Abstract
In this paper, we propose an improved power algorithm for finding maximal eigenvalues. Without any partition, we can get the maximal eigenvalue and show that the modified power algorithm is convergent for zero symmetric reducible nonnegative matrices. Numerical results are reported to demonstrate the effectiveness of the modified power algorithm. Finally, a modified algorithm is proposed to test the positive definiteness (positive semidefiniteness) of -matrices.
1. Introduction
The maximum eigenvalue problems of nonnegative matrices are important tools for matrix analysis and calculation and are widely used in many fields, such as principal component analysis, Markov chain [1], and stability of control systems [2–7]. Some researchers directly estimated the bounds of the maximum eigenvalues according to the nature of the nonnegative irreducible matrices [8–11]. Based on a geometric symmetrization of powers of matrix, Szyld [10] presented a sequence of lower bounds for the spectral radius. Dursun et al. [8] gave a sequence of upper bounds for maximal eigenvalue of a nonnegative matrix, which complemented results of [10]. Due to the complexity of the maximal eigenvalue problem, it is difficult to obtain exact solutions by means of estimation methods. Hence, some efficient algorithms for computing eigenpairs of nonnegative matrices have been proposed such as (improved) Arnoldi's algorithm [12, 13], (modified) diagonal transformation algorithm [14–17], and power algorithm [18–24]. Generally, (improved) Arnoldi’s method [12] is suitable for computing some selected eigenpairs of large asymmetrical matrices. The (modified) diagonal transformation algorithm is convergent for irreducible nonnegative matrices [15, 17]. Wood et al. [22, 23] proposed the convergent power method to calculate maximum eigenvalue of irreducible nonnegative matrices. For reducible matrices, the diagonal transformation algorithm and power algorithm may not be convergent [15, 22]. As we know, reducible matrices can be decomposed into some irreducible matrices. However, it is very costly to test the irreducibility and determine the partition for large-size matrices. Motivated by these observations, we want to establish a modified power algorithm and obtain the maximal eigenvalue of zero symmetric reducible nonnegative matrices without any partition.
This paper is organized as follows. In Section 2, we introduce important notation and recall fundamental results. In Section 3, we propose a modified power algorithm for computing the maximal eigenvalue of zero symmetric reducible nonnegative matrices. It is worth noting that the algorithm is convergent and does not require the partition for zero symmetric reducible nonnegative matrices. Numerical results are reported to demonstrate the effectiveness of the proposed algorithm. As an application, we can test the positive definiteness (positive semidefiniteness) of -matrices in Section 4.
2. Notation and Preliminaries
We start this section with some fundamental notion and properties of a nonnegative matrix, which are needed in the subsequent analysis.
Definition 1. (i) A matrix is called symmetric if (ii) A matrix is called zero symmetric if
Obviously, if a matrix is symmetric, then is zero symmetric. Conversely, the result may not hold.
Definition 2. A matrix is said to be reducible if there exists a permutation matrix such thatwhere , and is the zero matrix. If such a permutation matrix does not exist, is called irreducible.
Definition 3. Let be a nonnegative matrix of dimension and with . A submatrix of the matrix with index set is an -dimensional submatrix of consisting of elements defined as follows: where is the number of elements of .
Definition 4. We call the set of all eigenvalues of . Assume . Then the spectral radius of is denoted by
If is a nonnegative matrix, then is the maximal eigenvalue.
Let and . For any nonnegative column vector , we define by Now, we give the power algorithm [22, 23] of irreducible nonnegative matrices as follows.
Algorithm 5.
Step 1. Choose , , , and a unit matrix
Step 2. Compute Step 3. If , stop. Output the maximal eigenvalue . Otherwise, set and compute and go to Step 2.
For irreducible nonnegative matrices, we have the following convergent theorems.
Lemma 6 (Theorem 9 of [22]). If is an irreducible nonnegative matrix, then the matrix is primitive, where and is a unit matrix.
Lemma 7 (Theorem 8 of [22]). generated by Algorithm 5 converge to , if and only if the irreducible nonnegative matrix is primitive.
3. Modified Power Algorithm for the Maximal Eigenvalue of Zero Symmetric Nonnegative Matrices
In this section, we give the modified power algorithm for computing the maximal eigenvalue of zero symmetric reducible nonnegative matrices without any partition. Furthermore, we demonstrate that this algorithm is convergent by Theorem 10. We state the algorithm as follows.
Algorithm 8.
Step 1. Choose , and , , and
Step 2. Compute Step 3. If , stop. Output the maximal eigenvalue . Otherwise, set and compute and go to Step 2.
We show that the sequence generated by Algorithm 8 is convergent and the limit is the maximal eigenvalue for any zero symmetric nonnegative matrices. In what follows, we give the following lemma.
Lemma 9. Let be a zero symmetric reducible nonnegative matrix. Then, we have the following results:
(I) There exists a partition of such that each induced matrix , , either is irreducible or is a zero matrix
(II) For , there exists an permutation such that (III)
For a zero symmetric reducible nonnegative matrix , by Lemma 9, it holds that As we know, it is very costly to determine the partition for large-size reducible matrices. Fortunately, Algorithm 8 is not necessary to decompose the matrix and can compute the maximal eigenvalue of each block matrix.
Theorem 10. Suppose that is a zero symmetric reducible nonnegative matrix and the sequence is generated by Algorithm 8. Then, .
Proof. For the sequence , it holds that is monotonously decreasing and bounded to the below. According to Algorithm 8, it is easy to get . Now, we show that is a decreasing sequence. For any , by Algorithm 8, we haveSo, Since is nonnegative, we get For , one has which means that is a decreasing sequence. So, the sequence is convergent. Hence, there exists such that when Next, we show that . Without loss of generality, we assume that the matrix is stated as follows: where each block is square and either is primitive or is a unit matrix. It follows from Lemma 9 that We divide the proof into two parts.
Case 1. has a unique maximal eigenvalue. Without loss of generality, we assume Let be the maximum value of matrix generated by Algorithm 8. Let be the maximum value and eigenvector of and let be the maximum eigenvalue of Set Obviously, Taking into account that each block is primitive, from Lemma 7, we have Since is primitive and , from Algorithm 8, we obtain For , when , we have and, furthermore,On the other hand, since and , it holds that For , it follows from (24) that and, equivalently, It follows from that From Lemma 7, we obtain Case 2. has two maximal eigenvalues, that is, According to the above analysis, , when , we know For , we deduce or It follows from Lemma 7 that and, equivalently, When has more than two maximal eigenvalues, we repeat the above process and can obtain the same convergent conclusions.
Remark 11. We are able to obtain an eigenvector corresponding to as follows. Since the sequence is bounded, it has an accumulation point. Suppose that is an accumulation point of . Define bywhere By Lemma 9, is an eigenvector corresponding to when .
In the following, we report numerical results for Algorithm 8. In order to show this algorithm is efficient, we compare it with improved Arnoldi's algorithm in [13] and the modified diagonal algorithm in [17]. For Algorithm 8, we stop the iteration as long as , . (Detailed information is in Tables 1 and 2.)
All testing matrices are generated as follows: Give an integer and generate randomly three positive matrices . Let , , and . Then, define , where , other elements being zero. Clearly, is zero symmetric reducible. Algorithm 8 is implemented in MATLAB (R2011a) and all the numerical computations are conducted using an Intel 3.60 GHz computer with 8 GB of RAM. In Table 1, the CPU time is the average of instances for each . From Table 1, we can see that both algorithms can find maximal eigenvalues of the testing matrices but Algorithm 8 uses much less CPU time than improved Arnoldi's algorithm in [13], especially for large-size matrices.
Considering that the matrix is nonnegative and irreducible for the modified diagonal algorithm in [17], all testing matrices are positive matrices generated randomly in Table 2. Compared with the modified diagonal algorithm in [17], the Algorithm 8 has better execution efficiency in CPU time.
4. Applications
In this section, we give an application of Algorithm 8 for testing the positive definiteness (positive semidefiniteness) for large-size -matrices. We recall some definitions and notations about -matrices and -matrices.
Definition 12. A matrix is called a -matrix if it can be written as , where is a unit matrix and is a nonnegative matrix. Furthermore, if , then is said to be an -matrix; if , then is said to be a strong -matrix.
Let and Define a quadratic polynomial form as follows:
Definition 13. (i) is called positive definite if for all .
(ii) is called positive semidefinite if for all
As we know, for a symmetric matrix , is positive definite if and only if all of its eigenvalues are positive, and is positive semidefinite if and only if all of its eigenvalues are nonnegative. For an asymmetric -matrix , we can do the following transformation to verify whether is a positive definite (semidefinite) matrix.
For a -matrix , consider the following optimization problem: Clearly, we observe that is positive definite if and only if , and is positive semidefinite if and only if . Set Then, can be written as , where is a nonnegative matrix.
Given indices , where , let be the set of all permutations of the indices . For any , let
and define
Hence, is a symmetrized matrix of . For any , set Clearly, we have that . Therefore,
From the above analysis, we can propose the following algorithm for testing the positive definiteness (positive semidefiniteness) of -matrices.
Algorithm 14.
Step 1. Given a -matrix , compute If , then is not positive semidefinite and stop. Otherwise, go to the next step.
Step 2. Let and compute its symmetrized matrix . By Algorithm 8, compute the spectral radius .
Step 3. Let . If , is positive definite. If , is positive semidefinite. Otherwise, it is indefinite.
5. Conclusions
In this paper, a modified power algorithm was proposed for finding the maximal eigenvalue of zero symmetric nonnegative matrices. The modified power algorithm has the following important properties: (1) the convergence property is guaranteed for any zero symmetric nonnegative matrices and (2) it gets the maximal eigenvalue without any partition. Finally, we proposed Algorithm 14 to test the positive definiteness (positive semidefiniteness) of -matrices.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research is supported by the Natural Science Foundation of China (11671228) and the Natural Science Foundation of Shandong Province (ZR2016AM10).