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Positive eigenvector of nonlinear eigenvalue problem with a singular Mmatrix and NewtonSOR iterative solution
Journal of Inequalities and Applications volume 2016, Article number: 225 (2016)
Abstract
Some sufficient conditions are proposed in this paper such that the nonlinear eigenvalue problem with an irreducible singular Mmatrix has a unique positive eigenvector. Under these conditions, the NewtonSOR 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 Mmatrix, a nonsingular Mmatrix, and a general Mmatrix, respectively. Finally, a numerical example is given to illustrate that the NewtonSOR iterative method is superior to the Newton iterative method.
Introduction
In research of physics, BoseEinstein condensation of atoms near absolute zero temperature is modeled by a nonlinear GrossPitaevskii equation, see [1, 2], i.e.,
where V is a potential function. The discretization of this equation usually leads to the following nonlinear eigenvalue problem:
where \(A\in\mathbb{R}^{n\times n}\) is an irreducible Mmatrix, the function \(F(x)\) is diagonal, that is,
with the conditions that \(x_{i}>0\) and \(f_{i}(x_{i})>0\) for \(i=1,2,\ldots,n\).
In [3–7], some scholars studied the conditions that the nonlinear eigenvalue problem (3) with an irreducible nonsingular Mmatrix 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 Mmatrix 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 GrossPitaevskii equations have a nonsingular Mmatrix. Maybe some nonlinear eigenvalue problems have a singular Mmatrix or a Zmatrix. In this paper, we will mainly study the theory and solution of positive eigenvector of nonlinear eigenvalue problem with a singular Mmatrix. Some sufficient conditions will be proposed such that the nonlinear eigenvalue problem with an irreducible singular Mmatrix has a unique positive eigenvector. Meanwhile, the NewtonSOR 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 Mmatrix, a nonsingular Mmatrix, and a general Mmatrix, respectively.
The paper is organized as follows. Some notations and preliminary results as regards Mmatrices 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 Mmatrix. The NewtonSOR 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 Mmatrix, a nonsingular Mmatrix and a general Mmatrix, respectively. In Section 5, a numerical example is given to demonstrate the effectiveness of the NewtonSOR iterative method. Conclusions are given in Section 6.
Preliminaries
The following lemmas for the Mmatrix are useful and can be found in the literature, see [3, 6, 8–10], we present them here to make the paper selfcontained.
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 \(AB\geq0\).
Definition 2
A matrix \(A=(a_{ij})\in\mathbb{R}^{n\times n}\) is called a Zmatrix if \(a_{ij}\leq0\) for all \(i\neq j\).
We will use \(Z_{n}\) to denote the set of all \(n\times n\) Zmatrices.
Definition 3
A matrix \(A=(a_{ij})\in Z_{n}\) is called an Mmatrix if A can be expressed in the form \(A=sIB\), where \(B\geq0\), and \(s\geq\rho(B)\), the spectral radius of B. If \(s>\rho(B)\), A is called a nonsingular Mmatrix; if \(s=\rho(B)\), A is called a singular Mmatrix.
\(M_{n}\), \(M_{n}^{\bullet}\), and \(M_{n}^{0}\) will be used to denote the set of all \(n\times n\) Mmatrices, the set of all \(n\times n\) nonsingular Mmatrices, and the set of all \(n\times n\) singular Mmatrices, respectively. It is easy to see that
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])
Let \(S:R^{n}\rightarrow R^{n}\) be defined and continuous on an interval \([y,z]\) and let

(i)
\(y< S(y)< z\);

(ii)
\(y< S(z)< z\);

(iii)
\(y\leq x_{1}< x_{2}\leq z\) implies \(y< S(x_{1})< S(x_{2})< z\).
Then

(a)
the fixed point iteration \(x^{(k)}=S(x^{(k1)})\) with \(x^{(0)}=y\) is monotone increasing and converges: \(x^{(k)} \rightarrow x_{*}\), \(S(x_{*})=x_{*}\), \(y< x_{*}< z\);

(b)
the fixed point iteration \(x^{(k)}=S(x^{(k1)})\) 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 \(A=MN\) 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])
If \(A=MN\) is a regular splitting of the matrix \(A\in\mathbb {R}^{n\times n}\) with \(A^{1}\geq0\), then
Thus, the matrix \(M^{1}N\) is convergent, and the iterative method of \(Mx^{m+1}=Nx^{m}+k\), \(m\geq0\) converges for any initial vector \(x^{0}\).
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 Mmatrix.
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 (3), we let \(F(x)=D_{1}^{(x)}x\), where
is a positive diagonal matrix for \(x_{i}>0\), \(f_{i}(x_{i})>0\), \(i=1,\ldots ,n\). We reformulate (3) equivalently as follows:
Since \(\lambda\leq0\) and \(D_{1}^{(x)}\) is a positive diagonal matrix, it follows from Lemma 1 that \(A+D_{1}^{(x)}\lambda I\) is an irreducible nonsingular Mmatrix. Consequently, (6) has a zero solution. Thus, equation (3) has no positive solution for \(\lambda\leq0\). □
In what follows we study the solution of (3) when \(\lambda>0\).
Theorem 2
Let \(A\in M_{n}^{0}\) be irreducible. If \(\lambda>0\), and \(f_{i}(\cdot):(0,\infty)\rightarrow(0,\infty)\) is a \(C^{1}\) function satisfying the conditions:
for \(i=1,\ldots,n\), then the nonlinear eigenvalue problem has a positive solution. In addition, if
for any \(t>0\), then the positive solution is unique.
Proof
Since A is an irreducible singular Mmatrix, \(\mu=0\) is the smallest eigenvalue of A and there exists a positive vector \(p=(p_{1},p_{2},\ldots,p_{n})^{T}\) such that \(Ap=0\). According to (7), take \(\beta_{1}\) small enough such that \(\lambda(\beta_{1}p_{i})>f_{i}(\beta_{1}p_{i})\), \(i=1,\ldots,n\), and take \(\beta_{2}>\beta_{1}\) large enough so that \(\lambda(\beta _{2}p_{i})\leq f_{i}(\beta_{2}p_{i})\), \(i=1,\ldots,n\). This is possible in view of (7) if and only if \(\lambda\in (0,\infty)\). Take a positive number c such that
and let
To prove existence we show that \(S(x)\) satisfies conditions of Lemma 4 with \(y=\beta_{1}p\) and \(z=\beta_{2}p\). For condition (i) of Lemma 4, since
in combination with (9), we can fully show that
Since \((cI+A)^{1}\cdot c>0\), (10) is equivalent to
This follows immediately from the choices of \(\beta_{1}\) and \(\beta_{2}\). The conditions (ii) and (iii) can be verified in a similar way using the condition in (9) on c.
Now let \(x^{*}\) and \(x_{*}\) are two fixed points of S, and \(0< x_{*}\leq x^{*}\). That means
or equivalently that
Let us show that in this case \(x_{*}=x^{*}\). Premultiplying (11) and (12) by \(x_{*}^{T}\) and \(x^{*T}\), respectively, and subtracting we get
or equivalently
Since \(f_{i}(t)\) satisfies the condition (8),
for any \(i=1,2,\ldots,n\). Thus, the function \(\frac{f_{i}(t)}{t}\) is monotone increasing, and consequently, for any \(i=1,2,\ldots,n\),
Since all terms in (13) are nonnegative, the sum is zero if and only if
This implies the uniqueness of positive solution of (3). We completed the 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 GrossPitaevskii 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.
NewtonSOR iteration of positive eigenvector
The Newton iterative method was applied by Choi, Koltracht, and McKenna in [3] to solve numerically the nonlinear eigenvalue problem (3) with Stiltjes matrices. Later, Choi et al. in [4, 5] and Li et al. in [6, 7] used this iterative method to compute this problem with nonsingular Mmatrices, and they established some significant theoretical and numerical results. But each step in the Newton iterative method solves one linear system
where \(R'(x^{k})=A+F'(x^{k})\lambda I\) is nonsingular. If A is singular or illconditioned, the Newton iterative method can perform very bad. Thus, it is necessary to improve this method.
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 NewtonSOR iterative method. Next, the SOR iterative method will be introduced to construct the new iterative method.
For the linear equations
let \(A=DLU\), where \(D=\operatorname{diag}({a_{11},a_{2,2},\ldots,a_{nn}})\), and L and U are, respectively, strictly lower triangular and strictly upper triangular. We may then write the SOR iteration in the form
The quantity ω is called the relaxation factor, and \(0\leq\omega\leq1\). Clearly, (16) reduces to the GaussSeidel iteration when \(\omega=1\). For the equation
\([R'(x)]^{1}\) exists if A, \(F(x)\), and λ satisfy the conditions of Theorem 2, and the fixed point function of the Newton iteration scheme has the following form:
From (17)
where
can be decomposed as
where
Let \(D_{x}=D+D_{2}^{(x)}\lambda I\). Then
where \(D_{x}\), L, and U are diagonal, strictly lower triangular, and strictly upper triangular matrices, respectively. It follows from the SOR iteration scheme (16) and the fixed point function (17) of the Newton iteration scheme that the fixed point function of the NewtonSOR iteration scheme is as follows:
Assume that \(x^{k}\) has been determined. Then the NewtonSOR iteration scheme is given by
It follows from Theorem 5.1 in [8] that the SOR iterative method converges to the unique solution of (15) for any choice of the initial guess \(x^{(0)}\) if and only if \(\rho(H_{\omega})<1\), where \(H_{\omega}\) is the SOR iterative matrix.
In the remainder of this section, the convergence result of the NewtonSOR 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 NewtonSOR iterative matrix, i.e., the sequence \(\{x^{k}\}\) generated by the NewtonSOR iterative scheme (20) converges to the unique solution of (3) for any choice of the initial guess \(x^{(0)}\).
Proof
We only prove \(\rho(H^{k}_{\omega})<1\). According to the NewtonSOR iterative matrix
let \(M_{k}=D_{x^{k}}\omega L\) and \(N_{k}=(1\omega)D_{x^{k}}+\omega U\). Then \(H^{k}_{\omega}=M_{k}^{1}N_{k}\) and
It follows from Lemma 1 that \(A+D_{1}^{(x)}\) is an irreducible nonsingular Mmatrix, where \(D_{1}^{(x)}\) is defined in (5). From (8), \(D_{2}^{(x)}>D_{1}^{(x)}\), then \(A+D_{2}^{(x)}>A+D_{1}^{(x)}\). Lemma 2 shows the smallest eigenvalue of \(A+D_{2}^{(x)}\) is larger than \(A+D_{1}^{(x)}\). The PerronFrobenius theorem (see Theorem 2.7 in [10]) indicates that the positive solution of (3) exists if λ is the smallest eigenvalue of \(A+D_{1}^{(x)}\). Lemma 3 shows that \(A+D_{2}^{(x)}\lambda I\) is an irreducible nonsingular Mmatrix. So
is an irreducible nonsingular Mmatrix, the same for \(\omega R'(x^{k})\). It is easy to see that \(D_{x^{k}}\omega L\) is an irreducible nonsingular Mmatrix, then \((D_{x^{k}}\omega L)^{1}\geq0\). It follows that \((1\omega)D_{x^{k}}+\omega U\geq0\),
is a regular splitting of the matrix \(\omega R'(x^{k})\). It follows from Lemma 5 that \(\rho(M_{k}^{1}N_{k})=\rho(H_{\omega}^{k})<1\), i.e., the sequence \(\{x^{k}\}\) generated by the NewtonSOR iterative scheme (20) converges to the unique solution of (3) for any choice of the initial guess \(x^{(0)}\). This completes the 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 NewtonSOR iterative matrix, i.e., the sequence \(\{x^{k}\}\) generated by the NewtonSOR 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 NewtonSOR iterative matrix, i.e., the sequence \(\{x^{k}\}\) generated by the NewtonSOR iterative scheme (20) converges to the unique solution of (3) for any choice of the initial guess \(x^{(0)}\).
Proof
According to (4), the conclusion of this theorem is obtained immediately from Theorem 3 and Theorem 4. □
Numerical experiment
Now, we verify the convergence of NewtonSOR iterative solution by a numerically example. We start with the onedimensional prototype of the GrossPitaevskii equation,
where \(\nu(t)=t^{2}\). The finite difference method is applied to this equation, which leads to the matrix of the form
where \(\nu_{i}\) are the values of \(\nu(t)\) at the mesh points, h is the discretization stepsize. When h is small enough, A is a singular Mmatrix.
For simplicity, we truncate t on the interval \([0.5,0.5]\). Let \(n=100\), \(h=\frac{10}{n+1}\), and \(t_{i}=1+ih\), \(i=1,\ldots,n+1\). We chose the parameter values \(k=1\), \(\lambda=30\), \(\alpha=15\). We begin the iteration with \(x^{(0)}=\alpha p\), where p is a positive eigenvector of A. Let \(e_{1}^{(k)}=\frac{\x^{k}x^{k1}\_{2}}{\x^{k}\_{2}}\) and \(e_{2}^{(k)}=\Ax^{k}+F(x^{k})\lambda x^{k}\_{2}\). The iteration stops when \(e_{1}^{(k)}+e_{2}^{(k)}<\varepsilon\), \(\varepsilon=10^{5}\). We adopt the NewtonSOR iterative method and the Newton iterative method to compute the equation above on a PC computer by Matlab 7.0. The results on performing a numerical experiment are given in Tables 1 and 2.
The CPU time, iteration number, and values of \(e_{1}^{(k)}\) and \(e_{2}^{(k)}\) of the NewtonSOR method for different values of ω and the Newton method are given in Table 1 and Table 2, respectively. This experiment shows that the NewtonSOR 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 NewtonSOR has a much shorter CPU time, much less iterations, and much smaller error than Newton iterative method does. It is clearly illustrated that the NewtonSOR iterative method is superior to the Newton iterative method.
Conclusions
This article mainly studies the nonlinear eigenvalue problem with an irreducible singular Mmatrix 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 NewtonSOR 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 Mmatrix, a nonsingular Mmatrix, and a general Mmatrix, respectively. Finally, we give a numerical example to illustrate that the NewtonSOR iterative method is superior to the Newton iterative method.
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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).
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Zhang, C., Song, Y. & Luo, S. Positive eigenvector of nonlinear eigenvalue problem with a singular Mmatrix and NewtonSOR iterative solution. J Inequal Appl 2016, 225 (2016). https://doi.org/10.1186/s136600161169y
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MSC
 15A06
 15A15
 15A48
Keywords
 singular Mmatrix
 nonlinear eigenvalue problem
 positive eigenvector
 NewtonSOR iterative solution
 convergence