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# Computational Methods in Applied Mathematics

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Volume 18, Issue 2

# Spectral Analysis, Properties and Nonsingular Preconditioners for Singular Saddle Point Problems

Na Huang
• Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, P. R. China
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• Other articles by this author:
/ Chang-Feng Ma
/ Jun Zou
• Corresponding author
• Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, P. R. China
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• Other articles by this author:
Published Online: 2017-05-31 | DOI: https://doi.org/10.1515/cmam-2017-0006

## Abstract

We first derive some explicit bounds on the spectra of generalized non-symmetric singular or nonsingular saddle point matrices. Then we propose two new nonsingular preconditioners for solving generalized singular saddle point problems, and show that GMRES determines a solution without breakdown when applied to the resulting preconditioned systems with any initial guess. Furthermore, the detailed spectral properties of the preconditioned systems are analyzed. The nonsingular preconditioners are also applied to solve the singular finite element saddle point systems arising from the discretization of the Stokes problems to test their performance.

MSC 2010: 65F10; 65F50

## 1 Introduction

In this work we shall mainly study the spectral behavior and preconditioners of the generalized singular or nonsingular saddle point problems of the form

$\left(\begin{array}{cc}\hfill A\hfill & \hfill {B}^{T}\hfill \\ \hfill -B\hfill & \hfill C\hfill \end{array}\right)\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{c}\hfill f\hfill \\ \hfill -g\hfill \end{array}\right),$(1.1)

where $A\in {ℝ}^{n×n}$ is a symmetric positive definite matrix, $C\in {ℝ}^{m×m}$ is a symmetric positive semi-definite matrix, and B is a general matrix in ${ℝ}^{m×n}$, with $m\le n$ (possibly $m\ll n$); $f\in {ℝ}^{n}$ and $g\in {ℝ}^{m}$ are two given vectors.

Systems of the form (1.1) arise from a variety of scientific and engineering applications, such as computational fluid dynamics, optimal control problems, constrained optimizations, weighted least-squares formulations, mixed or hybrid finite element approximations of partial differential equations, electronic networks and computer graphics, boundary element discretization based on domain decomposition, and so on; see, e.g., [14, 11, 29, 20, 23, 21, 10, 22, 6] and the references therein.

$\mathcal{𝒜}=\left(\begin{array}{cc}\hfill A\hfill & \hfill {B}^{T}\hfill \\ \hfill -B\hfill & \hfill C\hfill \end{array}\right)$(1.2)

in (1.1), there are already quite many investigations available in literature; see [7, 8, 32, 34, 2, 3, 24]. Interesting estimates were established in [7] on spectral bounds of the complex eigenvalues of $\mathcal{𝒜}$, and further improved in [32] from the previous lower bound $\left({\rho }_{1}+{\mu }_{1}\right)/2$ to the lower bound $\mathrm{max}\left\{\left({\rho }_{1}+{\mu }_{1}\right)/2,{\mu }_{1}-\sqrt{{\delta }_{m}}\right\}$, where ${\rho }_{1}$ and ${\mu }_{1}$ are the minimum eigenvalues of A and C, respectively, and ${\delta }_{m}$ is the maximum eigenvalue of $B{B}^{T}$. But the matrix C is often singular or even vanishes in many applications, so we have ${\mu }_{1}=0$ and the improved estimates of [32] reduce to the original ones in [7]. The first results of this work are to improve the estimates in [32] for the important cases when matrix C is singular or zero. With the help of those improved results, we shall derive some explicit and sharp bounds on the spectra of the generalized non-symmetric singular saddle point matrices of the form (1.1). Then we propose new nonsingular preconditioners for solving the generalized singular saddle point problems, and establish the spectral bounds of the preconditioned systems.

The arrangement of this work is in the following order. We first present some improved estimates of the spectra for the generalized singular or nonsingular saddle point matrix $\mathcal{𝒜}$ in Section 2, then study more specific spectral properties of singular matrices $\mathcal{𝒜}$ in Section 3. We will then propose a new nonsingular preconditioner for the singular saddle point system in (1.1) in Section 4, and derive the spectral bounds of the preconditioned systems in Section 5. Finally, in Section 6, the nonsingular preconditioners are applied to solve the singular finite element saddle point systems arising from the discretization of the Stokes problems to demonstrate their effectiveness and efficiencies.

## Notation.

We introduce some notation that will be used in the subsequent analysis. For any matrix $H\in {ℝ}^{l×l}$, we shall often write its inverse, transpose, minimum and maximum eigenvalues, rank and null space as ${H}^{-1}$, ${H}^{T}$, ${\lambda }_{\mathrm{min}}\left[H\right]$, ${\lambda }_{\mathrm{max}}\left[H\right]$, $\mathrm{rank}\left[H\right]$ and $\mathrm{null}\left[H\right]$, respectively. Moreover, we write ${H}_{1}\sim {H}_{2}$ if ${H}_{1}$ is similar to ${H}_{2}$ (i.e., there exists an invertible matrix $T\in {ℝ}^{l×l}$ such that ${H}_{1}=T{H}_{2}{T}^{-1}$), and $H>0$ (resp. $H\ge 0$) if H is symmetric positive definite (resp. positive semi-definite). In addition, we use ${I}_{l}$ to denote the identity matrix of order l, and $\mathrm{Re}\left(\lambda \right)$ and $\mathrm{Im}\left(\lambda \right)$ the real and imaginary parts of any $\lambda \in ℂ$. And we shall denote the eigenvalues of matrices A, C, $B{B}^{T}$, and S respectively by ${\rho }_{1}\le {\rho }_{2}\le \mathrm{\cdots }\le {\rho }_{n}$, ${\mu }_{1}\le {\mu }_{2}\le \mathrm{\cdots }\le {\mu }_{m}$, ${\delta }_{1}\le {\delta }_{2}\le \mathrm{\cdots }\le {\delta }_{m}$, and ${\zeta }_{1}\le {\zeta }_{2}\le \mathrm{\cdots }\le {\zeta }_{m}$, where $S=C+B{A}^{-1}{B}^{T}$ is the Schur complement matrix.

## 2 Estimates of Complex Eigenvalues of Singular or Nonsingular Saddle Point Matrices

In this section, we make an effort to improve the following estimates of eigenvalues of matrix $\mathcal{𝒜}$ in [32], which will be important to our subsequent studies of the spectral distributions of matrix $\mathcal{𝒜}$ and its preconditioned system.

#### Theorem 2.1 ([32]).

Let $\mathcal{A}$ be the saddle point matrix of form (1.2), where A is symmetric positive definite and C is symmetric positive semi-definite. Then any complex eigenvalue λ of $\mathcal{A}$ with $\mathrm{Im}\mathit{}\mathrm{\left(}\lambda \mathrm{\right)}\mathrm{\ne }\mathrm{0}$ meets the following estimates:

$\mathrm{max}\left\{\frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right),{\mu }_{1}-\sqrt{{\delta }_{m}}\right\}\le \mathrm{Re}\left(\lambda \right)\le \mathrm{min}\left\{\frac{1}{2}\left({\rho }_{n}+{\mu }_{m}\right),{\mu }_{m}+\sqrt{{\delta }_{m}}\right\},$(2.1)(2.2)

It is often encountered in many applications that the matrix C is singular. In this case, Theorem 2.1 implies that the lower bound of the real part of any complex eigenvalue of $\mathcal{𝒜}$ reduces to $\left({\rho }_{1}+{\mu }_{1}\right)/2$, which deteriorates to the same result in [7]. In the next theorem we present some estimates that improve the ones in Theorem 2.1 for the important case that the matrix C is singular. There are no existing analytical tools to apply for this singular case. Instead, we choose to consider the eigensystem of the saddle point equation (1.1) directly, from where we solve the variable x in terms of the variable y and substitute into the second equation of the eigensystem. Then we separate the real and imaginary part of the quadratic form of the resultant equation, and achieve the desired estimates by using the important relation between the real and imaginary parts.

#### Theorem 2.2.

Let $\mathcal{A}$ be the saddle point matrix of form (1.2), where A is symmetric positive definite and C is symmetric positive semi-definite. Then any complex eigenvalue λ of $\mathcal{A}$ with $\mathrm{Im}\mathit{}\mathrm{\left(}\lambda \mathrm{\right)}\mathrm{\ne }\mathrm{0}$ meets the following estimates:

$\mathrm{max}\left\{\frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right),{\rho }_{1}-\sqrt{{\delta }_{m}}\right\}\le \mathrm{Re}\left(\lambda \right)\le \mathrm{min}\left\{\frac{1}{2}\left({\rho }_{n}+{\mu }_{m}\right),{\rho }_{n}+\sqrt{{\delta }_{m}}\right\},$

#### Proof.

Let $\lambda =a+b\mathrm{i}$ with $a\in ℝ$, $0\ne b\in ℝ$ and $\mathrm{i}=\sqrt{-1}$, and let $z={\left({x}^{T},{y}^{T}\right)}^{T}$ be an eigenvector corresponding to λ. Then $\mathcal{𝒜}z=\lambda z$, or in other words,

$Ax+{B}^{T}y=\lambda x,-Bx+Cy=\lambda y.$(2.3)

Clearly, $y\notin \mathrm{null}\left[{B}^{T}\right]$, otherwise (2.3) shows $Ax=\lambda x$. Then λ is an eigenvalue of A if $x\ne 0$. If $x=0$, then $y\ne 0$, and the second equality in (2.3) leads to $Cy=\lambda y$. Therefore $\lambda \in ℝ$ if $y\in \mathrm{null}\left[{B}^{T}\right]$. This contradicts the fact $b\ne 0$. By the first equality in (2.3), we have

$x={\left(\lambda I-A\right)}^{-1}{B}^{T}y.$

Together with the second equality in (2.3), this implies

$-B{\left(\lambda I-A\right)}^{-1}{B}^{T}y+Cy=\lambda y.$(2.4)

Let $A=PD{P}^{T}$ be the eigenvalue decomposition of A, where $P\in {ℝ}^{n×n}$ is an orthogonal matrix and $D=\mathrm{diag}\left({\rho }_{1},{\rho }_{2},\mathrm{\dots },{\rho }_{n}\right)$. Then (2.4) results in

$-BP{\left(\lambda I-D\right)}^{-1}{P}^{T}{B}^{T}y+Cy=\lambda y.$

Multiplying the above equality from the left with ${y}^{*}$ leads to

$-{y}^{*}BP{\left(\lambda I-D\right)}^{-1}{P}^{T}{B}^{T}y+{y}^{*}Cy=\lambda {y}^{*}y.$(2.5)

For any $1\le j\le n$, we have

${\left(\lambda -{\rho }_{j}\right)}^{-1}=\frac{1}{a+b\mathrm{i}-{\rho }_{j}}=\frac{a-{\rho }_{j}-b\mathrm{i}}{{\left(a-{\rho }_{j}\right)}^{2}+{b}^{2}}.$

${\left(\lambda I-D\right)}^{-1}=\mathrm{diag}\left(\frac{a-{\rho }_{1}-b\mathrm{i}}{{\left(a-{\rho }_{1}\right)}^{2}+{b}^{2}},\mathrm{\dots },\frac{a-{\rho }_{n}-b\mathrm{i}}{{\left(a-{\rho }_{n}\right)}^{2}+{b}^{2}}\right).$

The combination with (2.5) leads to

$-{y}^{*}BP\mathrm{diag}\left(\frac{a-{\rho }_{1}}{{\left(a-{\rho }_{1}\right)}^{2}+{b}^{2}},\mathrm{\dots },\frac{a-{\rho }_{n}}{{\left(a-{\rho }_{n}\right)}^{2}+{b}^{2}}\right){P}^{T}{B}^{T}y+{y}^{*}Cy=a{y}^{*}y$

and

${y}^{*}BP\mathrm{diag}\left(\frac{1}{{\left(a-{\rho }_{1}\right)}^{2}+{b}^{2}},\mathrm{\dots },\frac{1}{{\left(a-{\rho }_{n}\right)}^{2}+{b}^{2}}\right){P}^{T}{B}^{T}y={y}^{*}y.$(2.6)

Then we have

$a{y}^{*}y\le \left({\rho }_{n}-a\right){y}^{*}BP\mathrm{diag}\left(\frac{1}{{\left(a-{\rho }_{1}\right)}^{2}+{b}^{2}},\mathrm{\dots },\frac{1}{{\left(a-{\rho }_{n}\right)}^{2}+{b}^{2}}\right){P}^{T}{B}^{T}y+{y}^{*}Cy$$=\left({\rho }_{n}-a\right){y}^{*}y+{y}^{*}Cy,$

$a\le \frac{1}{2}\left({\rho }_{n}+\frac{{y}^{*}Cy}{{y}^{*}y}\right)\le \frac{1}{2}\left({\rho }_{n}+{\mu }_{m}\right).$(2.7)

Similarly, we deduce

$a\ge \frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right).$(2.8)

On the other hand, from (2.6), we have

$\underset{1\le j\le n}{\mathrm{min}}\left\{{\left(a-{\rho }_{j}\right)}^{2}\right\}<\frac{{y}^{*}B{B}^{T}y}{{y}^{*}y}\le {\delta }_{m},$

which implies

${\rho }_{1}-\sqrt{{\delta }_{m}}

The combination with (2.7)–(2.8) concludes the proof of the first result.

In the subsequence, we show the upper bounds for $|\mathrm{Im}\left(\lambda \right)|$. It follows from (2.6) that

$\underset{1\le j\le n}{\mathrm{min}}\left\{{\left(a-{\rho }_{j}\right)}^{2}\right\}+{b}^{2}\le \frac{{y}^{*}B{B}^{T}y}{{y}^{*}y}\le {\delta }_{m},$

which implies

$|b|\le \sqrt{{\delta }_{m}-\underset{1\le j\le n}{\mathrm{min}}\left\{{\left(a-{\rho }_{j}\right)}^{2}\right\}}\le \sqrt{{\delta }_{m}}.$(2.9)

If $\frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right)>{\rho }_{n}$, then, for any $1\le j\le n$, the first result implies

$a-{\rho }_{j}\ge \frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right)-{\rho }_{j}\ge \frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right)-{\rho }_{n}>0.$

Then

$\underset{1\le j\le n}{\mathrm{min}}{\left(a-{\rho }_{j}\right)}^{2}\ge {\left(\frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right)-{\rho }_{n}\right)}^{2}=\frac{{\left({\rho }_{1}+{\mu }_{1}-2{\rho }_{n}\right)}^{2}}{4}.$

Along with (2.9), this shows that

$|b|\le \sqrt{{\delta }_{m}-\frac{{\left({\rho }_{1}+{\mu }_{1}-2{\rho }_{n}\right)}^{2}}{4}}.$(2.10)

If $\frac{1}{2}\left({\rho }_{n}+{\mu }_{m}\right)<{\rho }_{1}$, then, for any $1\le j\le n$, the first result implies

$a-{\rho }_{j}\le \frac{1}{2}\left({\rho }_{n}+{\mu }_{m}\right)-{\rho }_{j}\le \frac{1}{2}\left({\rho }_{n}+{\mu }_{m}\right)-{\rho }_{1}<0.$

Then

$\underset{1\le j\le n}{\mathrm{min}}\left\{{\left(a-{\rho }_{j}\right)}^{2}\right\}\ge {\left(\frac{1}{2}\left({\rho }_{n}+{\mu }_{m}\right)-{\rho }_{1}\right)}^{2}=\frac{{\left({\rho }_{n}+{\mu }_{m}-2{\rho }_{1}\right)}^{2}}{4}.$

Along with (2.9), this shows that

$|b|\le \sqrt{{\delta }_{m}-\frac{{\left({\rho }_{n}+{\mu }_{m}-2{\rho }_{1}\right)}^{2}}{4}}.$(2.11)

The combination of (2.9), (2.10), and (2.11) concludes the proof of the second result. ∎

Since A is symmetric and positive definite, we have ${\rho }_{1}>0$. Theorem 2.2 then shows that our new estimates improve the previous estimates in Theorem 2.1 for the cases that C is singular, provided ${\rho }_{1}-\sqrt{{\delta }_{m}}>0$. The combination of Theorem 2.2 and Theorem 2.1 leads to the following estimates.

#### Theorem 2.3.

Under the same settings and conditions as in Theorem 2.2, any complex eigenvalue λ of $\mathcal{A}$ with $\mathrm{Im}\mathit{}\mathrm{\left(}\lambda \mathrm{\right)}\mathrm{\ne }\mathrm{0}$ meets the following estimates:

$\mathrm{max}\left\{\frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right),{\rho }_{1}-\sqrt{{\delta }_{m}},{\mu }_{1}-\sqrt{{\delta }_{m}}\right\}\le \mathrm{Re}\left(\lambda \right)\le \mathrm{min}\left\{\frac{1}{2}\left({\rho }_{n}+{\mu }_{m}\right),{\rho }_{n}+\sqrt{{\delta }_{m}},{\mu }_{m}+\sqrt{{\delta }_{m}}\right\},$(2.12)(2.13)

#### Proof.

The estimates of the real part of eigenvalues of $\mathcal{𝒜}$ follow directly from Theorems 2.1 and 2.2. The upper bound of $|\mathrm{Im}\left(\lambda \right)|$ follows in four different cases.

Case I: $\mathrm{max}\left\{{\rho }_{n},{\mu }_{m}\right\}<\frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right)$. Then ${\rho }_{n}<\frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right)$ and ${\mu }_{m}<\frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right)$. It follows from Theorem 2.2 and Theorem 2.1 that

$|\mathrm{Im}\left(\lambda \right)|\le \sqrt{{\delta }_{m}-\frac{{\left({\rho }_{1}+{\mu }_{1}-2{\rho }_{n}\right)}^{2}}{4}}\mathit{ }\text{and}\mathit{ }|\mathrm{Im}\left(\lambda \right)|\le \sqrt{{\delta }_{m}-\frac{{\left({\rho }_{1}+{\mu }_{1}-2{\mu }_{m}\right)}^{2}}{4}}.$

This shows that

$|\mathrm{Im}\left(\lambda \right)|\le \mathrm{min}\left\{\sqrt{{\delta }_{m}-\frac{{\left({\rho }_{1}+{\mu }_{1}-2{\rho }_{n}\right)}^{2}}{4}},\sqrt{{\delta }_{m}-\frac{{\left({\rho }_{1}+{\mu }_{1}-2{\mu }_{m}\right)}^{2}}{4}}\right\}$$=\sqrt{{\delta }_{m}-\frac{\mathrm{max}\left\{{\left({\rho }_{1}+{\mu }_{1}-2{\rho }_{n}\right)}^{2},{\left({\rho }_{1}+{\mu }_{1}-2{\mu }_{m}\right)}^{2}\right\}}{4}}$$=\sqrt{{\delta }_{m}-\frac{{\left({\rho }_{1}+{\mu }_{1}-2\mathrm{min}\left\{{\rho }_{n},{\mu }_{m}\right\}\right)}^{2}}{4}}.$

Case II: $\mathrm{min}\left\{{\rho }_{n},{\mu }_{m}\right\}<\frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right)\le \mathrm{max}\left\{{\rho }_{n},{\mu }_{m}\right\}$. If ${\rho }_{n}>{\mu }_{m}$, then ${\mu }_{m}<\frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right)$. Theorem 2.1 implies

$|\mathrm{Im}\left(\lambda \right)|\le \sqrt{{\delta }_{m}-\frac{{\left({\rho }_{1}+{\mu }_{1}-2{\mu }_{m}\right)}^{2}}{4}}=\sqrt{{\delta }_{m}-\frac{{\left({\rho }_{1}+{\mu }_{1}-2\mathrm{min}\left\{{\rho }_{n},{\mu }_{m}\right\}\right)}^{2}}{4}}.$

If ${\rho }_{n}\le {\mu }_{m}$, then ${\rho }_{n}<\frac{1}{2}\left({\rho }_{1}+{\mu }_{1}\right)$. Theorem 2.2 implies

$|\mathrm{Im}\left(\lambda \right)|\le \sqrt{{\delta }_{m}-\frac{{\left({\rho }_{1}+{\mu }_{1}-2{\rho }_{n}\right)}^{2}}{4}}=\sqrt{{\delta }_{m}-\frac{{\left({\rho }_{1}+{\mu }_{1}-2\mathrm{min}\left\{{\rho }_{n},{\mu }_{m}\right\}\right)}^{2}}{4}}.$

The combination of Case I and Case II leads to

Case III: $\mathrm{min}\left\{{\rho }_{1},{\mu }_{1}\right\}>\frac{1}{2}\left({\rho }_{n}+{\mu }_{m}\right)$. Then ${\rho }_{1}>\frac{1}{2}\left({\rho }_{n}+{\mu }_{m}\right)$ and ${\mu }_{1}>\frac{1}{2}\left({\rho }_{n}+{\mu }_{m}\right)$. Theorem 2.2 and Theorem 2.1 imply

$|\mathrm{Im}\left(\lambda \right)|\le \sqrt{{\delta }_{m}-\frac{{\left({\rho }_{n}+{\mu }_{m}-2{\rho }_{1}\right)}^{2}}{4}}\mathit{ }\text{and}\mathit{ }|\mathrm{Im}\left(\lambda \right)|\le \sqrt{{\delta }_{m}-\frac{{\left({\rho }_{n}+{\mu }_{m}-2{\mu }_{1}\right)}^{2}}{4}}.$

This is rewritten as

$|\mathrm{Im}\left(\lambda \right)|\le \sqrt{{\delta }_{m}-\frac{{\left({\rho }_{n}+{\mu }_{m}-2\mathrm{max}\left\{{\rho }_{1},{\mu }_{1}\right\}\right)}^{2}}{4}}.$

Case VI: $\mathrm{max}\left\{{\rho }_{1},{\mu }_{1}\right\}>\frac{1}{2}\left({\rho }_{n}+{\mu }_{m}\right)\ge \mathrm{min}\left\{{\rho }_{1},{\mu }_{1}\right\}$. By an analogous proof, we can prove the same result as that of Case III. The combination with Case III leads to

Collecting the results of the four cases above, we complete the proof. ∎

#### Remark 2.4.

In many cases, the estimates in (2.12) and (2.13) of Theorem 2.3 are sharper than the estimates in (2.1) and (2.2) of Theorem 2.1, respectively. Consider the following example of a non-symmetric saddle point matrix:

$\mathcal{𝒜}=\left(\begin{array}{ccccc}\hfill 6\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$

In this example, we have $m=2$ and $n=3$, and we can compute that ${\rho }_{1}={\rho }_{n}=6$, ${\mu }_{1}=0$, ${\mu }_{m}=5$ and ${\delta }_{m}=1$. Using MATLAB, we can find all the eigenvalues of $\mathcal{𝒜}$ as follows:

$5.5000+0.8660\mathrm{i},5.5000-0.8660\mathrm{i},0.1716,5.8284,6.0000.$

But the bounds (2.1)–(2.2) give the following estimates of an eigenvalue λ of $\mathcal{𝒜}$ with $\mathrm{Im}\left(\lambda \right)\ne 0$:

$3\le \mathrm{Re}\left(\lambda \right)\le 5.5,|\mathrm{Im}\left(\lambda \right)|\le 1,$

while the bounds (2.12)–(2.13) provide indeed clearly sharper estimates:

$5\le \mathrm{Re}\left(\lambda \right)\le 5.5,|\mathrm{Im}\left(\lambda \right)|\le 0.8660.$

## 3 Spectral Behavior of Singular Saddle Point Matrices

In recent years, several studies have been carried out to analyze the spectral properties of the saddle point matrix $\mathcal{𝒜}$ of form (1.2) when $\mathcal{𝒜}$ is nonsingular; see, e.g., [7, 8, 1]. But it is often encountered in many applications that matrix $\mathcal{𝒜}$ is singular. To the best of our knowledge, not much effort has been made yet in literature to study this important singular case. This is our major focus of this section. We shall first present some estimates of the rank of the singular saddle point matrix $\mathcal{𝒜}$, which help to determine the multiplicity of the eigenvalue 0. Then we will establish some important estimates of both real and complex parts of all eigenvalues of the singular matrix $\mathcal{𝒜}$. Those estimates are the essential tools for our analysis in Section 4 on the spectral behavior of the preconditioned systems for singular saddle point matrices by nonsingular preconditioners.

#### Theorem 3.1.

Let $\mathcal{A}$ be a saddle point matrix of form (1.2), where A is symmetric positive definite, C is symmetric positive semi-definite, and $\mathrm{rank}\mathit{}\mathrm{\left[}B\mathrm{\right]}\mathrm{=}p$ and $\mathrm{rank}\mathit{}\mathrm{\left[}C\mathrm{\right]}\mathrm{=}r$. Then the rank of matrix $\mathcal{A}$ can be estimated by

$n+\mathrm{max}\left\{r,p\right\}\le \mathrm{rank}\left[\mathcal{𝒜}\right]\le n+\mathrm{min}\left\{m,p+r\right\}.$

#### Proof.

A direct calculation verifies the identity

$\mathcal{𝒜}=\left(\begin{array}{cc}\hfill A\hfill & \hfill {B}^{T}\hfill \\ \hfill -B\hfill & \hfill C\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill 0\hfill \\ \hfill -B{A}^{-1}\hfill & \hfill {I}_{m}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill A\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill S\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill {A}^{-1}{B}^{T}\hfill \\ \hfill 0\hfill & \hfill {I}_{m}\hfill \end{array}\right).$

This shows that

$\mathrm{rank}\left[\mathcal{𝒜}\right]=\mathrm{rank}\left[\left(\begin{array}{cc}\hfill A\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill S\hfill \end{array}\right)\right]=\mathrm{rank}\left[A\right]+\mathrm{rank}\left[S\right]=n+\mathrm{rank}\left[S\right].$(3.1)

Since $S\ge C$ and $S\ge B{A}^{-1}{B}^{T}$, we have

$\mathrm{rank}\left[S\right]\ge \mathrm{max}\left\{\mathrm{rank}\left[C\right],\mathrm{rank}\left[B{A}^{-1}{B}^{T}\right]\right\}=\mathrm{max}\left\{r,p\right\}.$

Since A is symmetric and positive definite, it follows

$\mathrm{rank}\left[S\right]=\mathrm{rank}\left[C+B{A}^{-1}{B}^{T}\right]\le \mathrm{rank}\left[C\right]+\mathrm{rank}\left[B{A}^{-1}{B}^{T}\right]=r+p.$

This and (3.1) complete the proof. ∎

By Theorem 3.1, we know that $\mathrm{rank}\left[\mathcal{𝒜}\right]=n+m$ if $\mathrm{rank}\left[C\right]=m$ or $\mathrm{rank}\left[B\right]=m$. This implies immediately that $\mathrm{rank}\left[C\right], ${\mu }_{1}=0$ and $\mathrm{rank}\left[B\right] if $\mathcal{𝒜}$ is singular. This leads to the following estimate of the multiplicity of the eigenvalue 0 of $\mathcal{𝒜}$.

#### Theorem 3.2.

Under the same settings for the matrices $\mathcal{A}$, A and C as in Theorem 3.1, except that $\mathcal{A}$ is singular, the multiplicity $\mathrm{\ell }$ of eigenvalue 0 of $\mathcal{A}$ can be estimated by

$\mathrm{max}\left\{1,m-p-r\right\}\le \mathrm{\ell }\le m-\mathrm{max}\left\{r,p\right\}.$

For the real eigenvalues of the singular saddle point matrix $\mathcal{𝒜}$, we can get the following results by a similar argument as that used in [1, Proposition 2.2] and [31, Theorem 2.1].

#### Theorem 3.3.

Let $\mathcal{A}$, A and C be matrices as in Theorem 3.1, and S the Schur complement $S\mathrm{=}C\mathrm{+}B\mathit{}{A}^{\mathrm{-}\mathrm{1}}\mathit{}{B}^{T}$, with $\mathrm{0}\mathrm{<}q\mathrm{=}\mathrm{rank}\mathit{}\mathrm{\left[}S\mathrm{\right]}\mathrm{<}m$. Then any non-zero real eigenvalue λ of $\mathcal{A}$ can be estimated by

$\mathrm{min}\left\{{\rho }_{1},{\zeta }_{m-q+1}\right\}\le \lambda \le \mathrm{max}\left\{{\rho }_{n},{\mu }_{m}\right\}.$

The following results are direct consequences of Theorems 3.2 and 3.3.

#### Corollary 3.4.

Let $\mathcal{A}$ be the singular saddle point matrix of form (1.2), where A is symmetric positive definite and C is symmetric positive semi-definite. Assume that $\mathrm{rank}\mathit{}\mathrm{\left[}B\mathrm{\right]}\mathrm{=}p$, $\mathrm{rank}\mathit{}\mathrm{\left[}C\mathrm{\right]}\mathrm{=}r$ and $\mathrm{0}\mathrm{<}q\mathrm{=}\mathrm{rank}\mathit{}\mathrm{\left[}S\mathrm{\right]}$. Then we have the following estimates of the eigenvalues λ of $\mathcal{A}$:

• (1)

$\mathcal{𝒜}$ has at least $\mathrm{max}\left\{1,m-p-r\right\}$ and at most $m-\mathrm{max}\left\{r,p\right\}$ eigenvalues of 0.

• (2)

If $\lambda >0$ , then it has the bounds

$\mathrm{min}\left\{{\rho }_{1},{\zeta }_{m-q+1}\right\}\le \lambda \le \mathrm{max}\left\{{\rho }_{n},{\mu }_{m}\right\}.$

• (3)

If $\mathrm{Im}\left(\lambda \right)\ne 0$ , then

$\mathrm{max}\left\{\frac{1}{2}{\rho }_{1},{\rho }_{1}-\sqrt{{\delta }_{m}}\right\}\le \mathrm{Re}\left(\lambda \right)\le \mathrm{min}\left\{\frac{1}{2}\left({\rho }_{n}+{\mu }_{m}\right),{\rho }_{n}+\sqrt{{\delta }_{m}},{\mu }_{m}+\sqrt{{\delta }_{m}}\right\},$

If $C=0$, then Corollary 3.4 can be much simplified as stated in the following corollary.

#### Corollary 3.5.

Under the same assumptions as in Corollary 3.4, except that $C\mathrm{=}\mathrm{0}$, the following estimates of the eigenvalues λ of $\mathcal{A}$ hold:

• (1)

$\mathcal{𝒜}$ has $m-p$ eigenvalues of 0.

• (2)

If $\lambda >0$ and we set ${\delta }_{m+1}=+\mathrm{\infty }$ , then

$\mathrm{min}\left\{{\rho }_{1},\frac{{\delta }_{m-p+1}}{{\rho }_{n}}\right\}\le \lambda \le {\rho }_{n}.$

• (3)

If $\mathrm{Im}\left(\lambda \right)\ne 0$ , then

$|\mathrm{Im}\left(\lambda \right)|\le \sqrt{{\delta }_{m}-\frac{{\rho }_{1}^{2}}{4}}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }\mathrm{max}\left\{\frac{1}{2}{\rho }_{1},{\rho }_{1}-\sqrt{{\delta }_{m}}\right\}\le \mathrm{Re}\left(\lambda \right)\le \mathrm{min}\left\{\frac{1}{2}{\rho }_{n},\sqrt{{\delta }_{m}}\right\}.$

## 4 Nonsingular Preconditioners for Singular Saddle Point Matrices

Intensive efforts have been made for the studies of preconditioners for nonsingular saddle point systems of form (1.1); see, e.g., [7, 8, 28, 3, 13, 34, 21, 33, 5, 35, 19]. On the other hand, few results are available for the analysis on preconditioners for singular saddle point systems, and we are only aware of the results in [36], which considered the simple case with $C=0$ in (1.1). Moreover, only singular preconditioners were analyzed in [36] for the singular saddle point system (1.1), but which are not always reliable and reasonable, as the preconditioned systems may not be equivalent to the original system (1.1) if singular preconditioners are used. In this section, we present and study some nonsingular preconditioners $\mathcal{𝒦}$ for the saddle point matrix $\mathcal{𝒜}$ in (1.1) when $\mathcal{𝒜}$ is singular. We shall consider the preconditioners of the general form

$\mathcal{𝒦}=\left(\begin{array}{cc}\hfill {P}_{A}\hfill & \hfill {B}^{T}\hfill \\ \hfill -B\hfill & \hfill {P}_{C}\hfill \end{array}\right),$(4.1)

where ${P}_{A}$ and ${P}_{C}$ are symmetric positive definite approximations of A and C, respectively. Then we shall solve the following preconditioned system:

${\mathcal{𝒦}}^{-1}\mathcal{𝒜}z={\mathcal{𝒦}}^{-1}b,$(4.2)

or

$\mathcal{𝒜}{\mathcal{𝒦}}^{-1}\stackrel{^}{z}=b,\stackrel{^}{z}=\mathcal{𝒦}z,$(4.3)

where $z={\left({x}^{T},{y}^{T}\right)}^{T}$ and $b={\left({f}^{T},-{g}^{T}\right)}^{T}$. Since

${\mathcal{𝒦}}^{-1}=\left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill -{P}_{A}^{-1}{B}^{T}{\left({P}_{C}+B{P}_{A}^{-1}{B}^{T}\right)}^{-1}\hfill \\ \hfill 0\hfill & \hfill {\left({P}_{C}+B{P}_{A}^{-1}{B}^{T}\right)}^{-1}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {P}_{A}^{-1}\hfill & \hfill 0\hfill \\ \hfill B{P}_{A}^{-1}\hfill & \hfill {I}_{m}\hfill \end{array}\right),$

we need only to solve two subsystems ${P}_{A}x={r}_{x}$ and $\left({P}_{C}+B{P}_{A}^{-1}{B}^{T}\right)y={r}_{y}$ when applying the GMRES method coupled with the preconditioner $\mathcal{𝒦}$ to the saddle point problem (1.1). In addition, we may easily see that B is not of full row rank when $\mathcal{𝒜}$ is singular, so it is natural to require a symmetric positive definite ${P}_{C}$ in (4.1). By Theorem 3.1, we know that $\mathcal{𝒦}$ is nonsingular. Some simple choices of ${P}_{A}$ and ${P}_{C}$ can be, e.g., ${P}_{A}=\mathrm{diag}\left(A\right)$ and ${P}_{C}=\alpha I$ with a positive parameter α.

The preconditioners of form (4.1) were considered in [8] for the case when $\mathcal{𝒜}$ is nonsingular, and ${P}_{C}$ was taken to be the special matrix ${P}_{C}=B{P}_{A}^{-1}{B}^{T}-{P}_{S}$, where ${P}_{S}$ is a symmetric positive definite approximation of the Schur complement S. But we would emphasize that the singular case considered here is much more technical than the nonsingular system. One main technical difficulty one may encounter in solving a singular system by GMRES is that the iteration process may break down at some step. Some conditions were derived in [12] under which the GMRES iterates converge safely to a solution of a singular system. For our subsequent study of the convergence of the GMRES method for solving the preconditioned system (4.2), we introduce the following convergence result from [12, Theorem 2.4].

#### Theorem 4.1 ([12]).

GMRES determines a least-squares solution of the singular linear system $\mathcal{A}\mathit{}z\mathrm{=}b$ without breakdown for all b and initial guess ${z}_{\mathrm{0}}$ if and only if $\mathrm{null}\mathit{}\mathrm{\left[}\mathcal{A}\mathrm{\right]}\mathrm{=}\mathrm{null}\mathit{}\mathrm{\left[}{\mathcal{A}}^{T}\mathrm{\right]}$. Furthermore, if $\mathcal{A}\mathit{}z\mathrm{=}b$ is consistent and ${z}_{\mathrm{0}}\mathrm{\in }\mathcal{R}\mathit{}\mathrm{\left(}\mathcal{A}\mathrm{\right)}$, then the least-squares solution is the pseudoinverse solution, i.e., the minimizer in the Euclidean norm $\mathrm{\parallel }\mathrm{\cdot }{\mathrm{\parallel }}_{\mathrm{2}}$.

Using the above convergence theory, we can derive the following result when GMRES is applied for solving the singular saddle point problem (1.1).

#### Theorem 4.2.

Let $\mathcal{K}$ be the preconditioner defined in (4.1) and let the saddle point problem (1.1) be consistent. If $\mathrm{null}\mathit{}\mathrm{\left[}{B}^{T}\mathrm{\right]}\mathrm{\cap }\mathrm{null}\mathit{}\mathrm{\left[}C\mathrm{\right]}$ is the invariant subspace of ${P}_{C}$, then GMRES applied for the preconditioned system (4.2) with any initial guess ${z}_{\mathrm{0}}$ determines a solution of the saddle point problem (1.1) without breakdown. Furthermore, if ${z}_{\mathrm{0}}\mathrm{\in }\mathcal{R}\mathit{}\mathrm{\left(}{\mathcal{K}}^{\mathrm{-}\mathrm{1}}\mathit{}\mathcal{A}\mathrm{\right)}$, then the GMRES solution is the pseudoinverse solution.

#### Proof.

Firstly, we prove that $\mathrm{null}\left[{\mathcal{𝒦}}^{-1}\mathcal{𝒜}\right]=\mathrm{null}\left[{\mathcal{𝒜}}^{T}{\mathcal{𝒦}}^{-T}\right]$. Since the preconditioner $\mathcal{𝒦}$ is nonsingular, we have $\mathrm{null}\left[{\mathcal{𝒦}}^{-1}\mathcal{𝒜}\right]=\mathrm{null}\left[\mathcal{𝒜}\right]$. For any ${\left({x}^{T},{y}^{T}\right)}^{T}\in \mathrm{null}\left[\mathcal{𝒜}\right]$, it holds that

$Ax+{B}^{T}y=0,-Bx+Cy=0.$(4.4)

This shows $x=-{A}^{-1}{B}^{T}y$. Together with the second equality in (4.4), this implies ${y}^{T}B{A}^{-1}{B}^{T}y+{y}^{T}Cy=0$. Since both of the matrices $B{A}^{-1}{B}^{T}$ and C are symmetric semi-definite, we have ${B}^{T}y=0$ and $Cy=0$. Since $x=-{A}^{-1}{B}^{T}y=0$, the null space of $\mathcal{𝒜}$ can be expressed as

$\mathrm{null}\left[\mathcal{𝒜}\right]=\left\{{\left({0}^{T},{y}^{T}\right)}^{T},y\in \mathrm{null}\left[{B}^{T}\right]\cap \mathrm{null}\left[C\right]\right\}.$(4.5)

On the other hand, for any $z={\left({x}^{T},{y}^{T}\right)}^{T}\in \mathrm{null}\left[{\mathcal{𝒜}}^{T}{\mathcal{𝒦}}^{-T}\right]$, it follows ${\mathcal{𝒜}}^{T}{\mathcal{𝒦}}^{-T}z=0$. This shows that ${\mathcal{𝒦}}^{-T}z\in \mathrm{null}\left[{\mathcal{𝒜}}^{T}\right]$. Using the same argument as that for deriving $\mathrm{null}\left[\mathcal{𝒜}\right]$ in (4.5), we deduce

$\mathrm{null}\left[{\mathcal{𝒜}}^{T}\right]=\left\{{\left({0}^{T},{y}^{T}\right)}^{T},y\in \mathrm{null}\left[{B}^{T}\right]\cap \mathrm{null}\left[C\right]\right\}.$

Then there exists a vector $\stackrel{^}{y}\in \mathrm{null}\left[{B}^{T}\right]\cap \mathrm{null}\left[C\right]$ such that ${\mathcal{𝒦}}^{-T}z={\left({0}^{T},{\stackrel{^}{y}}^{T}\right)}^{T}$; in other words, $z={\mathcal{𝒦}}^{T}{\left({0}^{T},{\stackrel{^}{y}}^{T}\right)}^{T}$. Noticing the form (4.1) and the condition that $\mathrm{null}\left[{B}^{T}\right]\cap \mathrm{null}\left[C\right]$ is the invariant subspace of ${P}_{C}$, we deduce

$\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill {P}_{A}\hfill & \hfill -{B}^{T}\hfill \\ \hfill B\hfill & \hfill {P}_{C}\hfill \end{array}\right)\left(\begin{array}{c}\hfill 0\hfill \\ \hfill \stackrel{^}{y}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -{B}^{T}\stackrel{^}{y}\hfill \\ \hfill {P}_{C}\stackrel{^}{y}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill {P}_{C}\stackrel{^}{y}\hfill \end{array}\right)\in \mathrm{null}\left[\mathcal{𝒜}\right].$

This implies $\mathrm{null}\left[{\mathcal{𝒜}}^{T}{\mathcal{𝒦}}^{-T}\right]\subseteq \mathrm{null}\left[\mathcal{𝒜}\right]$. Since $\mathrm{rank}\left[{\mathcal{𝒜}}^{T}{\mathcal{𝒦}}^{-T}\right]=\mathrm{rank}\left[\mathcal{𝒜}\right]$, we derive

$\mathrm{null}\left[{\mathcal{𝒜}}^{T}{\mathcal{𝒦}}^{-T}\right]=\mathrm{null}\left[\mathcal{𝒜}\right]=\mathrm{null}\left[{\mathcal{𝒦}}^{-1}\mathcal{𝒜}\right].$

Together with Theorem 4.1 and the fact that (1.1) is consistent, this yields that GMRES determines a solution ${z}^{*}\in {ℝ}^{n+m}$ of the preconditioned system (4.2) without breakdown for all b and ${z}_{0}$. It is easy to see that ${z}^{*}$ is also a solution of the saddle point problem (1.1). Furthermore, if the initial vector ${z}_{0}\in \mathcal{ℛ}\left({\mathcal{𝒦}}^{-1}\mathcal{𝒜}\right)$, then ${z}^{*}$ is the pseudoinverse solution, which completes the proof of Theorem 4.2. ∎

#### Remark 4.3.

The condition that $\mathrm{null}\left[{B}^{T}\right]\cap \mathrm{null}\left[C\right]$ is the invariant subspace of ${P}_{C}$ is not restrictive. Actually, if we set ${P}_{C}$ as a scalar matrix, i.e., ${P}_{C}=\alpha {I}_{m}$ with $\alpha >0$, then this condition follows naturally.

Similarly to the results in Theorem 4.2, we have the following convergence for GMRES applied for the preconditioned system (4.3).

#### Theorem 4.4.

Let $\mathcal{K}$ be the preconditioner defined as in (4.1). If $\mathrm{null}\mathit{}\mathrm{\left[}{B}^{T}\mathrm{\right]}\mathrm{\cap }\mathrm{null}\mathit{}\mathrm{\left[}C\mathrm{\right]}$ is the invariant subspace of ${P}_{C}$, then GMRES applied for the preconditioned system (4.3) with any initial guess ${z}_{\mathrm{0}}$ determines a solution of the saddle point problem (1.1) without breakdown.

#### Proof.

Using a similar argument as that for Theorem 4.2, we can prove $\mathrm{null}\left[\mathcal{𝒜}{\mathcal{𝒦}}^{-1}\right]=\mathrm{null}\left[{\mathcal{𝒦}}^{-T}{\mathcal{𝒜}}^{T}\right]$. Then we know from Theorem 4.1 that GMRES determines a least-squares solution ${\stackrel{^}{z}}^{*}\in {ℝ}^{n+m}$ of the preconditioned system (4.3) without breakdown for all b and ${z}_{0}$. Since

$0={\left(\mathcal{𝒜}{\mathcal{𝒦}}^{-1}\right)}^{T}\left(b-\mathcal{𝒜}{\mathcal{𝒦}}^{-1}{\stackrel{^}{z}}^{*}\right)={\mathcal{𝒦}}^{-T}{\mathcal{𝒜}}^{T}\left(b-\mathcal{𝒜}{\mathcal{𝒦}}^{-1}{\stackrel{^}{z}}^{*}\right),$

we see ${\mathcal{𝒜}}^{T}\left(b-\mathcal{𝒜}{\mathcal{𝒦}}^{-1}{\stackrel{^}{z}}^{*}\right)=0$. This implies that ${z}^{*}={\mathcal{𝒦}}^{-1}{\stackrel{^}{z}}^{*}$ is a least-squares solution of the original system (1.1). ∎

#### Remark 4.5.

Unlike Theorem 4.2, Theorem 4.4 can be applied also to the case where system (1.1) is inconsistent.

Although Theorems 4.2 and 4.4 guarantee the convergence of GMRES when applied to the preconditioned system (4.2) and (4.3), respectively, its convergence rate depends strongly on the spectral properties of the preconditioned matrix. It will be our central task in the subsequent section to study the spectral properties of the preconditioned matrix. For these studies, we remark that the general case $C\ne 0$ considered here is much more technical than the simple case $C=0$ studied in [36], and the analysis of [36] clearly does not apply to the current case with $C\ne 0$. Moreover, unlike the results in [36], which gave only the estimates of the real part of the eigenvalues of the preconditioned system in terms of the spectrum of the preconditioned system for A in (1.1), we shall derive the lower and upper bounds of both the real and complex eigenvalues of the preconditioned matrix.

## 5 Spectral Estimates of the Preconditioned Matrix

There are two major analysis techniques that enable the study of the spectral behavior of the preconditioned matrix $\mathcal{𝒜}{\mathcal{𝒦}}^{-1}$ in (4.3). First, we manage to work out two effective similar transformations, which transform our preconditioned system $\mathcal{𝒜}{\mathcal{𝒦}}^{-1}$ into a matrix, see $\mathcal{ℳ}$ in (5.2) below, that has exactly the same block form as the original saddle point matrix $\mathcal{𝒜}$ in (1.2). Then we can apply our results of Section 3 for the saddle point matrix $\mathcal{𝒜}$ to derive the lower and upper bounds of the real and complex eigenvalues of the preconditioned system $\mathcal{𝒜}{\mathcal{𝒦}}^{-1}$. But this application is not straightforward. We thus have to establish the spectral estimates for each of the matrices $Q{Q}^{T}$, H and the Schur complement $W=:H+Q{A}_{p}^{-1}{Q}^{T}$ (see (5.2)–(5.4) below) associated with the new saddle point matrix $\mathcal{ℳ}$ in (5.2), and several delicate new techniques are introduced here.

In the remainder of this section, we will investigate the spectral properties of $\mathcal{𝒜}{\mathcal{𝒦}}^{-1}$ by using the results we developed in Section 3. To this end, we first define a matrix

$\mathcal{𝒫}=\left(\begin{array}{cc}\hfill {P}_{A}^{1/2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {P}_{C}^{1/2}\hfill \end{array}\right).$

We can directly verify that $\mathcal{𝒜}{\mathcal{𝒦}}^{-1}$ is similar to $\left({\mathcal{𝒫}}^{-1}\mathcal{𝒜}{\mathcal{𝒫}}^{-1}\right)\left(\mathcal{𝒫}{\mathcal{𝒦}}^{-1}\mathcal{𝒫}\right)$, i.e.,

$\left(\begin{array}{cc}\hfill {P}_{A}^{-1/2}A{P}_{A}^{-1/2}\hfill & \hfill {P}_{A}^{-1/2}{B}^{T}{P}_{C}^{-1/2}\hfill \\ \hfill -{P}_{C}^{-1/2}B{P}_{A}^{-1/2}\hfill & \hfill {P}_{C}^{-1/2}C{P}_{C}^{-1/2}\hfill \end{array}\right){\left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill {P}_{A}^{-1/2}{B}^{T}{P}_{C}^{-1/2}\hfill \\ \hfill -{P}_{C}^{-1/2}B{P}_{A}^{-1/2}\hfill & \hfill {I}_{m}\hfill \end{array}\right)}^{-1}.$

With the definitions

${A}_{p}={P}_{A}^{-1/2}A{P}_{A}^{-1/2},{C}_{p}={P}_{C}^{-1/2}C{P}_{C}^{-1/2},R={P}_{C}^{-1/2}B{P}_{A}^{-1/2},$(5.1)

the above observation is recast as

$\mathcal{𝒜}{\mathcal{𝒦}}^{-1}\sim \left(\begin{array}{cc}\hfill {A}_{p}\hfill & \hfill {R}^{T}\hfill \\ \hfill -R\hfill & \hfill {C}_{p}\hfill \end{array}\right){\left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill {R}^{T}\hfill \\ \hfill -R\hfill & \hfill {I}_{m}\hfill \end{array}\right)}^{-1}.$

This and the factorization

$\left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill {R}^{T}\hfill \\ \hfill -R\hfill & \hfill {I}_{m}\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill 0\hfill \\ \hfill -R\hfill & \hfill {I}_{m}+R{R}^{T}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill {R}^{T}\hfill \\ \hfill 0\hfill & \hfill {I}_{m}\hfill \end{array}\right)$

$\mathcal{𝒜}{\mathcal{𝒦}}^{-1}\sim \left(\begin{array}{cc}\hfill {A}_{p}\hfill & \hfill {R}^{T}\hfill \\ \hfill -R\hfill & \hfill {C}_{p}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill -{R}^{T}\hfill \\ \hfill 0\hfill & \hfill {I}_{m}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill 0\hfill \\ \hfill {\left({I}_{m}+R{R}^{T}\right)}^{-1}R\hfill & \hfill {\left({I}_{m}+R{R}^{T}\right)}^{-1}\hfill \end{array}\right)$$\sim \left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill 0\hfill \\ \hfill {\left({I}_{m}+R{R}^{T}\right)}^{-1}R\hfill & \hfill {\left({I}_{m}+R{R}^{T}\right)}^{-1}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {A}_{p}\hfill & \hfill {R}^{T}\hfill \\ \hfill -R\hfill & \hfill {C}_{p}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill -{R}^{T}\hfill \\ \hfill 0\hfill & \hfill {I}_{m}\hfill \end{array}\right)$$=\left(\begin{array}{cc}\hfill {A}_{p}\hfill & \hfill \left({I}_{n}-{A}_{p}\right){R}^{T}\hfill \\ \hfill -{\left({I}_{m}+R{R}^{T}\right)}^{-1}R\left({I}_{n}-{A}_{p}\right)\hfill & \hfill {\left({I}_{m}+R{R}^{T}\right)}^{-1}\left(2R{R}^{T}-R{A}_{p}{R}^{T}+{C}_{p}\right)\hfill \end{array}\right)$$\sim \left(\begin{array}{cc}\hfill {A}_{p}\hfill & \hfill {Q}^{T}\hfill \\ \hfill -Q\hfill & \hfill H\hfill \end{array}\right)=:\mathcal{ℳ},$(5.2)

where Q and H are given by

$Q={\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R\left({I}_{n}-{A}_{p}\right),$(5.3)$H={\left({I}_{m}+R{R}^{T}\right)}^{-1/2}\left(2R{R}^{T}-R{A}_{p}{R}^{T}+{C}_{p}\right){\left({I}_{m}+R{R}^{T}\right)}^{-1/2}.$(5.4)

Since $\mathcal{𝒜}{\mathcal{𝒦}}^{-1}$ and $\mathcal{ℳ}$ have the same eigenvalues, it suffices to estimate the eigenvalues of $\mathcal{ℳ}$. As $\mathcal{ℳ}$ has the same structure as the original saddle point matrix $\mathcal{𝒜}$ in (1.2), we can apply our results developed in Section 3 to investigate the spectral properties of $\mathcal{ℳ}$.

To do so, we write the eigenvalues of ${A}_{p}$, ${C}_{p}$, $R{R}^{T}$ and $Q{Q}^{T}$, respectively, as

$0<{\stackrel{^}{\rho }}_{1}\le {\stackrel{^}{\rho }}_{2}\le \mathrm{\cdots }\le {\stackrel{^}{\rho }}_{n}\mathrm{ }\text{and} 0\le {\stackrel{^}{\mu }}_{1}\le {\stackrel{^}{\mu }}_{2}\le \mathrm{\cdots }\le {\stackrel{^}{\mu }}_{m},$$0\le {\stackrel{^}{\zeta }}_{1}\le {\stackrel{^}{\zeta }}_{2}\le \mathrm{\cdots }\le {\stackrel{^}{\zeta }}_{m}\mathrm{ }\text{and} 0\le {\stackrel{^}{\delta }}_{1}\le {\stackrel{^}{\delta }}_{2}\le \mathrm{\cdots }\le {\stackrel{^}{\delta }}_{m}.$

Next, we will derive the spectral estimates for each of the matrices $Q{Q}^{T}$, H, and Schur complement W, so that we can apply the spectral bounds of Section 3 to the new saddle point matrix $\mathcal{ℳ}$ in (5.2).

Without loss of generality, we assume that ${\stackrel{^}{\rho }}_{n}<1$. If ${\stackrel{^}{\rho }}_{n}\ge 1$, a proper scaling of ${P}_{A}$ will bring us back to the case that ${\stackrel{^}{\rho }}_{n}<1$. Under this assumption, ${I}_{n}-{A}_{p}$ is symmetric positive definite. Since $R\left(2{I}_{n}-{A}_{p}\right){R}^{T}\ge 0$ and ${C}_{p}\ge 0$, we have

${\left({I}_{m}+R{R}^{T}\right)}^{1/2}H{\left({I}_{m}+R{R}^{T}\right)}^{1/2}\ge {C}_{p}\ge 0$

and

${\left({I}_{m}+R{R}^{T}\right)}^{1/2}H{\left({I}_{m}+R{R}^{T}\right)}^{1/2}\ge R\left(2{I}_{n}-{A}_{p}\right){R}^{T}\ge 0.$

This shows

$\mathrm{rank}\left[H\right]\ge \mathrm{max}\left\{\mathrm{rank}\left[R\left(2{I}_{n}-{A}_{p}\right){R}^{T}\right],\mathrm{rank}\left[{C}_{p}\right]\right\}=\mathrm{max}\left\{\mathrm{rank}\left[B\right],\mathrm{rank}\left[C\right]\right\}.$

Hence $H\ne 0$ if $B\ne 0$ or $C\ne 0$.

It is easy to verify that

$\mathrm{rank}\left[Q\right]=\mathrm{rank}\left[{P}_{C}^{-1/2}B{P}_{A}^{-1/2}\right]=\mathrm{rank}\left[B\right].$(5.5)

The combination of (5.1), (5.3), and (5.4) leads to

$W={\left({I}_{m}+R{R}^{T}\right)}^{-1/2}\left[2R{R}^{T}-R{A}_{p}{R}^{T}+{C}_{p}+R\left({I}_{n}-{A}_{p}\right){A}_{p}^{-1}\left({I}_{n}-{A}_{p}\right){R}^{T}\right]{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}$$={\left({I}_{m}+R{R}^{T}\right)}^{-1/2}\left({C}_{p}+R{A}_{p}^{-1}{R}^{T}\right){\left({I}_{m}+R{R}^{T}\right)}^{-1/2}$

and

${C}_{p}+R{A}_{p}^{-1}{R}^{T}={P}_{C}^{-1/2}\left(C+B{A}^{-1}{B}^{T}\right){P}_{C}^{-1/2}={P}_{C}^{-1/2}S{P}_{C}^{-1/2}.$

This implies

$\mathrm{rank}\left[W\right]=\mathrm{rank}\left[{C}_{p}+R{A}_{p}^{-1}{R}^{T}\right]=\mathrm{rank}\left[S\right].$

Together with (5.5), we are ready to establish the first major result in this section.

#### Lemma 5.1.

Let $\mathcal{A}$ be the singular saddle point matrix of form (1.2), where A is symmetric positive definite and C is symmetric and positive semi-definite, and let Q and H be the two matrices given in (5.3) and (5.4), respectively. Assume that $\mathrm{rank}\mathit{}\mathrm{\left[}B\mathrm{\right]}\mathrm{=}p$, $\mathrm{rank}\mathit{}\mathrm{\left[}C\mathrm{\right]}\mathrm{=}r$ and $\mathrm{0}\mathrm{<}q\mathrm{=}\mathrm{rank}\mathit{}\mathrm{\left[}S\mathrm{\right]}$. Then $\mathrm{rank}\mathit{}\mathrm{\left[}Q\mathrm{\right]}\mathrm{=}p$ and $\mathrm{max}\mathit{}\mathrm{\left\{}p\mathrm{,}r\mathrm{\right\}}\mathrm{\le }\mathrm{rank}\mathit{}\mathrm{\left[}W\mathrm{\right]}\mathrm{=}q\mathrm{\le }p\mathrm{+}r$, and the following estimates hold:

$0<\frac{{\stackrel{^}{\zeta }}_{m-p+1}{\left(1-{\stackrel{^}{\rho }}_{n}\right)}^{2}}{1+{\stackrel{^}{\zeta }}_{m-p+1}}\le {\stackrel{^}{\delta }}_{m-p+1}\le {\stackrel{^}{\delta }}_{m}\le \frac{{\stackrel{^}{\zeta }}_{m}{\left(1-{\stackrel{^}{\rho }}_{1}\right)}^{2}}{1+{\stackrel{^}{\zeta }}_{m}},$${\lambda }_{\mathrm{max}}\left[H\right]\le \mathrm{min}\left\{\frac{{\stackrel{^}{\zeta }}_{m}\left(2-{\stackrel{^}{\rho }}_{1}\right)}{1+{\stackrel{^}{\zeta }}_{m}}+{\stackrel{^}{\mu }}_{m},\mathrm{max}\left\{\left(2-{\stackrel{^}{\rho }}_{1}\right),{\stackrel{^}{\mu }}_{m}\right\}\right\},$${\lambda }_{m-q+1}\left[W\right]\ge \frac{1}{1+{\stackrel{^}{\zeta }}_{m}}{\lambda }_{m-q+1}\left[{C}_{p}+R{A}_{p}^{-1}{R}^{T}\right]>0.$

In particular, if $C\mathrm{=}\mathrm{0}$, then $\mathrm{rank}\mathit{}\mathrm{\left[}W\mathrm{\right]}\mathrm{=}p$ and

${\lambda }_{m-p+1}\left[W\right]\ge \frac{{\stackrel{^}{\zeta }}_{m-p+1}}{{\stackrel{^}{\rho }}_{n}\left(1+{\stackrel{^}{\zeta }}_{m-p+1}\right)}.$

#### Proof.

It is easy to verify that the eigenvalues of ${\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R{R}^{T}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}$ are

$\frac{{\stackrel{^}{\zeta }}_{j}}{1+{\stackrel{^}{\zeta }}_{j}},j=1,2,\mathrm{\dots },m.$

Since the function $\frac{x}{1+x}$ is monotonically increasing at $x\ge 0$, we have

${\lambda }_{\mathrm{max}}\left[{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R{R}^{T}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}\right]=\frac{{\stackrel{^}{\zeta }}_{m}}{1+{\stackrel{^}{\zeta }}_{m}}.$(5.6)

This, ${\stackrel{^}{\rho }}_{n}<1$ and (5.3) lead to

$Q{Q}^{T}={\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R{\left({I}_{n}-{A}_{p}\right)}^{2}{R}^{T}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}$$\le {\left(1-{\stackrel{^}{\rho }}_{1}\right)}^{2}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R{R}^{T}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}$$\le \frac{{\stackrel{^}{\zeta }}_{m}{\left(1-{\stackrel{^}{\rho }}_{1}\right)}^{2}}{1+{\stackrel{^}{\zeta }}_{m}}{I}_{m}.$(5.7)

Therefore,

${\stackrel{^}{\delta }}_{m}\le \frac{{\stackrel{^}{\zeta }}_{m}{\left(1-{\stackrel{^}{\rho }}_{1}\right)}^{2}}{1+{\stackrel{^}{\zeta }}_{m}}.$

On the other hand, (5.7) implies

$Q{Q}^{T}\ge {\left(1-{\stackrel{^}{\rho }}_{n}\right)}^{2}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R{R}^{T}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}\ge 0.$

This gives an estimate of the smallest positive eigenvalue of $Q{Q}^{T}$, namely

${\stackrel{^}{\delta }}_{m-p+1}\ge {\left(1-{\stackrel{^}{\rho }}_{n}\right)}^{2}{\lambda }_{m-p+1}\left[{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R{R}^{T}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}\right].$

Together with the fact that

${\lambda }_{m-p+1}\left[{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R{R}^{T}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}\right]=\frac{{\stackrel{^}{\zeta }}_{m-p+1}}{1+{\stackrel{^}{\zeta }}_{m-p+1}}>0,$

${\stackrel{^}{\delta }}_{m-p+1}\ge \frac{{\stackrel{^}{\zeta }}_{m-p+1}{\left(1-{\stackrel{^}{\rho }}_{n}\right)}^{2}}{1+{\stackrel{^}{\zeta }}_{m-p+1}}.$

The second result is proved in the sequel. From (5.4) and (5.6) we obtain

$H={\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R\left(2{I}_{n}-{A}_{p}\right){R}^{T}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}+{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}{C}_{p}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}$$\le \left(2-{\stackrel{^}{\rho }}_{1}\right){\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R{R}^{T}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}+{\stackrel{^}{\mu }}_{m}{\left({I}_{m}+R{R}^{T}\right)}^{-1}$$\le \left[\frac{{\stackrel{^}{\zeta }}_{m}\left(2-{\stackrel{^}{\rho }}_{1}\right)}{1+{\stackrel{^}{\zeta }}_{m}}+{\stackrel{^}{\mu }}_{m}\right]{I}_{m}$

and

$H\le \mathrm{max}\left\{\left(2-{\stackrel{^}{\rho }}_{1}\right),{\stackrel{^}{\mu }}_{m}\right\}\left[{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R{R}^{T}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}+{\left({I}_{m}+R{R}^{T}\right)}^{-1}\right]$$=\mathrm{max}\left\{\left(2-{\stackrel{^}{\rho }}_{1}\right),{\stackrel{^}{\mu }}_{m}\right\}{I}_{m}.$

This implies

${\lambda }_{\mathrm{max}}\left[H\right]\le \mathrm{min}\left\{\frac{{\stackrel{^}{\zeta }}_{m}\left(2-{\stackrel{^}{\rho }}_{1}\right)}{1+{\stackrel{^}{\zeta }}_{m}}+{\stackrel{^}{\mu }}_{m},\mathrm{max}\left\{\left(2-{\stackrel{^}{\rho }}_{1}\right),{\stackrel{^}{\mu }}_{m}\right\}\right\}.$

For any matrix $L\in {ℝ}^{m×m}$, we have $\mathrm{rank}\left[L{L}^{T}\right]=\mathrm{rank}\left[{L}^{T}L\right]$, and $L{L}^{T}$ and ${L}^{T}L$ have the same eigenvalues. This leads to

${\lambda }_{m-q+1}\left[W\right]={\lambda }_{m-q+1}\left[{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}\left({C}_{p}+R{A}_{p}^{-1}{R}^{T}\right){\left({I}_{m}+R{R}^{T}\right)}^{-1/2}\right]$$={\lambda }_{m-q+1}\left[{\left({C}_{p}+R{A}_{p}^{-1}{R}^{T}\right)}^{1/2}{\left({I}_{m}+R{R}^{T}\right)}^{-1}{\left({C}_{p}+R{A}_{p}^{-1}{R}^{T}\right)}^{1/2}\right]$$\ge \frac{1}{1+{\stackrel{^}{\zeta }}_{m}}{\lambda }_{m-q+1}\left[{C}_{p}+R{A}_{p}^{-1}{R}^{T}\right],$

which gives our desired estimate.

Particularly, if $C=0$, then ${C}_{p}=0$ leads to

$W={\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R{A}_{p}^{-1}{R}^{T}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}\ge \frac{1}{{\stackrel{^}{\rho }}_{n}}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R{R}^{T}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}.$

Therefore,

${\lambda }_{m-p+1}\left[W\right]\ge \frac{1}{{\stackrel{^}{\rho }}_{n}}{\lambda }_{m-p+1}\left[{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}R{R}^{T}{\left({I}_{m}+R{R}^{T}\right)}^{-1/2}\right]=\frac{{\stackrel{^}{\zeta }}_{m-p+1}}{{\stackrel{^}{\rho }}_{n}\left(1+{\stackrel{^}{\zeta }}_{m-p+1}\right)},$

which completes the proof. ∎

Introduce two constants

${\xi }_{m}=\sqrt{\frac{{\stackrel{^}{\zeta }}_{m}{\left(1-{\stackrel{^}{\rho }}_{1}\right)}^{2}}{1+{\stackrel{^}{\zeta }}_{m}}}\mathit{ }\text{and}\mathit{ }{\vartheta }_{m}=\mathrm{min}\left\{\frac{{\stackrel{^}{\zeta }}_{m}\left(2-{\stackrel{^}{\rho }}_{1}\right)}{1+{\stackrel{^}{\zeta }}_{m}}+{\stackrel{^}{\mu }}_{m},\mathrm{max}\left\{\left(2-{\stackrel{^}{\rho }}_{1}\right),{\stackrel{^}{\mu }}_{m}\right\}\right\}.$

The factorization

$\mathcal{ℳ}=\left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill 0\hfill \\ \hfill -Q{A}_{p}^{-1}\hfill & \hfill {I}_{m}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {A}_{p}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill W\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {I}_{n}\hfill & \hfill {A}_{p}^{-1}{Q}^{T}\hfill \\ \hfill 0\hfill & \hfill {I}_{m}\hfill \end{array}\right)$

leads to the following estimates directly from Corollary 3.4 and Lemma 5.1.

#### Theorem 5.2.

Let $\mathcal{A}$ and $\mathcal{K}$ be two saddle point matrices of the forms (1.2) and (4.1), respectively, where A is symmetric and positive definite and C is symmetric positive semi-definite, with $\mathrm{rank}\mathit{}\mathrm{\left[}B\mathrm{\right]}\mathrm{=}p$ and $\mathrm{rank}\mathit{}\mathrm{\left[}C\mathrm{\right]}\mathrm{=}r$. Then any eigenvalue λ of the preconditioning matrix $\mathcal{A}\mathit{}{\mathcal{K}}^{\mathrm{-}\mathrm{1}}$ fulfills the following estimates:

• (1)

$\mathcal{𝒜}{\mathcal{𝒦}}^{-1}$ has at least $\mathrm{max}\left\{1,m-p-r\right\}$ and at most $m-\mathrm{max}\left\{p,r\right\}$ eigenvalues of 0.

• (2)

If $\lambda >0$ and $\stackrel{^}{\eta }$ is the minimum non-zero eigenvalue of ${C}_{p}+R{A}_{p}^{-1}{R}^{T}$ , then

$\mathrm{min}\left\{{\stackrel{^}{\rho }}_{1},\frac{\stackrel{^}{\eta }}{1+{\stackrel{^}{\zeta }}_{m}}\right\}\le \lambda \le \mathrm{max}\left\{{\stackrel{^}{\rho }}_{n},{\vartheta }_{m}\right\},$

• (3)

If $\mathrm{Im}\left(\lambda \right)\ne 0$ , then

$\mathrm{max}\left\{\frac{{\stackrel{^}{\rho }}_{1}}{2},{\stackrel{^}{\rho }}_{1}-{\xi }_{m}\right\}\le \mathrm{Re}\left(\lambda \right)\le \mathrm{min}\left\{\frac{{\stackrel{^}{\rho }}_{n}+{\vartheta }_{m}}{2},{\stackrel{^}{\rho }}_{n}+{\xi }_{m},{\vartheta }_{m}+{\xi }_{m}\right\},$

The two simple inequalities

${\vartheta }_{m}=\mathrm{min}\left\{\frac{{\stackrel{^}{\zeta }}_{m}\left(2-{\stackrel{^}{\rho }}_{1}\right)}{1+{\stackrel{^}{\zeta }}_{m}}+{\stackrel{^}{\mu }}_{m},\mathrm{max}\left\{\left(2-{\stackrel{^}{\rho }}_{1}\right),{\stackrel{^}{\mu }}_{m}\right\}\right\}$$\le \mathrm{min}\left\{\left(2-{\stackrel{^}{\rho }}_{1}\right)+{\stackrel{^}{\mu }}_{m},\mathrm{max}\left\{\left(2-{\stackrel{^}{\rho }}_{1}\right),{\stackrel{^}{\mu }}_{m}\right\}\right\}$$=\mathrm{max}\left\{\left(2-{\stackrel{^}{\rho }}_{1}\right),{\stackrel{^}{\mu }}_{m}\right\}$

and

$2-{\stackrel{^}{\rho }}_{1}>{\stackrel{^}{\rho }}_{n}$

simplify the estimates of Theorem 5.2 as below.

#### Corollary 5.3.

Under the conditions of Theorem 5.2 on the singular saddle point matrix $\mathcal{A}$ and its nonsingular preconditioner $\mathcal{K}$, any eigenvalue λ of the preconditioning matrix $\mathcal{A}\mathit{}{\mathcal{K}}^{\mathrm{-}\mathrm{1}}$ fulfills the following estimates:

• (1)

If $\lambda >0$ and $\stackrel{^}{\eta }$ is the minimum non-zero eigenvalue of ${C}_{p}+R{A}_{p}^{-1}{R}^{T}$ , then

$\mathrm{min}\left\{{\stackrel{^}{\rho }}_{1},\frac{\stackrel{^}{\eta }}{1+{\stackrel{^}{\zeta }}_{m}}\right\}\le \lambda \le \mathrm{max}\left\{2-{\stackrel{^}{\rho }}_{1},{\stackrel{^}{\mu }}_{m}\right\}.$

• (2)

If $\mathrm{Im}\left(\lambda \right)\ne 0$ , then $|\mathrm{Im}\left(\lambda \right)|\le 1-{\stackrel{^}{\rho }}_{1}$ and

$\mathrm{max}\left\{\frac{1}{2}{\stackrel{^}{\rho }}_{1},2{\stackrel{^}{\rho }}_{1}-1\right\}\le \mathrm{Re}\left(\lambda \right)\le \mathrm{min}\left\{\frac{1}{2}\left({\stackrel{^}{\rho }}_{n}+\mathrm{max}\left\{\left(2-{\stackrel{^}{\rho }}_{1}\right),{\stackrel{^}{\mu }}_{m}\right\}\right),1+{\stackrel{^}{\rho }}_{n}-{\stackrel{^}{\rho }}_{1}\right\}.$

Corollary 5.3 shows that all eigenvalues of $\mathcal{𝒜}{\mathcal{𝒦}}^{-1}$ are located in a square with a side length of 2 if ${P}_{C}$ is chosen such that ${\stackrel{^}{\mu }}_{m}\le 2$.

Moreover, if we have $C=0$, then ${C}_{p}=0$, and ${\stackrel{^}{\mu }}_{m}=0$ and $r=0$ accordingly. In this case,

${\vartheta }_{m}=\mathrm{min}\left\{\frac{{\stackrel{^}{\zeta }}_{m}\left(2-{\stackrel{^}{\rho }}_{1}\right)}{1+{\stackrel{^}{\zeta }}_{m}},\mathrm{max}\left\{\left(2-{\stackrel{^}{\rho }}_{1}\right),0\right\}\right\}=\mathrm{min}\left\{\frac{{\stackrel{^}{\zeta }}_{m}\left(2-{\stackrel{^}{\rho }}_{1}\right)}{1+{\stackrel{^}{\zeta }}_{m}},\left(2-{\stackrel{^}{\rho }}_{1}\right)\right\}=\frac{{\stackrel{^}{\zeta }}_{m}\left(2-{\stackrel{^}{\rho }}_{1}\right)}{1+{\stackrel{^}{\zeta }}_{m}}.$

An application of this expression of ${\vartheta }_{m}$ in Theorem 5.2 verifies the following corollary.

#### Corollary 5.4.

Under the same conditions of Theorem 5.2 on the singular saddle point matrix $\mathcal{A}$ and its nonsingular preconditioner $\mathcal{K}$, except that we now assume $C\mathrm{=}\mathrm{0}$, the following holds:

• (1)

$\mathcal{𝒜}{\mathcal{𝒦}}^{-1}$ has $m-p$ eigenvalues at 0.

Moreover, any eigenvalue λ of the preconditioning matrix $\mathcal{A}\mathit{}{\mathcal{K}}^{\mathrm{-}\mathrm{1}}$ fulfills the following estimates:

• (2)

If $\lambda >0$ , then

$\mathrm{min}\left\{{\stackrel{^}{\rho }}_{1},\frac{{\stackrel{^}{\zeta }}_{m-p+1}}{{\stackrel{^}{\rho }}_{n}\left(1+{\stackrel{^}{\zeta }}_{m-p+1}\right)}\right\}\le \lambda \le \mathrm{max}\left\{{\stackrel{^}{\rho }}_{n},\frac{{\stackrel{^}{\zeta }}_{m}\left(2-{\stackrel{^}{\rho }}_{1}\right)}{1+{\stackrel{^}{\zeta }}_{m}}\right\}.$

• (3)

If $\mathrm{Im}\left(\lambda \right)\ne 0$ , then

$\mathrm{max}\left\{\frac{1}{2}{\stackrel{^}{\rho }}_{1},{\stackrel{^}{\rho }}_{1}-{\xi }_{m}\right\}\le \mathrm{Re}\left(\lambda \right)\le \mathrm{min}\left\{\frac{1}{2}\left({\stackrel{^}{\rho }}_{n}+\frac{{\stackrel{^}{\zeta }}_{m}\left(2-{\stackrel{^}{\rho }}_{1}\right)}{1+{\stackrel{^}{\zeta }}_{m}}\right),{\stackrel{^}{\rho }}_{n}+{\xi }_{m},\frac{{\stackrel{^}{\zeta }}_{m}\left(2-{\stackrel{^}{\rho }}_{1}\right)}{1+{\stackrel{^}{\zeta }}_{m}}+{\xi }_{m}\right\},$

It can be seen from Corollary 5.4 that the smaller the maximum eigenvalue ${\stackrel{^}{\zeta }}_{m}$ of $R{R}^{T}$ is, the more clustered all the eigenvalues of the preconditioned matrix $\mathcal{𝒜}{\mathcal{𝒦}}^{-1}$ are. And as $C=0$, we can choose ${P}_{C}=\alpha {I}_{m}$ with $\alpha >0$. Then $R{R}^{T}=\frac{1}{\alpha }B{P}_{A}^{-1}{B}^{T}$, which shows that ${\stackrel{^}{\zeta }}_{m}$ will be small for sufficiently large α. On the other hand, we can infer from Corollary 5.3 that the distribution of the eigenvalues of $\mathcal{𝒜}{\mathcal{𝒦}}^{-1}$ will be clustered even for large ${\stackrel{^}{\zeta }}_{m}$.

## 6 Numerical Experiments

In this section, we present a few numerical examples to test the performance of some nonsingular preconditioners in combination with the preconditioned restarted GMRES(50) method for solving the singular saddle point problems of form (1.1). All experiments were run in MATLAB R2013b on a PC with Intel(R) Core(TM) i5-5257U, CPU @2.70 GHz, 8.00 GB. For each example, we report the number of iterations, the CPU time and the relative residual, which are respectively denoted by “Iter”, “CPU” and “RES”. Let ${z}^{k}$ be the k-th approximate solution of the saddle point problem (1.1), then “RES” is defined by the relative residuals

$\mathrm{RES}:=\frac{{\parallel b-\mathcal{𝒜}{z}^{k}\parallel }_{2}}{{\parallel b\parallel }_{2}}.$

The initial guess ${z}^{0}$ is always set to 0 and the iteration is terminated when $\mathrm{RES}\le {10}^{-6}$.

Let $L=\mathrm{ichol}\left(A\right)$ be the incomplete Cholesky decomposition of the matrix A with droptol $=$ 1e-02. We set ${P}_{A}=L{L}^{T}$. Then it is easy to see that

$\mathcal{𝒦}=\left(\begin{array}{cc}\hfill L\hfill & \hfill 0\hfill \\ \hfill -B{L}^{-T}\hfill & \hfill {I}_{m}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {L}^{T}\hfill & \hfill {L}^{-1}{B}^{T}\hfill \\ \hfill 0\hfill & \hfill {P}_{C}+B{L}^{-T}{L}^{-1}{B}^{T}\hfill \end{array}\right).$

Since L is a sparse lower triangular matrix, then in each step, we just need to solve some lower triangular linear equations. In our computations, we shall compare the performance of the restarted GMRES(50) method, to be denoted by GMRES, and the preconditioned restarted GMRES(50) method coupled with the nonsingular preconditioner $\mathcal{𝒦}$ defined as in (4.1) with ${P}_{C}={I}_{m}/10,{I}_{m}, 10{I}_{m}$, to be denoted by ${I}_{m}/10$, ${I}_{m}$, $10{I}_{m}$, respectively.

#### Example 6.1.

This example arises from the discretization of the Stokes problem (see [37, Example 4.1]), where the block matrices in the saddle point system (1.1) take the form

$A=\left(\begin{array}{cc}\hfill I\otimes T+T\otimes I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill I\otimes T+T\otimes I\hfill \end{array}\right)\in {ℝ}^{2{l}^{2}×2{l}^{2}},$$B={\left(E,{b}_{1},{b}_{2}\right)}^{T}\in {ℝ}^{\left({l}^{2}+2\right)×2{l}^{2}},C=0\in {ℝ}^{\left({l}^{2}+2\right)×\left({l}^{2}+2\right)},$

where the matrices T and E and the vectors ${b}_{1}$ and ${b}_{2}$ are given by

$T=\frac{1}{{h}^{2}}\cdot \mathrm{tridiag}\left(-1,2,-1\right)\in {ℝ}^{l×l},E=\left(\begin{array}{c}\hfill I\otimes F\hfill \\ \hfill F\otimes I\hfill \end{array}\right)\in {ℝ}^{2{l}^{2}×{l}^{2}},$${b}_{1}=E\left(\begin{array}{c}\hfill e\hfill \\ \hfill 0\hfill \end{array}\right),{b}_{2}=E\left(\begin{array}{c}\hfill 0\hfill \\ \hfill e\hfill \end{array}\right),e={\left(1,1,\mathrm{\dots },1\right)}^{T}\in {ℝ}^{{l}^{2}/2},$

and $F=\frac{1}{h}\mathrm{tridiag}\left(-1,1,0\right)\in {ℝ}^{l×l}$, $h=1/\left(l+1\right)$, and $p=2{l}^{2}$, $q={l}^{2}+2$ and $n=p+q=3{l}^{2}+2$. As the matrix B is not of full row rank, the saddle point matrix $\mathcal{𝒜}$ in (1.2) is singular.

In this example, we choose the right-hand-side vector of the saddle point problem (1.1) so that its exact solution is given by the vector ${\left(1,1,\mathrm{\dots },1\right)}^{T}\in {R}^{n+m}$. The numerical results for Example 6.1 are listed in Table 1.

Table 1

Numerical results for Example 6.1.

#### Example 6.2 ([20]).

Consider the following Stokes system:

(6.1)(6.2)

where Ω is the square domain $\left(-1,1\right)×\left(0,1\right)$ in ${ℝ}^{2}$, the vector field $\stackrel{\to }{u}$ represents the velocity in Ω, p represents pressure, and the scalar $\nu >0$ is the kinematic viscosity. A Dirichlet no-flow condition is applied on the side and bottom boundaries. The non-zero horizontal velocity is set on the lid, namely $\partial u/\partial x=1$ on $\left[-1,1\right]×\left\{1\right\}$.

This example is a model of the flow in a square cavity with the lid moving from left to right. By the setting of the non-zero horizontal velocity, we know that the test problem is a leaky two-dimensional lid-driven cavity problem in the square domain. We discretize the Stokes system (6.1)–(6.2) respectively by the Q2-P1 and Q2-Q1 finite elements on some uniform grids and apply the IFISS software package developed by Elman, Ramage, and Silvester [20] to generate the discretized linear systems for the meshes of size $16×16$, $32×32$, $64×64$ and $128×128$. The resulting numerical results are listed in Tables 2 and 3.

Table 2

Numerical results for Example 6.2 discretized by the Q2-P1 finite elements.

Table 3

Numerical results for Example 6.2 discretized by the Q2-Q1 finite elements.

#### Example 6.3 ([20]).

This example still considers the Stokes system (6.1)–(6.2) as in Example 6.2. We set the non-zero horizontal velocity on the top part of the domain, namely $\partial u/\partial x=1$ on $\left(-1,1\right)×\left\{1\right\}$. The test problem describes a watertight cavity problem.

In this example, we still discretize the Stokes system (6.1)–(6.2) respectively by the Q2-P1 and Q2-Q1 finite elements on some uniform grids and apply the IFISS software package developed by Elman, Ramage, and Silvester [20] to generate the discretized linear systems for the meshes of size $16×16$, $32×32$, $64×64$ and $128×128$. The resulting numerical results are listed in Tables 4 and 5.

Table 4

Numerical results for Example 6.3 discretized by the Q2-P1 finite elements.

Table 5

Numerical results for Example 6.3 discretized by the Q2-Q1 finite elements.

#### Example 6.4 ([20]).

This example still considers the Stokes system (6.1)–(6.2) as in Example 6.2, but with the non-zero horizontal velocity on the top part of the domain, namely $\partial u/\partial x=1-{x}^{4}$ on $\left[-1,1\right]×\left\{1\right\}$. This test problem is a regularized cavity.

In this example, we still discretize the Stokes system (6.1)–(6.2) respectively by the Q2-P1 and Q2-Q1 finite elements on some uniform grids and apply the IFISS software package developed by Elman, Ramage, and Silvester [20] to generate the discretized linear systems for the meshes of size $16×16$, $32×32$, $64×64$ and $128×128$. The resulting numerical results are listed in Tables 6 and 7.

Table 6

Numerical results for Example 6.4 discretized by the Q2-P1 finite elements.

Table 7

Numerical results for Example 6.4 discretized by the Q2-Q1 finite elements.

We may observe from the numerical results shown in Tables 17 that our new preconditioner $\mathcal{𝒦}$ in (4.1) is effective and stable. In our computation, we set the symmetric and positive definite matrix ${P}_{C}$ as a scalar matrix, i.e., ${P}_{C}=\alpha {I}_{m}$ with $\alpha =\frac{1}{10}, 1, 10$. By comparing the numerical results of the three different cases, we can find that the performance of the preconditioner $\mathcal{𝒦}$ is not sensitive to the change of the parameter α or even the matrix ${P}_{C}$. Furthermore, with the dimension increased, the advantages on the required CPU time of the preconditioned GMRES(50) coupled with $\mathcal{𝒦}$ have become prominent gradually.

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Revised: 2017-04-01

Accepted: 2017-04-04

Published Online: 2017-05-31

Published in Print: 2018-04-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11071041

Award identifier / Grant number: N_CUHK437/16

The work of the first author was supported by National Postdoctoral Program for Innovative Talents (Grant No. BX201600182), China Postdoctoral Science Foundation (Grant No. 2016M600141) and partially by a Direct Grant for Research from CUHK. The work of the second author was supported by National Natural Science Foundation of China (Grant No. 11071041), Fujian Natural Science Foundation (Grant No. 2016J01005). The work of the third author was substantially supported by Hong Kong RGC grant (project 14306814) and National Natural Science Foundation (NSFC)/Research Grants Council (RGC) Joint Research Scheme 2016/17 (project N_CUHK437/16).

Citation Information: Computational Methods in Applied Mathematics, Volume 18, Issue 2, Pages 237–256, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840,

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