constant such thatwhere is a sufficiently large region containing and is a constant such thatIf and forthen there exists such that .
Remark 1. As pointed by Rall [23], one may take the region in Theorem 2 as . If is the Lipschitz constant for this region, then if and only if .
2.1. Main Result
For a square matrix if , then is called a symmetric matrix. For a matrix with entries , , define a subtranspose matrix with entries , . If , then is called a persymmetric matrix.
Let and , respectively, denote the symmetric and persymmetric matrices. Let and , respectively, represent the corresponding perturbation matrices, namely,
Let be the set of all distinct eigenvalues of computed by eig code in MATLAB. For each , the singular value decomposition of is
Assume that, for each , , where is a positive real number that is very close to zero.
If are all exact eigenvalues of , then, for each , the vectorsare linearly independent.
Therefore, we can make the following reasonable assumption.
Assumption 1. For each , the vectorsare linearly independent.
For , definewhere
Lemma 2. For each if , then the matrixis nonsingular.
Proof. This completes the proof.
For a persymmetric matrix , let be the set of all distinct real eigenvalues of computed by eig code in MATLAB. Let be the inverse identity matrix; then, is a symmetric matrix. For each , the singular value decomposition of is
Assume that, for each , , and then definewhere
For simplifying, we write to stand for both and and we write to denote both and . Then, define the constant by
Lemma 3. If , then, for each , the matrix is nonsingular.
Proof. If , for each ,From Lemma 1, we can deduce that, for each , the matrix is nonsingular.
For each , let and be, respectively, the first -rows and the last -rows of the solution of the following linear system:According to Cramer’s rule, we have the following easy lemma.
Lemma 4. For each , the matrix is a symmetric matrix.
Proposition 1. Suppose that . If holds for , then
Proof. Firstly, consider the case for the symmetric matrix. If , by Lemma 3, we know that, for each , the matrix is nonsingular. Next assume that is an arbitrary fixed integer from 1 to . If , thenNotice that the columns of are linearly independent. Hence, . Assume that ; then, there exists a nonzero vector such that the matrix is full rank, and then, for a nonzero vector , the following equalityholds, which leads to a conditions. Consequently, .
Consider the case of persymmetric matrix. According to the above conclusion, we know that . Apparently, .
Lemma 5. The following nonlinear system,is an underdetermined nonlinear system.
Proof. System (23) consists of variables, and equations. Sincethen the proof is completed.
In the following, assume that is an arbitrary fixed integer from 1 to . By Lemma 3, there exists a neighborhood with the center of the origin such that, for an arbitrary belonging to this neighborhood, the matrix is nonsingular. Moreover, in this neighborhood, each entry of the solution of system (19) has the partial derivative with respect to each variable.
Concerning symmetric matrix, for each pair with , , differentiating both sides of system (19) with respect to gives the following system:whereFor pairs and with , , , and , differentiating both sides of system (25) with respect to and , produces the following system:whereConsidering the persymmetric matrix, for each pair with and differentiating both sides of system (19) with respect to yields the following system:whereFor pairs and with , , , and , differentiating both sides of system (29) with respect to and , provides the following system:whereSolving (25) and (27) or (29) and (31) at can obtain Jacobi matrix and Hessian matrix .
Assumption 2. The Jacobi matrix is full rank.
Suppose that, for an index set , the following matrix,is nonsingular, and then define the following nonlinear system,by
Remark 2. By Theorem 2, we compute verified error bound when the zero vector is the solution of the nonlinear system (34).
For system (34), define the constants and in Theorem 2 byDefine an -dimensional interval vector with each entry as interval . Define the corresponding interval perturbation matrix by setting for and for .
By function, we solve interval linear systems (19), (25), (27) or (19), (29), (31) at , and then we can obtain an interval tensor that satisfiesDefine the Lipschitz constant in Theorem 2 by
3. Main Algorithm
We propose Algorithm 1 for computing verified error bounds such that a slightly perturbed matrix is guaranteed to possess an eigenvalue of geometric multiplicity q within the computed bounds if the algorithm is successful.
Theorem 3. If Algorithm 1 is successful, then there exists a perturbation matrix of the same type, whose entries satisfy the condition
are all exact real eigenvalues of matrix . Furthermore, for each , is the geometric multiplicity of eigenvalue .
|
Proof. If Algorithm 1 is successful, then by Theorem 2, we know that a perturbation matrix whose entries satisfy condition (39), such that . Thus, there exists an interval matrix whose entries satisfy condition (39), such that . Finally, it follows by Proposition 1 that, for each , .
4. Examples
In this section, we show the performance of Algorithm 1. The following experiments are carried out in Matlab R2012a with INTLAB V5 under Windows 7.
Example 1. Given a symmetric matrixthen by Algorithm 1, we get , , and .
Example 2. For a symmetric matrix,applying Algorithm 1 yields , , and .
Example 3. Given a symmetric matrixthen the computed results of Algorithm 1 are , , and .
Example 4. For an symmetric matrix , whose entries are uniformly distributed in the interval , Table 1 shows the radius computed by Algorithm 1 for different .
Example 5. Given a persymmetric matrixAlgorithm 1 outputs , , and .
Example 6. Given a persymmetric matrixthen by Algorithm 1, we obtain , , and .
Example 7. Given a persymmetric matrixthen by Algorithm 1, we obtain , , and .
5. Conclusion and Future Work
This paper demonstrated how to compute the validated and narrow error bounds for the symmetric and persymmetric matrices, such that there exists a perturbation matrix of the same type within the computed error bound whose exact real eigenvalues are the computed real eigenvalues. We will make further efforts to extend the results for other types of structured matrices.
Data Availability
The Matlab data used to support the findings of this study are included in the supplementary information files.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This research was supported in part by the Natural Science Foundation of Jilin Province under Grant no. 20180101345JC.
Supplementary Materials
Supplementary materials are the Matlab code of the examples, where “Function” folder is the main function, “Input” is the execution command of each example, and “Output” is the corresponding output result. (Supplementary Materials)