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On Minimization of Upper Bound for the Convergence Rate of the QHSS Iteration Method

  • Wen-Ting WuEmail author
Original Paper
  • 93 Downloads

Abstract

For an upper bound of the spectral radius of the QHSS (quasi Hermitian and skew-Hermitian splitting) iteration matrix which can also bound the contraction factor of the QHSS iteration method, we give its minimum point under the conditions which guarantee that the upper bound is strictly less than one. This provides a good choice of the involved iteration parameters, so that the convergence rate of the QHSS iteration method can be significantly improved.

Keywords

System of linear equations Non-Hermitian matrix QHSS iteration method Convergence rate 

Mathematics Subject Classification

65F10 65F20 65K05 90C25 15A06; CR: G1.3. 

1 Introduction

To solve the large sparse system of linear equations \(Ax=b\) with \(A \in {{\mathbb {C}}}^{n \times n}\) being the nonsingular coefficient matrix, \(x \in {{\mathbb {C}}}^{n}\) being the unknown vector, and \(b \in {{\mathbb {C}}}^{n}\) being the right-hand side, there are many classical iteration methods, such as Jacobi, Gauss–Seidel [25, 36], SOR [23, 33], SSOR [2, 3, 6, 30, 32], and CG [16]. But these methods theoretically converge to the unique solution of the linear system only for the coefficient matrix A being either Hermitian positive definite or strictly (irreducibly) diagonally dominant, and they may fail to compute an approximate solution to the linear system when the skew-Hermitian part of the matrix A is dominant over its Hermitian part; see, e.g., [1, 14, 21, 37] for more details.

There are many studies about the splitting iteration methods [22, 26, 27, 34], such as the additive and the multiplicative splitting iteration methods [4], and the splitting-MINRES methods [15]. Among them, the Hermitian and skew-Hermitian splitting (HSS) iteration method [11] is a useful and effective method which breaks this limitation and converges unconditionally to the unique solution of the linear system when the coefficient matrix A is positive definite (i.e., its Hermitian part is Hermitian positive definite). After the HSS iteration method, the PSS method, the AHSS method and the NSS method were proposed in [8, 10, 12] successively. For a review of these methods, we refer to [5]. The preconditioned variants of the HSS method could be found in [13, 20, 39]. And the preconditioning strategies based on the HSS were studied in [19, 24, 29, 35]. For more generalizations and developments about the HSS iteration method, we refer to [9, 17, 18, 38] and the references therein.

To avoid the weakness of the HSS iteration method that the number of iteration steps may grow quickly when either the number of unknowns or the norm of the skew-Hermitian part is increasing, Bai proposed the quasi-HSS (QHSS) iteration method [7] which can be used to quickly and stably solve the linear system when the coefficient matrix A is positive definite with a nearly singular Hermitian part or a strongly dominant skew-Hermitian part. Besides, the QHSS iteration method can be even employed to solve the linear system with the coefficient matrix A being non-Hermitian indefinite of only mild indefiniteness.

In [7], to describe the asymptotic convergence rate of the QHSS iteration method, Bai gave an upper bound of the spectral radius of the QHSS iteration matrix and discussed the conditions for guaranteeing that the upper bound is strictly less than one. But the minimum of this upper bound under the conditions which guarantee that the upper bound is strictly less than one is never mentioned.

Since the upper bound can also bound the contraction factor of the QHSS iteration method, finding its minimum point can help to provide a good choice of the involved iteration parameters and accelerate the convergence rate of the QHSS iteration method. In this paper, we discuss the problem that at which point does the upper bound reach its minimum under the conditions that guarantee the upper bound being strictly less than one. We find that the upper bound reaches its minimum with respect to the involved iteration parameter \(\alpha\) just at the geometric mean of the smallest and the largest eigenvalue of the matrix \(H_\omega = H+\omega S^*S\), where H and S are the Hermitian and the skew-Hermitian parts of the coefficient matrix A, respectively, \(S^*\) denotes the conjugate transpose of the matrix S, and \(\omega\) is a nonnegative parameter. And the numerical results show that this minimum point is a good choice of the involved iteration parameter \(\alpha\). The parameter \(\omega\) can also affect the asymptotic convergence rate of the QHSS iteration method significantly. We find that the minimum of the upper bound is still in connection with the parameter \(\omega\) and, to make it as small as possible, we should let the parameters \(\omega\) and \(\kappa (H_\omega )\) be small. Here \(\kappa (H_\omega )\) denotes the Euclidean condition number of the matrix \(H_\omega\).

The organization of this paper is as follows. In Sect. 2, we present the upper bound of the spectral radius of the QHSS iteration matrix, the conditions for guaranteeing that the upper bound is strictly less than one, and some necessary notations. In Sect. 3, we discuss the minimum of this upper bound under the conditions which guarantee that the upper bound is strictly less than one. The numerical results are reported in Sect. 4. Finally, in Sect. 5, we end the paper with a few remarks and conclusions.

2 Preliminaries

For a matrix \(X \in {{\mathbb {C}}}^{m \times n}\), \(X^*\) denotes its conjugate transpose, and \(\Vert X\Vert _2\) represents its Euclidean norm. If, in addition, X is square and nonsingular, we use \(\kappa (X)\) to indicate the Euclidean condition number of the matrix X. The Euclidean norm of the vector in \({{\mathbb {C}}}^n\) is the special case of that in \({{\mathbb {C}}}^{m \times n}\).

The QHSS iteration matrix is given by
$$\begin{aligned} L_\omega (\alpha ) = M_\omega (\alpha )^{-1}N_\omega (\alpha ), \end{aligned}$$
where
$$\begin{aligned} M_\omega (\alpha ) = \frac{1}{2\alpha }(I-\omega S)^{-1}(\alpha I+S)(\alpha I+H_\omega ), \end{aligned}$$
(1)
and
$$\begin{aligned} N_\omega (\alpha ) = \frac{1}{2\alpha }(I-\omega S)^{-1}[(\alpha I-S)(\alpha I-H_\omega )+2\omega \alpha SH], \end{aligned}$$
(2)
with I being the identity matrix, \(\omega\) and \(\alpha\) being the given nonnegative and positive constants, and
$$\begin{aligned} H=\frac{1}{2}(A+A^*), \quad S=\frac{1}{2}(A-A^*), \quad H_\omega = H+\omega S^*S. \end{aligned}$$
When the coefficient matrix \(A \in {{\mathbb {C}}}^{n \times n}\) is non-Hermitian positive definite, let \(\lambda _\omega ^{(\min )}\) and \(\lambda _\omega ^{(\max )}\) be the smallest and the largest eigenvalues of the matrix \(H_\omega\), respectively, and denote
$$\begin{aligned} {\text{\ae}} & = \min \{ \kappa (H),\kappa (H_{\omega } )\} ,\qquad \varpi = \sqrt {\lambda _{\omega }^{{(\min )}} \lambda _{\omega }^{{(\max )}} } , \\ \omega _{o} & = \frac{{\sqrt 2 }}{{\sqrt {{\text{\ae}}}\lambda _{\omega }^{{(\max )}} }} ,\quad \omega _{a} = \frac{{( {\lambda _{\omega }^{{(\max )}} +\left\| S \right\|_{2} })\omega _{d} }}{{\lambda _{\omega }^{{(\min )}} + \left\| S \right\|_{2} }},\quad \omega _{b} = \frac{{\varpi \omega _{o} }}{{\left\| S \right\|_{2} }},\\ \omega _{c} & = \sqrt {\omega _{p} \omega _{q} } ,\qquad \omega _{d} = \frac{{\lambda _{\omega }^{{(\min )}} \omega _{o} }}{{\left\| S \right\|_{2} }}, \\ \omega _{p} & = \frac{{(\left\|H_{\omega } \right\|_{2} + \left\| S \right\|_{2} )^{2} \omega _{d} }}{{( {\lambda _{\omega }^{{(\min )}} + \left\| S \right\|_{2} } )^{2} }},\qquad \omega _{q} = \frac{{(\varpi + \left\|H_{\omega } \right\|_{2} )^{2} \omega _{d} }}{{( {\lambda _{\omega }^{{(\min )}} + \varpi } )^{2} }}, \\ \end{aligned}$$
and
$$\alpha _{a} = \frac{{\lambda _{\omega }^{{(\min )}} }}{{2\omega }}\left( {\omega _{o} - \omega + \sqrt {(\omega _{o} - \omega )^{2} + \frac{{8\omega }}{{\sqrt {2{\text{\ae}}} \lambda _{\omega }^{{(\min )}}}}} } \right),\quad \alpha _{b} = \frac{{\left( {\omega _{o} \lambda _{\omega }^{{(\max )}} - \left\| S \right\|_{2} \omega } \right)\lambda _{\omega }^{{(\min )}} }}{{\left\| S \right\|_{2} \omega - \lambda _{\omega }^{{(\min )}} \omega _{o} }}.$$

In [7], Bai gave an upper bound about the spectral radius \(\rho (L_\omega (\alpha ))\) of the QHSS iteration matrix \(L_\omega (\alpha )\) and the conditions for guaranteeing that the upper bound is strictly less than one, when the coefficient matrix \(A \in {{\mathbb {C}}}^{n \times n}\) is a non-Hermitian positive definite matrix. We restate this result in the following theorem.

Theorem 2.1

Let\(A \in {{\mathbb {C}}}^{n \times n}\)be a non-Hermitian positive definite matrix, with\(H=\frac{1}{2}(A+A^*)\)and\(S=\frac{1}{2}(A-A^*)\)being its Hermitian and skew-Hermitian parts, respectively, and let\(\omega\)and\(\alpha\)be nonnegative and positive constants, respectively. Then, the spectral radius\(\rho (L_\omega (\alpha ))\)of the QHSS iteration matrix\(L_\omega (\alpha )\)is bounded by
$$\begin{aligned} f(\alpha ) :={\left\{ \begin{array}{ll} \frac{\lambda _\omega ^{(\max )}-\alpha }{\lambda _\omega ^{(\max )}+\alpha } + \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad for \quad \alpha \le \varpi \quad and \quad \alpha \le \Vert S\Vert _2, \\ \frac{\lambda _\omega ^{(\max )}-\alpha }{\lambda _\omega ^{(\max )}+\alpha } + \frac{\sqrt{2{\text{\ae}} }\omega \alpha \Vert H_\omega \Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad for \quad \alpha \le \varpi \quad and \quad \alpha \ge \Vert S\Vert _2, \\ \frac{\alpha -\lambda _\omega ^{(\min )}}{\alpha +\lambda _\omega ^{(\min )}} + \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad for \quad \alpha \ge \varpi \quad and \quad \alpha \le \Vert S\Vert _2, \\ \frac{\alpha -\lambda _\omega ^{(\min )}}{\alpha +\lambda _\omega ^{(\min )}} + \frac{\sqrt{2{\text{\ae}} }\omega \alpha \Vert H_\omega \Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad for \quad \alpha \ge \varpi \quad and \quad \alpha \ge \Vert S\Vert _2. \end{array}\right. } \end{aligned}$$
And it holds that
$$\begin{aligned} \rho (L_\omega (\alpha )) \le f(\alpha ) < 1, \end{aligned}$$
provided that the parameters\(\omega\)and\(\alpha\)satisfy the following conditions
  1. (a)

    when\(\Vert S\Vert _2 < \varpi\), for\(0 \le \omega <\omega _o\)the iteration parameter\(\alpha\)obeys\(\frac{\omega }{\omega _o}\Vert S\Vert _2< \alpha < \alpha _a\);

     
  2. (b)

    when\(\Vert S\Vert _2 \ge \varpi\), there are two cases distinguished as follows:

     
  3. (i)

    for\(0 \le \omega < \omega _a\), the iteration parameter\(\alpha\)obeys\(\frac{\omega }{\omega _o}\Vert S\Vert _2< \alpha < \alpha _a\);

     
  4. (ii)

    for\(\omega _a \le \omega < \omega _b\), the iteration parameter\(\alpha\)obeys\(\frac{\omega }{\omega _o}\Vert S\Vert _2< \alpha < \alpha _b\).

     

3 Minimization of the Upper Bound

We first give some inequalities which will be used later.

Lemma 3.1

When\(\Vert S\Vert _2 \ge \varpi\), from
$$\begin{aligned} \lambda _\omega ^{(\min )}\le \lambda _\omega ^{(\max )}= \Vert H_\omega \Vert _2, \end{aligned}$$
we can easily verify that
$$\begin{aligned} \omega _d \le \omega _p \le \omega _c \le \omega _q. \end{aligned}$$

From Theorem 2.1, it is known that we only need to discuss the cases that \(0 \le \omega < \omega _o\) when \(\Vert S\Vert _2 < \varpi\), and \(0 \le \omega < \omega _b\) when \(\Vert S\Vert _2 \ge \varpi\). For the minimum point of the upper bound \(f(\alpha )\) of \(\rho (L_\omega (\alpha ))\) under the conditions which guarantee that the upper bound is strictly less than one, we have the following result.

Theorem 3.1

The upper bound\(f(\alpha )\)of\(\rho (L_\omega (\alpha ))\)reaches its minimum at\(\alpha =\varpi\), for\(0 \le \omega < \omega _o\)when\(\Vert S\Vert _2 < \varpi\), and for\(0 \le \omega < \omega _b\) when\(\Vert S\Vert _2 \ge \varpi\), i.e.,
$$\begin{aligned} \min \limits _\alpha f(\alpha )= f(\varpi )= {\left\{ \begin{array}{ll} \frac{\lambda _\omega ^{(\max )}-\varpi }{\lambda _\omega ^{(\max )}+\varpi } + \frac{\sqrt{2{\text{\ae}} }\omega \varpi \Vert H_\omega \Vert _2}{\varpi +\Vert H_\omega \Vert _2}, &{} \quad for \quad 0 \le \omega< \omega _o \quad when \quad \Vert S\Vert _2< \varpi , \\ \frac{\varpi -\lambda _\omega ^{(\min )}}{\varpi +\lambda _\omega ^{(\min )}} + \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{\varpi +\Vert H_\omega \Vert _2}, &{} \quad for \quad 0 \le \omega < \omega _b \quad when \quad \Vert S\Vert _2 \ge \varpi . \end{array}\right. } \end{aligned}$$

Proof

Define
$$\begin{aligned} f(\alpha ) = {\left\{ \begin{array}{ll} f_1(\alpha ) :=\frac{\lambda _\omega ^{(\max )}-\alpha }{\lambda _\omega ^{(\max )}+\alpha } + \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad \text{ for }\quad \alpha \le \varpi \quad \text{ and }\quad \alpha \le \Vert S\Vert _2, \\ f_2(\alpha ) :=\frac{\lambda _\omega ^{(\max )}-\alpha }{\lambda _\omega ^{(\max )}+\alpha } + \frac{\sqrt{2{\text{\ae}} }\omega \alpha \Vert H_\omega \Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad \text{ for }\quad \alpha \le \varpi \quad \text{ and }\quad \alpha \ge \Vert S\Vert _2, \\ f_3(\alpha ) :=\frac{\alpha -\lambda _\omega ^{(\min )}}{\alpha +\lambda _\omega ^{(\min )}} + \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad \text{ for }\quad \alpha \ge \varpi \quad \text{ and }\quad \alpha \le \Vert S\Vert _2, \\ f_4(\alpha ) :=\frac{\alpha -\lambda _\omega ^{(\min )}}{\alpha +\lambda _\omega ^{(\min )}} + \frac{\sqrt{2{\text{\ae}} }\omega \alpha \Vert H_\omega \Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad \text{ for }\quad \alpha \ge \varpi \quad \text{ and }\quad \alpha \ge \Vert S\Vert _2. \end{array}\right. } \end{aligned}$$
For the case \(\Vert S\Vert _2 < \varpi\), it holds that
$$\begin{aligned} f(\alpha ) = {\left\{ \begin{array}{ll} f_1(\alpha ) :=\frac{\lambda _\omega ^{(\max )}-\alpha }{\lambda _\omega ^{(\max )}+\alpha } + \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad \text{ for }\quad \alpha \le \Vert S\Vert _2, \\ f_2(\alpha ) :=\frac{\lambda _\omega ^{(\max )}-\alpha }{\lambda _\omega ^{(\max )}+\alpha } + \frac{\sqrt{2{\text{\ae}} }\omega \alpha \Vert H_\omega \Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad \text{ for }\quad \Vert S\Vert _2 \le \alpha \le \varpi , \\ f_4(\alpha ) :=\frac{\alpha -\lambda _\omega ^{(\min )}}{\alpha +\lambda _\omega ^{(\min )}} + \frac{\sqrt{2{\text{\ae}} }\omega \alpha \Vert H_\omega \Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad \text{ for }\quad \alpha \ge \varpi . \end{array}\right. } \end{aligned}$$
Now, we discuss the monotonicity of the functions \(f_1(\alpha )\), \(f_2(\alpha )\) and \(f_4(\alpha )\).
First, we have
$$\begin{aligned} f_1^{\prime }(\alpha ) = -\frac{2\lambda _\omega ^{(\max )}}{( \lambda _\omega ^{(\max )}+\alpha ) ^2} - \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{(\alpha +\Vert H_\omega \Vert _2)^2} <0. \end{aligned}$$
Second, as
$$\begin{aligned} f_2^{\prime }(\alpha ) = -\frac{2\lambda _\omega ^{(\max )}}{(\lambda _\omega ^{(\max )}+\alpha )^2} + \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2^2}{(\alpha +\Vert H_\omega \Vert _2)^2}, \end{aligned}$$
by noticing \(\Vert H_\omega \Vert _2=\lambda _\omega ^{(\max )}\) and \(0 \le \omega < \omega _o\), we have
$$\begin{aligned} f_2^{\prime }(\alpha )= & {} -\frac{2\lambda _\omega ^{(\max )}}{(\lambda _\omega ^{(\max )}+\alpha )^2} + \frac{\sqrt{2{\text{\ae}} }\omega (\lambda _\omega ^{(\max )})^2}{(\alpha +\lambda _\omega ^{(\max )})^2} \\= & {} \frac{\sqrt{2{\text{\ae}} }\omega (\lambda _\omega ^{(\max )})^2-2\lambda _\omega ^{(\max )}}{(\alpha +\lambda _\omega ^{(\max )})^2} < 0. \end{aligned}$$
Finally, we have
$$\begin{aligned} f_4^{\prime }(\alpha ) = \frac{2\lambda _\omega ^{(\min )}}{(\alpha +\lambda _\omega ^{(\min )})^2} + \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2^2}{(\alpha +\Vert H_\omega \Vert _2)^2} > 0. \end{aligned}$$
From the continuity of \(f(\alpha )\), we know that \(f(\varpi )=\min \limits _\alpha f(\alpha )\) holds true for \(0 \le \omega < \omega _o\).
For the case \(\Vert S\Vert _2 \ge \varpi\), it holds that
$$\begin{aligned} f(\alpha ) = {\left\{ \begin{array}{ll} f_1(\alpha ) :=\frac{\lambda _\omega ^{(\max )}-\alpha }{\lambda _\omega ^{(\max )}+\alpha } + \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad \text{ for }\quad \alpha \le \varpi , \\ f_3(\alpha ) :=\frac{\alpha -\lambda _\omega ^{(\min )}}{\alpha +\lambda _\omega ^{(\min )}} + \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad \text{ for }\quad \varpi \le \alpha \le \Vert S\Vert _2, \\ f_4(\alpha ) :=\frac{\alpha -\lambda _\omega ^{(\min )}}{\alpha +\lambda _\omega ^{(\min )}} + \frac{\sqrt{2{\text{\ae}} }\omega \alpha \Vert H_\omega \Vert _2}{\alpha +\Vert H_\omega \Vert _2}, &{} \quad \text{ for }\quad \alpha \ge \Vert S\Vert _2. \end{array}\right. } \end{aligned}$$
Analogously to the case \(\Vert S\Vert _2 < \varpi\), we see that \(f_1^{\prime }(\alpha ) < 0\) and \(f_4^{'}(\alpha ) > 0\). Then, we look into the derivative of \(f_3(\alpha )\):
$$\begin{aligned} f_3^{'}(\alpha ) = \frac{2\lambda _\omega ^{(\min )}}{(\alpha +\lambda _\omega ^{(\min )})^2} - \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{(\alpha +\Vert H_\omega \Vert _2)^2}. \end{aligned}$$
We first discuss on what conditions does \(f_3^{'}(\alpha ) \ge 0\) hold. In fact, this inequality is equivalent to the inequality:
$$\begin{aligned} \left( \sqrt{2\lambda _\omega ^{(\min )}} -\sqrt{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}\right) \alpha \ge \sqrt{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2} \, \lambda _\omega ^{(\min )}-\sqrt{2\lambda _\omega ^{(\min )}} \, \Vert H_\omega \Vert _2. \end{aligned}$$
(3)
When \(\omega < \omega _d\), i.e.,
$$\begin{aligned} \sqrt{2\lambda _\omega ^{(\min )}}-\sqrt{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}>0, \end{aligned}$$
(4)
using the number
$$\begin{aligned} \alpha _o :=\frac{\sqrt{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2} \, \lambda _\omega ^{(\min )}-\sqrt{2\lambda _\omega ^{(\min )}} \, \Vert H_\omega \Vert _2}{\sqrt{2\lambda _\omega ^{(\min )}}-\sqrt{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}}, \end{aligned}$$
we know that the inequality (3) becomes \(\alpha \ge \alpha _o\). Since \(\alpha\) is also in the interval \(\varpi \le \alpha \le \Vert S\Vert _2\), the intersection of the two intervals is \(\varpi \le \alpha \le \Vert S\Vert _2\) for \(\alpha _o \le \varpi\), \(\alpha _o \le \alpha \le \Vert S\Vert _2\) for \(\varpi < \alpha _o \le \Vert S\Vert _2\), and empty for \(\alpha _o > \Vert S\Vert _2\).

From the inequality (4), we see that \(\alpha _o \le \Vert S\Vert _2\) is equivalent to \(\omega \le \omega _p\), and \(\alpha _o \le \varpi\) is equivalent to \(\omega \le \omega _q\). From Lemma 3.1, we can obtain \(\alpha _o \le \varpi \le \Vert S\Vert _2\) when \(\omega < \omega _d \le \omega _p \le \omega _q\). Then, it follows that the inequality (3) is satisfied if \(\varpi \le \alpha \le \Vert S\Vert _2\) holds true for \(\omega < \omega _d\).

When \(\omega > \omega _d\), i.e.,
$$\begin{aligned} \sqrt{2\lambda _\omega ^{(\min )}}-\sqrt{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}<0, \end{aligned}$$
(5)
the inequality (3) becomes \(\alpha \le \alpha _o\). Since \(\alpha\) is also in the interval \(\varpi \le \alpha \le \Vert S\Vert _2\), the intersection of the two intervals is \(\varpi \le \alpha \le \Vert S\Vert _2\) for \(\alpha _o \ge \Vert S\Vert _2\), \(\varpi \le \alpha \le \alpha _o\) for \(\varpi \le \alpha _o < \Vert S\Vert _2\), and empty for \(\alpha _o < \varpi\).

From the inequality (5), we know that \(\alpha _o \ge \varpi\) is equivalent to \(\omega \le \omega _q\), and \(\alpha _o \ge \Vert S\Vert _2\) is equivalent to \(\omega \le \omega _p\). From Lemma 3.1, we can obtain \(\alpha _o < \varpi\) when \(\omega > \omega _q\), \(\varpi \le \alpha _o < \Vert S\Vert _2\) when \(\omega _p < \omega \le \omega _q\), and \(\alpha _o \ge \Vert S\Vert _2\) when \(\omega _d < \omega \le \omega _p\). Then, it follows that the inequality (3) is satisfied if \(\varpi \le \alpha \le \Vert S\Vert _2\) holds true for \(\omega _d < \omega \le \omega _p\), or \(\varpi \le \alpha \le \alpha _o\) holds true for \(\omega _p < \omega \le \omega _q\). And there does not exist \(\alpha\) such that the inequality (3) is satisfied for \(\omega > \omega _q\).

When \(\omega = \omega _d\), i.e.,
$$\begin{aligned} \sqrt{2\lambda _\omega ^{(\min )}}-\sqrt{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}=0, \end{aligned}$$
the inequality (3) becomes
$$\begin{aligned} \sqrt{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2} \, \lambda _\omega ^{(\min )}-\sqrt{2\lambda _\omega ^{(\min )}} \, \Vert H_\omega \Vert _2 \le 0. \end{aligned}$$
It is obviously true. Then, we can claim that the inequality (3) is satisfied if \(\varpi \le \alpha \le \Vert S\Vert _2\) holds true for \(\omega = \omega _d\).
From the above arguments, we can conclude that \(f_3^{'}(\alpha ) \ge 0\) if the parameters \(\omega\) and \(\alpha\) satisfy the following conditions:
  1. (a)

    for \(0 \le \omega \le \omega _p\), the parameter \(\alpha\) obeys \(\varpi \le \alpha \le \Vert S\Vert _2\);

     
  2. (b)

    for \(\omega _p < \omega \le \omega _q\), the parameter \(\alpha\) obeys \(\varpi \le \alpha \le \alpha _o\).

     
Analogously, we can obtain that \(f_3^{\prime }(\alpha ) < 0\) if the parameters \(\omega\) and \(\alpha\) satisfy the following conditions:
  1. (a’)

    for \(\omega _p < \omega \le \omega _q\), the parameter \(\alpha\) obeys \(\alpha _o < \alpha \le \Vert S\Vert _2\);

     
  2. (b’)

    for \(\omega > \omega _q\), the parameter \(\alpha\) obeys \(\varpi \le \alpha \le \Vert S\Vert _2\).

     
Thereby, it follows from the continuity of \(f(\alpha )\) that
$$\begin{aligned} {\left\{ \begin{array}{ll} f(\varpi )=\min \limits _\alpha f(\alpha ),&{} \quad \text{ for }\quad 0 \le \omega \le \omega _p, \\ \min \{f(\varpi ), f(\Vert S\Vert _2)\}=\min \limits _\alpha f(\alpha ),&{} \quad \text{ for }\quad \omega _p < \omega \le \omega _q, \\ f(\Vert S\Vert _2)=\min \limits _\alpha f(\alpha ),&{} \quad \text{ for }\quad \omega > \omega _q. \end{array}\right. } \end{aligned}$$

Now, we compare the values \(f(\varpi )\) and \(f(\Vert S\Vert _2)\) when \(\omega _p < \omega \le \omega _q\).

From the definition of \(f(\alpha )\), for \(\Vert S\Vert _2 \ge \varpi\) we have
$$\begin{aligned} f(\varpi )=\frac{\varpi -\lambda _\omega ^{(\min )}}{\varpi +\lambda _\omega ^{(\min )}} +\frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{\varpi +\Vert H_\omega \Vert _2}, \end{aligned}$$
and
$$\begin{aligned} f(\Vert S\Vert _2)=\frac{\Vert S\Vert _2-\lambda _\omega ^{(\min )}}{\Vert S\Vert _2+\lambda _\omega ^{(\min )}} +\frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{\Vert S\Vert _2+\Vert H_\omega \Vert _2}. \end{aligned}$$
Hence,
$$\begin{aligned} &f(\varpi )-f(\Vert S\Vert _2) \\ &= \frac{\varpi -\lambda _\omega ^{(\min )}}{\varpi +\lambda _\omega ^{(\min )}} +\frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{\varpi +\Vert H_\omega \Vert _2} -\frac{\Vert S\Vert _2-\lambda _\omega ^{(\min )}}{\Vert S\Vert _2+\lambda _\omega ^{(\min )}} -\frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{\Vert S\Vert _2+\Vert H_\omega \Vert _2} \\ &= \left( \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{(\varpi +\Vert H_\omega \Vert _2)(\Vert S\Vert _2+\Vert H_\omega \Vert _2)} -\frac{2\lambda _\omega ^{(\min )}}{(\varpi +\lambda _\omega ^{(\min )})(\Vert S\Vert _2+\lambda _\omega ^{(\min )})}\right) (\Vert S\Vert _2-\varpi ). \end{aligned}$$
By Lemma 3.1 we have \(f(\varpi ) \le f(\Vert S\Vert _2)\) holds true for \(\omega _p < \omega \le \omega _c\), while \(f(\varpi ) \ge f(\Vert S\Vert _2)\) holds true for \(\omega _c < \omega \le \omega _q\).
Finally, we can obtain the following conclusion:
$$\begin{aligned} {\left\{ \begin{array}{ll} f(\varpi )=\min \limits _\alpha f(\alpha ),&{} \quad \text{ for }\quad 0 \le \omega \le \omega _c, \\ f(\Vert S\Vert _2)=\min \limits _\alpha f(\alpha ),&{} \quad \text{ for }\quad \omega > \omega _c. \end{array}\right. } \end{aligned}$$
Since \(\omega _b \le \omega _c\), \(f(\varpi )=\min \limits _\alpha f(\alpha )\) holds true for \(0 \le \omega < \omega _b\).

The parameter \(\omega\) can also significantly affect the asymptotic convergence rate of the QHSS iteration method. A suitable choice of \(\omega\) can substantially diminish the negative influence from the skew-Hermitian matrix S, especially when the matrix S is strongly dominant over the Hermitian matrix H.

In the QHSS iteration method, \(\omega\) is used to reformulate the linear system \(Ax=b\) into the quasi-normal equation
$$\begin{aligned} (I+\omega S^*)(H+S)x=(I+\omega S^*)b. \end{aligned}$$
As Bai mentioned in [7], the role of the matrix \(I+\omega S^*\) is to normalize the linear system \(Ax=b\), so the parameter \(\omega\) should be chosen reasonably small such that the matrix \(I+\omega S^*\) is well conditioned (e.g., close to I) as far as possible. It was also mentioned in [7] that compared with the classical normal equation, the normalizing effect of this quasi-normal equation is significant, provided that the parameter \(\omega\) is small enough.
When \(\Vert S\Vert _2 < \varpi\), for \(0 \le \omega < \omega _o\) it holds that
$$\begin{aligned} \min \limits _\alpha f(\alpha )= & {} \frac{\lambda _\omega ^{(\max )}-\varpi }{\lambda _\omega ^{(\max )}+\varpi } + \frac{\sqrt{2{\text{\ae}} }\omega \varpi \Vert H_\omega \Vert _2}{\varpi +\Vert H_\omega \Vert _2} \nonumber \\= & {} \frac{\sqrt{\kappa (H_\omega )}-1}{\sqrt{\kappa (H_\omega )}+1} + \frac{\sqrt{2{\text{\ae}} \lambda _\omega ^{(\max )}\lambda _\omega ^{(\min )}}\sqrt{\kappa (H_\omega )}}{\sqrt{\kappa (H_\omega )}+1}\omega , \end{aligned}$$
(6)
and when \(\Vert S\Vert _2 \ge \varpi\), for \(0 \le \omega < \omega _b\) it holds that
$$\begin{aligned} \min \limits _\alpha f(\alpha )= & {} \frac{\varpi -\lambda _\omega ^{(\min )}}{\varpi +\lambda _\omega ^{(\min )}} + \frac{\sqrt{2{\text{\ae}} }\omega \Vert H_\omega \Vert _2\Vert S\Vert _2}{\varpi +\Vert H_\omega \Vert _2} \nonumber \\= & {} \frac{\sqrt{\kappa (H_\omega )}-1}{\sqrt{\kappa (H_\omega )}+1} + \frac{\sqrt{2{\text{\ae}} }\Vert S\Vert _2\sqrt{\kappa (H_\omega )}}{\sqrt{\kappa (H_\omega )}+1}\omega . \end{aligned}$$
(7)
Since the first terms in the right-hand sides of the equalities (6) and (7) are the same for both cases and monotonically increasing with respect to \(\kappa (H_\omega )\), and the second terms in the right-hand sides of the equalities (6) and (7) depend on the parameter \(\omega\), we find that to minimize \(\min \limits _\alpha f(\alpha )\), a relatively small \(\omega\) should be taken to make \(\kappa (H_\omega )\) be small.
If \(\kappa (H_\omega )\) has a constant upper bound \(\tau _1\), i.e., \(1\le \kappa (H_\omega )\le \tau _1\), and \(\lambda _\omega ^{(\max )}\) has a constant upper bound \(\tau _2\), i.e., \(0<\lambda _\omega ^{(\max )}\le \tau _2\), then it follows from the equalities (6) and (7) that when \(\Vert S\Vert _2 < \varpi\), for \(0 \le \omega < \omega _o\) it holds that
$$\begin{aligned} \min \limits _\alpha f(\alpha )\le & {} \frac{\sqrt{\tau _1}-1}{\sqrt{\tau _1}+1} +\frac{\sqrt{2\min \{\kappa (H), \tau _1\}}\lambda _\omega ^{(\max )}}{\sqrt{\kappa (H_\omega )}+1}\omega \\\le & {} \frac{\sqrt{\tau _1}-1}{\sqrt{\tau _1}+1} +\frac{\sqrt{2\min \{\kappa (H), \tau _1\}}\tau _2}{2}\omega , \end{aligned}$$
and when \(\Vert S\Vert _2 \ge \varpi\), for \(0 \le \omega < \omega _b\) it holds that
$$\begin{aligned} \min \limits _\alpha f(\alpha ) \le \frac{\sqrt{\tau _1}-1}{\sqrt{\tau _1}+1} +\frac{\sqrt{2\min \{\kappa (H), \tau _1\}}\Vert S\Vert _2\sqrt{\tau _1}}{\sqrt{\tau _1}+1}\omega . \end{aligned}$$
Hence, we find that to make \(\min \limits _\alpha f(\alpha )\) as small as possible, we should also let \(\omega\) be small.
When \(\omega =0\), the QHSS iteration method automatically reduces to the HSS iteration method. For \(0 \le \omega < \omega _o\) when \(\Vert S\Vert _2 < \varpi\), and for \(0 \le \omega < \omega _b\) when \(\Vert S\Vert _2 \ge \varpi\), it holds that
$$\begin{aligned} \min \limits _\alpha f(\alpha )= \frac{\sqrt{\kappa (H)}-1}{\sqrt{\kappa (H)}+1}, \end{aligned}$$
(8)
which is the same as the convergence rate of the conjugate gradient method [25] applied to a linear system with the Hermitian positive definite coefficient matrix H.
If the parameter \(\omega\) is chosen appropriately such that \(\kappa (H_\omega )\) is smaller than \(\kappa (H)\), it holds that
$$\begin{aligned} \frac{\sqrt{\kappa (H_\omega )}-1}{\sqrt{\kappa (H_\omega )}+1} < \frac{\sqrt{\kappa (H)}-1}{\sqrt{\kappa (H)}+1}. \end{aligned}$$
Besides, if \(\omega\) is small enough, the right-hand sides of the equalities (6) and (7) will be both smaller than the right-hand side of the equality (8). This implies that the QHSS iteration method could be much faster than the HSS iteration method if the parameters \(\omega\) and \(\alpha\) are chosen appropriately.

Remark 3.1

Besides finding the minimum point for the upper bound of the spectral radius of the QHSS iteration matrix, we can also look for some other ways to obtain good and easily computable values for the parameters \(\alpha\) and \(\omega\). Since the QHSS iteration method is induced from the matrix splitting \(A=M_\omega (\alpha )-N_\omega (\alpha )\) with \(M_\omega (\alpha )\) and \(N_\omega (\alpha )\) defined as in (1) and (2), respectively, like that done in [28], we could choose \((\alpha , \omega )\) as the minimum point for the function \(\Vert {\tilde{N}}(\alpha , \omega )\Vert _F^2\), with \({\tilde{N}}(\alpha , \omega )=2\alpha N_\omega (\alpha )\).

4 Experimental Results

In this section, we test the QHSS iteration method when the linear system is discretized from a time-dependent two-dimensional fractional convection–diffusion equation [7] and show that \(\varpi\) is a good value for the parameter \(\alpha\).

The time-dependent two-dimensional fractional convection–diffusion equation which the linear system is discretized from is of the form
$$\begin{aligned} \frac{\partial u(x, y, t)}{\partial t}= & {} \left( \frac{\partial ^\mu u(x, y, t)}{\partial _+ x^\mu } +\frac{\partial ^\mu u(x, y, t)}{\partial _- x^\mu }\right) +\left( \frac{\partial ^\nu u(x, y, t)}{\partial _+ y^\nu } +\frac{\partial ^\nu u(x, y, t)}{\partial _- y^\nu }\right) \\&+q\frac{\hbox{e}^{x+y}}{x+y}\left( x\frac{\partial u(x, y, t)}{\partial x} +y\frac{\partial u(x, y, t)}{\partial y}\right) +f(x, y, t), \\&(x, y)\in \Omega \triangleq (0, 2)\times (0, 2) \triangleq (0, 2)^2 \quad \text{ and } \quad t\in (0, 1), \end{aligned}$$
with the boundary and initial conditions
$$\begin{aligned} \left\{ \begin{array}{ll} u(x, y, t)=0, &{} \text{ for } \quad (x, y)\in \partial \Omega \quad \text{ and } \quad 0<t\le 1,\\ u(x, y, 0)=x^2(2-x^2)y^2(2-y^2), &{} \text{ for } \quad (x, y)\in \Omega . \end{array} \right. \end{aligned}$$
Let \(\Delta t\) be the temporal stepsize, \(h_x\) and \(h_y\) be the spatial stepsizes in the x- and the y-direction, \(N_x=\frac{2}{h_x}-1\) and \(N_y=\frac{2}{h_y}-1\) be the numbers of spatial grid points, and
$$\begin{aligned} {\left\{ \begin{array}{ll} u^m=(u_{1,1}^m, u_{2,1}^m, \ldots , u_{N_x,1}^m, \ldots , u_{1,N_y}^m, \ldots , u_{N_x,N_y}^m)^\mathrm{{T}},\\ f^m=(f_{1,1}^m, f_{2,1}^m, \ldots , f_{N_x,1}^m, \ldots , f_{1,N_y}^m, \ldots , f_{N_x,N_y}^m)^\mathrm{{T}}, \end{array}\right. } \end{aligned}$$
with \(u_{i,j}^m\approx u(ih_x,jh_y,m\Delta t)\), \(f_{i,j}^m=f(ih_x,jh_y,m\Delta t)\), \(i=1, 2, \ldots , N_x\), \(j=1, 2, \ldots , N_y\) and \((\cdot )^\mathrm{{T}}\) denoting the transpose of the corresponding vector or matrix. Then, from [7], we have that the system of the linear equations can be written as
$$\begin{aligned} A^m u^m=\hbar u^{m-1}+h_x^\mu h_y^\nu f^m, \end{aligned}$$
where
$$\begin{aligned} A^m = \hbar I+h_y^\nu \left[ I \otimes T_x^\mu +I \otimes (T_x^\mu )^\mathrm{{T}}\right] +h_x^\mu \left[ T_y^\nu \otimes I+(T_y^\nu )^\mathrm{{T}} \otimes I\right] -B \end{aligned}$$
is the coefficient matrix and \(\hbar =\frac{h_x^\mu h_y^\nu }{\Delta t}\) is a positive constant. Here, \(\otimes\) represents the Kronecker product symbol,
$$\begin{aligned} T_x^\mu = -\left( \begin{array}{ccccc} g_1^{(\mu )} &{} g_0^{(\mu )} &{} 0 &{} \cdots &{} 0 \\ g_2^{(\mu )} &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ g_3^{(\mu )} &{} \ddots &{} \ddots &{} \ddots &{} 0 \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} g_0^{(\mu )} \\ g_{N_x}^{(\mu )} &{} \cdots &{} g_3^{(\mu )} &{} g_2^{(\mu )} &{} g_1^{(\mu )} \end{array} \right) \quad \text{ and } \quad T_y^\nu = -\left( \begin{array}{ccccc} g_1^{(\nu )} &{} g_0^{(\nu )} &{} 0 &{} \cdots &{} 0 \\ g_2^{(\nu )} &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ g_3^{(\nu )} &{} \ddots &{} \ddots &{} \ddots &{} 0 \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} g_0^{(\nu )} \\ g_{N_y}^{(\nu )} &{} \cdots &{} g_3^{(\nu )} &{} g_2^{(\nu )} &{} g_1^{(\nu )} \end{array} \right) \end{aligned}$$
are Toeplitz matrices with
$$\begin{aligned} g_0^{(\beta )}=1 \quad \text{ and } \quad g_k^{(\beta )}=(-1)^k\frac{\beta (\beta -1)\cdots (\beta -k+1)}{k!}, \quad k=1, 2, \ldots , \quad \beta =\mu , \nu , \end{aligned}$$
and \(B=\mathrm{Tridiag}(B_2, B_1, B_3)\) is the block-tridiagonal matrix having \(B_1\) on its diagonal and \(B_2\) and \(B_3\) on its subdiagonals, with \(B_1=\mathrm{Diag}(B_{1,j})\), \(B_2=\mathrm{Diag}(B_{2,j})\), and \(B_3=\mathrm{Diag}(B_{3,j})\) being block-diagonal matrices of the block elements \(B_{1,j}=\mathrm{tridiag}({\underline{b}}_{1,j},0,{\bar{b}}_{1,j})\) for \(j=1, 2, \ldots , N_y\), \(B_{2,j}=\mathrm{diag}(b_{2,j})\) for \(j=2, 3, \ldots , N_y\), and \(B_{3,j}=\mathrm{diag}(b_{3,j})\) for \(j=1, 2, \ldots , N_y-1\), and \(B_{1,j}\), \(B_{2,j}\), and \(B_{3,j}\) being either tridiagonal or diagonal matrices of the elements
$$\begin{aligned} \left\{ \begin{array}{ll} {\underline{b}}_{1,j}(i)=-\frac{q}{2}h_x^\mu h_y^\nu \frac{\hbox{e}^{(i+1)h_x+jh_y}}{(i+1)h_x+jh_y}(i+1), &{} i=1, 2, \ldots , N_x-1, \\ {\bar{b}}_{1,j}(i)=\frac{q}{2}h_x^\mu h_y^\nu \frac{\hbox{e}^{ih_x+jh_y}}{ih_x+jh_y}i, &{} i=1, 2, \ldots , N_x-1, \\ b_{2,j}(i)=-\frac{q}{2}h_x^\mu h_y^\nu \frac{\hbox{e}^{ih_x+jh_y}}{ih_x+jh_y}j, &{} i=1, 2, \ldots , N_x, \\ b_{3,j}(i)=\frac{q}{2}h_x^\mu h_y^\nu \frac{\hbox{e}^{ih_x+jh_y}}{ih_x+jh_y}j, &{} i=1, 2, \ldots , N_x. \end{array} \right. \end{aligned}$$

Thus, in terms of the linear system \(Ax=b\), we have \(A=A^m\), \(x=u^m\), and \(b=\hbar u^{m-1}+h_x^\mu h_y^\nu f^m\). We find that the skew-Hermitian part of \(A^m\) is only contained in the matrix B, and when \(q>0\) is increasing, the norm of the skew-Hermitian part of \(A^m\) is increasing, too. In the implementations, for the fixed positive integers \(M_t\) and \(N_t\), we choose \(\Delta t=\frac{1}{N_t}\) and \(N_x=N_y=M_t\), and we set f(xyt) to be the function such that the exact solution of the linear system is given by \(x_*=(1, -1, 1, -1, \ldots , (-1)^{n-1})^\mathrm{{T}}\), with \(n=M_t^2\).

All experiments are carried out using MATLAB (version R2017b) on a personal computer with 2.60 GHz central processing unit (Intel(R) Core(TM) i7-6700HQ CPU), 8.00 GB memory and Windows operating system (Windows 10). All computations are started from the initial vector \(x^{(0)}=0\), and terminated once the stopping criterion \(\Vert b-Ax^{(k)}\Vert _2 < 10^{-6} \times \Vert b\Vert _2\) is satisfied.

We test the QHSS iteration method in two cases. One is \(M_t=N_t\), and the other is \(\Delta _t=\frac{1}{N_t}\) and \(h_x=h_y=\frac{2}{M_t+1}\) for different \(M_t\) and/or \(N_t\); see [31]. Tables 1, 2, 3 and 4 and Figs. 1, 2, 3 and 4 are the numerical results for the first case \(M_t=N_t\), while Tables 5, 6, 7 and 8 and Figs. 5, 6, 7 and 8 are the numerical results for the second case \(\Delta _t=\frac{1}{N_t}\) and \(h_x=h_y=\frac{2}{M_t+1}\), for different \(M_t\) and/or \(N_t\).

In Tables 1 and 3, we list the experimentally computed optimal values \(\omega _{\mathrm{{exp}}}\) and \(\alpha _{\mathrm{{exp}}}\) of the parameters \(\omega\) and \(\alpha\) that minimize the numbers of iteration steps of the QHSS method. We also give the geometric mean of the smallest and the largest eigenvalue of \(H_{\omega _{\mathrm{{exp}}}}\), i.e., \(\varpi\) in these two tables. From these tables, we see that when \(\mu\), \(\nu\), \(M_t=N_t\) and q are fixed, the distance between the values of \(\alpha _{\mathrm{{exp}}}\) and \(\varpi\) is very small. They are even the same when \(M_t=N_t=2^5\), \(q=10\); \(M_t=N_t=2^5\), \(q=15\); \(M_t=N_t=2^6\), \(q=10\); and \(M_t=N_t=2^7\), \(q=20\) in Table 1.

In Tables 2 and 4, we report the number of iteration steps (“IT”) and the elapsed computing time in seconds (“CPU”) of the QHSS iteration method with the parameters of \(\omega =\omega _{\mathrm{{exp}}}\), \(\alpha =\alpha _{\mathrm{{exp}}}\) and \(\omega =\omega _{\mathrm{{exp}}}\), \(\alpha =\varpi\) given in Tables 1 and 3. We find that the numbers of iteration steps of the QHSS iteration method with the parameters \(\omega =\omega _{\mathrm{{exp}}}\), \(\alpha =\alpha _{\mathrm{{exp}}}\) and \(\omega =\omega _{\mathrm{{exp}}}\), \(\alpha =\varpi\) are very close to each other, so are the CPU times. We can also see that the number of iteration steps of the QHSS method is decreasing with respect to q when \(\mu\), \(\nu\) and \(M_t=N_t\) are fixed except for the case when \(\mu =1.5\), \(\nu =1.3\) and \(M_t=N_t=2^5\).

We depict the curves of \(\rho (L_\omega (\alpha ))\) and \(f(\alpha )\) versus \(\alpha\) when \(M_t=N_t=2^5\), \(\mu =1.5, \nu =1.5, q=5\) in Fig. 1 and when \(M_t=N_t=2^5\), \(\mu =1.5, \nu =1.3, q=20\) in Fig. 3. The curves of the IT and the CPU versus \(\alpha\) of the QHSS iteration method are depicted in Figs.  2 and 4. The parameter \(\omega\) is taken to be \(\omega _{\mathrm{{exp}}}\) in these figures. We find that \(f(\alpha )\) is a tight upper bound of \(\rho (L_\omega (\alpha ))\), and the IT and the CPU reach their minimums around the minimum point \(\varpi\) of \(f(\alpha )\). Thus, the minimum point \(\varpi\) of \(f(\alpha )\) is a good choice of the parameter \(\alpha\).
Table 1

Parameters for QHSS when \(\mu =1.5\), \(\nu =1.5\), \(M_t=N_t\)

\(M_t\)

Index

q

5

10

15

20

25

\(2^5\)

\(\omega _{\mathrm{{exp}}}\)

0.002 0

0.001 5

0.001 1

0.000 5

0.000 5

\(\alpha _{\mathrm{{exp}}}\)

0.050 0

0.051 8

0.056 1

0.060 0

0.068 0

\(\varpi\)

0.046 4

0.051 8

0.056 1

0.059 8

0.063 1

\(2^6\)

\(\omega _{\mathrm{{exp}}}\)

0.001 0

0.002 0

0.001 0

0.000 5

0.001 0

\(\alpha _{\mathrm{{exp}}}\)

0.013 5

0.013 7

0.015 5

0.016 5

0.017 5

\(\varpi\)

0.012 7

0.013 7

0.014 5

0.015 3

0.016 0

\(2^7\)

\(\omega _{\mathrm{{exp}}}\)

0.002 0

0.001 0

0.001 0

0.002 0

0.000 5

\(\alpha _{\mathrm{{exp}}}\)

0.004 1

0.004 2

0.004 2

0.004 0

0.004 3

\(\varpi\)

0.003 6

0.003 7

0.003 9

0.004 0

0.004 1

Table 2

IT and CPU of QHSS when \(\mu =1.5\), \(\nu =1.5\), \(M_t=N_t\)

\(M_t\)

\(\alpha\)

Index

q

5

10

15

20

25

\(2^5\)

\(\alpha _{\mathrm{{exp}}}\)

IT

17

14

12

11

11

CPU

0.172 0

0.203 0

0.182 0

0.172 0

0.187 0

\(\varpi\)

IT

18

14

12

12

12

CPU

0.188 0

0.203 0

0.182 0

0.188 0

0.203 0

\(2^6\)

\(\alpha _{\mathrm{{exp}}}\)

IT

23

21

18

16

15

CPU

3.167 0

3.056 0

2.802 0

2.899 0

2.854 0

\(\varpi\)

IT

24

21

19

17

16

CPU

3.257 0

3.056 0

2.865 0

2.973 0

2.958 0

\(2^7\)

\(\alpha _{\mathrm{{exp}}}\)

IT

27

27

26

25

23

CPU

68.486 0

66.648 0

64.679 0

64.382 0

62.897 0

\(\varpi\)

IT

30

29

27

25

24

CPU

71.851 0

69.132 0

66.772 0

64.382 0

64.100 0

Table 3

Parameters for QHSS when \(\mu =1.5\), \(\nu =1.3\), \(M_t=N_t\)

\(M_t\)

Index

q

5

10

15

20

25

\(2^5\)

\(\omega _{\mathrm{{exp}}}\)

0.002 0

0.001 0

0.001 0

0.000 5

0.009 0

\(\alpha _{\mathrm{{exp}}}\)

0.075 0

0.080 0

0.092 0

0.100 0

0.165 0

\(\varpi\)

0.069 3

0.076 9

0.083 2

0.088 5

0.109 9

\(2^6\)

\(\omega _{\mathrm{{exp}}}\)

0.002 0

0.002 0

0.001 0

0.003 0

0.001 0

\(\alpha _{\mathrm{{exp}}}\)

0.023 0

0.024 5

0.026 7

0.028 1

0.030 0

\(\varpi\)

0.021 3

0.022 9

0.024 3

0.025 6

0.026 8

\(2^7\)

\(\omega _{\mathrm{{exp}}}\)

0.003 0

0.002 0

0.003 0

0.002 0

0.001 0

\(\alpha _{\mathrm{{exp}}}\)

0.007 4

0.007 5

0.007 7

0.007 9

0.008 3

\(\varpi\)

0.006 8

0.007 1

0.007 3

0.007 6

0.007 8

Table 4

IT and CPU of QHSS when \(\mu =1.5\), \(\nu =1.3\), \(M_t=N_t\)

\(M_t\)

\(\alpha\)

Index

q

5

10

15

20

25

\(2^5\)

\(\alpha _{\mathrm{{exp}}}\)

IT

15

13

11

11

14

CPU

0.177 0

0.182 0

0.167 0

0.173 0

0.197 0

\(\varpi\)

IT

16

14

13

13

22

CPU

0.192 0

0.199 0

0.183 0

0.185 0

0.247 0

\(2^6\)

\(\alpha _{\mathrm{{exp}}}\)

IT

20

18

16

15

15

CPU

2.969 0

2.875 0

2.938 0

2.860 0

2.922 0

\(\varpi\)

IT

22

19

18

17

16

CPU

3.156 0

2.922 0

3.141 0

3.047 0

3.032 0

\(2^7\)

\(\alpha _{\mathrm{{exp}}}\)

IT

24

24

23

22

21

CPU

63.194 0

62.397 0

62.054 0

60.772 0

61.210 0

\(\varpi\)

IT

26

25

24

23

23

CPU

65.131 0

63.366 0

63.116 0

61.334 0

63.638 0

Fig. 1

Picture of \(\rho (L_\omega (\alpha ))\) and \(f(\alpha )\) versus \(\alpha\) when \(M_t=N_t=2^5\), \(\mu =1.5, \nu =1.5, q=5\)

Fig. 2

Pictures of IT (left) and CPU (right) of QHSS versus \(\alpha\) when \(M_t=N_t=2^5\), \(\mu =1.5, \nu =1.5, q=5\)

Fig. 3

Picture of \(\rho (L_\omega (\alpha ))\) and \(f(\alpha )\) versus \(\alpha\) when \(M_t=N_t=2^5\), \(\mu =1.5, \nu =1.3, q=20\)

Fig. 4

Pictures of IT (left) and CPU (right) of QHSS versus \(\alpha\) when \(M_t=N_t=2^5\), \(\mu =1.5, \nu =1.3, q=20\)

For the second case \(\Delta _t=\frac{1}{N_t}\) and \(h_x=h_y=\frac{2}{M_t+1}\), for different \(M_t\) and/or \(N_t\) we obtain similar numerical results.

In Tables 5 and 7, we list the experimentally computed optimal values \(\omega _{\mathrm{{exp}}}\) and \(\alpha _{\mathrm{{exp}}}\) of the parameters \(\omega\) and \(\alpha\) that minimize the numbers of iteration steps of the QHSS method, and the geometric mean of the smallest and the largest eigenvalue of \(H_{\omega _{\mathrm{{exp}}}}\), i.e., \(\varpi\). From these tables, we can also see that when \(\mu\), \(\nu\), \(M_t\), \(N_t\) and q are fixed, the distance between the values of \(\alpha _{\mathrm{{exp}}}\) and \(\varpi\) is very small. They are even the same when \(M_t=2^5\), \(N_t=2^4\), \(q=5\) in Table 5; \(M_t=2^5\), \(N_t=2^6\), \(q=5, 10, 15, 20\); \(M_t=2^5\), \(N_t=2^7\), \(q=10, 15, 20\); and \(M_t=2^5\), \(N_t=2^8\), \(q=15, 25\) in Table 7. From Table 5, we find that when \(\mu\)=1.5, \(\nu\)=1.5, \(N_t=2^4\), for a fixed q, \(\alpha _{\mathrm{{exp}}}\) and \(\varpi\) are decreasing with respect to the increase of \(M_t\), i.e., the increase of the size of the coefficient matrix of the linear system \(Ax=b\). From Table 7, we find that when \(\mu =1.5\), \(\nu =1.5\), \(M_t=2^5\), for a fixed q, \(\alpha _{\mathrm{{exp}}}\) and \(\varpi\) are increasing with respect to the increase of \(N_t\).

In Tables 6 and 8, we report the number of iteration steps (“IT”) and the elapsed computing time in seconds (“CPU”) of the QHSS iteration method with the parameters of \(\omega =\omega _{\mathrm{{exp}}}\), \(\alpha =\alpha _{\mathrm{{exp}}}\) and \(\omega =\omega _{\mathrm{{exp}}}\), \(\alpha =\varpi\) given in Tables 5 and 7. We can also find that the numbers of iteration steps of the QHSS iteration method with the parameters \(\omega =\omega _{\mathrm{{exp}}}\), \(\alpha =\alpha _{\mathrm{{exp}}}\) and \(\omega =\omega _{\mathrm{{exp}}}\), \(\alpha =\varpi\) are very close to each other, so are the CPU times. From Table 6, we find that when \(\mu\)=1.5, \(\nu\)=1.5, \(N_t=2^4\), for a fixed q, the IT and the CPU of the QHSS iteration method with the parameters of \(\omega =\omega _{\mathrm{{exp}}}\), \(\alpha =\alpha _{\mathrm{{exp}}}\) or \(\omega =\omega _{\mathrm{{exp}}}\), \(\alpha =\varpi\) are increasing with respect to the increase of \(M_t\), i.e., the increase of the size of the coefficient matrix of the linear system \(Ax=b\). From Table 8, we find that when \(\mu\)=1.5, \(\nu\)=1.5, \(M_t=2^5\), for a fixed q, the IT of the QHSS iteration method with the parameters of \(\omega =\omega _{\mathrm{{exp}}}\), \(\alpha =\alpha _{\mathrm{{exp}}}\) or \(\omega =\omega _{\mathrm{{exp}}}\), \(\alpha =\varpi\) is decreasing with respect to the increase of \(N_t\).

We depict the curves of \(\rho (L_\omega (\alpha ))\) and \(f(\alpha )\) versus \(\alpha\) when \(M_t=2^5\), \(N_t=2^4\), \(\mu =1.5, \nu =1.5, q=5\) in Fig. 5 and when \(M_t=2^5\), \(N_t=2^6\), \(\mu =1.5, \nu =1.5, q=20\) in Fig. 7. The curves of the IT and the CPU versus \(\alpha\) of the QHSS iteration method are depicted in Figs. 6 and 8. The parameter \(\omega\) is taken to be \(\omega _{\mathrm{{exp}}}\) in these figures. We find that \(f(\alpha )\) is a tight upper bound of \(\rho (L_\omega (\alpha ))\), and the IT and the CPU reach their minimums around the minimum point \(\varpi\) of \(f(\alpha )\).
Table 5

Parameters for QHSS when \(\mu =1.5\), \(\nu =1.5\), \(N_t=2^4\)

\(M_t\)

Index

q

5

10

15

20

25

\(2^5\)

\(\omega _{\mathrm{{exp}}}\)

0.002 0

0.001 0

0.001 0

0.002 0

0.001 0

\(\alpha _{\mathrm{{exp}}}\)

0.038 9

0.049 0

0.055 0

0.056 0

0.082 2

\(\varpi\)

0.038 9

0.045 1

0.050 0

0.054 1

0.057 7

\(2^6\)

\(\omega _{\mathrm{{exp}}}\)

0.001 0

0.002 0

0.002 0

0.001 0

0.001 0

\(\alpha _{\mathrm{{exp}}}\)

0.009 4

0.011 1

0.012 6

0.013 9

0.014 6

\(\varpi\)

0.008 5

0.009 9

0.011 0

0.012 0

0.012 9

\(2^7\)

\(\omega _{\mathrm{{exp}}}\)

0.001 0

0.002 0

0.001 0

0.000 5

0.003 0

\(\alpha _{\mathrm{{exp}}}\)

0.002 3

0.002 5

0.002 9

0.003 3

0.003 8

\(\varpi\)

0.001 8

0.002 1

0.002 4

0.002 6

0.002 8

Table 6

IT and CPU of QHSS when \(\mu =1.5\), \(\nu =1.5\), \(N_t=2^4\)

\(M_t\)

\(\alpha\)

Index

q

5

10

15

20

25

\(2^5\)

\(\alpha _{\mathrm{{exp}}}\)

IT

19

14

12

12

10

CPU

0.056 0

0.049 0

0.046 0

0.047 0

0.046 0

\(\varpi\)

IT

19

15

13

13

13

CPU

0.056 0

0.058 0

0.062 0

0.063 0

0.063 0

\(2^6\)

\(\alpha _{\mathrm{{exp}}}\)

IT

30

23

20

18

17

CPU

1.016 0

0.938 0

0.984 0

0.906 0

0.874 0

\(\varpi\)

IT

31

25

23

21

20

CPU

1.062 0

0.984 0

1.078 0

0.984 0

0.937 0

\(2^7\)

\(\alpha _{\mathrm{{exp}}}\)

IT

47

37

32

28

25

CPU

23.713 0

21.073 0

19.815 0

18.901 0

18.370 0

\(\varpi\)

IT

52

42

38

35

33

CPU

26.572 0

22.198 0

21.338 0

21.792 0

21.386 0

Table 7

Parameters for QHSS when \(\mu =1.5\), \(\nu =1.5\), \(M_t=2^5\)

\(N_t\)

Index

q

5

10

15

20

25

\(2^6\)

\(\omega _{\mathrm{{exp}}}\)

0.003 0

0.001 0

0.000 5

0.001 0

0.002 0

\(\alpha _{\mathrm{{exp}}}\)

0.059 3

0.063 6

0.067 2

0.070 3

0.093 0

\(\varpi\)

0.059 3

0.063 6

0.067 2

0.070 3

0.073 1

\(2^7\)

\(\omega _{\mathrm{{exp}}}\)

0.001 0

0.000 5

0.002 0

0.001 0

0.003 0

\(\alpha _{\mathrm{{exp}}}\)

0.083 0

0.084 1

0.086 9

0.089 3

0.097 0

\(\varpi\)

0.081 0

0.084 1

0.086 9

0.089 3

0.091 6

\(2^8\)

\(\omega _{\mathrm{{exp}}}\)

0.002 0

0.001 0

0.003 0

0.000 5

0.001 0

\(\alpha _{\mathrm{{exp}}}\)

0.123 0

0.122 0

0.121 8

0.130 0

0.125 3

\(\varpi\)

0.117 6

0.119 9

0.121 8

0.123 6

0.125 3

Table 8

IT and CPU of QHSS when \(\mu =1.5\), \(\nu =1.5\), \(M_t=2^5\)

\(N_t\)

\(\alpha\)

Index

q

5

10

15

20

25

\(2^6\)

\(\alpha _{\mathrm{{exp}}}\)

IT

15

13

11

10

9

CPU

0.063 0

0.047 0

0.046 0

0.047 0

0.046 0

\(\varpi\)

IT

15

13

11

10

12

CPU

0.063 0

0.047 0

0.046 0

0.047 0

0.062 0

\(2^7\)

\(\alpha _{\mathrm{{exp}}}\)

IT

11

11

10

9

9

CPU

0.046 0

0.044 0

0.041 0

0.037 0

0.042 0

\(\varpi\)

IT

12

11

10

9

10

CPU

0.047 0

0.044 0

0.041 0

0.037 0

0.045 0

\(2^8\)

\(\alpha _{\mathrm{{exp}}}\)

IT

8

8

8

7

8

CPU

0.042 0

0.045 0

0.046 0

0.046 0

0.047 0

\(\varpi\)

IT

9

9

8

8

8

CPU

0.044 0

0.047 0

0.046 0

0.047 0

0.047 0

Fig. 5

Picture of \(\rho (L_\omega (\alpha ))\) and \(f(\alpha )\) versus \(\alpha\) when \(M_t=2^5\), \(N_t=2^4\), \(\mu =1.5, \nu =1.5, q=5\)

Fig. 6

Pictures of IT (left) and CPU (right) of QHSS versus \(\alpha\) when \(M_t=2^5\), \(N_t=2^4\), \(\mu =1.5, \nu =1.5, q=5\)

Fig. 7

Picture of \(\rho (L_\omega (\alpha ))\) and \(f(\alpha )\) versus \(\alpha\) when \(M_t=2^5\), \(N_t=2^6\), \(\mu =1.5, \nu =1.5, q=20\)

Fig. 8

Pictures of IT (left) and CPU (right) of QHSS versus \(\alpha\) when \(M_t=2^5\), \(N_t=2^6\), \(\mu =1.5, \nu =1.5, q=20\)

5 Remarks and Conclusions

For solving the large sparse non-Hermitian positive definite linear systems, the QHSS iteration method [7] is a good choice, especially for the coefficient matrix with a nearly singular Hermitian part H or a strongly dominant skew-Hermitian part S. The parameters \(\alpha\) and \(\omega\) affect the asymptotic convergence rate of the QHSS iteration method significantly. To find a good choice of these parameters such that the convergence rate of the QHSS iteration method could be relatively fast is still a challenging problem. In [7], Bai derived an upper bound about the spectral radius of the QHSS iteration matrix which can also bound the contraction factor of the QHSS iteration method, and gave the conditions for guaranteeing that the upper bound is strictly less than one.

In this paper, we give the minimum point of this upper bound under the conditions which guarantee that the upper bound is strictly less than one. From the experimental results, we see that this provides a good choice for the parameter \(\alpha\) involved in the QHSS iteration method. And we find that to make the minimum of the upper bound as small as possible, we should let the parameter \(\omega\) and the Euclidean condition number of the matrix \(H_\omega\) be small. To obtain the value of the minimum point \(\varpi\), we should know the smallest and the largest eigenvalue of the matrix \(H_\omega\) or, at least, their sharp bounds, which are not easily computable in practical applications. Besides finding the minimum point for the upper bound of the spectral radius of the QHSS iteration matrix, to look for some other ways, like those done in [9, 10, 18, 28], to obtain good and easily computable values for the parameters \(\alpha\) and \(\omega\) is still a valuable topic to be studied in the future. And deriving tighter bounds for the convergence rate or the contraction factor of the QHSS iteration method should be also a valuable topic.

Notes

Acknowledgements

The author is thankful to the referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript of this paper. Supported by the National Natural Science Foundation (No. 11671393), China.

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Copyright information

© Shanghai University 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

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