AbstractAn approximation to the exact derivative leads to perturbed fixed slope iterations in the context of Inexact Newton methods. We prove an a posteriori convergence theorem for such an algorithm, and show an application to nonlinear differential boundary value problems. The abstract setting is a complex Banach space

Ahues, Mario

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Applied Mathematics Letters 18 (2005) 375–380
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A note on perturbed fixed slope iterations
Mario Ahues
Équipe d’Analyse Numérique (EA3058), Université de Saint-Étienne, 23 rue Docteur Paul Michelon,
42023 Saint-Étienne, France
Received 1 January 2004; accepted 1 March 2004
Abstract
An approximation to the exact derivative leads to perturbed fixed slope iterations in the context of Inexact
Newton methods. We prove an a posteriori convergence theorem for such an algorithm, and show an application to
nonlinear differential boundary value problems. The abstract setting is a complex Banach space.
© 2004 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear problem; Newton fixed slope method; A posteriori error bounds
1. The fixed slope iteration
Let us consider the following abstract formulation of a nonlinear equation: given a nonempty open set
Ω in a Banach space X whose norm is denoted by ‖ · ‖,
find x∞ ∈ Ω such that F(x∞) = 0, (1)
where F : Ω → X is a Fréchet differentiable (most probably nonlinear) operator.
We recall that the standard Fixed slope iteration for solving (1) is defined by
ξ0 ∈ Ω,
ξk+1 := ξk − F ′(ξ0)−1 F(ξk) for all k ≥ 0,
(2)
where F is supposed to have an invertible Fréchet derivative at ξ0.
E-mail address: Mario.Ahues@univ-st-etienne.fr.
0893-9659/$ - see front matter © 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.aml.2004.03.010
376 M. Ahues / Applied Mathematics Letters 18 (2005) 375–380
Let L(X) denote the Banach algebra of bounded linear operators from X into itself. Its norm is also
denoted by ‖ · ‖.
The following a posteriori result gives sufficient conditions for the existence and uniqueness of a solu-
tion of (1) at a vicinity of ξ0, and for the convergence of the sequence defined by (2) towards it (cf. [1,2]).
Theorem 1. Suppose that Ω , F, ξ0 ∈ Ω ,  > 0, µ > 0 and c > 0 satisfy
1. 4µc ≤ 1,
2. Ω includes the closed neighborhood N0 := {ξ ∈ X : ‖ξ − ξ0‖ ≤ r}, where
r := 1 −
√
1 − 4µc
2µ
,
3. F ′(ξ) exists for all ξ ∈ N0,
4. F ′(ξ0) is invertible, and ‖F ′(ξ0)−1‖ ≤ µ,
5. ‖ξ1 − ξ0‖ ≤ c,
6.  is a Lipschitz constant for F ′ : N0 → L(X).
Then (1) has a unique solution x∞ in N0 and the sequence defined by (2) converges to x∞ at the
following rate: for all k ≥ 0,
‖ξk − x∞‖ ≤ cβ
k
1 − β ,
where
β := 1
2
(1 −√1 − 4µc) ∈
[
0,
1
2
[
.
2. A perturbed fixed slope iteration
A perturbed Fixed slope iteration for solving (1) may be defined as
x0 ∈ Ω,
xk+1 := xk − B−1F(xk) for all k ≥ 0,
(3)
where B ∈ L(X) is invertible and sufficiently close to F ′(x0). The following a posteriori result gives
sufficient conditions for the existence and uniqueness of a solution of (1) at a vicinity of x0, and for the
convergence of the sequence defined by (3) to it.
Theorem 2. Suppose that Ω , F, B, x0 ∈ Ω , δ ≥ 0,  > 0, m > 0 and c > 0 satisfy
1. mδ < 1 and 4mc ≤ (1 − mδ)2,
2. Ω includes the closed ball B0 := {x ∈ X : ‖x − x0‖ ≤ }, where
 := 1 − mδ −
√
(1 − mδ)2 − 4mc
2m
, (4)
3. F ′(x) exists for all x ∈ B0,
4. ‖F ′(x0) − B‖ ≤ δ, B ∈ L(X) is invertible and ‖B−1‖ ≤ m,
5. ‖x1 − x0‖ ≤ c,
6.  is a Lipschitz constant for F ′ : B0 → L(X).
M. Ahues / Applied Mathematics Letters 18 (2005) 375–380 377
Then (1) has a unique solution x∞ in B0 and the sequence defined by (3) converges to x∞ at the
following rate: for all k ≥ 0,
‖xk − x∞‖ ≤ cγ
k
1 − γ ,
where
γ := 1
2
(1 + mδ −
√
(1 − mδ)2 − 4mc) ∈ [0, 1[.
Proof. For x ∈ B0, we set:
G(x) := x − B−1F(x),
E(x) := F(x0) + B(x − x0) − F(x).
We remark that x∞ = G(x∞) if and only if F(x∞) = 0. We will apply the Fixed point Banach theorem
to G. Let us prove that B0 is invariant under G and that G : B0 → X is a contraction. Notice that
E(x0) = 0,
and that, for all x ∈ B0,
E ′(x) = B − F ′(x0) + F ′(x0) − F ′(x),
‖E ′(x)‖ ≤ ‖x − x0‖ + δ ≤  + δ,
‖E(x)‖ = ‖E(x) − E(x0)‖ ≤ ( + δ)‖x − x0‖ ≤ ( + δ),
G(x) − x0 = B−1(E(x) − F(x0)).
Hence
‖G(x) − x0‖ ≤ m( + δ) + c = .
Also, for all x ∈ B0,
G ′(x) = I − B−1F ′(x) = B−1(B − F ′(x0) + F ′(x0) − F ′(x)) = B−1E ′(x),
so
‖G ′(x)‖ ≤ m( + δ) = γ < 1
for all x ∈ B0, which proves that G : B0 → X is a contraction. The convergence rate is a well known
result of successive approximation fixed point iterations. 
3. An application
Let L be a linear closed unbounded operator whose domain D is included in a complex Banach space
X normed by ‖ · ‖. Suppose that L has a bounded inverse T . Let N be a nonlinear Fréchet differentiable
operator such that N ′ is locally Lipschitz. For y ∈ X , we consider the abstract nonlinear boundary value
problem
Find x∞ ∈ D such that Lx∞ + N(x∞) = y,
or equivalently,
Find x∞ ∈ X such that x∞ + T N(x∞) = T y.
378 M. Ahues / Applied Mathematics Letters 18 (2005) 375–380
To enter the abstract setting of the previous sections, define the nonlinear Fréchet differentiable operator
F : X → X, x 	→ F(x) := x + T (N(x) − y).
Clearly, the first Fréchet derivative of F is defined by
F ′(x)h = h + T N ′(x)h for all x ∈ X, h ∈ X.
Let x0 ∈ X be given. Define B := I so that m = 1. Then
‖F ′(x0) − B‖ = ‖T N ′(x0)‖ ≤ δ := ‖T ‖ ‖N ′(x0)‖,
‖x1 − x0‖ = ‖B−1F(x0)‖ ≤ c := ‖x0‖ + ‖T ‖ ‖N(x0)‖ + ‖T ‖ ‖y‖.
Let ν be a Lipschitz constant for N ′ on a closed ball centered at x0 with radius r . Then
 := ν‖T ‖
is a Lipschitz constant for F ′ on the same ball. Suppose that
r ≥  := 1 − δ −
√
(1 − δ)2 − 4c
2
,
so  is a Lipschitz constant for F ′ on B0. Now we are in position to check if the hypotheses of Theorem 2
are satisfied or not. Iterations (3) read
xk+1 := T (y − N(xk)), k ≥ 0,
or equivalently
Lxk+1 := y − N(xk), xk+1 ∈ D, k ≥ 0.
As an example, let us consider a nonlinear differential problem (cf. [1,3]) in the space X := C0([0, 1],C)
of all complex-valued continuous functions defined on [0, 1], equipped with the uniform convergence
norm. Given y ∈ X , the nonlinear differential boundary value problem
Find x ∈ C2([0, 1],C) such that −x ′′ + x2 = y, x(0) = x(1) = 0, (5)
is equivalent to the nonlinear integral equation
x(s) +
∫ 1
0
κ(s, t) x(t)2 dt = w(s) :=
∫ 1
0
κ(s, t) y(t) dt for all s ∈ [0, 1],
where κ is the Green kernel defined by
κ(s, t) :=
{
s(1 − t), if 0 ≤ s ≤ t ≤ 1,
t (1 − s), otherwise.
The nonlinear operator F : X → X is then
F(x)(s) := x(s) +
∫ 1
0
κ(s, t) x(t)2 dt − w(s) for all s ∈ [0, 1] and all x ∈ X.
Choose the constant function
x0(s) := 12 for all s ∈ [0, 1], (6)
M. Ahues / Applied Mathematics Letters 18 (2005) 375–380 379
as the starting point of the sequence of iterations defined by (3), where B := I so m = 1. We remark that
N ′(x0) = I too. Hence
F ′(x0)h(s) =
∫ 1
0
κ(s, t) h(t) dt + h(s) for all s ∈ [0, 1] and all h ∈ X,
and
‖F ′(x0) − B‖ ≤ δ := max
s∈[0,1]
∫ 1
0
|κ(s, t)| dt = 1
8
.
Also
‖F ′(u) − F ′(v)‖ ≤ 2 δ‖u − v‖ = 1
4
‖u − v‖ for all u, v ∈ X,
and
‖x1 − x0‖ = ‖B−1F(x0)‖ ≤ c := 12 +
1
4
δ + 1
8
‖y‖.
We conclude from Theorem 2 that if ‖y‖ ≤ 1.875, then the sequence defined by (3) with starting point
(6) converges to the unique solution of (5) in the closed ball with radius
 := 1.750 −
√
1.875 − ‖y‖
2
centered at the starting point. The iterations read as
x0(s) := 12 for all s ∈ [0, 1],
xk+1(s) :=
∫ 1
0
κ(s, t)( y(t) − xk(t)2 ) dt for all s ∈ [0, 1].
Consider for example
y(s) := 1 + 1
4
(1 − s)2 s2, x∞(s) := 12 s(1 − s) for all s ∈ [0, 1].
Then, up to seven significant digits, ‖y‖ = 1.015625 and  = 1.094495.
4. Remarks and bibliographical comments
We remark that the condition ‖F ′(x0) − B‖ ≤ δ < 1m ≤ 1‖B−1‖ implies that F ′(x0) is invertible,
although its inverse is neither needed in computations nor used in the proof of the previous theorem, and
that each one of the six hypotheses of Theorem 2 reduces to the corresponding hypothesis of Theorem 1
when δ = 0, and the same holds for its conclusion, since δ = 0 implies γ = β. Hence the proof of
Theorem 2 carried out with δ = 0 gives a proof of Theorem 1.
The following figure shows the Neperian logarithm of the relative error corresponding to the first ten
iterates in the case of the application developed in Section 3. As predicted by theory, convergence is
linear.
380 M. Ahues / Applied Mathematics Letters 18 (2005) 375–380
The most interesting aspects of perturbed fixed slope iterations are the following:
– Theoretically speaking, they handle in an abstract setting the error in computations associated to the
exact fixed slope algorithm.
– Computationally speaking, they preserve the linear convergence predicted by the exact fixed slope
algorithm, although the rate itself may be weaker.
– They reduce the complexity of the classical Newton–Kantorovich method in which the Fréchet
derivative of the nonlinear operator F must be computed at each iterate, and with it, a linear problem
must be solved for the next iterate. Certainly, this computational advantage has a price: the quadratic
rate of convergence becomes linear.
A spectral application of Theorem 2 is developed in [4].
References
[1] M. Ahues, Calcul Scientifique, Textbook edited by Université Jean Monnet de Saint-Étienne, France, 2000.
[2] R.F. Curtain, A.J. Pritchard, Functional Analysis in Modern Applied Mathematics, Academic Press, London, 1977.
[3] R.S. Varga, Functional Analysis and Approximation Theory in Numerical Analysis, SIAM, Philadelphia, 1991.
[4] M. Ahues, A. Largillier, F. Dias d’Almeida, Paulo B. Vasconcelos, Spectral refinement on quasidiagonal matrices, Linear
Algebr. Appl. doi:10.1016/j.laa.2003.12.004.


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