Abstract
In 1986, I. Păvăloiu [6] has considered a Banach space and the fixed point problem \[x=\lambda D\left( x\right) +y, \qquad D:X\rightarrow X \ \textrm{nonlinear},\ \lambda\in {\mathbb R},\ y\in X \ \textrm{given}\]written in the equivalent form \(F(x):=x -\lambda D\left( x\right) -y=0\) and solved by the general quasi-Newton method\[x_{n+1}=x_n-A(x_n) \left[ x_n-\lambda D(x_n) -y\right] ,\qquad n=0,1,\ldots\]Semilocal convergence results were obtained, ensuring linear convergence of these iterates. Further results were obtained for the iterates \[x_{n+1}=x_n-[I+\lambda D^\prime(x_n)] \left[x_n+\lambda D(x_n) -y\right] ,\qquad n=0,1,\ldots\] In this note, we analyze the local convergence of these iterates, and, using the Ostrowski local attraction theorem, we give some sufficient conditions such that the iterates converge locally either linearly or with higher convergence orders. The local convergence results require fewer differentiability assumptions for \(D\).
Authors
Keywords
nonlinear equations in Banach spaces; inexact Newton method; quasi-Newton method; Ostrowski local attraction theorem; local convergence; convergence order.
References
About this paper
Cite this paper as:
E. Cătinaş, On the convergence of some quasi-Newton iterates studied by I. Păvăloiu, J. Numer. Anal. Approx. Theory, 44 (2015) no. 1, pp. 38-41.
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2457-6794
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2501-059X
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[1] E. Catinas, On the superlinear convergence of the successive approximations method, J. Optim. Theory Appl., 113 (2002) no. 3, pp. 473-485. http://dx.doi.org/10.1023/A:1015304720071
[2] E. Catinas, The inexact, inexact perturbed and quasi-Newton methods are equivalent models, Math. Comp., 74 (2005) no. 249, pp. 291-301, http://dx.doi.org/10.1090/S0025-5718-04-01646-1
[3] E. Catinas, On the convergence orders , manuscript.
[4] Diaconu, A., Pavaloiu, I., Sur quelque methodes iteratives pour la resolution des equations operationnel les, Rev. Anal. Numer. Theor. Approx., vol. 1, 45-61 (1972),
https://ictp.acad.ro/jnaat/journal/article/view/1972-vol1-art3
[5] J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several Variables , Academic Press, New York, 1970.
[6] I. Pavaloiu, La convergence de certaines methodes iteratives pour resoudre certaines equations operationnelles, Seminar on functional analysis and numerical methods, Preprint(1986), pp. 127-132 (in French).
[7] I. Pavaloiu, A unified treatment of the modified Newton and chord methods, Carpathian J. Math. 25 (2009) no. 2, pp. 192-196.
.
Paper (preprint) in HTML form
ON THE CONVERGENCE OF SOME QUASI-NEWTON ITERATES STUDIED BY I. PĂVĂLOIU
Abstract
I. Păvăloiu has considered a Banach space XX and the problem x=lambda D(x)+y quad(D:X rarr X,lambda inR,y in X" given ")x=\lambda D(x)+y \quad(D: X \rightarrow X, \lambda \in \mathbb{R}, y \in X \text { given }) written in the equivalent form F(x):=x-lambda D(x)-y=0F(x):=x-\lambda D(x)-y=0 and solved by the general quasi-Newton method x_(k+1)=x_(k)-A(x_(k))[x_(k)-lambda D(x_(k))-y],quad k=0,1,dotsx_{k+1}=x_{k}-A\left(x_{k}\right)\left[x_{k}-\lambda D\left(x_{k}\right)-y\right], \quad k=0,1, \ldots
Semilocal convergence results were obtained, ensuring linear convergence of this method. Further results were obtained for the iterates: x_(k+1)=x_(k)-[I+lambdaD^(')(x_(k))][x_(k)-lambda D(x_(k))-y],quad k=0,1,dotsx_{k+1}=x_{k}-\left[I+\lambda D^{\prime}\left(x_{k}\right)\right]\left[x_{k}-\lambda D\left(x_{k}\right)-y\right], \quad k=0,1, \ldots
In this note, we analyze the local convergence of these iterates, and, using the Ostrowski local attraction theorem, we give some sufficient conditions such that the iterates converge locally either linearly or with higher convergence orders. The local convergence results require fewer differentiability assumptions for DD.
MSC 2010. 65H10.
Keywords. quasi-Newton methods, Ostrowski local attraction theorem, local convergence.
Keywords. quasi-Newton methods, Ostrowski local attraction theorem, local convergence.
1. INTRODUCTION
In [6], Păvăloiu has considered a Banach space ( X,||*||X,\|\cdot\| ), a nonlinear mapping D:X rarr XD: X \rightarrow X, a parameter lambda inR\lambda \in \mathbb{R}, an element y in Xy \in X and the equation (arising from certain integral equations)
{:(1)x=lambda D(x)+y:}\begin{equation*}
x=\lambda D(x)+y \tag{1}
\end{equation*}
solved by the following iterations:
{:(2)x_(k+1)=x_(k)-A(x_(k))[x_(k)-lambda D(x_(k))-y]","quad k=0","1","dots","x_(0)in E sube X",":}\begin{equation*}
x_{k+1}=x_{k}-A\left(x_{k}\right)\left[x_{k}-\lambda D\left(x_{k}\right)-y\right], \quad k=0,1, \ldots, x_{0} \in E \subseteq X, \tag{2}
\end{equation*}
where A(x):E rarr EA(x): E \rightarrow E is a linear continuous mapping (i.e., A(x)inL(X)A(x) \in \mathcal{L}(X) ), for each x in Ex \in E.
Denoting
{:(3)F(x)=x-lambda D(x)-y:}\begin{equation*}
F(x)=x-\lambda D(x)-y \tag{3}
\end{equation*}
the above iterations can be written as
x_(k+1)=x_(k)-A(x_(k))F(x_(k)),quad k=0,1,dots;x_{k+1}=x_{k}-A\left(x_{k}\right) F\left(x_{k}\right), \quad k=0,1, \ldots ;
in a subsequent paper, Părăloiu [7] has analyzed the above iterations for general mappings FF, not necessarily given by (3).
x_(k+1)=x_(k)-A(x_(k))F(x_(k)),quad k=0,1,dots;x_{k+1}=x_{k}-A\left(x_{k}\right) F\left(x_{k}\right), \quad k=0,1, \ldots ;
in a subsequent paper, Părăloiu [7] has analyzed the above iterations for general mappings FF, not necessarily given by (3).
The following semilocal convergence results were obtained.
Theorem 1. [6] If the mappings DD and A(x)A(x), the initial approximation x_(0)x_{0} and the real number r > 0r>0 satisfy the following conditions:
i. the mapping DD admits Fréchet derivatives of order one and two on the ball S=S(x_(0),r)S=S\left(x_{0}, r\right);
ii. ||A(x)|| <= beta\|A(x)\| \leq \beta, for each x in Sx \in S, and some beta > 0\beta>0;
iii. ||I-F^(')(x)A(x)|| <= alpha\left\|I-F^{\prime}(x) A(x)\right\| \leq \alpha, for each x in Sx \in S, and some alpha > 0\alpha>0;
iv. ||D^('')(x)|| <= M//|lambda|\left\|D^{\prime \prime}(x)\right\| \leq M /|\lambda|, for each x in Sx \in S, and some M > 0M>0;
v. (betarho_(0))/(1-d_(0)) <= r\frac{\beta \rho_{0}}{1-d_{0}} \leq r, where rho_(0)=||F(x_(0))||,d_(0)=(Mbeta^(2)rho_(0))/(2)+alpha\rho_{0}=\left\|F\left(x_{0}\right)\right\|, d_{0}=\frac{M \beta^{2} \rho_{0}}{2}+\alpha;
vi. d_(0) < 1d_{0}<1,
then the sequence (x_(k))_(k >= 0)\left(x_{k}\right)_{k \geq 0} given by (2) converges: x^(**)=lim_(k rarr oo)x_(k)x^{*}=\lim _{k \rightarrow \infty} x_{k}, with F(x^(**))=0F\left(x^{*}\right)=0. The following estimations hold:
Theorem 1. [6] If the mappings DD and A(x)A(x), the initial approximation x_(0)x_{0} and the real number r > 0r>0 satisfy the following conditions:
i. the mapping DD admits Fréchet derivatives of order one and two on the ball S=S(x_(0),r)S=S\left(x_{0}, r\right);
ii. ||A(x)|| <= beta\|A(x)\| \leq \beta, for each x in Sx \in S, and some beta > 0\beta>0;
iii. ||I-F^(')(x)A(x)|| <= alpha\left\|I-F^{\prime}(x) A(x)\right\| \leq \alpha, for each x in Sx \in S, and some alpha > 0\alpha>0;
iv. ||D^('')(x)|| <= M//|lambda|\left\|D^{\prime \prime}(x)\right\| \leq M /|\lambda|, for each x in Sx \in S, and some M > 0M>0;
v. (betarho_(0))/(1-d_(0)) <= r\frac{\beta \rho_{0}}{1-d_{0}} \leq r, where rho_(0)=||F(x_(0))||,d_(0)=(Mbeta^(2)rho_(0))/(2)+alpha\rho_{0}=\left\|F\left(x_{0}\right)\right\|, d_{0}=\frac{M \beta^{2} \rho_{0}}{2}+\alpha;
vi. d_(0) < 1d_{0}<1,
then the sequence (x_(k))_(k >= 0)\left(x_{k}\right)_{k \geq 0} given by (2) converges: x^(**)=lim_(k rarr oo)x_(k)x^{*}=\lim _{k \rightarrow \infty} x_{k}, with F(x^(**))=0F\left(x^{*}\right)=0. The following estimations hold:
||x^(**)-x_(k)|| <= (betad_(0)^(k)rho_(0))/(1-d_(0)),quad k=0,1,dots\left\|x^{*}-x_{k}\right\| \leq \frac{\beta d_{0}^{k} \rho_{0}}{1-d_{0}}, \quad k=0,1, \ldots
When ||lambdaD^(')(x)|| < 1\left\|\lambda D^{\prime}(x)\right\|<1, it is known that the operator I-lambdaD^(')(x)I-\lambda D^{\prime}(x) is invertible, with (I-lambdaD^(')(x))^(-1)=I+lambdaD^(')(x)+lambda^(2)D^(')(x)^(2)+dots\left(I-\lambda D^{\prime}(x)\right)^{-1}=I+\lambda D^{\prime}(x)+\lambda^{2} D^{\prime}(x)^{2}+\ldots Păvăloiu has considered the operator A(x)A(x) as being given by the first two terms of this expansion, obtaining the following iterates
{:(4)x_(k+1)=x_(k)-(I+lambdaD^(')(x_(k)))[x_(k)-lambda D(x_(k))-y]","quad k=0","1","dots",":}\begin{equation*}
x_{k+1}=x_{k}-\left(I+\lambda D^{\prime}\left(x_{k}\right)\right)\left[x_{k}-\lambda D\left(x_{k}\right)-y\right], \quad k=0,1, \ldots, \tag{4}
\end{equation*}
and the following result.
Theorem 2. [6 If the mapping DD, the initial approximation x_(0)x_{0} and the real number r > 0r>0 satisfy the following assumptions:
i. the mapping DD admits Fréchet derivatives of order one and two for each x in S=S(x_(0),r)x \in S=S\left(x_{0}, r\right);
ii. ||D^(')(x)|| <= b\left\|D^{\prime}(x)\right\| \leq b, for each x in Sx \in S;
iii. ||D^('')(x)|| <= M//|lambda|\left\|D^{\prime \prime}(x)\right\| \leq M /|\lambda|, for each x in Sx \in S;
iv. 2-Mrho_(0) > 02-M \rho_{0}>0, where rho_(0)=||x_(0)-lambda D(x_(0))-y||\rho_{0}=\left\|x_{0}-\lambda D\left(x_{0}\right)-y\right\|;
v. (rho_(0)(1+|lambda|b))/(1-d_(0)) <= r\frac{\rho_{0}(1+|\lambda| b)}{1-d_{0}} \leq r, where d_(0)=M((1+|lambda|b)^(2))/(2)rho_(0)+lambda^(2)b^(2)d_{0}=M \frac{(1+|\lambda| b)^{2}}{2} \rho_{0}+\lambda^{2} b^{2};
vi. |lambda| <= (2-Mrho_(0))/(b(2+Mrho_(0)))|\lambda| \leq \frac{2-M \rho_{0}}{b\left(2+M \rho_{0}\right)},
then the sequence given by (4) converges to a solution x^(**)x^{*} of equation (1) and the following estimates hold:
Theorem 2. [6 If the mapping DD, the initial approximation x_(0)x_{0} and the real number r > 0r>0 satisfy the following assumptions:
i. the mapping DD admits Fréchet derivatives of order one and two for each x in S=S(x_(0),r)x \in S=S\left(x_{0}, r\right);
ii. ||D^(')(x)|| <= b\left\|D^{\prime}(x)\right\| \leq b, for each x in Sx \in S;
iii. ||D^('')(x)|| <= M//|lambda|\left\|D^{\prime \prime}(x)\right\| \leq M /|\lambda|, for each x in Sx \in S;
iv. 2-Mrho_(0) > 02-M \rho_{0}>0, where rho_(0)=||x_(0)-lambda D(x_(0))-y||\rho_{0}=\left\|x_{0}-\lambda D\left(x_{0}\right)-y\right\|;
v. (rho_(0)(1+|lambda|b))/(1-d_(0)) <= r\frac{\rho_{0}(1+|\lambda| b)}{1-d_{0}} \leq r, where d_(0)=M((1+|lambda|b)^(2))/(2)rho_(0)+lambda^(2)b^(2)d_{0}=M \frac{(1+|\lambda| b)^{2}}{2} \rho_{0}+\lambda^{2} b^{2};
vi. |lambda| <= (2-Mrho_(0))/(b(2+Mrho_(0)))|\lambda| \leq \frac{2-M \rho_{0}}{b\left(2+M \rho_{0}\right)},
then the sequence given by (4) converges to a solution x^(**)x^{*} of equation (1) and the following estimates hold:
||x^(**)-x_(k)|| <= ((1+|lambda|b)d_(0)^(k)rho_(0))/(1-d_(0)),quad k=0,1,dots\left\|x^{*}-x_{k}\right\| \leq \frac{(1+|\lambda| b) d_{0}^{k} \rho_{0}}{1-d_{0}}, \quad k=0,1, \ldots
Remark 3. We note that the assumptions of the above results require the existence of the second derivative of DD, and also that the smaller d_(0)d_{0} (i.e., the smaller |lambda|,b,M|\lambda|, b, M and rho_(0)\rho_{0} ), the faster is the convergence of sequence (4).
2. LOCAL CONVERGENCE
In order to analyze the local convergence of the considered iterates, we shall use the Ostrowski local attraction theorem, which offers sharp general conditions ensuring the local convergence. We shall consider for simplicity that X=R^(n)X=\mathbb{R}^{n}, with ||*||\|\cdot\| an arbitrary given norm, though the results hold in Banach spaces (see, e.g., [5, NR 10.1-3.]).
Theorem 4 (Ostrowski local attraction theorem). [5, Th. 10.1.3] Suppose that G:Omega subR^(n)rarrR^(n)G: \Omega \subset \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} has a fixed point x^(**)in int(Omega)x^{*} \in \operatorname{int}(\Omega) and is differentiable at x^(**)x^{*}. If the spectral radius of G^(')(x^(**))G^{\prime}\left(x^{*}\right) satisfies
rho(G^(')(x^(**)))=sigma < 1,\rho\left(G^{\prime}\left(x^{*}\right)\right)=\sigma<1,
then x^(**)x^{*} is a point of attraction of the successive approximations x_(k+1)=G(x_(k))x_{k+1}=G\left(x_{k}\right), k >= 0k \geq 0, i.e., there exists an open neighborhood V sube OmegaV \subseteq \Omega of x^(**)x^{*} such that AAx_(0)in V\forall x_{0} \in V, the successive approximations given above all lie in Omega\Omega and converge to x^(**)x^{*}.
Remark 5. The classical book of Ortega and Rheinboldt also contains completions to this result (see [5, Ch. 10]), in the sense that the spectral radius sigma\sigma yields the "worst" ( r-r- )convergence factor among the sequences converging to the fixed point: when sigma!=0\sigma \neq 0, the convergence of the (whole) process is not faster than linear (though, theoretically, there may exist sequences converging at least rr-superlinearly), while when sigma=0\sigma=0, all the sequences converge at least rr-superlinearly. This result was refined by us in [1], where we have shown that x_(k)rarrx^(**)qx_{k} \rightarrow x^{*} q-superlinearly iff G^(')(x^(**))G^{\prime}\left(x^{*}\right) has a zero eigenvalue and, starting from a certain step, x^(**)-x_(k)x^{*}-x_{k} are corresponding eigenvectors. This implies that no qq-superlinear convergence may occur when G^(')(x^(**))G^{\prime}\left(x^{*}\right) has no zero eigenvalue.
The above result can be applied to method (2) if we notice that the derivative of x-A(x)F(x)x-A(x) F(x) has a simple form at the fixed point x^(**)x^{*}, the following auxiliary result being similar to (Lemma) 10.2.1 in [5].
Lemma 6. Suppose that F:Omega subR^(n)rarrR^(n)F: \Omega \subset \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} is differentiable at a point x^(**)in int(Omega)x^{*} \in \operatorname{int}(\Omega) for which F(x^(**))=0F\left(x^{*}\right)=0. Let A:Omega_(0)rarrL(R^(n))A: \Omega_{0} \rightarrow \mathcal{L}\left(\mathbb{R}^{n}\right) be defined on an open neighborhood Omega_(0)sube Omega\Omega_{0} \subseteq \Omega of x^(**)x^{*} and continuous at x^(**)x^{*}. Then the mapping G:S rarrR^(n)G: S \rightarrow \mathbb{R}^{n},
G(x)=x-A(x)F(x)G(x)=x-A(x) F(x)
is differentiable at x^(**)x^{*} and
G^(')(x^(**))=I-A(x^(**))F^(')(x^(**)).G^{\prime}\left(x^{*}\right)=I-A\left(x^{*}\right) F^{\prime}\left(x^{*}\right) .
Proof. The proof is elementary:
{:[||G(x)-G(x^(**))-[I-A(x^(**))F^(')(x^(**))](x-x^(**))||=],[=||(A(x)-A(x^(**)))F(x)+A(x^(**))[F(x)-F(x^(**))-F^(')(x^(**))(x-x^(**))]||],[=o(||x-x^(**)||)","quad" as "x rarrx^(**)]:}\begin{aligned}
& \left\|G(x)-G\left(x^{*}\right)-\left[I-A\left(x^{*}\right) F^{\prime}\left(x^{*}\right)\right]\left(x-x^{*}\right)\right\|= \\
& =\left\|\left(A(x)-A\left(x^{*}\right)\right) F(x)+A\left(x^{*}\right)\left[F(x)-F\left(x^{*}\right)-F^{\prime}\left(x^{*}\right)\left(x-x^{*}\right)\right]\right\| \\
& =o\left(\left\|x-x^{*}\right\|\right), \quad \text { as } x \rightarrow x^{*}
\end{aligned}
Now we can state the main results of this note. First, consider iterations (2).
Theorem 7. Let D:R^(n)rarrR^(n),y inR^(n),lambda inR,x^(**)D: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}, y \in \mathbb{R}^{n}, \lambda \in \mathbb{R}, x^{*} a solution of F(x):=x-lambda D(x)-y=0F(x):= x-\lambda D(x)-y=0, and the mapping AA is defined on an open neighborhood EE of x^(**),A:E rarrL(R^(n))x^{*}, A: E \rightarrow \mathcal{L}\left(\mathbb{R}^{n}\right). If DD is differentiable at x^(**),Ax^{*}, A is continuous at x^(**)x^{*} and
rho(I-A(x^(**))(I-lambdaD^(')(x^(**))) < 1:}\rho\left(I-A\left(x^{*}\right)\left(I-\lambda D^{\prime}\left(x^{*}\right)\right)<1\right.
then x^(**)x^{*} is a point of attraction for the method (2).
The proof is an immediate application of Lemma 6 and Theorem 4.
The conditions are much simpler for the case of the second method.
Theorem 8. Let D:R^(n)rarrR^(n),y inR^(n),lambda inR,x^(**)D: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}, y \in \mathbb{R}^{n}, \lambda \in \mathbb{R}, x^{*} a solution of F(x):=x-lambda D(x)-y=0F(x):= x-\lambda D(x)-y=0. If the mapping DD is differentiable on an open neighborhood of x^(**)x^{*}, with D^(')D^{\prime} continuous at x^(**)x^{*}, and
The proof is an immediate application of Lemma 6 and Theorem 4.
The conditions are much simpler for the case of the second method.
Theorem 8. Let D:R^(n)rarrR^(n),y inR^(n),lambda inR,x^(**)D: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}, y \in \mathbb{R}^{n}, \lambda \in \mathbb{R}, x^{*} a solution of F(x):=x-lambda D(x)-y=0F(x):= x-\lambda D(x)-y=0. If the mapping DD is differentiable on an open neighborhood of x^(**)x^{*}, with D^(')D^{\prime} continuous at x^(**)x^{*}, and
|lambda|rho(D^(')(x^(**))) < 1|\lambda| \rho\left(D^{\prime}\left(x^{*}\right)\right)<1
then x^(**)x^{*} is a point of attraction for the method (4).
Proof. By Lemma 6 we get
Proof. By Lemma 6 we get
G^(')(x^(**))=I-(I+lambdaD^(')(x^(**)))(I-lambdaD^(')(x^(**)))=lambda^(2)D^(')(x^(**))^(2),G^{\prime}\left(x^{*}\right)=I-\left(I+\lambda D^{\prime}\left(x^{*}\right)\right)\left(I-\lambda D^{\prime}\left(x^{*}\right)\right)=\lambda^{2} D^{\prime}\left(x^{*}\right)^{2},
whence, by Theorem 4, the conclusion follows.
The same observations as in Remark 5 apply.
The same observations as in Remark 5 apply.
REFERENCES
[1] E. Cătinaş, On the superlinear convergence of the successive approximations method, J. Optim. Theory Appl., 113 (2002) no. 3, pp. 473-485. 뜬
[2] E. Cătinaş, The inexact, inexact perturbed and quasi-Newton methods are equivalent models, Math. Comp., 74 (2005) no. 249, pp. 291-301. ㄸ
[3] E. Cătinaş, On the convergence orders, manuscript.
[4] Diaconu, A., Păvăloiu, I., Sur quelque méthodes itératives pour la resolution des équations opérationnelles, Rev. Anal. Numér. Theor. Approx., vol. 1, 45-61 (1972). ㄸ
[5] J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970.
[6] I. Păvăloiu, La convergence de certaines méthodes itératives pour résoudre certaines equations operationnelles, Seminar on functional analysis and numerical methods, Preprint no. 1 (1986), pp. 127-132 (in French).
[7] I. Păvăloiu, A unified treatment of the modified Newton and chord methods, Carpathian J. Math. 25 (2009) no. 2, pp. 192-196.
[2] E. Cătinaş, The inexact, inexact perturbed and quasi-Newton methods are equivalent models, Math. Comp., 74 (2005) no. 249, pp. 291-301. ㄸ
[3] E. Cătinaş, On the convergence orders, manuscript.
[4] Diaconu, A., Păvăloiu, I., Sur quelque méthodes itératives pour la resolution des équations opérationnelles, Rev. Anal. Numér. Theor. Approx., vol. 1, 45-61 (1972). ㄸ
[5] J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970.
[6] I. Păvăloiu, La convergence de certaines méthodes itératives pour résoudre certaines equations operationnelles, Seminar on functional analysis and numerical methods, Preprint no. 1 (1986), pp. 127-132 (in French).
[7] I. Păvăloiu, A unified treatment of the modified Newton and chord methods, Carpathian J. Math. 25 (2009) no. 2, pp. 192-196.
Received by the editors: October 3, 2015.
- *"T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania, e-mail: ecatinas@ictp.acad.ro.