Recurrence relations for rational cubic methods I: The Halley method ...
Computing
Computing 44, 169-184 (1990)
9 by Springer-Verlag1990
Recurrence Relations for Rational Cubic Methods I: The Halley Method v. Candela* and A. Marquina**, Valencia Received August 16, 1988; revised June 2, 1989
Abstract - - Zusammenfassung Recurrence Relations for Rational Cubic Methods I: The Halley Method. In this paper we present a system of a priori error bounds for the Halley method in Banach spaces. Our theorem supplies sufficient conditions on the initial point to ensure the convergence of Halley iterates, by means of a system of "recurrence relations", analogous to those given for the Newton method by Kantorovich, improving previous results by D6ring [4]. The error bounds presented are optimal for second degree polynomials. Other rational cubic methods, as the Chebyshev method, will be treated in a subsequent paper.
AMS Subject Classification (1980): Primary: 65J15 Key words: Third order iterative methods, a priori error bounds, non-linear equations Rekursions~Beziehungen far rationale kubisebe Verfahren I: Das Halley-Verfahren. Wir betrachten ein System yon a priori Fehlerabsch~itzungen fiir die Konvergenz des Halley-Verfahrens in Banachr~iumen. Unsere S~itze geben hinreichende Bedingungen an den Startwert, welche die Konvergenz der HalleyIteration sichern. Sie bestehen aus einem System rekursiver Beziehungen, ~ihnlich den Bedingungen yon Kantorovich f'tir das Newton-Verfahren. Weitere rationale kubische Verfahren werden in einer kiinftigen Arbeit untersucht.
1. Introduction The Kantorovich theorem for the Newton method is of fundamental importance in the study of nonlinear equations in Banach spaces, since it provides sufficient conditions (i.e. the so called "Kantorovich assumptions") on the pivot in order to ensure the convergence of the method through a system of error bounds for the distance to the solution from each iterate (see Kantorovich and Akilov [71, Ortega [101, Tapia [14], Miel [8] and [91, Davies and Dawson [3], Ehrman [51, Potra and Ptak [121 and Yamamoto [16]). The Halley iteration for real functions, recently analyzed by several authors (see [11, [2] and [6]), is defined forf(x) by * This paper is part of the PhD dissertation, realized under the direction of the second named author. ** Supported in part by C.A.I.C.Y.T. GR85-0035. University of Valencia (Spain).
170
V. Candela and A. Marquina
H(x) : =
x
f(x) -
1 .....
f(x)
if(x) - ~ J tx) f ~ ) requiring the existence of f '(x) and if(x), the first and second derivatives off, in the domain o f f . The iteration was defined by Halley to solve the equation f(x) = 0, as a correction of the Newton iteration. The Halley iteration for Banach spaces has been developed by several authors. Among the contributions to this topic the paper by D/Sring [4] is the deepest one, where an extensive reference list can be found. In order to define the Halley iteration in Banach spaces we shall consider, along this paper, the following notations and general assumptions: i) X and Y are real or complex Banach spaces. ii) D subset of X is an open convex domain. iii) L(X, Y) (resp. L(X, X)) is the space of bounded linear operators from X into Y (resp. from X into X), with the operator norm topology. iv) F from D into Y is an operator (non-linear) such that there exist both first and second Fr6chet derivatives, F'(x) and F"(x), respectively, for every x in D. v) All solutions xo of the equation F(x) = O, Xo in O, are isolated, i.e., if F(xo) = O, then there will exist r > 0 such that if0 < 1Ix - xol[ < r, then F(x) r 0. vi) If x in D is such that there exists the bounded inverse operator F'(x) -1, of F'(x) as a linear mapping from X into Y, i.e. F'(x) -~ in L(Y, X), then we will define the following linear operator on X:
T(x) := ~ F'(x)-*F"(x)F'(x)-IF(x) vii) For x in D, we define the Halley iteration,
H(x) := x - [I - T(x)]-*F'(x)-lF(x) if F'(x) -1 exists in L(Y, X) and I - T(x) has an inverse in L(X, X). In this paper we establish general conditions on a point x o in D, under which the sequence of iterates xN+, := H(xN), N = 0, 1, 2 . . . . . is well defined and converges to a point x* in D, with F(x*) = 0. In order to obtain this result, we construct, from two parameters a and b, a system of "recurrence relations", consisting of four sequences of positive real numbers, which yields an increasing sequence t N > O, N = 0, 1, 2. . . . . converging to t > 0, such that it will majorize the sequence x n, i.e. I[x - xN[I -< t -- t N, N = 0, 1, 2. . . . and [Ix - xN[I = t - t N if F(x) is a second degree polynomial, provided Xo is chosen in a suitable way. As a consequence, we obtain, in a natural and simple way, a priori error bounds, improving previous results given by D6ring. We illustrate through examples the results obtained. The next section deals with the general "recurrence relations" that will appear later.
Recurrence Relations for Rational Cubic M e t h o d s I: The Halley Method
171
2. General Reeurrenee Relations Let a and b be real numbers such that b > 0, 0 < a < 2/3. We set ao:=1;
co : = l ;
b~
a
d~
2
(0)
and, for N > O, a N + l "--
oN+1 := aN+1
aN 1 - aasdN + (1 - bN)aN~-
(1)
l
d3
(2)
a
bN+l := ~aN+lcN+I
(3)
cN+1 dN+l .-- 1 -- bN+ 1
(4)
We shall define from (4), r N := do + dl + "'" + dN, and r = lira rN, if the limit exists. We denote by 9t(a, b) := {an, c N, b N, dN} the system of sequences defined by (0), (1), (2), (3) and (4), constructed from a and b. We call the equations defining the sequences of 9t(a, b) "recurrence relations". We say that the system 91(a, b) is positive if a n >_ 1, for all N. A positive system is said to be stable if there is a constant M _> 1 such that a N G M for all N. If 91(a, b) is positive then it will be clear from (1) that 1 aN+l = 1 -- ar N
(5)
A positive system 91(a, b) is convergent if there exist limN rN = r. From the initial relations, it follows by induction that for N = 0, 1, 2 . . . . cx 2bx a a x d s = aaN1 -- bN = 1 - b N
(6)
and, as consequence, we have 2.1. Proposition: 91(a, b) is positive i f and only i f b N < 1/3, N = 0, 1, 2 . . . . . Our next goal is to provide sufficient conditions on the constants a and b in order 91(a, b) be a positive, stable and convergent system. 2.2. Proposition: L e t a = 0 and b >_ O. T h e n (i) 91(0, b) is positive and stable. (ii) 91(0, b) is convergent i f and only i f b < 6.
172
V. Candela and A. Marquina
Proof: Since a = 0 then a s = 1 for all N, and (i) is satisfied. On the other hand, bN = 0 for all N, therefore du = cu = au-6cN-t = -6dN-1 = \ 6 J
'
do =
for N > 1. Thus, ~(0, b) is convergent if and only if b < 6.
[]
In order to study the general case a > 0, we state without proof the following simple lemma. 2.3. Lemma: Let x in [0, 1/3[ and y > 1 be real numbers. We define the real functions X3
f ( x ) := ( l ~ x ) 2
o(x,y) :--- f(x)
1 ~ y(1 - x~
(7) (8)
where K is a nonnegative real constant. Then we have the following properties:
(i) f(x) from [0, 1/3[ into N is strictly increasing, f(O) = 0, f(1/4) = 1/4, f ( x ) < x, x in ]0, 1/4[. (ii) f'(x), derivative o f f , is strictly increasino, f'(O) = 0 and there exists x o in ]0, 1/4[ such that f ' ( x o ) = 1. (iii) I f K > 0 and y > 0, the function g(x, y) will be strictly increasing in x e [0, 1/3[ and g(O, y) = O. (iv) Under the hypothesis of (iii) g'~(x, y) is strictly increasing in x ~ [0, 1/3[ and
ox(0, y) = 0. r
(v) For y >_ 1, f ( x ) < g(x, y) < g(x, 1). 2.4. Theorem: Let ~(a, b) be a system with 0 < a < 2/3, b >_ O. We set K := 2b/(3a2). Then
O) bN+l = (1 -- 3bs) 2 1 + as(1 _ bs)
(9)
(ii) ~(a, b) is positive if and only if bs ~- 1/4 for all N. In particular, a ~_ 1/2 is a necessary condition for ~(a, b) to be positive. (iii) aN+1 =
kOo( 1 + 1 --2b~3bk)~
(10)
(iv) I f N(a, b) is positive, then in order that ~l(a, b) be stable it is necessary and sufficient that ~ = o bk converges.
Rectirrence Relations for Rational Cubic Methods I: The Halley Method
173
(v) Positivity and stability imply convergence. (vi) If b = O, ~(a, O) will be positive if and only if it is convergent, and positivity will be equivalent to a < 1/2; ~ ( a , 0) is stable if and only if a < 1/2.
Proof: (i) (9) follows from the definition of bN+1, by using (6): bN+l = a
aN+lcu+l =
(a )3
:(;
\
2(
aN "~21-b (a)Zl 3 1 -- aaNdN ) L6 + aN(1 -- bN) ~ dN
i2611
-2-~-N ~ 2 1~ 3a 2aN(l_bN)-
1 --bN}
: ( 1 - - - 3 b N ) 2 l +aN(~--bN ~
Analogously, we put aN+ ~ as function function of a N and bN: aN
1
aN+l - 1 -- aaNdN -- aN
1
2bN = 1-- bN
1-bN 3bN
aN 1 -
(11)
Thus, if we consider the real function 1--X
h(x,y) := y 1 - 3 ~ '
x e [0,89
y >_ 1
(12)
and the function g defined in L e m m a 2.3, we will generate the sequences bN and a n as iterates of g and h, respectively, through
xN+I = g(xN, YN) YN+I = h(xN, YN)
(13)
by choosing the pivot (Xo, Yo) with x o = bo = a/2, Yo = ao = 1, and according to (7) and (11), bN+l = xN+l and aN+l = YN+I, for N = 0, 1, 2 . . . . (ii) Let us suppose ~(a, b) positive. F r o m Proposition 2.1, bN < 1/3, N = 0, 1, 2 . . . . . We assume that there exists N(0) such that bNto) > 1/4. We set 20 = bNto) and 2N+1 = f(2N) where f is the function defined by (7). Since f ( x ) > x, then 20 < 21 < 22 < "'". F r o m the m e a n value t h e o r e m there exists r with 1/4 < ~ < x and f(x) - 1/4 = f ' ( ~ ) (x - 1/4), whence 2N+ 1 - 2 N > 8 (2N - 1/4) _> 820 - 2 > 0, thus proving the existence of N(1) such that 2N(1) -> 1/3. According to L e m m a 2.3, 21 = f(2o) = f(bmo)) < 9(bN(o), aN(o)) = bN(o)+l. Since f is increasing, 22 = f(21) = f(bs(o)+l) 1/3, which is a contradiction. P r o p o s i t o n 2.1 shows the converse. (iii) (10) follows easily from (11) by recurrence. (iv) Since ~(a, b) is positive, according to (ii), bN < 1/4 for all N, so
2bN
2bN
2bN < 1 -- 3 ~ < 1 -- 88 -- 8bN.
(14)
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V. Candela and A. Marquina
Since the sequence aN is bounded if and only if the infinite product
converges, then, by (14), this occurs if and only if ~ = o bk converges. (v) If ~(a, b) is positive and stable, then there exists M > 1 such that 1 < aN < M for all N. From (ii) and (6) it follows that 2bN --< dN < 8bN a-M --~-a'
N = 0, 1,2,...
co
(15) 8b k
Thus, series k=o ~ d k converges, since it is dominated by k=O ~-a' which is convergent bythe(iv). (vi) Let us suppose b = 0. Then K = 0 and, from (9), bN+1 = f(bN), where f is the function defined by (7). From Lemma 2.3 and (ii), ~(a, 0) is positive if and only if a _< 1/2, and ~(a, 0) is stable if and only if a < 1/2, by applying (iv). Thus (v) implies that ~(a,0) will converge if a < 1/2. If a = 1/2 then bN = 1/4, N = 0, 1, 2 . . . . and, therefore, a s = 1-[ 1 + = 1+ k=O 1 -- 3 b J ~=o
4
= 3 N.
(16)
Thus, it follows by induction that cN
4
dN = 1 -- bN - 3N+I
and N(a,0) will be convergent.
[]
The next theorem will analyze the dynamic behaviour of the sequence bN. 2.5. Theorem: Let a > O. Let 9l(a,b) be a system with b >_ O. We set K = 2b/(3a2). Then (i) 9~(a, b) is positive if and only if for N = O, 1, 2 . . . . K < an
(1
-
bN)3(1 -- 4bN)
(17)
4bna
(ii) I f ~R(a, b) is positive then 9t(a, b) will be stable if and only if lim NbN exists and limN bN = 0. (iii) I f ~R(a, b) is positive and stable then there will exist N O such that bo < bl < " " -< bNo and for N >_ N O we have
(18)
Recurrence Relations for Rational Cubic Methods I: The Halley Method
bN > bN+~
(19)
K
(1 - bn)(1 - 2bN)(1 -- 4bn)
an
b2
--
(1 - bN)(1 -- 2bN)(1 -- 4bN) b2
(22)
In this case the sequence bN is increasing and lim n b n = 1/4. (v) Let 9~(a, b) be positive and unstable. Then 9~(a, b) will be convergent and the
sequence a n will satisfy the following estimates. 2a
in
1 2b N
1
1 + _2 - - - ~ a 3
a 1 -- bn3 n
< an -< < du
0, a < 1/2, it is easily seen that there is a real number b with 0 < b < 6 such that ~R(a,b) is positive and unstable. The next proposition shows that a system ~R(a,b) with 0 < a < 1/2, satisfying the estimate (21) produces the sequence dN with cubic R-order, (see the estimate (27) below), (see also Ortega-Rheinboldt ([11], Chapter 9, Definition 9.2.5, p. 290) and Potra and Ptak [13]): 2.6. Proposition:
Let a be a real number such that 0 < a < 1/2. Let b ~ 0 be such that
b _< 3(1 - a)(2 - a)(1 - 2a). We set 7 := b2/bl. Then, for N >_ 1 bN+l 1, it follows that b 2 - g(b 1, a~) < g(bt, 1) _< g(b o, 1) = b~ ifb > 0,and b 2 = f ( b l ) < f(bo) = bl ifb = 0, thus < 1, and, therefore, estimate (26) is satisfied. F r o m (15) and (26), (27) and (28) follow. [] In practice, Proposition 2.6 is used by means of checking the inequality (21) and by computing approximately the geometric subseries appearing in (28), in order to obtain a priori error bounds for the Halley iterates.
3. Convergence of the Halley Method Now, the main theorem states that the convergence of the Halley m e t h o d is controlled by certain "recurrence relations".
3.1. Theorem: Let X and Y be Banach spaces. Let D be an open convex subset of X. Let F be from D into Y an operator twice Frdchet differentiable on D. Let us suppose that O) F" from D into L(X, L ( X , Y)) is a uniformly bounded operator on D i.e. there exists a positive real number K 2 such that IJF"(x)J[ _< K 2 ,
x ~D
(29)
(ii) F" is Lipschitz continuous on D, i.e. there exists a positive real number K 2 such that
[[V"(x) -- F"(y)[I
Computing 44, 169-184 (1990)
9 by Springer-Verlag1990
Recurrence Relations for Rational Cubic Methods I: The Halley Method v. Candela* and A. Marquina**, Valencia Received August 16, 1988; revised June 2, 1989
Abstract - - Zusammenfassung Recurrence Relations for Rational Cubic Methods I: The Halley Method. In this paper we present a system of a priori error bounds for the Halley method in Banach spaces. Our theorem supplies sufficient conditions on the initial point to ensure the convergence of Halley iterates, by means of a system of "recurrence relations", analogous to those given for the Newton method by Kantorovich, improving previous results by D6ring [4]. The error bounds presented are optimal for second degree polynomials. Other rational cubic methods, as the Chebyshev method, will be treated in a subsequent paper.
AMS Subject Classification (1980): Primary: 65J15 Key words: Third order iterative methods, a priori error bounds, non-linear equations Rekursions~Beziehungen far rationale kubisebe Verfahren I: Das Halley-Verfahren. Wir betrachten ein System yon a priori Fehlerabsch~itzungen fiir die Konvergenz des Halley-Verfahrens in Banachr~iumen. Unsere S~itze geben hinreichende Bedingungen an den Startwert, welche die Konvergenz der HalleyIteration sichern. Sie bestehen aus einem System rekursiver Beziehungen, ~ihnlich den Bedingungen yon Kantorovich f'tir das Newton-Verfahren. Weitere rationale kubische Verfahren werden in einer kiinftigen Arbeit untersucht.
1. Introduction The Kantorovich theorem for the Newton method is of fundamental importance in the study of nonlinear equations in Banach spaces, since it provides sufficient conditions (i.e. the so called "Kantorovich assumptions") on the pivot in order to ensure the convergence of the method through a system of error bounds for the distance to the solution from each iterate (see Kantorovich and Akilov [71, Ortega [101, Tapia [14], Miel [8] and [91, Davies and Dawson [3], Ehrman [51, Potra and Ptak [121 and Yamamoto [16]). The Halley iteration for real functions, recently analyzed by several authors (see [11, [2] and [6]), is defined forf(x) by * This paper is part of the PhD dissertation, realized under the direction of the second named author. ** Supported in part by C.A.I.C.Y.T. GR85-0035. University of Valencia (Spain).
170
V. Candela and A. Marquina
H(x) : =
x
f(x) -
1 .....
f(x)
if(x) - ~ J tx) f ~ ) requiring the existence of f '(x) and if(x), the first and second derivatives off, in the domain o f f . The iteration was defined by Halley to solve the equation f(x) = 0, as a correction of the Newton iteration. The Halley iteration for Banach spaces has been developed by several authors. Among the contributions to this topic the paper by D/Sring [4] is the deepest one, where an extensive reference list can be found. In order to define the Halley iteration in Banach spaces we shall consider, along this paper, the following notations and general assumptions: i) X and Y are real or complex Banach spaces. ii) D subset of X is an open convex domain. iii) L(X, Y) (resp. L(X, X)) is the space of bounded linear operators from X into Y (resp. from X into X), with the operator norm topology. iv) F from D into Y is an operator (non-linear) such that there exist both first and second Fr6chet derivatives, F'(x) and F"(x), respectively, for every x in D. v) All solutions xo of the equation F(x) = O, Xo in O, are isolated, i.e., if F(xo) = O, then there will exist r > 0 such that if0 < 1Ix - xol[ < r, then F(x) r 0. vi) If x in D is such that there exists the bounded inverse operator F'(x) -1, of F'(x) as a linear mapping from X into Y, i.e. F'(x) -~ in L(Y, X), then we will define the following linear operator on X:
T(x) := ~ F'(x)-*F"(x)F'(x)-IF(x) vii) For x in D, we define the Halley iteration,
H(x) := x - [I - T(x)]-*F'(x)-lF(x) if F'(x) -1 exists in L(Y, X) and I - T(x) has an inverse in L(X, X). In this paper we establish general conditions on a point x o in D, under which the sequence of iterates xN+, := H(xN), N = 0, 1, 2 . . . . . is well defined and converges to a point x* in D, with F(x*) = 0. In order to obtain this result, we construct, from two parameters a and b, a system of "recurrence relations", consisting of four sequences of positive real numbers, which yields an increasing sequence t N > O, N = 0, 1, 2. . . . . converging to t > 0, such that it will majorize the sequence x n, i.e. I[x - xN[I -< t -- t N, N = 0, 1, 2. . . . and [Ix - xN[I = t - t N if F(x) is a second degree polynomial, provided Xo is chosen in a suitable way. As a consequence, we obtain, in a natural and simple way, a priori error bounds, improving previous results given by D6ring. We illustrate through examples the results obtained. The next section deals with the general "recurrence relations" that will appear later.
Recurrence Relations for Rational Cubic M e t h o d s I: The Halley Method
171
2. General Reeurrenee Relations Let a and b be real numbers such that b > 0, 0 < a < 2/3. We set ao:=1;
co : = l ;
b~
a
d~
2
(0)
and, for N > O, a N + l "--
oN+1 := aN+1
aN 1 - aasdN + (1 - bN)aN~-
(1)
l
d3
(2)
a
bN+l := ~aN+lcN+I
(3)
cN+1 dN+l .-- 1 -- bN+ 1
(4)
We shall define from (4), r N := do + dl + "'" + dN, and r = lira rN, if the limit exists. We denote by 9t(a, b) := {an, c N, b N, dN} the system of sequences defined by (0), (1), (2), (3) and (4), constructed from a and b. We call the equations defining the sequences of 9t(a, b) "recurrence relations". We say that the system 91(a, b) is positive if a n >_ 1, for all N. A positive system is said to be stable if there is a constant M _> 1 such that a N G M for all N. If 91(a, b) is positive then it will be clear from (1) that 1 aN+l = 1 -- ar N
(5)
A positive system 91(a, b) is convergent if there exist limN rN = r. From the initial relations, it follows by induction that for N = 0, 1, 2 . . . . cx 2bx a a x d s = aaN1 -- bN = 1 - b N
(6)
and, as consequence, we have 2.1. Proposition: 91(a, b) is positive i f and only i f b N < 1/3, N = 0, 1, 2 . . . . . Our next goal is to provide sufficient conditions on the constants a and b in order 91(a, b) be a positive, stable and convergent system. 2.2. Proposition: L e t a = 0 and b >_ O. T h e n (i) 91(0, b) is positive and stable. (ii) 91(0, b) is convergent i f and only i f b < 6.
172
V. Candela and A. Marquina
Proof: Since a = 0 then a s = 1 for all N, and (i) is satisfied. On the other hand, bN = 0 for all N, therefore du = cu = au-6cN-t = -6dN-1 = \ 6 J
'
do =
for N > 1. Thus, ~(0, b) is convergent if and only if b < 6.
[]
In order to study the general case a > 0, we state without proof the following simple lemma. 2.3. Lemma: Let x in [0, 1/3[ and y > 1 be real numbers. We define the real functions X3
f ( x ) := ( l ~ x ) 2
o(x,y) :--- f(x)
1 ~ y(1 - x~
(7) (8)
where K is a nonnegative real constant. Then we have the following properties:
(i) f(x) from [0, 1/3[ into N is strictly increasing, f(O) = 0, f(1/4) = 1/4, f ( x ) < x, x in ]0, 1/4[. (ii) f'(x), derivative o f f , is strictly increasino, f'(O) = 0 and there exists x o in ]0, 1/4[ such that f ' ( x o ) = 1. (iii) I f K > 0 and y > 0, the function g(x, y) will be strictly increasing in x e [0, 1/3[ and g(O, y) = O. (iv) Under the hypothesis of (iii) g'~(x, y) is strictly increasing in x ~ [0, 1/3[ and
ox(0, y) = 0. r
(v) For y >_ 1, f ( x ) < g(x, y) < g(x, 1). 2.4. Theorem: Let ~(a, b) be a system with 0 < a < 2/3, b >_ O. We set K := 2b/(3a2). Then
O) bN+l = (1 -- 3bs) 2 1 + as(1 _ bs)
(9)
(ii) ~(a, b) is positive if and only if bs ~- 1/4 for all N. In particular, a ~_ 1/2 is a necessary condition for ~(a, b) to be positive. (iii) aN+1 =
kOo( 1 + 1 --2b~3bk)~
(10)
(iv) I f N(a, b) is positive, then in order that ~l(a, b) be stable it is necessary and sufficient that ~ = o bk converges.
Rectirrence Relations for Rational Cubic Methods I: The Halley Method
173
(v) Positivity and stability imply convergence. (vi) If b = O, ~(a, O) will be positive if and only if it is convergent, and positivity will be equivalent to a < 1/2; ~ ( a , 0) is stable if and only if a < 1/2.
Proof: (i) (9) follows from the definition of bN+1, by using (6): bN+l = a
aN+lcu+l =
(a )3
:(;
\
2(
aN "~21-b (a)Zl 3 1 -- aaNdN ) L6 + aN(1 -- bN) ~ dN
i2611
-2-~-N ~ 2 1~ 3a 2aN(l_bN)-
1 --bN}
: ( 1 - - - 3 b N ) 2 l +aN(~--bN ~
Analogously, we put aN+ ~ as function function of a N and bN: aN
1
aN+l - 1 -- aaNdN -- aN
1
2bN = 1-- bN
1-bN 3bN
aN 1 -
(11)
Thus, if we consider the real function 1--X
h(x,y) := y 1 - 3 ~ '
x e [0,89
y >_ 1
(12)
and the function g defined in L e m m a 2.3, we will generate the sequences bN and a n as iterates of g and h, respectively, through
xN+I = g(xN, YN) YN+I = h(xN, YN)
(13)
by choosing the pivot (Xo, Yo) with x o = bo = a/2, Yo = ao = 1, and according to (7) and (11), bN+l = xN+l and aN+l = YN+I, for N = 0, 1, 2 . . . . (ii) Let us suppose ~(a, b) positive. F r o m Proposition 2.1, bN < 1/3, N = 0, 1, 2 . . . . . We assume that there exists N(0) such that bNto) > 1/4. We set 20 = bNto) and 2N+1 = f(2N) where f is the function defined by (7). Since f ( x ) > x, then 20 < 21 < 22 < "'". F r o m the m e a n value t h e o r e m there exists r with 1/4 < ~ < x and f(x) - 1/4 = f ' ( ~ ) (x - 1/4), whence 2N+ 1 - 2 N > 8 (2N - 1/4) _> 820 - 2 > 0, thus proving the existence of N(1) such that 2N(1) -> 1/3. According to L e m m a 2.3, 21 = f(2o) = f(bmo)) < 9(bN(o), aN(o)) = bN(o)+l. Since f is increasing, 22 = f(21) = f(bs(o)+l) 1/3, which is a contradiction. P r o p o s i t o n 2.1 shows the converse. (iii) (10) follows easily from (11) by recurrence. (iv) Since ~(a, b) is positive, according to (ii), bN < 1/4 for all N, so
2bN
2bN
2bN < 1 -- 3 ~ < 1 -- 88 -- 8bN.
(14)
174
V. Candela and A. Marquina
Since the sequence aN is bounded if and only if the infinite product
converges, then, by (14), this occurs if and only if ~ = o bk converges. (v) If ~(a, b) is positive and stable, then there exists M > 1 such that 1 < aN < M for all N. From (ii) and (6) it follows that 2bN --< dN < 8bN a-M --~-a'
N = 0, 1,2,...
co
(15) 8b k
Thus, series k=o ~ d k converges, since it is dominated by k=O ~-a' which is convergent bythe(iv). (vi) Let us suppose b = 0. Then K = 0 and, from (9), bN+1 = f(bN), where f is the function defined by (7). From Lemma 2.3 and (ii), ~(a, 0) is positive if and only if a _< 1/2, and ~(a, 0) is stable if and only if a < 1/2, by applying (iv). Thus (v) implies that ~(a,0) will converge if a < 1/2. If a = 1/2 then bN = 1/4, N = 0, 1, 2 . . . . and, therefore, a s = 1-[ 1 + = 1+ k=O 1 -- 3 b J ~=o
4
= 3 N.
(16)
Thus, it follows by induction that cN
4
dN = 1 -- bN - 3N+I
and N(a,0) will be convergent.
[]
The next theorem will analyze the dynamic behaviour of the sequence bN. 2.5. Theorem: Let a > O. Let 9l(a,b) be a system with b >_ O. We set K = 2b/(3a2). Then (i) 9~(a, b) is positive if and only if for N = O, 1, 2 . . . . K < an
(1
-
bN)3(1 -- 4bN)
(17)
4bna
(ii) I f ~R(a, b) is positive then 9t(a, b) will be stable if and only if lim NbN exists and limN bN = 0. (iii) I f ~R(a, b) is positive and stable then there will exist N O such that bo < bl < " " -< bNo and for N >_ N O we have
(18)
Recurrence Relations for Rational Cubic Methods I: The Halley Method
bN > bN+~
(19)
K
(1 - bn)(1 - 2bN)(1 -- 4bn)
an
b2
--
(1 - bN)(1 -- 2bN)(1 -- 4bN) b2
(22)
In this case the sequence bN is increasing and lim n b n = 1/4. (v) Let 9~(a, b) be positive and unstable. Then 9~(a, b) will be convergent and the
sequence a n will satisfy the following estimates. 2a
in
1 2b N
1
1 + _2 - - - ~ a 3
a 1 -- bn3 n
< an -< < du
0, a < 1/2, it is easily seen that there is a real number b with 0 < b < 6 such that ~R(a,b) is positive and unstable. The next proposition shows that a system ~R(a,b) with 0 < a < 1/2, satisfying the estimate (21) produces the sequence dN with cubic R-order, (see the estimate (27) below), (see also Ortega-Rheinboldt ([11], Chapter 9, Definition 9.2.5, p. 290) and Potra and Ptak [13]): 2.6. Proposition:
Let a be a real number such that 0 < a < 1/2. Let b ~ 0 be such that
b _< 3(1 - a)(2 - a)(1 - 2a). We set 7 := b2/bl. Then, for N >_ 1 bN+l 1, it follows that b 2 - g(b 1, a~) < g(bt, 1) _< g(b o, 1) = b~ ifb > 0,and b 2 = f ( b l ) < f(bo) = bl ifb = 0, thus < 1, and, therefore, estimate (26) is satisfied. F r o m (15) and (26), (27) and (28) follow. [] In practice, Proposition 2.6 is used by means of checking the inequality (21) and by computing approximately the geometric subseries appearing in (28), in order to obtain a priori error bounds for the Halley iterates.
3. Convergence of the Halley Method Now, the main theorem states that the convergence of the Halley m e t h o d is controlled by certain "recurrence relations".
3.1. Theorem: Let X and Y be Banach spaces. Let D be an open convex subset of X. Let F be from D into Y an operator twice Frdchet differentiable on D. Let us suppose that O) F" from D into L(X, L ( X , Y)) is a uniformly bounded operator on D i.e. there exists a positive real number K 2 such that IJF"(x)J[ _< K 2 ,
x ~D
(29)
(ii) F" is Lipschitz continuous on D, i.e. there exists a positive real number K 2 such that
[[V"(x) -- F"(y)[I
