An improvement of the infinity norm bound for the inverse of $\{P_{1},P_{2}\}$-Nekrasov matrices

Abstract

A new upper bound for the infinity norm for the inverse of $\{P_{1},P _{2}\}$-Nekrasov matrices is given. It is proved that the upper bound is sharper than those in Cvetković et al. (Open Math. 13:96–105, 2015) and than well-known Varah’s bound for strictly diagonally dominant matrices. Numerical examples are given to illustrate the corresponding results.

1 Introduction

By $\mathbb{C}^{n\times n}(\mathbb{R}^{n\times n})$ we denote the set of all complex (real) matrices of order n. A matrix $A=[a_{ij}] \in \mathbb{C}^{n\times n}$ is called an H-matrix if its comparison matrix $\langle A\rangle =[m_{ij}]\in \mathbb{R}^{n\times n}$ defined by

$$ m_{ij}= \textstyle\begin{cases} \vert a_{ii} \vert , &i=j, \\ - \vert a_{ij} \vert , &i\neq j, \end{cases} $$

is a nonsingular M-matrix, i.e., $\langle A\rangle ^{-1}\geq 0$ [1, 8, 20].

It is well known that H-matrices are widely used in many subjects such as numerical algebra, the control system, mathematical physics, economics, and dynamical system theory [1, 2, 4, 20]. An important problem among them is to find upper bounds for the infinity norm of the inverse of H-matrices, because it can be used to the convergence analysis of matrix splitting and matrix multi-splitting iterative methods for solving large sparse systems of linear equations [18], as well as linear complementarity problems [10,11,12,13, 19]. For example, when solving linear systems in practice, it is important to have an economical method for estimating the condition number $\kappa (A)$ of the matrix of coefficients, which shows how ‘ill’ the systems could be. Here, the condition number is defined in the following way:

$$ \kappa (A)= \Vert A \Vert \cdot \bigl\Vert A^{-1} \bigr\Vert , $$

as the product of a matrix norm and a norm of its inverse. Hence, it can be useful to determine the upper bound for the norm of the inverse matrix without calculating the inverse.

In 1975, Varah provided a simple and elegant upper bound for the infinity norm of the inverse of strictly diagonally dominant (SDD) matrices as one of the most important subclass of H-matrices. Here a matrix $A=[a_{ij}]\in \mathbb{C}^{n\times n}$ is said to be an SDD matrix if, for each $i\in N:=\{1,2,\ldots ,n\}$,

$$ \vert a_{ii} \vert >r_{i}(A), $$

where $r_{i}(A)=\sum_{j\neq i} |a_{ij}|$.

Theorem 1

([17])

If $A=[a_{ij}]\in \mathbb{C}^{n\times n}$ is SDD, then

$$ \bigl\Vert A^{-1} \bigr\Vert _{\infty }\leq \frac{1}{\min_{i\in N} ( \vert a_{ii} \vert -r _{i}(A) )}. $$

(1)

Bound (1) is usually called Varah’s bound and works only for SDD matrices. Moreover, when the class of involved matrices is a wider subclass of H-matrices, such as doubly strictly diagonally dominant (DSDD) matrices, S-SDD matrices, weakly chained diagonally dominant matrices, Nekrasov matrices, S-Nekrasov matrices, and DZ-type matrices, upper bounds for $\|A^{-1}\|_{\infty }$ are derived, which sometimes are tighter in the SDD case, see [3, 5, 7, 9, 10, 15, 16, 21] and the references therein. Recently, Cvetković et al. [6] presented two upper bounds for $\|A^{-1}\|_{\infty }$ involved with $\{P_{1},P_{2}\}$-Nekrasov matrices, which are only dependent on the entries of the matrix A.

In this paper, we give a new upper bound for the infinity norm of the inverse of $\{P_{1},P_{2}\}$-Nekrasov matrices. It is shown by the comparison theorems that the new bound improves corresponding bounds of Cvetković et al. (2015) for $\{P_{1},P_{2}\}$-Nekrasov matrices and improves well-known Varah’s bound for strictly diagonally dominant matrices. The tested numerical examples show that the new bound is tighter than those derived recently.

2 Main results

First, some notation and definitions are listed. Given a matrix $A=[a_{ij}]\in \mathbb{C}^{n\times n}$, denote

$$\begin{aligned}& d(A)=\bigl( \vert a_{11} \vert ,\ldots , \vert a_{nn} \vert \bigr)^{T}; \end{aligned}$$

(2)

$$\begin{aligned}& z_{1}(A)=1,\qquad z_{i}(A)=\sum _{j=1}^{i-1} \frac{ \vert a_{ij} \vert }{ \vert a_{jj} \vert }z_{j}(A)+1, \quad i=2,3,\ldots ,n; \end{aligned}$$

(3)

and

$$ h_{1}(A)=\sum_{j\neq 1} \vert a_{1j} \vert , \qquad h_{i}(A)=\sum _{j=1} ^{i-1}\frac{ \vert a_{ij} \vert }{ \vert a_{jj} \vert }h_{j}(A)+\sum _{j=i+1}^{n} \vert a_{ij} \vert , \quad i=2,3,\ldots ,n. $$

(4)

Next, we recall the concept of $\{P_{1}, P_{2}\}$-Nekrasov matrices.

Definition 1

([14])

A matrix $A=[a_{ij}]\in \mathbb{C}^{n\times n}$ is called a Nekrasov matrix if, for each $i\in N$,

$$ \vert a_{ii} \vert >h_{i}(A). $$

Motivated by Definition 1, Cvetković et al. in [6] presented the following new subclass of H-matrices, called $\{P_{1},P_{2}\}$-Nekrasov matrices, which contains Nekrasov matrices.

Definition 2

([6])

Given two permutation matrices $P_{1}$ and $P_{2}$, a matrix $A=[a_{ij}]\in \mathbb{C}^{n\times n}$, $n\geq 2$, is called a $\{P_{1},P_{2}\}$-Nekrasov matrix if

$$ d(A)>\min \bigl\{ h^{P_{1}}(A),h^{P_{2}}(A)\bigr\} , $$

where

$$ h^{P_{k}}(A)=P_{k}h\bigl(P_{k}^{T}AP_{k} \bigr),\quad k=1,2, $$

with

$$ h\bigl(P_{k}^{T}AP_{k}\bigr)= \bigl[h_{1}\bigl(P_{k}^{T}AP_{k}\bigr), \ldots ,h_{n}\bigl(P_{k} ^{T}AP_{k} \bigr) \bigr]^{T}, $$

in which $h_{i}(P_{k}^{T}AP_{k})$, $i\in N$, being defined as (4).

Remark here that if $P_{1}=P_{2}=I$, where I is an identity matrix, then $h^{P_{1}}(A)=h^{P_{2}}(A)=h(A)$, which implies that a Nekrasov matrix is a $\{P_{1},P_{2}\}$-Nekrasov matrix for $P_{1}=P_{2}=I$. In addition, note that for any permutation matrix P, the matrix $P^{T}AP$ has the same set of diagonal entries as does A and, moreover, the same set of row sums as does A. Hence, if A is an SDD matrix, then $|a_{ii}|>r_{i}(A)>h_{i}^{P_{k}}(A)$ holds for all $i\in N$, which means that an SDD matrix is a $\{P_{1},P_{2}\}$-Nekrasov matrix for any $\{P_{1},P_{2}\}$.

Next, we recall two upper bounds for the infinity norm of the inverse of $\{P_{1},P_{2}\}$-Nekrasov matrices which are given by Cvetković et al. in [6].

Theorem 2

([6])

Given a set of permutation matrices $\{P_{1},P_{2}\}$, let $A=[a_{ij}] \in \mathbb{C}^{n\times n}$ be a $\{P_{1},P_{2}\}$-Nekrasov matrix. Then

$$ \bigl\Vert A^{-1} \bigr\Vert _{\infty }\leq \frac{\max_{i\in N} (\frac{z_{i} ^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert } )}{\min_{i\in N} (1- \min \{\frac{h_{i}^{P_{1}}(A)}{ \vert a_{ii} \vert },\frac{h_{i}^{P_{2}}(A)}{ \vert a _{ii} \vert } \} )}, $$

(5)

and

$$ \bigl\Vert A^{-1} \bigr\Vert _{\infty }\leq \frac{\max_{i\in N} ({z_{i}^{P _{k_{i}}}(A)} )}{\min_{i\in N} ({ \vert a_{ii} \vert }-\min \{{h_{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)} \} )}, $$

(6)

where $z^{P_{k_{i}}}(A)=P_{k_{i}}z(P_{k_{i}}^{T}AP_{k_{i}})=[z_{i} ^{P_{k_{i}}}(A),\ldots ,z_{n}^{P_{k_{i}}}(A)]^{T}$ with $z(P_{k_{i}} ^{T}AP_{k_{i}})$ being defined as (3), $h_{i}^{P_{1}}(A)$ and $h_{i}^{P_{2}}(A)$ are given by Definition 2, and for each index i, the corresponding index $k_{i}\in \{1,2\}$ is chosen in such a way that

$$ \min \bigl\{ {h_{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)} \bigr\} ={h_{i}^{P_{k_{i}}}(A)}. $$

In what follows, we give a new upper bound for the infinity norm of the inverse of $\{P_{1},P_{2}\}$-Nekrasov matrices. Before that, some lemmas and notation which will be used later are listed.

Given a matrix $A=[a_{ij}]\in \mathbb{C}^{n\times n}$, by $A=D-L- U$ we denote the standard splitting of A into its diagonal $(D)$, strictly lower $(-L)$, and strictly upper $(-U)$ triangular parts, and $|A|=[|a_{ij}|]$.

Given a $\{P_{1},P_{2}\}$-Nekrasov matrix $A=[a_{ij}]\in \mathbb{C} ^{n\times n}$, $n\geq 2$, we recall two special matrices $C\in \mathbb{C}^{n\times n}$ and $\widetilde{C}\in \mathbb{C}^{n\times n}$ as follows:

$$ C := \begin{bmatrix} C(1) \\ C(2) \\ \vdots \\ C(n) \end{bmatrix} \quad \mbox{and}\quad \widetilde{C}:= \begin{bmatrix} \widetilde{C}(1) \\ \widetilde{C}(2) \\ \vdots \\ \widetilde{C}(n) \end{bmatrix}, $$

(7)

where

$$ C(i)=e_{i}^{T}P_{k_{i}} \bigl( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert \bigr)^{-1} \vert U _{k_{i}} \vert P_{k_{i}}^{T} $$

and

$$ \widetilde{C}(i)=e_{i}^{T}P_{k_{i}} \bigl( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert \bigr) ^{-1}P_{k_{i}}^{T}, $$

with $e_{i}=(0,\ldots ,1,\ldots ,0)^{T}$ and for each index i, the corresponding index $k_{i}\in \{1,2\}$ is chosen in the same way given in Theorem 2.

Lemma 1

([6])

Let $A=[a_{ij}]\in \mathbb{C}^{n\times n}$, $n\geq 2$, be a $\{P_{1},P_{2}\}$-Nekrasov matrix, then the matrix $I-C$ is an SDD matrix, where I is the identify matrix and C is defined as in (7).

Lemma 2

([1])

Let $A=[a_{ij}]\in \mathbb{C}^{n\times n}$ be a nonsingular H-matrix. Then

$$ \bigl\vert A^{-1} \bigr\vert \leq \langle A\rangle ^{-1}. $$

Lemma 3

([6])

Given any $A=[a_{ij}]\in \mathbb{C}^{n\times n}$, $n\geq 2$, with $a_{ii}\neq 0$ for all $i\in N$, and given a permutation matrix $P\in \mathbb{R}^{n\times n}$, then

$$ h_{i}^{P}(A)= \vert a_{ii} \vert \bigl[P \bigl( \vert \widetilde{D} \vert - \vert \widetilde{L} \vert \bigr)^{-1} \vert \widetilde{U} \vert e \bigr]_{i}, $$

where $e=(1,1,\ldots ,1)^{T}$ and $P^{T}AP=\widetilde{D}- \widetilde{L}-\widetilde{U}$ is the standard splitting of the matrix $P^{T}AP$.

The following lemma will be used in the proof of Theorem 3.

Lemma 4

Let $A=[a_{ij}]\in \mathbb{C}^{n\times n}$ with $a_{ii}\neq 0$ for all $i\in N$, and $P\in \mathbb{R}^{n\times n}$ is a permutation matrix. Then

$$ z^{P}(A)= \vert D \vert \bigl[P\bigl( \vert \widetilde{D} \vert - \vert \widetilde{L} \vert \bigr)^{-1} \bigr]e, $$

where $z^{P}(A)=Pz(P^{T}AP)$ with $z(P^{T}AP)$ being defined as (3), $P^{T}AP=\widetilde{D}-\widetilde{L}-\widetilde{U}$ is the standard splitting of the matrix $P^{T}AP$, and $e=(1,1,\ldots ,1)^{T}$.

Proof

Let $x:=(\widetilde{D}-\widetilde{L})^{-1}e=(x_{1},x_{2},\ldots ,x _{n})^{T}$. Then

$$ e= (\widetilde{D}-\widetilde{L} )x, $$

i.e.,

$$ \widetilde{D}x=\widetilde{L}x+e. $$

(8)

By (8), we have

$$ \vert \widetilde{a}_{11} \vert x_{1}=1,\qquad \vert \widetilde{a}_{ii} \vert x_{i}=1+\sum _{j=1}^{i-1} \vert a_{ij} \vert x_{j},\quad i=2,\ldots ,n, $$

which implies that

$$ \vert \widetilde{D} \vert \bigl( \vert \widetilde{D} \vert - \vert \widetilde{L} \vert \bigr)^{-1}e=z\bigl(P^{T}AP\bigr). $$

Therefore,

$$ P \vert \widetilde{D} \vert \bigl( \vert \widetilde{D} \vert - \vert \widetilde{L} \vert \bigr)^{-1}e=Pz\bigl(P^{T}AP\bigr). $$

Note that $|\widetilde{D}|=P^{T}|D|P$ and $P^{T}P=I$, we can see that

$$ \vert D \vert \bigl[P\bigl( \vert \widetilde{D} \vert - \vert \widetilde{L} \vert \bigr)^{-1} \bigr]e=z^{P}(A). $$

This completes the proof. □

Now, we give the main result of this paper by Lemmas 1, 2, 3, and 4.

Theorem 3

Let $A=[a_{ij}]\in \mathbb{C}^{n\times n}$, $n\geq 2$, be a $\{P_{1},P _{2}\}$-Nekrasov matrix. Then

$$ \bigl\Vert A^{-1} \bigr\Vert _{\infty }\leq \max_{i\in N}\frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a _{ii} \vert -\min\{{h_{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)}\}}, $$

(9)

where $z_{i}^{P_{k_{i}}}(A)$ and $h_{i}^{P_{k_{i}}}(A)$, $i \in N$, $k _{i}\in \{1,2\}$ are given by Theorem 2.

Proof

Since $A=[a_{ij}]\in \mathbb{C}^{n\times n}$ is a $\{P_{1},P_{2}\}$-Nekrasov matrix, then from Lemma 1 we have

$$ B:=I-C $$

is an SDD matrix, where C is given by (7). By the proof of Theorem 3.1 (see[6]), we have that, for a fixed $k\in \{1,2 \}$,

$$ I-P_{k} \bigl( \vert D_{k} \vert - \vert L_{k} \vert \bigr)^{-1} \vert U_{k} \vert P_{k}^{T}=P_{k} \bigl( \vert D _{k} \vert - \vert L_{k} \vert \bigr)^{-1}P_{k}^{T} \langle A\rangle . $$

(10)

By (7) and (10), we have

$$ B:=I-C=\widetilde{C}\langle A\rangle , $$

which implies that

$$ \langle A\rangle ^{-1}=B^{-1}\widetilde{C}=B^{-1} \triangle \cdot \triangle ^{-1} \widetilde{C}, $$

where

$$ \triangle =\operatorname{diag}(\delta _{1},\delta _{2}, \ldots ,\delta _{n}),\quad \delta _{i}>0, i=1,2,\ldots ,n. $$

Since a $\{P_{1},P_{2}\}$-Nekrasov matrix is an H-matrix, we have from Lemma 2 that

$$ \bigl\Vert A^{-1} \bigr\Vert _{\infty }\leq \bigl\Vert \langle A\rangle ^{-1} \bigr\Vert _{\infty }\leq \bigl\Vert B^{-1}\triangle \bigr\Vert _{\infty }\cdot \bigl\Vert \triangle ^{-1} \widetilde{C} \bigr\Vert _{\infty }. $$

(11)

First, we estimate $\|\triangle ^{-1}\widetilde{C}\|_{\infty }$. Because $|D_{k_{i}}|-|L_{k_{i}}|$ for $k_{i}\in \{1,2\}$ is an M-matrix, so we can take a positive diagonal matrix $\triangle =\operatorname{diag}(\delta _{1}, \delta _{2},\ldots ,\delta _{n})$ with

$$ \delta _{i}=\bigl[P_{k_{i}} \bigl( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert \bigr)^{-1}e \bigr]_{i} \quad \mbox{for all } i\in N. $$

(12)

Note that

$$ \widetilde{C}:= \begin{bmatrix} \widetilde{C}(1) \\ \widetilde{C}(2) \\ \vdots \\ \widetilde{C}(n) \end{bmatrix}\in \mathbb{C}^{n\times n}, $$

where

$$ \widetilde{C}(i)=e_{i}^{T}P_{k_{i}} \bigl( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert \bigr) ^{-1}P_{k_{i}}^{T},\quad k_{i}\in \{1,2\}. $$

It follows from (12) that

$$\begin{aligned} \bigl\Vert \triangle ^{-1} \widetilde{C} \bigr\Vert _{\infty } =&\max_{i\in N} \bigl\{ \bigl[ \triangle ^{-1} \widetilde{C}e\bigr]_{i} \bigr\} =\max _{i\in N} \biggl\{ \frac{[P_{k_{i}} ( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert )^{-1}P _{k_{i}}^{T}e]_{i}}{\delta _{i}} \biggr\} \\ =&\max_{i\in N} \biggl\{ \frac{[P_{k_{i}} ( \vert D_{k_{i}} \vert - \vert L _{k_{i}} \vert )^{-1}e]_{i}}{\delta _{i}} \biggr\} \\ =&1. \end{aligned}$$

(13)

Next, we estimate $\|B^{-1}\triangle \|_{\infty }$. Observe that $B:=I-C$, where C is given by (7). Then, for each $i\in N$, we have

$$\begin{aligned}{} [Be]_{i} =&\bigl[(I-C)e\bigr]_{i} =1-\bigl[P_{k_{i}} \bigl( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert \bigr) ^{-1} \vert U_{k_{i}} \vert P_{k_{i}}^{T}e \bigr]_{i} \\ =&1-\bigl[P_{k_{i}} \bigl( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert \bigr)^{-1} \vert U_{k_{i}} \vert e \bigr]_{i} \\ =&1-\frac{h_{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert }\quad (\mbox{by Lemma 3}) \\ =&1-\min \biggl\{ \frac{h_{i}^{P_{1}}(A)}{ \vert a_{ii} \vert },\frac{h_{i}^{P _{2}}(A)}{ \vert a_{ii} \vert } \biggr\} . \end{aligned}$$

By Lemma 1, we have B is an SDD matrix. Since $\triangle ^{-1}$ is a positive diagonal matrix, it holds that $\triangle ^{-1}B$ is also an SDD matrix. Hence, applying Varah’s bound (1), we have

$$\begin{aligned} \bigl\Vert B^{-1}\triangle \bigr\Vert _{\infty } \leq& \frac{1}{\min_{i\in N}( \triangle ^{-1}Be)_{i}}= \frac{1}{\min_{i\in N}[\frac{1}{\delta _{i}}\cdot (Be)_{i}]} =\max_{i\in N} \frac{\delta _{i}}{1- \min \{\frac{h_{i}^{P_{1}}(A)}{ \vert a_{ii} \vert },\frac{h_{i}^{P_{2}}(A)}{ \vert a _{ii} \vert }\}} \\ =&\max_{i\in N}\frac{\delta _{i} \vert a_{ii} \vert }{ \vert a_{ii} \vert -\min \{h _{i}^{P_{1}}(A),h_{i}^{P_{2}}(A)\}}. \end{aligned}$$

By Lemma 4, we have

$$ z_{i}^{P_{k_{i}}}(A)= \vert a_{ii} \vert \bigl[P_{k_{i}}\bigl( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert \bigr)^{-1}e\bigr]_{i},\quad i\in N, $$

which together with (12) implies that

$$\begin{aligned} \delta _{i} \vert a_{ii} \vert =z_{i}^{P_{k_{i}}}(A), \quad i\in N. \end{aligned}$$

Hence,

$$ \bigl\Vert B^{-1}\triangle \bigr\Vert _{\infty }\leq \max_{i\in N}\frac{\delta _{i} \vert a_{ii} \vert }{ \vert a_{ii} \vert -\min \{h_{i}^{P_{1}}(A),h_{i}^{P_{2}}(A)\}} =\max _{i\in N}\frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert -\min \{h_{i} ^{P_{1}}(A),h_{i}^{P_{2}}(A)\}}, $$

(14)

where $k_{i}\in \{1,2\}$ is chosen in such a way that

$$ \min \bigl\{ {h_{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)} \bigr\} ={h_{i}^{P_{k_{i}}}(A)}\quad \mbox{for each }i \in N. $$

Now, the conclusion follows from (11), (13), and (14). □

The following comparison theorem shows that bound (9) of Theorem 3 is better than bounds (5) and (6) of Theorem 2 (bound (8) of Theorem 3.1 and bound (10) of Theorem 3.2 in [6]).

Theorem 4

Let $A=[a_{ij}]\in \mathbb{C}^{n\times n}$, $n\geq 2$, be a $\{P_{1},P _{2}\}$-Nekrasov matrix. Then

$$ \max_{i\in N}\frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert -\min \{{h _{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)}\}}\leq \frac{\max_{i\in N} (\frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert } )}{\min_{i \in N} (1-\min \{\frac{h_{i}^{P_{1}}(A)}{ \vert a_{ii} \vert },\frac{h _{i}^{P_{2}}(A)}{ \vert a_{ii} \vert } \} )} $$

(15)

and

$$ \max_{i\in N}\frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert -\min \{{h _{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)}\}}\leq \frac{\max_{i\in N} ({z_{i}^{P_{k_{i}}}(A)} )}{\min_{i\in N} ({ \vert a _{ii} \vert }-\min \{{h_{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)}\} )}. $$

(16)

Furthermore, equality in (15) holds if and only if, for certain $l\in N$, the following two conditions hold:

$$ \max_{i\in N} \biggl(\frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert } \biggr)= \frac{z _{l}^{P_{k_{l}}}(A)}{ \vert a_{ll} \vert } $$

and

$$ \max_{i\in N} \biggl(\min \biggl\{ \frac{h_{i}^{P_{1}}(A)}{ \vert a _{ii} \vert }, \frac{h_{i}^{P_{2}}(A)}{ \vert a_{ii} \vert } \biggr\} \biggr)=\min \biggl\{ \frac{h _{l}^{P_{1}}(A)}{ \vert a_{ll} \vert }, \frac{h_{l}^{P_{2}}(A)}{ \vert a_{ll} \vert } \biggr\} . $$

Similarly, equality in (16) holds if and only if, for certain $l\in N$, the following two conditions hold:

$$ \max_{i\in N} \bigl(z_{i}^{P_{k_{i}}}(A) \bigr)=z_{l}^{P_{k _{l}}}(A) $$

(17)

and

$$ \min_{i\in N} \bigl({ \vert a_{ii} \vert }-\min \bigl\{ {h_{i}^{P_{1}}(A)},{h _{i}^{P_{2}}(A)} \bigr\} \bigr)={ \vert a_{ll} \vert }-\min \bigl\{ {h_{l}^{P_{1}}(A)},{h _{l}^{P_{2}}(A)}\bigr\} . $$

(18)

Proof

It is easy to see that inequality in (15) holds, and the inequality in (16) also holds if we use the following relation:

$$ \max_{i\in N}\frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert -\min \{{h _{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)}\}}=\max_{i\in N} \frac{\frac{z _{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert }}{1-\min \{\frac{h_{i}^{P_{1}}(A)}{ \vert a _{ii} \vert },\frac{h_{i}^{P_{2}}(A)}{ \vert a_{ii} \vert } \}}. $$

Next, we prove that the case of equality in (16) holds if and only if (17) and (18) hold. Suppose that

$$ \max_{i\in N}\frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert -\min \{{h _{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)}\}}=\frac{\max_{i\in N} ({z_{i}^{P_{k_{i}}}(A)} )}{\min_{i\in N} ({ \vert a _{ii} \vert }-\min \{{h_{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)}\} )}. $$

Note that

$$ \max_{i\in N}\frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert -\min \{{h _{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)}\}}=\frac{z_{l}^{P_{k_{l}}}(A)}{ \vert a _{ll} \vert -\min \{{h_{l}^{P_{1}}(A)},{h_{l}^{P_{2}}(A)}\}}\quad \mbox{for some } l \in N. $$

Therefore,

$$ \frac{\max_{i\in N} ({z_{i}^{P_{k_{i}}}(A)} )}{ \min_{i\in N} ({ \vert a_{ii} \vert }-\min \{{h_{i}^{P_{1}}(A)},{h _{i}^{P_{2}}(A)}\} )}=\frac{z_{l}^{P_{k_{l}}}(A)}{ \vert a_{ll} \vert - \min \{{h_{l}^{P_{1}}(A)},{h_{l}^{P_{2}}(A)}\}}. $$

(19)

Since

$$ { \vert a_{ll} \vert -\min \bigl\{ {h_{l}^{P_{1}}(A)},{h_{l}^{P_{2}}(A)} \bigr\} }\geq \min_{i\in N} \bigl({ \vert a_{ii} \vert }-\min \bigl\{ {h_{i}^{P_{1}}(A)},{h_{i} ^{P_{2}}(A)}\bigr\} \bigr), $$

it follows from (21) that

$$ z_{l}^{P_{k_{l}}}(A)\geq \max_{i\in N} \bigl(z_{i}^{P_{k_{i}}}(A) \bigr), $$

which implies (17) holds, and thus (18) holds from (17) and (19).

Conversely, if conditions (17) and (18) hold for some $l\in N$, then we have

$$\begin{aligned} \frac{\max_{i\in N} ({z_{i}^{P_{k_{i}}}(A)} )}{ \min_{i\in N} ({ \vert a_{ii} \vert }-\min \{{h_{i}^{P_{1}}(A)},{h _{i}^{P_{2}}(A)}\} )} =&\frac{z_{l}^{P_{k_{l}}}(A)}{{ \vert a_{ll} \vert }- \min \{{h_{l}^{P_{1}}(A)},{h_{l}^{P_{2}}(A)}\}} \\ \leq &\max_{i\in N}\frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert - \min \{{h_{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)}\}} \\ \leq &\frac{\max_{i\in N} ({z_{i}^{P_{k_{i}}}(A)} )}{ \min_{i\in N} ({ \vert a_{ii} \vert }-\min \{{h_{i}^{P_{1}}(A)},{h _{i}^{P_{2}}(A)}\} )}, \end{aligned}$$

this implies that the equality in (16) holds. The equality in (15) can also be proved in a similar way. The proof is completed. □

Since an SDD matrix is a $\{P_{1},P_{2}\}$-Nekrasov matrix, by Theorem 3, a new upper bound for $\|A^{-1}\|_{\infty }$ when A is an SDD matrix can be obtained. As expected, the following theorem shows that this new bound works better than Varah’s bound of Theorem 1.

Theorem 5

Let $A=[a_{ij}]\in \mathbb{C}^{n\times n}$ be an SDD matrix. Then, for any set of permutation matrices $\{P_{1},P_{2}\}$,

$$ \bigl\Vert A^{-1} \bigr\Vert _{\infty }\leq \max_{i\in N}\frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a _{ii} \vert -\min \{{h_{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)} \}}. $$

(20)

Furthermore,

$$ \max_{i\in N}\frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert -\min \{{h _{i}^{P_{1}}(A)},{h_{i}^{P_{2}}(A)} \}}\leq \frac{1}{\min_{i\in N}\{ \vert a_{ii} \vert -r_{i}(A)\}}. $$

(21)

Proof

Since an SDD matrix is a $\{P_{1},P_{2}\}$-Nekrasov matrix for any permutation matrices $P_{1}$ and $P_{2}$, so (20) directly follows from Theorem 2. We next prove that (21) holds.

Let $d:=d(A)e=|D|e$ and $P_{k_{i}}^{T}AP_{k_{i}}=D_{k_{i}}-U_{k_{i}}-L _{k_{i}}$ be the standard splitting of the matrix $P_{k_{i}}^{T}AP _{k_{i}}$ for $k_{i}\in \{1,2\}$. Then, by Lemma 3, Lemma 4, and $\langle P_{k_{i}}^{T}AP_{k_{i}}\rangle =|D_{k_{i}}|-|U_{k_{i}}|-|L _{k_{i}}|$, we have

$$\begin{aligned} d-h^{P_{k_{i}}}(A) =& \vert D \vert e-P_{k_{i}} \vert D_{k_{i}} \vert \bigl( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert \bigr)^{-1} \vert U _{k_{i}} \vert e \\ =&P_{k_{i}} \vert D_{k_{i}} \vert P_{k_{i}}^{T}e-P_{k_{i}} \vert D_{k_{i}} \vert \bigl( \vert D_{k_{i}} \vert - \vert L _{k_{i}} \vert \bigr)^{-1} \vert U_{k_{i}} \vert e \\ =&P_{k_{i}} \vert D_{k_{i}} \vert e-P_{k_{i}} \vert D_{k_{i}} \vert \bigl( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert \bigr)^{-1} \vert U _{k_{i}} \vert e \\ =&P_{k_{i}} \vert D_{k_{i}} \vert \bigl(I-\bigl( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert \bigr)^{-1} \vert U_{k_{i}} \vert \bigr)e \\ =&P_{k_{i}} \vert D_{k_{i}} \vert \bigl( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert \bigr)^{-1} \cdot \bigl\langle P_{k_{i}} ^{T}AP_{k_{i}}\bigr\rangle e \\ \geq &\min_{i\in N}\bigl\{ \bigl\langle P_{k_{i}}^{T}AP_{k_{i}} \bigr\rangle e\bigr\} _{i}\cdot P _{k_{i}} \vert D_{k_{i}} \vert \bigl( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert \bigr)^{-1}e \\ =&\min_{i\in N}\bigl\{ \bigl\langle P_{k_{i}}^{T}AP_{k_{i}} \bigr\rangle e\bigr\} _{i}\cdot \vert D \vert P _{k_{i}} \bigl( \vert D_{k_{i}} \vert - \vert L_{k_{i}} \vert \bigr)^{-1}e \\ =&\min_{i\in N}\bigl\{ \bigl\langle P_{k_{i}}^{T}AP_{k_{i}} \bigr\rangle e\bigr\} _{i}\cdot z^{P _{k_{i}}}(A), \end{aligned}$$

which implies that

$$\begin{aligned} \vert a_{ii} \vert -h_{i}^{P_{k_{i}}}(A) \geq &\min _{i\in N}\bigl\{ \bigl\langle P_{k_{i}} ^{T}AP_{k_{i}} \bigr\rangle e\bigr\} _{i}\cdot z_{i}^{P_{k_{i}}}(A) \\ =&\min_{i\in N} \bigl\{ \bigl\vert \bigl(P_{k_{i}}^{T}AP_{k_{i}} \bigr)_{ii} \bigr\vert -r_{i}\bigl(P _{k_{i}}^{T}AP_{k_{i}} \bigr) \bigr\} \cdot z_{i}^{P_{k_{i}}}(A), \quad i\in N. \end{aligned}$$

(22)

It is easy to see that, for a given permutation matrix $P_{k_{i}}$, the matrix $P_{k_{i}}^{T}AP_{k_{i}}$ has the same set of diagonal entries as does A and the same set of row (and column) sums as does A. Therefore,

$$ \min_{i\in N} \bigl\{ \bigl\vert \bigl(P_{k_{i}}^{T}AP_{k_{i}} \bigr)_{ii} \bigr\vert -r_{i}\bigl(P _{k_{i}}^{T}AP_{k_{i}} \bigr) \bigr\} =\min_{i\in N} \bigl\{ \vert a_{ii} \vert -r _{i}(A) \bigr\} , $$

which together with (22) implies that

$$ \frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert -\min \{h_{i}^{P_{1}}(A),h_{i} ^{P_{2}}(A)\}}\leq \frac{z_{i}^{P_{k_{i}}}(A)}{ \vert a_{ii} \vert -h_{i}^{P_{k _{i}}}(A)}\leq \frac{1}{\min_{i\in N}( \vert a_{ii} \vert -r_{i}(A))},\quad i \in N. $$

This completes the proof. □

3 Numerical examples

In this section, we give the numerical example to show that bound (9) in Theorem 3 improves bounds (5) and (6) of Theorem 2, and Varah’s bound of Theorem 1.

Example 1

Consider the following matrices in [6]:

$$\begin{aligned}& A_{1} = \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} 12 &2 &0 &0 &0 &0 &0 &0 &0 &0 &0 & 0 \\ 2 &12 &2 &0 &0 &0 &0 &0 &0 &0 &0 & 0 \\ 0 &0 &12 &0 &1 &0 &0 &0 &0 &0 &0 & 0 \\ 1 &0 &1 &8 &0 &0 &0 &0 &0 &1 &0 & 0 \\ 0 &0 &0 &1 &12 &1 &0 &0 &2 &2 &0 & 0 \\ 0 &0 &2 &2 &2 &12 &0 &0 &0 &0 &0 & 0 \\ 0 &0 &0 &0 &0 &0 &12 &2 &0 &0 &0 & 0 \\ 0 &0 &0 &0 &0 &0 &2 &114 &2 &0 & 0&0 \\ 0 &0 &0 &0 &0 &1 &0 &0 &14 &0 &1 & 0 \\ 0 &0 &0 &0 &2 &2 &1 &0 &1 &814 &0 & 0 \\ 0 &0 &0 &0 &0 &0 &0 &0 &0 &3 & 8&1 \\ 0 &0 &0 &0 &0 &0 &0 &0 &2 &2 & 2&8 \end{array}\displaystyle \right ), \\& A_{2}= \begin{pmatrix} 7 &-2 &1 &-2 &0 &0 &0 &0 &0 &0 \\ -1 &7 &0 &0 &0 &0 &0 &0 &0 &0 \\ 0 &1 &8 &4 &1 &-2 &0 &0 &0 &0 \\ -2 &0 &1 &7 &0 &0 &0 &0 &0 &0 \\ 0 &0 &0 &1 &8 &1 &0 &0 &0 &0 \\ 0 &0 &2 &2 &2 &7 &0 &0 &0 &0 \\ 0 &0 &0 &0 &0 &0 &6 &2 &0 &0 \\ -2 &0 &0 &0 &0 &0 &2 &8 &0 &0 \\ 0 &-2 &0 &0 &0 &1 &0 &0 &5 &0 \\ 0 &0 &-2 &0 &0 &-1 &0 &-1 &0 &8 \end{pmatrix}, \\& A_{3}= \begin{pmatrix} -1.5 &-0.1 &0 &-0.1 &0 &0 \\ -0.1 & 2 &-0.1 &-1.9 &0 &0 \\ 0 &-0.1 &23 &-0.1 &-0.1 &-0.1 \\ 0 & 0 &-0.5 &44 &0 &0 \\ 0 & 0 &0 &-0.1 &44 &-0.4 \\ 0 & 0 &-0.5 &0 &-1 &1 \end{pmatrix}. \end{aligned}$$

Obviously, $A_{1}$ is an SDD matrix, and thus it is a $\{P_{1},P _{2}\}$-Nekrasov matrix for any set of permutations $\{P_{1},P_{2}\}$. As reported in [6], $A_{2}$ is a Nekrasov matrix and $A_{3}$ is neither SDD nor Nekrasov matrix, but they are both $\{P_{1},P_{2}\}$-Nekrasov matrices for choosing identical permutation $P_{1}$ and counteridentical permutation $P_{2}$. Hence, by bound (1) of Theorem 1, bounds (5) and (6) of Theorem 2, and bound (9) of Theorem 3, we can compute the upper bounds for the infinity norm of the inverse of $A_{i}$, $i=1,2,3$, which are shown in Table 1 (in Table 1 we call bounds (5) and (6) $\{P_{1},P_{2}\}$-Nek I and $\{P_{1},P_{2}\}$-Nek II).

Table 1 The upper bounds for $\|A_{i}^{-1}\|_{\infty }$, $i=1,2,3$

It can be seen from Table 1 that bound (9) in Theorem 3 is better than Varah’s bound for strictly diagonally dominant matrices, and it is also better than (5) and (6) in Theorem 2 (Theorem 3.1 and Theorem 3.2 in [6]) for $\{P_{1},P_{2}\}$-Nekrasov matrices.

4 Conclusions

In this paper, we presented a new upper bound for the infinity norm of the inverse of $\{P_{1},P_{2}\}$-Nekrasov matrices and proved that the new bound improves those bounds obtained in [6] for $\{P_{1},P_{2}\}$-Nekrasov matrices and well-known Varah’s bound for strictly diagonally dominant matrices. Numerical examples were included to illustrate the corresponding results.