J. Numer. Anal. Approx. Theory, vol. 46 (2017) no. 1, pp. 25–37 ictp.acad.ro/jnaat
ON NEWTON’S METHOD FOR SUBANALYTIC EQUATIONS
IOANNIS K. ARGYROS∗and SANTHOSH GEORGE†
Abstract. We present local and semilocal convergence results for Newton’s method in order to approximate solutions of subanalytic equations. The local convergence results are given under weaker conditions than in earlier studies such as [9], [10], [14], [15], [24], [25], [26], resulting to a larger convergence ball and a smaller ratio of convergence. In the semilocal convergence case contravariant conditions not used before are employed to show the convergence of Newton’s method. Numerical examples illustrating the advantages of our approach are also presented in this study.
MSC 2010. 65H10, 65G99, 65K10, 47H17, 49M15.
Keywords. Newton’s methods, convergence ball, local-semilocal convergence, subanalytic functions.
1. INTRODUCTION
In this study we are concerned with the problem of approximating a solution x∗ of the equation
(1) F(x) = 0,
whereF is a continuous mapping from a subsetD of Rn intoRn.
Many problems in computational sciences and other disciplines can be brought in a form like (1) using mathematical modeling [3], [7], [8], [9], [14], [16], [17], [22], [24]– [28]. In general the solutions of equation (1) can not be found in closed form. Therefore iterative methods are used for obtaining approximate solutions of (1). In Numerical Functional Analysis, for finding solution x∗ of equation (1) is essentially connected to variants of Newton’s method. Newton’s method is defined by
(2) xk+1=xk−F0(xk)−1F(xk), for each k= 0,1,2, . . . ,
where x0 is an initial point and F is a continuously Fr´echet differentiable function on D, i.e., F is a smooth function.
∗Cameron University, Department of Mathematicsal Sciences, Lawton, OK 73505, USA, e-mail: [email protected].
†National Institute of Technology Karnataka, Department of Mathematical and Compu- tational Sciences, India-575 025, e-mail: [email protected].
The study about convergence matter of iterative procedures is usually based on two types: semi-local and local convergence analysis. The semi-local con- vergence matter is, based on the information around an initial point, to give conditions ensuring the convergence of the iterative procedure; while the lo- cal one is, based on the information around a solution, to find estimates of the radii of convergence balls. There exist many studies which deal with the local and semilocal convergence analysis of Newton’s methods (2) under var- ious Lipschitz-type conditions on F0. We refer the reader to [1–28] and the references therein for this type of results.
However, in many interesting applications F is not a smooth function [3], [7], [8], [23], [24], [26], [28]. In particular, we are interested in the case when F is a semismooth function. Then, we define Newton’s method
(3) xk+1 =xk−Λ(xk)−1F(xk), for each, k= 0,1,2, . . . ,
where x0 ∈ Rn is an initial point and Λ(xk) ∈ ∂F(xk) the generalized Ja- cobian of F as defined by Clarke [14]. We present local as well as semilo- cal convergence results under weaker conditions than in earlier studies such as [9], [10], [14], [15], [24], [25], [26]. In the case of local convergence, our convergence ball is larger and the ratio of convergence smaller than before [9], [10], [14], [15], [24]– [26]. These advantages are also obtained under weaker hypotheses. This type of improved convergence results are important in com- putational Mathematics, since this way we have a wider choice of initial guesses and we compute less iterates in order to obtain a desired error tolerance.
The rest of the paper is organized as follows: In order to make the paper as self contained as possible, we provide the definitions of semismooth, semi- analytic and subanalytic functions as well as earlier results in Section 2. The semilocal and local convergence analysis of Newton’s method is given in Sec- tion 3. Finally the numerical examples illustrating the theoretical results are given in the concluding Section 4.
2. SEMISMOOTH, SEMIANALYTIC AND SUBANALYTIC FUNCTIONS
In order to make the paper as self contained as possible we state some standard definitions and results. In [24], Milfflin introduced the concept of semismoothness for functionals, later in [26], L.Qi and J. Sun extended this concept for functions of several variable. In fact they showed that semismooth- ness is equivalent to the uniform convergence of directional derivatives in all directions.
Definition 1. (see [14, p. 70])Let F : Rn → Rn be a locally Lipschitz continuous function. The limiting Jacobian of F at x∈Rn is defined as
∂◦F(x) ={A∈L(Rn,Rm) :∃uk∈DF;F0(uk)→A, k→ ∞}
where DF denotes the point of differentiability ofF.The Clarke Jacobian ofF at x∈Rn denoted ∂F(x) is the subset of X∗ dual of X, defined as the closed convex hull of ∂◦F(x).
Definition 2. [26]We say that F is semismooth at x∈Rn if F is locally Lipschitzian at x and
lim
V∈F0(x+th0),h0→h, t↓0{V h0} exists for any h∈Rn.
Note that convex functions and smooth functions are semismooth functions, Further the product and sums of semismooth functions are semismooth func- tions (see [10]). Moreover, semismoothness of F implies
lim
h0→h, t→0
F(x+th0)−F(x)
t = lim
V∈F0(x+th0),h0→h, t↓0{V h0}.
Definition 3. [15] A subset X of Rn is semianalytic if for each a∈ Rn there is a neighbourhoohU of a and real analytic functionsfi,j onU such that
X∩U =
r
[
i=1 si
\
j=1
{x∈U|fi,jεi,j0}
where εi,j ∈ {<, >,=}.
Remark4. Xis said to be semianalytic ifX=Rnandfi,j are polynomials.
Definition 5. [15] A subset X of Rn is subanalytic if for each a ∈ Rn admits a neighborhoodU such thatX∩U is a projection of a relatively compact semianalytic set: there is a semianalytic bounded set A in Rn+p such that X∩U =Q(A) where Q:Rn+p →Rn is the projection.
Definition 6. Let Xbe a subset of Rn.A function F :X →Rn is semian- alytic (resp. subanalytic) if its graph is semianalytic (resp. subanalytic).
It can be seen that the class of semianalytic (resp. subanalytic) sets are closed under elementary set operations, further the closure, the interior and the connected components of a semianalytic (resp. subanalytic) set are semi- analytic(resp. subanalytic). But, the image of a bounded semianalytic set by a semianalytic functions is not stable under algebraic operations (see [23], [13]).
That is why the subanalytic functions are introduced. If X is a subanalytic and relatively compact set the image ofXby a subanalytic function is suban- alytic (see [23], [9]). Further, if F and g are subanalytic continuous functions defined on a compact subanalytic set K then F+g is subanalytic.
For examples and properties of subanalytic or semianalytic functions we refer the interested reader to [1], [14], [16], [17], [24], [25], [27], [28]. The following Propositions and Remark can be found in [10]
Proposition7. [10] IfF :X⊂Rn→Rnis a subanalytic locally Lipschitz mapping then for all x∈X
kF(x+d)−F(x)−F0(x;d)k=ox(kdk).
Remark 8. [28] A subanalytic functiont→oc(t) admits a Puiseux devel- opment; so there exist a constant c >0,a real number > 0 and a rational number γ > 0 such that kF(x+d) −F(x)−F0(x;d)k = ckdkγ whenever kdk ≤.
Proposition9. [10] LetF :Rn→Rn be locally Lipschitz and subanalytic, there exists a positive rational number γ such that:
kF(y)−F(x)−Λ(y)(y−x)k=Cxky−xk1+γ
where y is close to x, Λ(y) is any element of ∂F(y) and Cx is a positive constant.
3. CONVERGENCE
We present semilocal and local convergence results for Newton’s method.
First, we present a semilocal result for Newton’s method. LetU(x, ρ),U¯(x, ρ) denote the open and closed balls in Rn with centerxand of radius ρ >0.
Theorem 10. Let F :D⊆Rn→Rn is locally Lipschitz subanalytic on D.
For any Λ(x)∈∂F(x), x∈D,Λ(x) nonsingular;
(1) kΛ(x0)−1F(x0)≤η, for somex0∈D;
kΛ(y)−1[F(y)−F(x)−Λ(y)(y−x)]k ≤ Kky−xk1+γ, (2)
for each x, y∈D and some γ≥0;
kΛ(y)−1(Λ(y)−Λ(x))(y−x)k ≤ Mky−xk1+γ0, (3)
for each x, y∈D and some γ0 ≥0;
(4) 0≤α:=Kηγ+M ηγ0 <1 and for
r=1−αη (5)
U¯(x∗, r)⊆D.
Then, sequence{xk}generated by Newton’s method(3)is well defined, remains in U(x0, r) for each k= 0,1,2, . . . and converges to x∗ ∈U¯(x0, r) of equation F(x) = 0.Moreover, the following estimates hold:
(6) kxk+1−xkk ≤αkxk−x∗k, for each k= 0,1,2, . . . and
(7) kxk−x∗k ≤ 1−ααkη, for each k= 0,1,2, . . . .
Proof. It follows from (1), (4), (5) and Newton’s method fork= 0 that kx1−x0k=kF0(x0)−1F(x0)k ≤η≤ 1−αη =r.
Hence, x1 ∈ U¯(x0, r). Using Newton’s method (3) for k = 1, we get the approximation
F(x1) = F(x1)−F(x0)−Λ(x0)(x1−x0)
= [F(x1)−F(x0)−Λ(x1)(x1−x0)] + [Λ(x1)−Λ(x0)](x1−x0).
(8)
Then, since x1 ∈D we have that Λ(x1)−1 ∈L(Y, X). In view of (1), (2), (3), (4) and (8) we get that
kx2−x1k = kΛ(x1)−1F(x1)k
≤ kΛ(x1)−1(F(x1)−F(x0)−Λ(x1)(x1−x0))k +kΛ(x1)−1(Λ(x1)−Λ(x0))(x1−x0)k
≤ Kkx1−x0k1+γ+Mkx1−x0k1+γ0
= (Kkx1−x0kγ+Mkx1−x0kγ0)kx1−x0k
≤ (Kηγ+M ηγ0)kx1−x0k=αkx1−x0k and
kx2−x0k ≤ kx2−x1k+kx1−x0k ≤(α+ 1)kx1−x0k
= 1−α1−α2kx1−x0k ≤ 1−α1−α2η ≤r, (9)
which shows that (6) holds for k= 1 and x2 ∈U¯(x0, r).
Let us assume that (6) holds for all i ≤ k and xi ∈ U¯(x0, r). Then, by simply usingxi−1, xi in place of x0, x1 in (8)–(9) we get that
kxi+1−xik ≤αkxi−xi−1k, so
kxi+1−x0k ≤ 1−α1−αi+1kx1−x0k ≤ 1−α1−αi+1η≤r,
which complete the induction for (6) and xi+1 ∈ U¯(x0, r). It follows that sequence{xk}is complete inRnand as such it converges to somex∗ ∈U¯(x0, r) (since ¯U(x0, r) is a closed set). By lettingi→ ∞in the estimate
kΛ(xi)−1F(xi)k=kxi+1−xik ≤αi+1η
and since Λ(xi)−1 ∈L(Y, X),we get that F(x∗) = 0.We also have that kxk+i−xkk ≤ kxk+i−xk+i−1|+kxk+i−1−xk+i−2k+. . .+kxk+1−xkk
≤ (αk+i−1+αk+i−2+. . .+αk)kx1−x0k
= αk1−α1−αikx1−x0k ≤αk1−α1−αiη.
(10)
By letting i→ ∞in (10) we obtain (7).
Remark11. (a) Condition (3) does not necessarily imply that Λ is Lipschitz and cannot be avoided if you want to show convergence.
(b) If you use
(11) kΛ(y)−1k ≤K1
kF(y)−F(x)−Λ(y)(y−x)k ≤ K2ky−xk1+γ, (12)
for each x, y∈D, and some γ ≥0;
k(Λ(y)−Λ(x))(y−x)k ≤ M1ky−xk1+γ0, (13)
for each x, y∈D,and some γ0 ≥0,then, we have the estimates kΛ(y)−1[F(y)−F(x)−Λ(y)(y−x)]k ≤ kΛ(y)−1k
×kF(y)−F(x)−Λ(y)(y−x)k
≤ K1K2ky−xk1+γ
kΛ(y)−1[Λ(y)−Λ(x)](y−x)k ≤ kΛ(y)−1kk[Λ(y)−Λ(x)](y−x)k
≤ K1M1ky−xk1+γ0.
Set K = K1K2, M = K1M1. If (1), (2) are replaced by (11), (12) and (13), then the conclusion of Theorem 10 hold in this stronger though setting.
(c) Notice that due to the estimate
kΛ(y)−1[Λ(y)−Λ(x)](y−x)k ≤ kΛ(y)−1[Λ(y)−Λ(x)]kky−xk, M in (3) can be chosen to be an upper bound on kΛ(y)−1[Λ(y)−Λ(x)]k.That is
kΛ(y)−1[Λ(y)−Λ(x)]k ≤M.
Notice also thatkΛ(y)−Λ(x)k ≤M2 holds if e.g. Λ is continuous. Then, we can choose e.g. M =K1M2 forγ0 = 0.
(d) Ifγ0 =γ = 0 and (1)-(3) are given in non-invariant form, then Theorem 10 reduces to the corresponding one in [26]. Otherwise, our Theorem 10 is an extension of the one in [26]. Moreover, it is an improvement in the case γ0=γ = 0,since our results are given in affine invariant form. The advantages of affine over non-affine invariant form results are well known in the literature [17].
(e) It was shown in [10] that if F :D ⊆Rn →Rn is locally Lipschitz and subanalytic, then (12) always holds. Therefore, (2) holds forK =K1K2.
Next, we present a local convergence result for Newton’s method.
Theorem 12. Suppose F : D ⊆ Rn → Rn is locally Lipschitz and sub- analytic; there exists a regular point x∗ ∈ D such that F(x∗) = 0; for any Λ(x)∈∂F(x), x∈D,Λ(x) is nonsingular;
kΛ(y)−1[F(y)−F(x∗)−Λ(y)(y−x∗)]k ≤ λky−x∗k1+β (14)
for each x, y∈D and some λ >0, β >0,and for R= min{1λ,λ1β}
U¯(x0, R)⊆D.
Then, sequence {xn} generated by Newton’s method (3) converges to x∗ pro- vided that x0 ∈ U(x∗, R). Moreover, the following estimates hold for each n= 0,1,2, . . .
(15) kxn+1−x∗k ≤λkxn−x∗k1+β ≤ kx0−x∗k< R and
(16) kxn+1−x∗k ≤λ−1β(λkx0−x∗k)(1+β)n+1.
Proof. We have that x1 ∈ U(x∗, R) by the choice of R. Then, using the estimate
x2−x∗ =−Λ(x1)−1[F(x1)−F(x∗)−Λ(x1)(x1−x∗)]
and (14), we get that
kx2−x∗k = kΛ(x1)−1[F(x1)−F(x∗)−Λ(x1)(x1−x∗)]k
≤ λkx1−x∗k1+β ≤ kx1−x∗k< R
by the choice of R, which shows (15) and (16) for n = 0. Suppose that (15) and (16) hold for each k≤n.Then, we have that
(17) xk+1−x∗ =−Λ(xk)−1[F(xk)−F(x∗)−Λ(xk)(xk−x∗)]
so by (14) and (17) we get that
kxk+1−x∗k = kΛ(xk)−1[F(xk)−F(x∗)−Λ(xk)(xk−x∗)]k
≤ λkxk−x∗k1+β ≤λ(λkxk−1−x∗k1+β)1+β
≤ λλ1+β(kxk−1−x∗k)(1+β)2 ...
= λ
(1+β)k+1+...+(1+β)+1
β kx0−x∗k(1+β)k+1
= λ−β1(λkx0−x∗k)k+1
which shows (15), (16) for alln and that limk→∞xk=x∗. Remark 13. Condition (14) certainly holds if replaced by the stronger (18) kΛ(y)−1[F(y)−F(x)−Λ(y)(y−x)]k ≤λ1ky−xk1+β, for each x, y∈D.
In this case however,
(19) λ≤λ1
holds in general and λλ1 can be arbitrarily large [3, 7, 8]. Moreover, if λ1 =λ and (14) is replaced by (19), then our result reduces to the corresponding one in [26]. Otherwise (i.e., if λ < λ1), it constitutes an improvement with advantages:
(i) (14) is weaker than (18). That is (18) implies (14) but (14) does not necessarily imply (18).
(ii) If λ < λ1,the new error bounds on the distances kxn−x∗k are tighter and the ratio of convergence smaller. That means in practice fewer iterates are required to achieve a given error tolerance. Hence, the applicability of Newton’s method is expanded under less computational cost. Notice also that the computation of constant λ requires the computation of constant λ1 as a
special case (see, Example 4.2).
Next, we present the corresponding semilocal and local convergence results under contravariant conditions [3, 6, 7, 17, 25].
Theorem 14. Suppose: F :D⊆Rn→Rn is locally Lipschitz subanalytic;
for ant Λ(x)∈∂F(x), x∈D,Λ(x) is nonsingular;
(20) kF(y)−F(x)−Λ(y)(y−x)k ≤µ0kΛ(x)(y−x)k1+γ for some µ0 >0, γ >0 and each x, y∈D;
(21) k(Λ(y)−Λ(x))(y−x)k ≤µ1kΛ(x)(y−x)k1+γ for some µ1 >0 and eachx, y∈D;Define the setQ by
Q={x∈D:kF(x)kγ < 1+γµ },
and let Q be bounded, whereµ=µ0+µ1;x0 ∈D is such that kF(x0)k ≤ξ
and
ξµγ1 <(1 +γ)γ1
(i.e., x0 ∈Q). Then, sequence{xk}generated for x0 ∈Q by Newton’s method (3)is well defined, remains inQfor eachn= 0,1,2, . . .and converges to some x∗ ∈ Q such that F(x∗) = 0. Moreover, sequence {F(xk)} converges to zero and satisfies
kF(xk+1)k ≤µkF(xk)k1+γ, for each k= 0,1,2, . . .
Proof. By hypothesis x0 ∈ Q. Suppose xk ∈ Q. Then, using Newton’s method (3) we get the approximation
F(xk+1 = [F(xk+1)−F(xk)−Λ(xk+1)(xk+1−xk)]
(22)
+[Λ(xk+1)−Λ(xk)](xk+1−xk).
Using (20), (21), (22) we get in turn that
kF(xk+1k = k[F(xk+1)−F(xk)−Λ(xk+1)(xk+1−xk)]k +[Λ(xk+1)−Λ(xk)](xk+1−xk)
≤ µ0kΛ(xk)(xk+1−xk)k1+γ+µ1kΛ(xk)(xk+1−xk)k1+γ
= µkΛ(xk)(xk+1−xk)k1+γ. Since xk∈Q, we have that
kF(xk)kγ < 1+γµ .
Therefore, we get that
kF(xk+1k ≤ µkF(xkk1+γ< µkF(xµkk =kF(xkk.
Notice also that
kF(xk+1kγ<kF(xkkγ < 1+γµ . We also have the implication
xk+1∈Q⇒ {xk} ⊆Q.
Set
sk=µ1γkF(xk)k.
Then, we have in turn
sk+1 ≤ 1+γ1 s1+γk sk+1 ≤ 1+γ1 s1+γk
≤ . . .≤ 1+γ1 (1+γ1 )
(1+γ)[(1+γ)k−1]
γ s(1+γ)0 1+k
= (1 +γ)1γ( s0
(1+γ)γ1
)(1 +γ)k+1. But we have that
s0 =ξµγ1 <(1 +γ)γ1.
Hence, we obtain limk→∞sk = 0, which imply limk→∞kF(xk)k= 0. The set Qis bounded, so there exists an accumulation pointx∗∈Qof sequence{xk}
such thatF(x∗) = 0.
IfF is Fr´echet differentiable andDis a convex set, then due to the estimate (23) F(xk+1) =
Z 1 0
[F0(xk+θ(xk+1−xk))−F0(xk)](xk+1−xk)dθ
by repeating the proof of Theorem 14 using (23), we arrive at the follow- ing semilocal convergence result for Newton’s method (2) under contravariant conditions.
Theorem15. Suppose :F :D⊆Rn→Rn is Fr´echet-differentiable; for any x∈D, F0(x)−1 ∈L(Rn);
k(F0(y)−F0(x))(y−x)k ≤µ1kF0(x)(y−x)k1+γ for some µ1 >0 and eachx, y∈D;Define the setQ1 by
Q1 ={x∈D:kF(x)kγ< 1+γµ
1 };
x0 ∈Dis such that
kf(x0)k ≤ξ and
ξµ
1 γ
1 <(1 +γ)1γ.
Then, sequence{xk}generated by Newton’s method (2) is well defined, remains in Qfor each n= 0,1,2, . . . and converges to some x∗ ∈Qsuch that F(x∗) = 0. Moreover, sequence{F(xk)} converges to zero and satisfies
kF(xk+1)k ≤µ1kF(xk)k1+γ, for eachk= 0,1,2, . . . .
Remark 16. If γ = 1,Theorem 15 reduces to the corresponding Theorem in [26]. However, there are examples where γ 6= 1 (see Example 17). Then, in
this case the results in [26] cannot apply.
4. NUMERICAL EXAMPLES
We present numerical examples to illustrate the theoretical results. First, we present an example under contravariant conditions.
Example 17. Let X =Y =R2, D ={x= (x1, x2) : 12 ≤xi ≤2, i= 1,2}
equipped with the max-norm and defineF onD forx= (x1, x2) by
F(x) =
x
3 2
1 −2x1+x2
x1+x
3 2
2 −2x2
.
Then, the Fr´echet-derivative is given by
F0(x) =
3 2x
1 2
1 −2 1
1 32x
1 2
2 −2
.
Therefore, for any x, y∈D,we have in turn that kF0(x)−F0(y)k =
3 2(x
1 2
1 −y
1 2
1) 0
0 32(x
1 2
2 −y
1 2
2)
= 32max{|x
1 2
1 −y
1 2
1|,|x
1 2
2 −y
1 2
2|}
≤ 32max{|x1−y1|12,|x2−y2|12}
≤ 32[max{|x1−y1|,|x2−y2|}]12 = 32|x−y|12. We also have thatkF0(x)k ≤2 for eachx∈D.Then, we get that
k(F0(x)−F0(y))(x−y)k ≤ 3
√ 2
8 kF0(x)(x−y)k32. Therefore, we can choose γ = 12, µ1= 3
√ 2 8 and Q1 ={x∈D:kF(x)k ≤8}.
The, conclusions of Theorem 15 hold and Newton’s method converges tox∗ =
(1,1).
Next, we present an example for the local convergence case.
Example 18. LetX=Y =R, D=U(0,1) and define functionF onDby
(24) F(x) =ex−1.
Then, we have that x∗ = 0.Using (24) we get in turn that F0(y)−1(F(y)−F(x∗)−F0(y)(y−x∗)) =e−y(ey−1−eyy)
=1−y−(1−y+y2!2 −y3!3 + y4!4 −. . .)
=(2!1 −3!y +y4!2 −. . .)y2 so,
kF0(y)−1(F(y)−F(x∗)−F0(y)(y−x∗))k = |2!1 −3!y +y4!2 −. . .||y|2
≤ (2!1 −|y|3! + |y|4!2 −. . .)|y|2
≤ (2!1 −3!1 + 4!1 −. . .)|y|2
= (e−2)|y|2. Hence, we can chooseλ=e−2 and β= 1.Moreover, we have that
F0(y)−1(F(y)−F(x)−F0(y)(y−x)) =
= e−y(ey−1−ex+ 1−ey(y−x))
= 1−(y−x)−ex−y
= 1 +x−y−[1 + (x−y) + (x−y)2! +(x−y)3! 2 +(x−y)4! 3 −. . .]
= (2!1 +x−y3! +(x−y)4! 2 +. . .)(x−y)2 so,
kF0(y)−1(F(y)−F(x)−F0(y)(y−x))k=|2!1 +x−y3! +(x−y)4! 2 +. . .||y−x|2
≤(2!1 +|x−y|3! +|x−y|4! 2 +. . .)|y−x|2
≤(2!1 +|x−y|3! +|x−y|4! 2 +. . .)|y−x|2
≤(2!1 +3!2 +24!2 +. . .)|y−x|2
=(e2−3)|y−x|2.
Hence, we can chooseλ1=e2−3 and β = 1.Therefore, we obtain R1= λ1
1 = 0.227839421<1.392211191 = 1λ =R.
That is, we deduce that the new convergence ball is larger and the ratio of convergence smaller than the old convergence ball and the old ratio of convergence. Finally, the convergence of Newton’s method is guaranteed by
Theorem 12 provided that x0 ∈U(x0, R).
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Received by the editors: September 23, 2014.
