From r05zw@cx1.cyf.gov.pl Fri Nov 17 03:50:30 1995 Return-Path: Received: from cx1.cyf.gov.pl by clinch.math.niu.edu (4.1/SMI-4.1) id AA08749; Fri, 17 Nov 95 03:50:11 CST Received: from localhost by cx1.cyf.gov.pl (8.6.4/11.0) id KAA03644; Fri, 17 Nov 1995 10:50:45 +0100 Date: Fri, 17 Nov 1995 10:50:45 +0100 From: r05zw@cx1.cyf.gov.pl (Zbigniew Woznicki) Message-Id: <199511170950.KAA03644@cx1.cyf.gov.pl> To: dattab@math.niu.edu X-Status: Status: RO Dear Biswa, Do you remember that in Hamburg you promissed me to send your new book??? In the mean time I wrote a new work which as a LaTeX file is enclosed below. As you can see, the paper is more a criticism of (parts of) Axelsson's book than it is a research paper. Per haps, it may looks as an attack on Axelsson which in my feeling is careless, especially about giving others credit. However, writting this manuscript I thought mainly on reader's benefits, especially I would like to caution of potential readers of the Axelsson's book. Any comments on your part would certainly be greatly appreciated. Today, my daughter Agnieszka stopped to be a student and she became Master of Science in biological sciences. Best regards also from Agnieszka, Zbyszek ENCLOSURE (LaTeX file): \documentstyle[11pt]{article} \topmargin=-15mm \evensidemargin=-0.5mm \oddsidemargin=-0.5mm \textwidth=155mm \textheight=235mm \newcommand{\RR}{{\rm I}\!{\rm R}} \title{\Large \bf Axelsson's Results Versus the Splitting Principles} \author{{\bf Zbigniew I. Wo\'znicki}\\ Institute of Atomic Energy\\ 05--400 Otwock-\'Swierk, Poland\\ E-mail: r05zw@cx1.cyf.gov.pl} \date{November 1995} \setlength{\parindent}{1cm} \begin{document} \maketitle \begin{abstract} Axelsson in Chapter 6 of his recent book "Iterative Solution Methods" (1994, Cambridge Univ.Press) discusses convergence aspects for basic iterative methods where some essential results are disregarded. The material of this chapter, related with convergent splittings and comparison theorems, is plagued by erroneous results which may suggest that the author of the book is not entirely awared of the splitting principle. Moreover, some results are given with rather avaricious biographical (and not always strict) information. Since this book is recommended as a valuable resource for education needs, therefore it seems to be desirable to comment some of these results in a more detailed form; where considerations are restricted only to two sections of Chapter 6. \end{abstract} \begin{center}{\large \bf 1. Comments to Section 6.2 in [1]} \end{center} %\section{Introduction} In [1,p.213] Axelsson consideres the question of convergence of the basic iterative method $C(x^{l+1} - x^{l}) =b - Ax^{l}$ or \[ Cx^{l+1} = Rx^{l} + b, \;\; l=0,1, \ldots \] where $A=C-R$ is a splitting of $A$, with the following definition \begin{quote} {\bf Definition 6.6} {\em Let $C,R\in \RR ^{n\times n}$. Then $A=C-R$ is called:} \vspace*{-2mm} \end{quote} \begin{description} \item[{\em \ \ (a)}] a {\em regular splitting} (Varga, 1962) if $C$ is monotone and $R \geq 0$. \vspace*{-3mm} \item[{\em \ \ (b)}] a {\em weak regular splitting} (Ortega and Rheinboldt, 1970) if $C$ is monotone and \\ $C^{-1}R \geq 0$. \vspace*{-3mm} \item[{\em \ \ (c)}] a {\em nonnegative splitting} (Beauwens, 1979, and Song, 1991) if $C$ is nonsingular and \\ $C^{-1}R~\geq~0$. \vspace*{-3mm} \item[{\em \ \ (d)}] a {\em convergent splitting} if $C$ is nonsingular and $\rho (C^{-1}R) < 1$. \end{description} It should be remarked that the definition given in item {\em (b)} does not correspond to the original definition introduced by Ortega and Rheinboldt [6]. According to them $A=C-R$ is a weak regular splitting of $A$ if $C$ is monotone, $C^{-1}R \geq 0$ and $RC^{-1} \geq 0$. The last condition is often ignored also by other authors [4,5,7] but as will be shown in Section 3 just this condition has the essential meaning for comparison theorems. On the same page Theorem 6.16 summarizing well known results is given together with the proof because this theorem was stated by Beauwens without the proof [2]. As can be seen in [2], there is a sufficient information on the proof. At the end of this section an erroneous example of a weak regular splitting of an $2 \times 2$ matrix $A$ is given in the item {\em (c)} of Example 6.6, which is used in the next section for showing that the result proven by the author for regular splittings does not carry over to the case of weak regular splittings defined by Axelsson. \newpage \begin{center}{\large \bf 2. Comments to Section 6.3 in [1]} \end{center} In this section [1,p.215] some comparison results, based mainly on a paper by Beauwens [2] with reference to works [3,5] and the author's thesis [8], are given. As a matter of strict fact, it should be mentioned that the author's name is incorrectly written in the book, that is, Axelsson writes Wo\'zni\v{c}ki instead of Wo\'znicki and the author's thesis has been issued not in Warszawa but in Warzaw (see [1,p.251]). There are not either the Czech letter "\v{c}" in Polish alphabet or a town "Warzaw" in Poland. In this special spelling of names the author is not isolated, for instance, Csordas (the co-author of [3]) is written by Axelsson as Czordas. In page 216 [1] two classical comparison theorems are quoted: Let $A=C_{i}-R_{i}$, $i=1,2$ be two regular splittings. Then \begin{equation} \varrho (C_{1}^{-1}R_{1}) \leq \varrho (C_{2}^{-1}R_{2}) \label{1} \end{equation} if either (Varga, 1962) \begin{equation} R_{1} \leq R_{2} \label{2} \end{equation} or if (Wo\'znicki, 1973) \begin{equation} C_{1}^{-1} \geq C_{2}^{-1}. \label{3} \end{equation} At this point, Axelsson notices that the result of Wo\'znicki does not carry over to the case of weak regular splittings (defined in the book), and for illustrating his observation he compares an iterative method with the direct one in the following example: \\ Let \begin{eqnarray} A = C_{1}-R_{1} = \left[ \begin{array}{rr} 1 & -(1+ \varepsilon) \\ - \frac{1}{2} & (1 - \varepsilon) \end{array} \right] \; - \; \left[ \begin{array}{rr} 0 & -\varepsilon \\ 0 & \varepsilon \end{array} \right] \end{eqnarray} where $0 < \varepsilon < \frac{1}{3}$, and let $A = C_{2}-R_{2}$, where $C_{2} = A$ and $R_{2} = 0$. Then \begin{eqnarray*} C_{1}^{-1} = \frac{2}{1-3\varepsilon} \left[ \begin{array}{cc} 1 & (1+ \varepsilon) \\ \frac{1}{2} & 1 \end{array} \right], \; \; {\rm while} \; \; \; C_{2}^{-1} = A^{-1} = 2 \left[ \begin{array}{cc} 1 & 1 \\ \frac{1}{2} & 1 \end{array} \right] \end{eqnarray*}, \\ so $C_{1}^{-1} \geq C_{2}^{-1}$ but $\varrho (C_{1}^{-1}R_{1}) > \varrho (C_{2}^{-1}R_{2})$, because $\varrho (C_{2}^{-1}R_{2}) = 0$. \vspace*{3mm} It would be nice if a typing error, which may occur even with a very careful proof- -reading, could be found in the above example taken from the mentioned previously item {\em (c)} in Example 6.6. Unfortunately, there are three essential errors in this example. The first splitting is not the splitting of \begin{eqnarray} A = \left[ \begin{array}{rr} 1 & -1 \\ -\frac{1}{2} & 1 \end{array} \right], \end{eqnarray} moreover, $C_{1}^{-1}$ is erroneous because \begin{eqnarray} (C_{1}^{-1})^{-1} = \frac{1-3\varepsilon}{1-\varepsilon} \left[ \begin{array}{cc} 1 & -(1+ \varepsilon) \\ -\frac{1}{2} & 1 \end{array} \right] \neq C_{1} \end{eqnarray} If both matrices $C_{1}$ and $R_{1}$ are chosen as in (4), then they represent the weak regular splittig of the matrix \begin{eqnarray} \bar{A} = \left[ \begin{array}{rc} 1 & -1 \\ -\frac{1}{2} & 1-2\varepsilon \end{array} \right] \; \; {\rm where} \; \; \bar{A}^{-1} = \frac{2}{1-4\varepsilon}\left[ \begin{array}{cc} 1 -2\varepsilon & 1 \\ \frac{1}{2} & 1 \end{array} \right] \end{eqnarray} is a positive matrix for all $\varepsilon < \frac{1}{4}$. Then, with $0< \varepsilon < \frac{1}{4}$, \begin{eqnarray} C_{1}^{-1} = \frac{2}{1-3\varepsilon} \left[ \begin{array}{cc} (1 - \varepsilon) & (1+ \varepsilon) \\ \frac{1}{2} & 1 \end{array} \right] \leq C_{2}^{-1}=\bar{A}^{-1} \end{eqnarray} and the inequality (1) is satisfied, that is, $1 > \varrho (C_{1}^{-1}R_{1})= \frac{\varepsilon}{1-3\varepsilon} > \varrho (C_{2}^{-1}R_{2})=0$. \vspace*{3mm} However the most glaring error is a trial in constructing a convergent weak regular splitting of $A=C-R$ in which $C^{-1} \geq A^{-1} \geq 0$ as is in the case of the first splitting. Evidently for each convergent weak regular splitting \begin{equation} A^{-1} = (C-R)^{-1} = (I-C^{-1}R)^{-1}C^{-1} = \sum_{k=0}^{\infty}(C^{-1}R)^{k}C^{-1} \geq C^{-1} \geq 0, \end{equation} in particularity $A^{-1} = C^{-1}$ when $R=0$, which shows us that a weak regular splitting of $A=C-R$ can be convergent if and only if the nonsingular matrix $A$ is monotone, and conversely. Hence, it can be concluded that for $R_{1} \neq 0$ and $C_{2} = A$, not only the result of Wo\'znicki carries over the case of weak regular splittings, but the inequality \begin{equation} \varrho (C_{1}^{-1}R_{1}) \geq \varrho (C_{2}^{-1}R_{2}) \end{equation} holds for each splitting of $A=C_{1}-R_{1}$ if $A$ and $C_{1}$ are nonsingular matrices. Thus, it is impossible to construct a convergent weak regular splitting of $A$, if $A$ is not a monotone matrix. Does Axelsson see about it? \vspace*{2mm} In this section two following theorems are given. \begin{quote} {\bf Theorem 6.21} {\em Let $A=C_{1}-R_{1}=C_{2}-R_{2}$ be two convergent splittings where $C_{i}^{-1}R_{i}$ are nonnegative. Let $G_{i}=A_{i}^{-1}R_{i}$, $i=1,2$. Then \vspace*{-7mm} \begin{quote} \item[(a)] $G_{i} \geq 0$, $i=1,2$, and \\ \vspace*{-5mm} \item[(b)] $(G_{2} - G_{1})G_{1}x \geq 0$, where $x \geq 0$ is the Perron vector of $G_{1}$, implies \\ $\varrho (C_{1}^{-1}R_{1}) \leq \varrho (C_{2}^{-1}R_{2})$. \end{quote}} \end{quote} \begin{quote} {\bf Theorem 6.22} {\em Let $A=C_{1}-R_{1}=C_{2}-R_{2}$ be two convergent splittings of $A$. Then the following holds: \vspace*{-7mm} \begin{quote} \item[(a)] $R_{2} \geq R_{1} \geq 0$, and $C_{i}$, $i=1,2$ are monotone (i.e., the splittings are regular), then $C_{1}^{-1} \geq C_{2}^{-1}$. \\ \vspace*{-5mm} \item[(b)] If $C_{1}^{-1} \geq C_{2}^{-1}$ and $R_{1}x \geq 0$, then $(C_{1}^{-1} - C_{2}^{-1})R_{1}x \geq 0$, where $x$ is the Perron vector of $G_{1}=A^{-1}R_{1}$. \\ \vspace*{-5mm} \item[(c)] If $(C_{1}^{-1} - C_{2}^{-1})R_{1}x \geq 0$, where $x$ is the Perron vector of $G_{1}$, and if $A=C_{i}-R_{i}$, $i=1,2$ are weak regular splittings, then \\ $\varrho (C_{1}^{-1}R_{1}) \leq \varrho (C_{2}^{-1}R_{2})$. \end{quote}} \end{quote} Theorem 6.21 is supposedly an extension of a theorem in Beauwens [2]. As can be seen in [10] the Beauwens' results fail and the same result has been already earlierly shown in a simple example in the Song's paper [7] which is just quoted in the Axelsson's book. \vspace*{2mm} Theorem 6.22 is an extension of theorems in Beauwens [2] (again those incorrect results) as well as in Wo\'znicki [8], and Csordas and Varga [3]. %\newpage The result that the inequality $R_{2} \geq R_{1} \geq 0$ implies the inequality $C_{1}^{-1} \geq C_{2}^{-1}$ has been stated many years ago for regular splittings [8,3]. Since the proof of this trivial result is given in the book, it seems that Axelsson discovered this implication independently. \newpage However, as is shown in [8], $C_{1}^{-1} \geq C_{2}^{-1}$ may not imply $R_{2} \geq R_{1} \geq 0$, which means that $C_{1}^{-1} \geq C_{2}^{-1}$ is a weaker condition than $R_{2} \geq R_{1} \geq 0$. Probably Axelsson did not yet discover that the converse statement need not be true, therefore, he included "his result" in the item {\em (a)} of Theorem 6.22 as an extension of the Wo\'znicki's theorem proven just with the weaker hypothesis $C_{1}^{-1} \geq C_{2}^{-1}$. Extraordinarily! Since Axelsson restricted item {\em (a)} to "his developments" for regular splittings, it seems, he did not yet notice that the inequality $R_{2} \geq R_{1} \not \geq 0$ implies the inequality $C_{1}^{-1} \geq C_{2}^{-1}$ when both splittings are convergent weak regular ones but it should be again emphasizes that the reverse implication may fail [10]. The applicability of both above theorems is accompanied by an increased complexity in the verification of their conditions. It is necessary to know additionally the Perron vector of $G_{1}=A^{-1}R_{1}$, which may be a more cumbersome process than the solution of the iterative problem. However, it should be mentioned that each eigenvector of $G_{1}$ is as well the same eigenvector of $C_{1}^{-1}R_{1}$ because both matrices commute, which holds for each splitting of an arbitrary nonsingular matrix $A$ if and only if $C$ is a nonsingular matrix [10]. Indeed, from the definition of the splitting of $A=C-R$, it follows that \begin{equation} C^{-1} = (A+R)^{-1} = A^{-1}(I+RA^{-1})^{-1} = (I+A^{-1}R)^{-1}A^{-1} \end{equation} or \begin{equation} A^{-1} = C^{-1} + C^{-1}RA^{-1} = C^{-1} + A^{-1}RC^{-1}. \end{equation} Hence \begin{equation} C^{-1}RA^{-1}R = A^{-1}RC^{-1}R \end{equation} Axelsson, perhaps endeavouring to the originality and generality of his results, uses just the Perron eigenvector of $A^{-1}R_{1}$ instead of $C_{1}^{-1}R_{1}$ in the item {\em (c)} of Theorem 6.22. Such a suspection is also confirmed by items {\em (a)} and {\em (c)} which hold when $A$ is a monotone matrix but the condition $A^{-1} \geq 0$ is unnecessary only for item {\em (b)} and its existence in this theorem is a most enigmatic matter constraining the following questions. Which needs of the user of this theorem can be met by item {\em (b)}? Why $x$ must be the Perron vector of $G_{1}$, if there is no an assumption on $G_{1}$ and the inequality $(C_{1}^{-1} - C_{2}^{-1})R_{1}x \geq 0$ may be even satisfied for regular splitting when a vector $x$ has some negative componets. Removing item {\em (b)} implies the validity of this theorem if and only if $A^{-1} \geq 0$. Thus, is item {\em (b)} a trick for keeping only a general character of Theorem 6.22? \vspace*{2mm} In the book both above theorems are followed with the change of denotation by \begin{quote} {\bf Corollary 6.22'} {\em Let $A=M_{1}-N_{1}=M_{2}-N_{2}$ be weak regular splittings. Then \[ \varrho (M_{1}^{-1}N_{1}) \leq \varrho (M_{2}^{-1}N_{2}) \] if any of the following holds: \vspace*{-7mm} \begin{quote} \item[(a)] $N_{2} \geq N_{1} \geq 0$. %\vspace*{-5mm} \item[(b)] $M_{1}^{-1} \geq M_{2}^{-1}$, $N_{1} \geq 0$. %\vspace*{-5mm} \item[(c)] $M_{1}^{-1} \geq M_{2}^{-1}$, $N_{2} \geq 0$. \end{quote}} \end{quote} The above corollary is presented in the book as the Axelsson's result. As discussed before, a weak regular splitting of $A=M-N$ can be convergent, if and only if $A^{-1} \geq 0$ and therefore, this corollary is exactly equivalent to the Elsner's lemma [4] in which the above three items are given with the assumption that $A^{-1} \geq 0$. Thus, this corollary is either an example of manipulation of results obtained by other authors or it indicates evidently that Axelsson, abanding the condition $A^{-1} \geq 0$, is not awared of the equivalence between a convergent weak regular splitting of $A=M-N$ and the monotonicity of the natrix $A$. \newpage \begin{center}{\large \bf 3. Further Discussion} \end{center} The mentioned result of Wo\'znicki has been proven for regular splittings, where the strict inequality in (3) implies the strict inequality in (1) and the proof can be found in [10]. However, as was earlierly (before the Axelsson's observation) noticed by other authors [4,5], this result nay not hold for weak regular splittings. In order to better recognize this aspect and the usefulness of Axelsson's theorems, let us consider for the following example of the matrix \begin{eqnarray} A = \left[ \begin{array}{rrr} 1 & -1 & 0 \\ 0 & 1 & -1 \\ -1 & 0 & 2 \end{array} \right] \; \; {\rm where} \; \; A^{-1} = \left[ \begin{array}{rrr} 2 & 2 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 1 \end{array} \right] \end{eqnarray} some splittings of $A=M_{i}-N_{i}$ [10]. \begin{eqnarray} M_{1} = \left[ \begin{array}{rrr} 1 & -1 & 0 \\ 0 & 1 & -\frac{3}{2} \\ -1 & 0 & 3 \end{array} \right], \; \; N_{1} = \left[ \begin{array}{rrr} 0 & 0 & 0 \\ 0 & 0 & -\frac{1}{2} \\ 0 & 0 & 1 \end{array} \right] \; \; {\rm where} \end{eqnarray} \vspace*{-2mm} \begin{eqnarray} M^{-1}_{1} = \left[ \begin{array}{rrr} 2 & 2 & 1 \\ 1 & 2 & 1 \\ \frac{2}{3} & \frac{2}{3} & \frac{2}{3} \end{array} \right], \; \; M^{-1}_{1}N_{1} = \left[ \begin{array}{rrr} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \frac{1}{3} \end{array} \right], \; \; A^{-1}N_{1} = \left[ \begin{array}{rrr} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \frac{1}{2} \end{array} \right], \end{eqnarray} where $\varrho (M_{1}^{-1}N_{1}) = \frac{1}{3}$, $\varrho (A^{-1}N_{1}) = \frac{1}{2}$ and the corresponding eigenvector (Perron vector) \\ $x_{1}^{T} = [0 \; 0 \; 1]^{T}$. \begin{eqnarray} M_{2} = \left[ \begin{array}{rrr} 1 & -1 & 0 \\ \vspace*{1mm} 0 & \frac{4}{5} & -\frac{6}{5} \\ \vspace*{1mm} -1 & \frac{2}{5} & \frac{12}{5} \end{array} \right], \; \; N_{2} = \left[ \begin{array}{rrr} 0 & 0 & 0 \\ \vspace*{1mm} 0 & -\frac{1}{5} & -\frac{1}{5} \\ \vspace*{1mm} 0 & \frac{2}{5} & \frac{2}{5} \end{array} \right] \; \; {\rm where} \end{eqnarray} \vspace*{-2mm} \begin{eqnarray} M^{-1}_{2} = \left[ \begin{array}{rrr} 2 & 2 & 1 \\ 1 & 2 & 1 \\ \frac{2}{3} & \frac{1}{2} & \frac{2}{3} \end{array} \right], \; \; M^{-1}_{2}N_{2} = \left[ \begin{array}{rrr} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & \frac{1}{6} & \frac{1}{6} \end{array} \right], \; \; A^{-1}N_{2} = \left[ \begin{array}{rrr} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & \frac{1}{5} & \frac{1}{5} \end{array} \right], \end{eqnarray} where $\varrho (M_{2}^{-1}N_{2}) = \frac{1}{6}$, $\varrho (A^{-1}N_{2}) = \frac{1}{5}$ and the corresponding eigenvector (Perron vector) \\ $x_{2}^{T} = [0 \; 0 \; 1]^{T}$. \begin{eqnarray} M_{3} = \left[ \begin{array}{rrr} 1 & -1 & 0\\ \vspace*{1mm} -\frac{4}{23} & \frac{19}{23} & -\frac{20}{23} \\ \vspace*{1mm} -\frac{14}{23} & \frac{9}{23} & \frac{45}{23} \end{array} \right], \; \; N_{3} = \left[ \begin{array}{rrr} 0 & 0 & 0\\ \vspace*{1mm} -\frac{4}{23} & -\frac{4}{23} & \frac{3}{23} \\ \vspace*{1mm} \frac{9}{23} & \frac{9}{23} & -\frac{1}{23} \end{array} \right] \; \; {\rm where} \end{eqnarray} \vspace*{-2mm} \begin{eqnarray} M^{-1}_{3} = \left[ \begin{array}{rrr} \vspace*{1mm} \frac{9}{5} & \frac{9}{5} & \frac{4}{5} \\ \vspace*{1mm} \frac{4}{5} & \frac{9}{5} & \frac{4}{5} \\ \vspace*{1mm} \frac{2}{5} & \frac{1}{5} & \frac{3}{5} \end{array} \right], \; \; M_{3}^{-1}N_{3} = \left[ \begin{array}{rrr} \vspace*{1mm} 0 & 0 & \frac{1}{5} \\ \vspace*{1mm} 0 & 0 & \frac{1}{5} \\ \vspace*{1mm} \frac{1}{5} & \frac{1}{5} & 0 \end{array} \right], \; \; A^{-1}N_{3} = \left[ \begin{array}{rrr} \vspace*{1mm} \frac{1}{23} & \frac{1}{23} & \frac{5}{23} \\ \vspace*{1mm} \frac{1}{23} & \frac{1}{23} & \frac{5}{23} \\ \vspace*{1mm} \frac{5}{23} & \frac{5}{23} & \frac{2}{23} \end{array} \right], \end{eqnarray} where $\varrho (M_{3}^{-1}N_{3}) = \frac{\sqrt{2}}{5}$, $\varrho (A^{-1}N_{3}) = \frac{2+5\sqrt{2}}{2}$ and the corresponding eigenvector (Perron vector) $x_{3}^{T} = [1 \; 1 \; \sqrt{2}]^{T}$. As can be seen, the above splittings are weak regular in the sense of the Axelsson's definition but not in the sense of the Ortega and Rheinboldt's definition which follows from (25). In this case $M_{1}^{-1} \geq M_{2}^{-1}$ and $M_{1}^{-1} > M_{3}^{-1}$, while $\varrho (M_{1}^{-1}N_{1}) > \varrho (M_{2}^{-1}N_{2})$ and $\varrho (M_{1}^{-1}N_{1}) > \varrho (M_{3}^{-1}N_{3})$ which are correct examples showing that the result of Wo\'znicki does not carry over the case of weak regular splittings defined is such a way. But for the second and third splitting with $M_{2}^{-1} > M_{3}^{-1}$ this result holds because $\varrho (M_{2}^{-1}N_{2}) < \varrho (M_{3}^{-1}N_{3})$. In the case of Theorems 6.21 and 6.22 it is possible to compare only two first splittings but with comparing the first with third splitting and the second with third splitting both theorems are useless because, as can be easily verified, the hypotheses of these theorems are not fulfiled. Is it an accident validity of the above results? If not, then it seems to be an interesting to ask to the question, why both theorems are not an extension of the result of Wo\'znicki holding in the above examples for the second and third splitting? The result of Varga (2) obtained for regular splitting holds as well for the case of weak regular splittings because with $A^{-1} \geq 0$, as the necessary condition for existing a convergent weak regular splitting, the Varga's condition \begin{equation} N_{2} \geq N_{1} \end{equation} implies \begin{equation} A^{-1}N_{2} = (I-M^{-1}_{2}N_{2})^{-1}M^{-1}_{2}N_{2} \geq A^{-1}N_{1} = (I-M^{-1}_{1}N_{1})^{-1}M^{-1}_{1}N_{1} \geq 0 \end{equation} hence, it is evident that [4,10] \begin{equation} \varrho (M^{-1}_{1}N_{1}) \leq \varrho (M^{-1}_{2}N_{2}). \end{equation} The inequality \begin{equation} M^{-1}_{1} \geq M_{2}^{-1} \ge 0, \end{equation} implying the inequality (23) for regular splittings, as a weaker hypothesis than (21) is not a sufficient condition in this case of weak regular splittings and therefore, it is necessary to use additional assumptions for ensuring the inequality (23), as can be seen in other works [4,5,7,9]. \vspace*{2mm} In the last decade a renewed interest with comparison theorems, proven for different types of splittings and assumptions, is observed in the literature [3,4,5,7,10]. The Varga's definition of regular splitting became the standard terminology in the literature, whereas other splittings are usually defined as a matter of author's taste. The specification of splittings preferable by the author is given in the following definition [10]. \begin{quote} {\bf Definition A} {\em The decomposition $A=M-N$ is called:} \vspace*{-2mm} \end{quote} \begin{description} \item[{\em \ \ (a)}] a {\em regular splitting of $A$} if $M^{-1} \geq 0$ and $N \geq 0$. \vspace*{-3mm} \item[{\em \ \ (b)}] a {\em nonnegative splitting of $A$} if $M^{-1} \geq 0$, $M^{-1}N \geq 0$ and $NM^{-1} \geq 0$. \vspace*{-3mm} \item[{\em \ \ (c)}] a {\em weak nonnegative splitting of $A$} if $M^{-1} \geq 0$ and either $M^{-1}N \geq 0$ (the {\em first type}) or $NM^{-1} \geq 0$ (the {\em second type}). \vspace*{-3mm} \item[{\em \ \ (d)}] a {\em weak splitting of $A$} if $M$ is nonsingular and either $M^{-1}N \geq 0$ (the {\em first type}) or $NM^{-1} \geq 0$ (the {\em second type}). In particular a given weak splitting can be both types. \end{description} The definition assumed in item {\em (b)} is equivalent to the definition of a weak regular splitting of A due to Ortega and Rheinboldt [6]. However, as was mentioned in Section 1, $M^{-1} \geq 0$ and only $M^{-1}N \geq 0$ (without the condition $NM^{-1} \geq 0$) is defined as a weak regular splitting of $A=M-N$ by others authors as well as by Axelsson, but in this case it is necessary to use additional assumptions in comparison theorems. It should be remarked that the use of the Ortega and Rheinboldt's terminology "weak regular" in item {\em (b)} causes a confusion with using a splitting name in item {\em (c)} therefore, it seems that assuming the term "nonnegative" allows us to avoid this confusion. The definition of the weak splitting of $A$ for the first type case has been introduced by Marek and Szyld [5]. When $A$ is a monotone matrix, then each weak nonnegative splitting of $A$ is convergent and many comparison theorems are proven in [10] under the progressively weaker but natural conditions, such as $N_{2} \geq N_{1}$, $M^{-1}_{1} \geq M^{-1}_{2} \geq 0$ or $A^{-1}N_{2}A^{-1} \geq A^{-1}N_{1}A^{-1} \geq 0$, as well as hypotheses with an increased complexity in its verification. The following theorem summarizes some results from [10] \begin{quote} {\bf Theorem B} {\em Let $A=M_{1}-N_{1}=M_{2}-N_{2}$ be two weak nonnegative splittings of $A$ but of different type, that is, either $M^{-1}_{1}N_{1} \geq 0$ and $N_{2}M^{-1}_{2} \geq 0$ or $N_{1}M^{-1}_{1} \geq 0$ and $M^{-1}_{2}N_{2} \geq 0$ where $A^{-1} \geq (>) \; 0$. If $M_{1}^{-1} \geq (>) \; M_{2}^{-1}$, then} \[ \varrho (M_{1}^{-1}N_{1}) \leq (<) \; \varrho (M_{2}^{-1}N_{2}). \] \end{quote} Since a nonnegative splitting is a weak nonnegative splitting, it is evident that the above theorem holds for nonnegative splittings or if only one of them is a nonnegative splitting. \vspace*{2mm} Referring back to the considered splittings of the matrix (14), we have \begin{eqnarray*} N_{1}M^{-1}_{1} = \left[ \begin{array}{rrr} 0 & 0 & 0 \\ \vspace*{1mm} -\frac{1}{3} & -\frac{1}{3} & -\frac{1}{3} \\ \vspace*{1mm} \frac{2}{3} & \frac{2}{3} & \frac{2}{3} \end{array} \right], \; \; N_{2}M^{-1}_{2} = \left[ \begin{array}{rrr} 0 & 0 & 0 \\ \vspace*{1mm} -\frac{1}{3} & -\frac{1}{2} & -\frac{1}{3} \\ \vspace*{1mm} \frac{2}{3} & 1 & \frac{2}{3} \end{array} \right] \end{eqnarray*} \vspace*{-2mm} \begin{eqnarray} {\rm and} \; \; N_{3}M^{-1}_{3} = \left[ \begin{array}{rrr} 0 & 0 & 0 \\ \vspace*{1mm} -\frac{2}{5} & -\frac{3}{5} & -\frac{1}{5} \\ \vspace*{1mm} 1 & \frac{7}{5} & \frac{3}{5} \end{array} \right] \end{eqnarray} The inspection of matrices $M^{-1}_{i}N_{i}$ and $N_{i}M^{-1}_{i}$ shows us that all three splittings are weak nonnegative of the first type and Theorems B can not be used. \vspace*{2mm} Let us consider two another splittings of the matrix (14) [10]. \begin{eqnarray} M_{4} = \left[ \begin{array}{rrr} \vspace*{1mm} 2 & -2 & 0 \\ \vspace*{1mm} 0 & 2 & -1 \\ \vspace*{1mm} -2 & 1 & 4 \end{array} \right], \; \; N_{4} = \left[ \begin{array}{rrr} \vspace*{1mm} 1 & -1 & 0 \\ \vspace*{1mm} 0 & 1 & -1 \\ \vspace*{1mm} -1 & 1 & 2 \end{array} \right] \; \; {\rm where} \; \; M^{-1}_{4} = \left[ \begin{array}{rrr} \vspace*{1mm} \frac{5}{6} & \frac{2}{3} & \frac{1}{3} \\ \vspace*{1mm} \frac{1}{3} & \frac{2}{3} & \frac{1}{3} \\ \vspace*{1mm} \frac{1}{3} & \frac{1}{6} & \frac{1}{3} \end{array} \right], \end{eqnarray} \vspace*{-2mm} \begin{eqnarray} M_{4}^{-1}N_{4} = \left[ \begin{array}{rrr} \vspace*{1mm} \frac{1}{2} & \frac{1}{6} & 0 \\ \vspace*{1mm} 0 & \frac{2}{3} & 0 \\ \vspace*{1mm} 0 & \frac{1}{6} & \frac{1}{2} \end{array} \right], \; \; A^{-1}N_{4} = \left[ \begin{array}{rrr} \vspace*{1mm} 1 & 1 & 0 \\ \vspace*{1mm} 0 & 2 & 0 \\ \vspace*{1mm} 0 & 1 & 1 \end{array} \right], \; \; N_{4}M^{-1}_{4} = \left[ \begin{array}{rrr} \vspace*{1mm} \frac{1}{2} & 0 & 0 \\ \vspace*{1mm} 0 & \frac{1}{2} & 0 \\ \vspace*{1mm} \frac{1}{6} & \frac{1}{3} & \frac{2}{3} \end{array} \right], \end{eqnarray} where $\varrho (M_{4}^{-1}N_{4}) = \frac{2}{3}$, $\varrho (A^{-1}N_{4}) = 2$ and the corresponding eigenvector (Perron vector) \\ $x_{4}^{T} = [1 \; 1 \; 1]^{T}$. \begin{eqnarray} M_{5} = \left[ \begin{array}{rrr} \vspace*{1mm} 2 & -2 & 0 \\ \vspace*{1mm} 0 & 1 & -1 \\ \vspace*{1mm} -\frac{3}{2} & 1 & 2 \end{array} \right], \; \; N_{5} = \left[ \begin{array}{rrr} \vspace*{1mm} 1 & -1 & 0 \\ \vspace*{1mm} 0 & 0 & 0 \\ \vspace*{1mm} -\frac{1}{2} & 1 & 0 \end{array} \right] \; \; {\rm where} \; \; M^{-1}_{5} = \left[ \begin{array}{rrr} \vspace*{1mm} 1 & \frac{4}{3} & \frac{2}{3} \\ \vspace*{1mm} \frac{1}{2} & \frac{4}{3} & \frac{2}{3} \\ \vspace*{1mm} \frac{1}{2} & \frac{1}{3} & \frac{2}{3} \end{array} \right], \end{eqnarray} \vspace*{-2mm} \begin{eqnarray} M_{5}^{-1}N_{5} = \left[ \begin{array}{rrr} \vspace*{1mm} \frac{2}{3} & -\frac{1}{3} & 0 \\ \vspace*{1mm} \frac{1}{6} & \frac{1}{6} & 0 \\ \vspace*{1mm} \frac{1}{6} & \frac{1}{6} & 0 \end{array} \right], \; \; A^{-1}N_{5} = \left[ \begin{array}{rrr} \vspace*{1mm} \frac{3}{2} & -1 & 0 \\ \vspace*{1mm} \frac{1}{2} & 0 & 0 \\ \vspace*{1mm} \frac{1}{2} & 0 & 0 \end{array} \right], \; \; N_{5}M^{-1}_{4} = \left[ \begin{array}{rrr} \vspace*{1mm} \frac{1}{2} & 0 & 0 \\ 0 & 0 & \\ \vspace*{1mm} 0 & \frac{2}{3} & \frac{1}{3} \vspace*{1mm} \end{array} \right], \end{eqnarray} where $\varrho (M_{5}^{-1}N_{5}) = \frac{1}{2}$, $\varrho (A^{-1}N_{5}) = 1$ and the corresponding eigenvector (Perron vector) \\ $x_{5}^{T} = [1 \; \frac{1}{2} \; \frac{1}{2}]^{T}$. \vspace*{2mm} As can be noticed the fourth splitting is nonnegative and the fifth is weak nonnegative but the second type. Since $M_{1}^{-1}, M_{2}^{-1}, M_{3}^{-1} > M_{4}^{-1}$, then by Theorem B it follows \begin{equation} \varrho (M_{1}^{-1}N_{1}), \varrho (M_{2}^{-1}N_{2}), \varrho (M_{3}^{-1}N_{3}) < \varrho (M_{4}^{-1}N_{4}) \end{equation} In this case the hypotheses of Theorems 6.21 and 6.22 are satisfied which implies the above inequality however, without the strict sign of inequality. Taking in considerations the fifth spliting, we have $M_{1}^{-1},M_{2}^{-1} \geq M_{5}^{-1}$, $M_{3}^{-1} > M_{5}^{-1}$ and $M_{5}^{-1} > M_{4}^{-1}$ which by Theorem B gives us \begin{equation} \varrho (M_{1}^{-1}N_{1}), \varrho (M_{2}^{-1}N_{2}) \leq \varrho (M_{5}^{-1}N_{5}), \end{equation} \begin{equation} \varrho (M_{3}^{-1}N_{3}) < \varrho (M_{5}^{-1}N_{5}), \end{equation} and \begin{equation} \varrho (M_{5}^{-1}N_{5}) < \varrho (M_{4}^{-1}N_{4}) \end{equation} respectively. Since $M_{5}^{-1}N_{5}$ is not a nonnegative matrix both Theorems 6.21 and 6.22 are can not be used for making a comparison of spectral radius of this splitting with remaining ones. Thus, the above results show expressively that the second condition $NM^{-1}$ (introduced originally by Ortega and Rheinboldt and overlooked often by other authors) in nonnegative splittings has the essential meaning for comparison theorems. Moreover, weak nonnegative splittings extend the convergence analysis to a class of linear system problems with iteration matrices $M^{-1}N$ which some entries may also be negative. \vspace*{2mm} It is necessary to say that there are many comparison theorems based on natural conditions, that is, such conditions which are easy for verification in many applications as, for instance, the case when both weak nonnegative splittings of a symmetric matrix $A$ (appearing very often in applications) are the same type [10] \begin{quote} {\bf Theorem C} {\em Let $A=M_{1}-N_{1}=M_{2}-N_{2}$ be two weak nonnegative splittings of a symmetric matrix $A$ where $A^{-1} \geq (>) \; 0$. If $M_{1}^{-1} \geq (>) \; M_{2}^{-1}$ and at least one of $M_{1}$ and $M_{2}$ is a symmetric matrix, then} \[ \varrho (M_{1}^{-1}N_{1}) \leq (<) \; \varrho (M_{2}^{-1}N_{2}). \] \end{quote} or the special category of natural conditions important in applications represented, for instance, by the following theorem [10] \begin{quote} {\bf Theorem D} {\em Let $A=M_{1}-N_{1}=M_{2}-N_{2}$ be two weak nonnegative splittings of $A$ but of the same type, that is, either $M^{-1}_{1}N_{1} \geq 0$ and $M^{-1}_{2}N_{2} \geq 0$ or $N_{1}M^{-1}_{1} \geq 0$ and $N_{2}M^{-1}_{2} \geq 0$ where $A^{-1} \geq 0$. If $N_{2}^{T} \geq N_{1}$, then} \[ \varrho (M_{1}^{-1}N_{1}) \leq \varrho (M_{2}^{-1}N_{2}). \] \end{quote} allowing us to choose a more efficient splitting of $A$ in the Gauss-Seidel method when $A$ is an unsymmetric matrix, where of course, $N_{2}^{T}$ is a transpose matrix. Similarly, as in the case of Theorem B, it is not difficult to show examples in which both above theorems hold, while the hypotheses of Theorems 6.21 and 6.22 are not satisfied. But this leads to the conclusion that Theorems 6.21 and 6.22 (with the hypotheses cumbersome for verification) are a doubtful value from the viewpoint of their helpfulness. To say nothing of many errors and inaccuracies noticed in these two important sections of Chapter 6, it seems that the disregarding of comparison theorems with natural conditions (like Theorems B, C and D given here for illustrative reasons or similar to them), as basic tools in the convergence analysis of iterative solutions, mainly depreciates the book devoted just to such a topic. \begin{center}{\large \bf References} \end{center} \begin{description} \item[{\em \ \ \ 1.}] O.Axelsson, "Iterative Solution Methods", Cambridge Univ.Press (1994) \item[{\em \ \ \ 2.}] R.Beauwens, "Factorization Iterative Methods, M-operators and H-operators", Numer. Math., 31(1979), 335-357 \item[{\em \ \ \ 3.}] G.Csordas and R.S.Varga, "Comparison of regular splittings of matrices", Numer. Math., 44(1984), 23-35 \item[{\em \ \ \ 4.}] L.Elsner, "Comparisons of Weak Regular Splittings and Multisplitting Methods", Numer. Math., 56(1989), 283-289 \item[{\em \ \ \ 5.}] I.Marek and D.B.Szyld, "Comparison Theorems for Weak Splittings of bounded operators", Numer. Math., 58(1990), 389-397 \item[{\em \ \ \ 6.}] J.M.Ortega and W.Rheinboldt, "Iterative Solution of Nonlinear Equations in Several Variables", Academic Press (1970) \item[{\em \ \ \ 7.}] Y.Song, "Comparisons of nonnegative splittings of matrices", Lin. Alg. Appl., 154-156 (1991), 433-455 \item[{\em \ \ \ 8.}] Z.I.Wo\'znicki, "Two-sweep iterative methods for solving large linear systems and their application to the numerical solution of multi-group, multi-dimensional neutron diffusion equations", Doctoral Dissertation, Rep. No.1447-CYFRONET-PM-A, Institute of Nuclear Research, Swierk-Otwock, Warszawa, Poland (1973) \item[{\em \ \ \ 9.}] Z.I.Wo\'znicki, "AGA two-sweep iterative method and their application in critical reactor calculations", Nukleonika 9(1978), 941-968 \item[{\em \ \ 10.}] Z.I.Wo\'znicki, "Nonnegative Splitting Theory", Japan J.Industr. Appl. Math., 10(1994), 289-342 \end{description} \end{document}