Positive eigenvector of nonlinear eigenvalue problem with a singular M-matrix and Newton-SOR iterative solution
- Cheng-yi Zhang^{1}Email author,
- Yao-yan Song^{1} and
- Shuanghua Luo^{1}
https://doi.org/10.1186/s13660-016-1169-y
© Zhang et al. 2016
Received: 27 April 2016
Accepted: 7 September 2016
Published: 15 September 2016
Abstract
Some sufficient conditions are proposed in this paper such that the nonlinear eigenvalue problem with an irreducible singular M-matrix has a unique positive eigenvector. Under these conditions, the Newton-SOR iterative method is proposed for numerically solving such a positive eigenvector and some convergence results on this iterative method are established for the nonlinear eigenvalue problems with an irreducible singular M-matrix, a nonsingular M-matrix, and a general M-matrix, respectively. Finally, a numerical example is given to illustrate that the Newton-SOR iterative method is superior to the Newton iterative method.
Keywords
singular M-matrix nonlinear eigenvalue problem positive eigenvector Newton-SOR iterative solution convergenceMSC
15A06 15A15 15A481 Introduction
In [3–7], some scholars studied the conditions that the nonlinear eigenvalue problem (3) with an irreducible nonsingular M-matrix has a unique positive eigenvector, applied the Newton iterative method to solve numerically this problem, and established some significant theoretical and numerical results. It is shown in [3–7] that the main contributions were made to the nonlinear eigenvalue problem as follows: (i) any number greater than the smallest positive eigenvalue of the nonsingular M-matrix is an eigenvalue of the nonlinear eigenvalue problems; (ii) the corresponding positive eigenvector is unique, and (iii) the Newton iterative method is convergent for numerically solving the positive eigenvector.
However, not all nonlinear eigenvalue problems from the discretization of various Gross-Pitaevskii equations have a nonsingular M-matrix. Maybe some nonlinear eigenvalue problems have a singular M-matrix or a Z-matrix. In this paper, we will mainly study the theory and solution of positive eigenvector of nonlinear eigenvalue problem with a singular M-matrix. Some sufficient conditions will be proposed such that the nonlinear eigenvalue problem with an irreducible singular M-matrix has a unique positive eigenvector. Meanwhile, the Newton-SOR iterative method will be proposed under these conditions for numerically solving such a positive eigenvector, and some convergence results on this iterative method will be established for the nonlinear eigenvalue problems with an irreducible singular M-matrix, a nonsingular M-matrix, and a general M-matrix, respectively.
The paper is organized as follows. Some notations and preliminary results as regards M-matrices are given in Section 2. The existence and uniqueness of a positive eigenvector are studied in Section 3 for the nonlinear eigenvalue problem (3) with an irreducible singular M-matrix. The Newton-SOR iterative method is proposed in Section 4 for numerically solving such a positive eigenvector and some convergence results on this iterative method are established for the nonlinear eigenvalue problems with an irreducible singular M-matrix, a nonsingular M-matrix and a general M-matrix, respectively. In Section 5, a numerical example is given to demonstrate the effectiveness of the Newton-SOR iterative method. Conclusions are given in Section 6.
2 Preliminaries
The following lemmas for the M-matrix are useful and can be found in the literature, see [3, 6, 8–10], we present them here to make the paper self-contained.
Definition 1
A matrix \(A=(a_{ij})\in\mathbb{R}^{n\times n}\) is called nonnegative if \(a_{ij}\geq0\) for all \(i,j\in\langle n\rangle=\{ 1,2,\ldots,n\}\).
We write \(A\geq0\) if A is nonnegative. Let \(A\geq0\) and \(B\geq0\), we write \(A\geq B\) if \(A-B\geq0\).
Definition 2
A matrix \(A=(a_{ij})\in\mathbb{R}^{n\times n}\) is called a Z-matrix if \(a_{ij}\leq0\) for all \(i\neq j\).
We will use \(Z_{n}\) to denote the set of all \(n\times n\) Z-matrices.
Definition 3
A matrix \(A=(a_{ij})\in Z_{n}\) is called an M-matrix if A can be expressed in the form \(A=sI-B\), where \(B\geq0\), and \(s\geq\rho(B)\), the spectral radius of B. If \(s>\rho(B)\), A is called a nonsingular M-matrix; if \(s=\rho(B)\), A is called a singular M-matrix.
Lemma 1
(see [8])
If \(A\in M_{n}^{0}\), then \(A+D\in M_{n}^{\bullet}\) for each positive diagonal matrix D.
Lemma 2
(see [6])
If \(A=(a_{ij})\in M_{n}^{\bullet}\), and \(B=(b_{ij})\in Z_{n}\) satisfies \(a_{ij}\leq b_{ij}\), \(i,j=1,\ldots,n\), then \(B\in M_{n}^{\bullet}\), and hence, \(B^{-1}\leq A^{-1}\) and \(\mu _{A}\leq\mu_{B}\), where \(\mu_{A}\) and \(\mu_{B}\) are the smallest eigenvalues of A and B, respectively. In addition, if A is irreducible and \(A\neq B\), then \(B^{-1}< A^{-1}\) and \(\mu_{A}<\mu_{B}\).
Lemma 3
(see [6])
Let \(A\in M_{n}^{\bullet}\) and μ be the smallest positive eigenvalue of A. Then, for any \(\nu\leq\mu\), \(A-\nu I\in M_{n}\).
Lemma 4
(see [3])
- (i)
\(y< S(y)< z\);
- (ii)
\(y< S(z)< z\);
- (iii)
\(y\leq x_{1}< x_{2}\leq z\) implies \(y< S(x_{1})< S(x_{2})< z\).
- (a)
the fixed point iteration \(x^{(k)}=S(x^{(k-1)})\) with \(x^{(0)}=y\) is monotone increasing and converges: \(x^{(k)} \rightarrow x_{*}\), \(S(x_{*})=x_{*}\), \(y< x_{*}< z\);
- (b)
the fixed point iteration \(x^{(k)}=S(x^{(k-1)})\) with \(x^{(0)}=z\) is monotone decreasing and converges: \(x^{(k)} \rightarrow x^{*}\), \(S(x^{*})=x^{*}\), \(y< x^{*}< z\);
- (c)
if x is a fixed point of S in \([y,z]\) then \(x_{*}\leq x\leq x^{*}\);
- (d)
S has a unique fixed point in \([y,z]\) if and only if \(x_{*}=x^{*}\).
Definition 4
(see [10])
A splitting \(A=M-N\) of \(A\in\mathbb{R}^{n\times n}\) is called a regular splitting of the matrix A if M is nonsingular with \(M^{-1}\geq0\) and \(N\geq0\).
Lemma 5
(see [10])
3 Existence and uniqueness of positive eigenvector
In this section, we study the existence and uniqueness of positive eigenvector of nonlinear eigenvalue problem (3) with an irreducible singular M-matrix.
Theorem 1
Let \(A\in M_{n}^{0}\) be irreducible. Then, for any \(\lambda\leq0\), the nonlinear eigenvalue problem (3) has no positive solution.
Proof
In what follows we study the solution of (3) when \(\lambda>0\).
Theorem 2
Proof
Remark 1
It is easy to see that the function \(F(x)=(f_{1}(t),f_{2}(t),\ldots ,f_{n}(t))^{T}\) in the Gross-Pitaevskii equation with \(f_{i}(t)=t^{3}\) for \(i=1,2,\ldots,n\) satisfies all conditions of Theorem 2. Thus, this equation has a positive solution.
4 Newton-SOR iteration of positive eigenvector
On the other hand, many iterative methods are proposed for linear systems; see [11–13]. Among these methods, SOR iterative method is a more effective one. In the following the Newton iterative method will be improved to propose a new iterative method - the Newton-SOR iterative method. Next, the SOR iterative method will be introduced to construct the new iterative method.
In the remainder of this section, the convergence result of the Newton-SOR iterative method will be established for the nonlinear eigenvalue problem (3).
Theorem 3
If A, λ, and \(F(x)=[f_{1}(x_{1}),f_{2}(x_{2}),\ldots,f_{n}(x_{n})]^{T}\) in (3) satisfy all conditions of Theorem 2, then \(\rho(H^{k}_{\omega})<1\), where \(H^{k}_{\omega}=(D_{x^{k}}-\omega L)^{-1}[(1-\omega)D_{x^{k}}+\omega U]\) is the Newton-SOR iterative matrix, i.e., the sequence \(\{x^{k}\}\) generated by the Newton-SOR iterative scheme (20) converges to the unique solution of (3) for any choice of the initial guess \(x^{(0)}\).
Proof
Theorem 4
Let \(A\in M_{n}^{\bullet}\). If λ and \(F(x)=[f_{1}(x_{1}),f_{2}(x_{2}),\ldots,f_{n}(x_{n})]^{T}\) in (3) satisfy all conditions of Theorem 2, then \(\rho(H^{k}_{\omega })<1\), where \(H^{k}_{\omega}=(D_{x^{k}}-\omega L)^{-1}[(1-\omega )D_{x^{k}}+\omega U]\) is the Newton-SOR iterative matrix, i.e., the sequence \(\{x^{k}\}\) generated by the Newton-SOR iterative scheme (20) converges to the unique solution of (3) for any choice of the initial guess \(x^{(0)}\).
Proof
Similar to the proof of Theorem 3, the conclusion of this theorem is obtained immediately from Lemma 1, Lemma 2, Lemma 3, and Lemma 5. □
Theorem 5
Let \(A\in M_{n}\). If λ and \(F(x)=[f_{1}(x_{1}),f_{2}(x_{2}),\ldots ,f_{n}(x_{n})]^{T}\) in (3) satisfy all conditions of Theorem 2, then \(\rho(H^{k}_{\omega})<1\), where \(H^{k}_{\omega }=(D_{x^{k}}-\omega L)^{-1}[(1-\omega)D_{x^{k}}+\omega U]\) is the Newton-SOR iterative matrix, i.e., the sequence \(\{x^{k}\}\) generated by the Newton-SOR iterative scheme (20) converges to the unique solution of (3) for any choice of the initial guess \(x^{(0)}\).
5 Numerical experiment
Iteration number, CPU time, and values of \(\pmb{e_{1}^{(k)}}\) and \(\pmb{e_{2}^{(k)}}\) for different values of ω
ω | 0.25 | 1 | 1.5 | 2.1 | 2.45 |
---|---|---|---|---|---|
CPU time | 2.5498 | 0.4931 | 0.2930 | 0.1685 | 0.4252 |
Iter. | 664 | 137 | 76 | 50 | 103 |
\(e_{1}^{(k)}\) | 7.7452 × 10^{−12} | 3.1752 × 10^{−11} | 4.9257 × 10^{−11} | 1.5896 × 10^{−10} | 2.5597 × 10^{−10} |
\(e_{2}^{(k)}\) | 9.9186 × 10^{−7} | 8.7991 × 10^{−7} | 8.1715 × 10^{−7} | 7.6891 × 10^{−7} | 8.8627 × 10^{−7} |
Iteration number, CPU time, and values of \(\pmb{e_{1}^{(k)}}\) and \(\pmb{e_{2}^{(k)}}\) of the Newton method
Iter. | CPU time | \(\boldsymbol{e_{1}^{(k)}}\) | \(\boldsymbol{e_{2}^{(k)}}\) |
---|---|---|---|
93 | 4.5745 | 4.0768 × 10^{−11} | 8.9977 × 10^{−7} |
The CPU time, iteration number, and values of \(e_{1}^{(k)}\) and \(e_{2}^{(k)}\) of the Newton-SOR method for different values of ω and the Newton method are given in Table 1 and Table 2, respectively. This experiment shows that the Newton-SOR method has the shortest CPU time, the least iterations, and the minimal error when \(\omega=2.1\). By comparing Table 1 and Table 2, we find that Newton-SOR has a much shorter CPU time, much less iterations, and much smaller error than Newton iterative method does. It is clearly illustrated that the Newton-SOR iterative method is superior to the Newton iterative method.
6 Conclusions
This article mainly studies the nonlinear eigenvalue problem with an irreducible singular M-matrix and proposes some sufficient conditions such that a positive eigenvector of this problem exists and is unique. Under these conditions, we improve the SOR iterative method to construct the Newton-SOR iterative method for numerically solving such a positive eigenvector, meanwhile, we establish some convergence results on this iterative method for the nonlinear eigenvalue problems with an irreducible singular M-matrix, a nonsingular M-matrix, and a general M-matrix, respectively. Finally, we give a numerical example to illustrate that the Newton-SOR iterative method is superior to the Newton iterative method.
Declarations
Acknowledgements
The work was supported by the National Natural Science Foundations of China (11601409, 11201362, and 11271297), the Natural Science Foundations of Shaanxi Province of China (2016JM1009) and the Science Foundation of the Education Department of Shaanxi Province of China (14JK1305).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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