Abstract
We have recently characterized the q-quadratic convergence of the perturbed successive approximations. For a particular choice of the parameters, these sequences resulted as accelerated iterations toward a fixed point. We give here a Kantorovich-type result, which provides sufficient conditions ensuring the convergence of the accelerated iterates.
Authors
Keywords
fixed point; successive approximations; accelerated successive approximations; nonlinear system of equations in Rn; inexact Newton method; perturbed Newton method; ; local convergence; convergence order.
References
Cite this paper as:
E. Cătinaş, Sufficient convergence conditions for certain accelerated successive approximations. In: Mache D.H., Szabados J., de Bruin M.G. (eds) Trends and Applications in Constructive Approximation. ISNM International Series of Numerical Mathematics, vol 151, pp. 71-75, 2005. Birkhäuser Basel
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Trends and Applications in Constructive Approximation. ISNM International Series of Numerical Mathematics, vol 151.
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Birkhäuser Basel
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978-3-7643-7124-1
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978-3-7643-7356-6
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Sufficient convergence conditions for certain accelerated successive approximations
Abstract
We have recently characterized the qq-quadratic convergence of the perturbed successive approximations. For a particular choice of the parameters, these sequences resulted as accelerated iterations toward a fixed point.
We give here a Kantorovich-type result, which provides sufficient conditions ensuring the convergence of the accelerated iterates.
1 Introduction
Let (X,||*||)(X,\|\cdot\|) be a Banach space and G:Omega sube X rarr OmegaG: \Omega \subseteq X \rightarrow \Omega a nonlinear mapping having x^(**)in int Omegax^{*} \in \operatorname{int} \Omega as fixed point:
x^(**)=G(x^(**))x^{*}=G\left(x^{*}\right)
We are interested in the qq-quadratic convergence toward x^(**)x^{*} of the sequences of successive approximation type. Recall that an arbitrary sequence (y_(k))_(k >= 0)sub X\left(y_{k}\right)_{k \geq 0} \subset X converges ( qq-)quadratically to its limit bar(y)in X\bar{y} \in X if [11], [12], [13]
i n f{alpha in[1,+oo):Q_(alpha){y_(k)}=+oo}=2\inf \left\{\alpha \in[1,+\infty): Q_{\alpha}\left\{y_{k}\right\}=+\infty\right\}=2
where
Q_(alpha){y_(k)}={[0","," if "y_(k)= bar(y)","" for all but finitely many "k],[l i m s u p_({:[k rarr oo],[+oo","]:})(||y_(k+1)-( bar(y))||)/(||y_(k)-( bar(y))||^(alpha),)," if "y_(k)!= bar(y)","" for all but finitely many "k],[," otherwise "]:}Q_{\alpha}\left\{y_{k}\right\}= \begin{cases}0, & \text { if } y_{k}=\bar{y}, \text { for all but finitely many } k \\ \limsup _{\substack{k \rightarrow \infty \\+\infty,}} \frac{\left\|y_{k+1}-\bar{y}\right\|}{\left\|y_{k}-\bar{y}\right\|^{\alpha},} & \text { if } y_{k} \neq \bar{y}, \text { for all but finitely many } k \\ & \text { otherwise }\end{cases}
In the case when 0 < Q_(2){y_(k)} < +oo0<Q_{2}\left\{y_{k}\right\}<+\infty, one obtains the well known estimate of the form
||y_(k+1)-( bar(y))|| <= (Q_(p){y_(k)}+epsi)||y_(k)-( bar(y))||^(2),quad" for all "k >= k_(0)\left\|y_{k+1}-\bar{y}\right\| \leq\left(Q_{p}\left\{y_{k}\right\}+\varepsilon\right)\left\|y_{k}-\bar{y}\right\|^{2}, \quad \text { for all } k \geq k_{0}
(in the sense that for all epsi > 0\varepsilon>0 there exists k_(0) >= 0k_{0} \geq 0 such the above inequalities hold).
The successive approximations converging quadratically to x^(**)x^{*} are characterized by the following result.
Theorem 1.1 [6] Assume that GG is differentiable on a neighborhood DD of x^(**)x^{*}, with the derivative G^(')G^{\prime} Lipschitz continuous:
||G^(')(x)-G^(')(y)|| <= L||x-y||,quad AA x,y in D.\left\|G^{\prime}(x)-G^{\prime}(y)\right\| \leq L\|x-y\|, \quad \forall x, y \in D .
Suppose further that for a certain initial approximation x_(0)in Dx_{0} \in D, the successive approximations
x_(k+1)=G(x_(k)),quad k >= 0x_{k+1}=G\left(x_{k}\right), \quad k \geq 0
converge to x^(**)x^{*}, and I-G^(')(x_(k))I-G^{\prime}\left(x_{k}\right) are invertible starting from a certain step.
Then the convergence is with order 2 if and only if G^(')G^{\prime} has a zero eigenvalue and, starting from a certain step, the corrections x_(k+1)-x_(k)x_{k+1}-x_{k} are corresponding eigenvectors:
Then the convergence is with order 2 if and only if G^(')G^{\prime} has a zero eigenvalue and, starting from a certain step, the corrections x_(k+1)-x_(k)x_{k+1}-x_{k} are corresponding eigenvectors:
G^(')(x^(**))(x_(k+1)-x_(k))=0,quad AA k >= k_(0)G^{\prime}\left(x^{*}\right)\left(x_{k+1}-x_{k}\right)=0, \quad \forall k \geq k_{0}
This condition holds equivalently iff the errors x_(k)-x^(**)x_{k}-x^{*} are corresponding eigenvectors:
G^(')(x^(**))(x_(k)-x^(**))=0,quad AA k >= k_(0)G^{\prime}\left(x^{*}\right)\left(x_{k}-x^{*}\right)=0, \quad \forall k \geq k_{0}
or iff
x_(k)inx^(**)+KerG^(')(x^(**)),quad AA k >= k_(0)x_{k} \in x^{*}+\operatorname{Ker} G^{\prime}\left(x^{*}\right), \quad \forall k \geq k_{0}
This result implies that if G^(')G^{\prime} has no eigenvalue 0 , there exists no sequence of successive approximations converging to x^(**)x^{*} with order 2.^(1)2 .{ }^{1} In such a case, one may choose to consider for some (delta_(k))_(k >= 0)sub X\left(\delta_{k}\right)_{k \geq 0} \subset X the perturbed successive approximations
{:(1)x_(k+1)=G(x_(k))+delta_(k)","quad k >= 0:}\begin{equation*}
x_{k+1}=G\left(x_{k}\right)+\delta_{k}, \quad k \geq 0 \tag{1}
\end{equation*}
Their quadratic convergence is characterized by the following result, which does not require the existence of the eigenvalue 0 .
Theorem 1.2 [6] Suppose that GG satisfies the assumptions of Theorem 1.1, and that the sequence (1) of perturbed successive approximations converges to x^(**)x^{*}. Then the convergence is with qq-order 2 iff
{:(2)||G^(')(x_(k))(x_(k)-G(x_(k)))+(I-G^(')(x_(k)))delta_(k)||=O(||x_(k)-G(x_(k))||^(2))","" as "k rarr oo.:}\begin{equation*}
\left\|G^{\prime}\left(x_{k}\right)\left(x_{k}-G\left(x_{k}\right)\right)+\left(I-G^{\prime}\left(x_{k}\right)\right) \delta_{k}\right\|=O\left(\left\|x_{k}-G\left(x_{k}\right)\right\|^{2}\right), \text { as } k \rightarrow \infty . \tag{2}
\end{equation*}
In [5] we have shown that if we write
delta_(k)=(I-G^(')(x_(k)))^(-1)(G^(')(x_(k))(G(x_(k))-x_(k))+gamma_(k))\delta_{k}=\left(I-G^{\prime}\left(x_{k}\right)\right)^{-1}\left(G^{\prime}\left(x_{k}\right)\left(G\left(x_{k}\right)-x_{k}\right)+\gamma_{k}\right)
with (gamma_(k))_(k >= 0)sub X\left(\gamma_{k}\right)_{k \geq 0} \subset X, then condition
gamma_(k)=O(||x_(k)-G(x_(k))||^(2)),quad" as "k rarr oo,\gamma_{k}=O\left(\left\|x_{k}-G\left(x_{k}\right)\right\|^{2}\right), \quad \text { as } k \rightarrow \infty,
is equivalent to (2).
We have also noticed in [5] that, under the assumption ||G^(')(x)|| <= q < 1\left\|G^{\prime}(x)\right\| \leq q<1 for all xx in a certain neighborhood of x^(**)x^{*}, and for a given K > 0K>0, a natural choice (implied by the Banach lemma) for delta_(k)\delta_{k} is:
delta_(k)=(I+cdots+G^(')(x_(k))^(i_(k)))G^(')(x_(k))(G(x_(k))-x_(k))\delta_{k}=\left(I+\cdots+G^{\prime}\left(x_{k}\right)^{i_{k}}\right) G^{\prime}\left(x_{k}\right)\left(G\left(x_{k}\right)-x_{k}\right)
with i_(k)i_{k} such that
{:(3)(q^(ik)+2)/(1-q) <= K||x_(k)-G(x_(k))||:}\begin{equation*} \frac{q^{i k}+2}{1-q} \leq K\left\|x_{k}-G\left(x_{k}\right)\right\| \tag{3} \end{equation*}
gamma_(k)=O(||x_(k)-G(x_(k))||^(2)),quad" as "k rarr oo,\gamma_{k}=O\left(\left\|x_{k}-G\left(x_{k}\right)\right\|^{2}\right), \quad \text { as } k \rightarrow \infty,
is equivalent to (2).
We have also noticed in [5] that, under the assumption ||G^(')(x)|| <= q < 1\left\|G^{\prime}(x)\right\| \leq q<1 for all xx in a certain neighborhood of x^(**)x^{*}, and for a given K > 0K>0, a natural choice (implied by the Banach lemma) for delta_(k)\delta_{k} is:
delta_(k)=(I+cdots+G^(')(x_(k))^(i_(k)))G^(')(x_(k))(G(x_(k))-x_(k))\delta_{k}=\left(I+\cdots+G^{\prime}\left(x_{k}\right)^{i_{k}}\right) G^{\prime}\left(x_{k}\right)\left(G\left(x_{k}\right)-x_{k}\right)
with i_(k)i_{k} such that
{:(3)(q^(ik)+2)/(1-q) <= K||x_(k)-G(x_(k))||:}\begin{equation*} \frac{q^{i k}+2}{1-q} \leq K\left\|x_{k}-G\left(x_{k}\right)\right\| \tag{3} \end{equation*}
When applying Theorem 1.2 to characterize the quadratic convergence of the resulted sequence
{:(4)x_(k+1)=G(x_(k))+(I+cdots+G^(')(x_(k))^(i_(k)))G^(')(x_(k))(G(x_(k))-x_(k))","quad k >= 0:}\begin{equation*}
x_{k+1}=G\left(x_{k}\right)+\left(I+\cdots+G^{\prime}\left(x_{k}\right)^{i_{k}}\right) G^{\prime}\left(x_{k}\right)\left(G\left(x_{k}\right)-x_{k}\right), \quad k \geq 0 \tag{4}
\end{equation*}
with i_(k)i_{k} given by (3), we must assume that this sequence converges to the fixed point x^(**)x^{*}. But is this assumption reasonable? The purpose of this note is to show that under certain natural conditions the sequence converges to x^(**)x^{*}, so the answer is positive.
2 Main result
First of all, we remark that the fixed point problem is equivalent to solving
F(x)=0," with "F(x)=x-G(x),F(x)=0, \text { with } F(x)=x-G(x),
for which the Newton method generates the iterates
{:[(5)s_(k)^(N)=-F^(')(x_(k))^(-1)F(x_(k))],[x_(k+1)=x_(k)+s_(k)^(N)","quad k=0","1","dots]:}\begin{align*}
s_{k}^{N} & =-F^{\prime}\left(x_{k}\right)^{-1} F\left(x_{k}\right) \tag{5}\\
x_{k+1} & =x_{k}+s_{k}^{N}, \quad k=0,1, \ldots
\end{align*}
In this setting, iterations (4) may be rewritten as
{:[(6)x_(k+1)=x_(k)+(I+G^(')(x_(k))+dots+G^(')(x_(k))^(i_(k)+1))(G(x_(k))-x_(k))],[:=x_(k)+s_(k)","quad k=0","1","dots],[" with "i_(k)" s.t. "(q^(i_(k)+2))/(1-q) <= K||F(x_(k))||]:}\begin{align*}
x_{k+1} & =x_{k}+\left(I+G^{\prime}\left(x_{k}\right)+\ldots+G^{\prime}\left(x_{k}\right)^{i_{k}+1}\right)\left(G\left(x_{k}\right)-x_{k}\right) \tag{6}\\
& :=x_{k}+s_{k}, \quad k=0,1, \ldots \\
& \text { with } i_{k} \text { s.t. } \frac{q^{i_{k}+2}}{1-q} \leq K\left\|F\left(x_{k}\right)\right\|
\end{align*}
i.e., as quasi-Newton iterations (see, e.g., [11], [8], [7]).
We obtain the following sufficient Kantorovich-type conditions for the convergence to x^(**)x^{*} of these iterates.
We obtain the following sufficient Kantorovich-type conditions for the convergence to x^(**)x^{*} of these iterates.
Theorem 2.1 Assume that GG is differentiable on the domain Omega\Omega, with G^(')G^{\prime} bounded on Omega\Omega by
{:(7)||G^(')(x)|| <= q < 1","quad AA x in Omega:}\begin{equation*}
\left\|G^{\prime}(x)\right\| \leq q<1, \quad \forall x \in \Omega \tag{7}
\end{equation*}
and Lipschitz continuous:
||G^(')(x)-G^(')(y)|| <= L||x-y||,quad AA x,y in Omega\left\|G^{\prime}(x)-G^{\prime}(y)\right\| \leq L\|x-y\|, \quad \forall x, y \in \Omega
Let x_(0)in Omegax_{0} \in \Omega and K > 0K>0 be chosen such that
{:(8)nu=((L)/(2(1-q)^(2))+K(1+q))||F(x_(0))|| < 1",":}\begin{equation*}
\nu=\left(\frac{L}{2(1-q)^{2}}+K(1+q)\right)\left\|F\left(x_{0}\right)\right\|<1, \tag{8}
\end{equation*}
and suppose that bar(B)_(r)(x_(0))={x in X:||x-x_(0)|| <= r}sube Omega\bar{B}_{r}\left(x_{0}\right)=\left\{x \in X:\left\|x-x_{0}\right\| \leq r\right\} \subseteq \Omega for
r=(1)/((1-nu)(1-q))||F(x_(0))||r=\frac{1}{(1-\nu)(1-q)}\left\|F\left(x_{0}\right)\right\|
Then the elements of the sequence defined by (6) remain in the ball bar(B)_(r)(x_(0))\bar{B}_{r}\left(x_{0}\right) and converge to a fixed point x^(**)x^{*} of GG, which is unique in this ball. According to Theorem 1.2, the convergence is quadratic.
Proof. Recall first [11, 3.2.12] that the Lipschitz hypothesis on G^(')G^{\prime} implies that
||G(y)-G(x)-G^(')(x)(y-x)|| <= (L)/(2)||y-x||^(2),quad AA x,y in Omega\left\|G(y)-G(x)-G^{\prime}(x)(y-x)\right\| \leq \frac{L}{2}\|y-x\|^{2}, \quad \forall x, y \in \Omega
while (7) attracts the existence of (I-G^(')(x))^(-1)=I+G^(')(x)+dots+G^(')(x)^(k)+dots\left(I-G^{\prime}(x)\right)^{-1}=I+G^{\prime}(x)+\ldots+G^{\prime}(x)^{k}+\ldots and the bound
||(I-G^(')(x))^(-1)|| <= (1)/(1-q),quad AA x in Omega\left\|\left(I-G^{\prime}(x)\right)^{-1}\right\| \leq \frac{1}{1-q}, \quad \forall x \in \Omega
Our hypotheses imply the following inequalities:
||s_(0)|| <= (1)/(1-q)||F(x_(0))||\left\|s_{0}\right\| \leq \frac{1}{1-q}\left\|F\left(x_{0}\right)\right\|
i.e., x_(1)in bar(B)_(r)(x_(0))x_{1} \in \bar{B}_{r}\left(x_{0}\right) and also
{:[||F(x_(1))||=||F(x_(1))-F(x_(0))-F^(')(x_(0))s_(0)^(N)||quad(" by "(5))],[ <= ||F(x_(1))-F(x_(0))-F^(')(x_(0))s_(0)||+||F^(')(x_(0))(s_(0)^(N)-s_(0))||],[ <= ||G(x_(1))-G(x_(0))-G^(')(x_(0))s_(0)||+(1+q)||s_(0)^(N)-s_(0)||],[ <= (L)/(2)||s_(0)||^(2)+(1+q)||G^(')(x_(0))^(i_(0)+2)(I+G^(')(x_(0))+dots)F(x_(0))||],[ <= (L)/(2)(1+q+dots+q^(i_(0)+1))^(2)||F(x_(0))||^(2)+(q^(i_(0)+2))/(1-q)(1+q)||F(x_(0))||],[ <= (L(1-q^(i_(0)+2))^(2))/(2(1-q)^(2))||F(x_(0))||^(2)+K(1+q)||F(x_(0))||^(2)],[ <= nu||F(x_(0))||]:}\begin{aligned}
\left\|F\left(x_{1}\right)\right\| & =\left\|F\left(x_{1}\right)-F\left(x_{0}\right)-F^{\prime}\left(x_{0}\right) s_{0}^{N}\right\| \quad(\text { by }(5)) \\
& \leq\left\|F\left(x_{1}\right)-F\left(x_{0}\right)-F^{\prime}\left(x_{0}\right) s_{0}\right\|+\left\|F^{\prime}\left(x_{0}\right)\left(s_{0}^{N}-s_{0}\right)\right\| \\
& \leq\left\|G\left(x_{1}\right)-G\left(x_{0}\right)-G^{\prime}\left(x_{0}\right) s_{0}\right\|+(1+q)\left\|s_{0}^{N}-s_{0}\right\| \\
& \leq \frac{L}{2}\left\|s_{0}\right\|^{2}+(1+q)\left\|G^{\prime}\left(x_{0}\right)^{i_{0}+2}\left(I+G^{\prime}\left(x_{0}\right)+\ldots\right) F\left(x_{0}\right)\right\| \\
& \leq \frac{L}{2}\left(1+q+\ldots+q^{i_{0}+1}\right)^{2}\left\|F\left(x_{0}\right)\right\|^{2}+\frac{q^{i_{0}+2}}{1-q}(1+q)\left\|F\left(x_{0}\right)\right\| \\
& \leq \frac{L\left(1-q^{i_{0}+2}\right)^{2}}{2(1-q)^{2}}\left\|F\left(x_{0}\right)\right\|^{2}+K(1+q)\left\|F\left(x_{0}\right)\right\|^{2} \\
& \leq \nu\left\|F\left(x_{0}\right)\right\|
\end{aligned}
In an analogous fashion, we obtain by induction that for all k >= 2k \geq 2
{:[||F(x_(k))|| <= ((L)/(2(1-q)^(2))+K(1+q))||F(x_(k-1))||^(2)],[ <= nu||F(x_(k-1))||],[vdots],[ <= nu^(k)||F(x_(0))||]:}\begin{aligned}
\left\|F\left(x_{k}\right)\right\| & \leq\left(\frac{L}{2(1-q)^{2}}+K(1+q)\right)\left\|F\left(x_{k-1}\right)\right\|^{2} \\
& \leq \nu\left\|F\left(x_{k-1}\right)\right\| \\
& \vdots \\
& \leq \nu^{k}\left\|F\left(x_{0}\right)\right\|
\end{aligned}
{:[||x_(k)-x_(k-1)||=||s_(k-1)|| <= (1)/(1-q)||F(x_(k-1))|| <= (nu^(k-1))/(1-q)||F(x_(0))||],[||x_(k)-x_(0)|| <= ||x_(k)-x_(k-1)||+dots+||x_(1)-x_(0)|| <= (1)/((1-nu)(1-q))||F(x_(0))||=r]:}\begin{aligned}
\left\|x_{k}-x_{k-1}\right\| & =\left\|s_{k-1}\right\| \leq \frac{1}{1-q}\left\|F\left(x_{k-1}\right)\right\| \leq \frac{\nu^{k-1}}{1-q}\left\|F\left(x_{0}\right)\right\| \\
\left\|x_{k}-x_{0}\right\| & \leq\left\|x_{k}-x_{k-1}\right\|+\ldots+\left\|x_{1}-x_{0}\right\| \leq \frac{1}{(1-\nu)(1-q)}\left\|F\left(x_{0}\right)\right\|=r
\end{aligned}
It follows that
{:[||x_(k+m)-x_(k)|| <= ||x_(k+m)-x_(k+m-1)||+dots+||x_(k+1)-x_(k)||],[ <= (nu^(k+m-1)+dots+nu^(k))/((1-q))||F(x_(0))||],[ <= (nu^(k))/((1-nu)(1-q))||F(x_(0))||]:}\begin{aligned}
\left\|x_{k+m}-x_{k}\right\| & \leq\left\|x_{k+m}-x_{k+m-1}\right\|+\ldots+\left\|x_{k+1}-x_{k}\right\| \\
& \leq \frac{\nu^{k+m-1}+\ldots+\nu^{k}}{(1-q)}\left\|F\left(x_{0}\right)\right\| \\
& \leq \frac{\nu^{k}}{(1-\nu)(1-q)}\left\|F\left(x_{0}\right)\right\|
\end{aligned}
which shows that (x_(k))_(k >= 0)\left(x_{k}\right)_{k \geq 0} is a Cauchy sequence, and therefore converges to a certain x^(**)in bar(B)_(r)(x_(0))x^{*} \in \bar{B}_{r}\left(x_{0}\right). By the definition and continuity of F,x^(**)F, x^{*} is a fixed point of GG, which is unique in bar(B)_(r)(x_(0))\bar{B}_{r}\left(x_{0}\right) (and also in Omega\Omega ) since GG is a contraction.
We note that condition (8) contains certain natural demands: ||F(x_(0))||\left\|F\left(x_{0}\right)\right\| is sufficiently small (which holds, e.g., when x_(0)x_{0} is sufficiently close to x^(**)x^{*} ), qq is sufficiently small (in accordance with the results in [6]), the Lipschitz constant LL is sufficiently small (the graph of GG is close to a constant in case X=RX=R ) and KK is sufficiently small (the linear systems are solved with increasingly precision, the iterates approaching to those given by the Newton method - see the classical results of Dennis and Moré [8]).
Acknowledgement: The author would like to thank Professor Detlef Mache for his kind support regarding the participation to the IBoMat04 Conference.
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"T. Popoviciu" Institute of Numerical Analysis (Romanian Academy)
P.O. Box 68-1
[2] C. Brezinski, Dynamical systems and sequence transformations, J. Phys. A, 34 (2001), pp. 10659-10669.
[3] C. Brezinski and J.-P. Chehab, Nonlinear hybrid procedures and fixed point iterations, Numer. Funct. Anal. Optimiz., 19 (1998), pp. 465-487.
[4] E. Cătinaş, Inexact perturbed Newton methods and applications to a class of Krylov solvers, J. Optim. Theory Appl., 108 (2001), 543-570.
[5] E. Cătinaş, On accelerating the convergence of the successive approximations method, Rev. Anal. Numér. Théor. Approx., 30 (2001) no. 1, pp. 3-8.
[6] E. Cătinas, On the superlinear convergence of the successive approximations method, J. Optim. Theory Appl., 113 (2002) no. 3, pp. 473-485
[7] E. Cătinaş, The inexact, inexact perturbed and quasi-Newton methods are equivalent models, Math. Comp., (www.ams.org/mcom) to appear.
[8] J. E. Dennis, Jr., and J. J. Moré, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comp., 28 (1974), 549560.
[9] Şt. Măruşter, Quasi-nonexpansivity and two classical methods for solving nonlinear equations, Proc. AMS, 62 (1977), 119-123.
[10] I. Moret, On the behaviour of approximate Newton methods, Computing, 37 (1986), pp. 185-193.
[11] J. M. Ortega, and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
[12] F. A. Potra, On QQ-order and RR-order of convergence, J. Optim. Theory Appl., 63 (1989), 415-431.
[13] W. C. Rheinboldt, Methods for Solving Systems of Nonlinear Equations, SIAM, Philadelphia, 1998.
[14] D. A. Smith, W. F. Ford and A. Sidi, Extrapolation methods for vector sequences, SIAM Rev., 29 (1987), pp. 199-233.
"T. Popoviciu" Institute of Numerical Analysis (Romanian Academy)
P.O. Box 68-1
- ^(1){ }^{1} In this case, the successive approximations cannot converge faster than qq-linearly [6].