AN INTEGRAL - INTERPOLATORY ITERATIVE METHOD FOR THE SOLUTION OF NON-LINEAR SCALAR EQUATIONS

This paper deals with the iterative solution of non-linear equations f(x) = 0. We consider integral information on f which is given by f(x Q ),f•(x Q ),...,f (s) ( Xq ) and J °f(t)dt. We define an inter-y polatory-integral method which uses integral information and which has maximal order of convergence equal to s+3. Since the maximal order of iterations which use f(x Q ),...,f (s} ( x ) is equal to s+1 , the additional information given by the integral J f(t)dt increases the order by two.


INTRODUCTION
We consider the solution of the nonlinear scalar equation where f is a complex function of complex variable.
In most papers which deal with stationary iterative methods for (1.1) it is assumed we know the standard information for f (Wozniakowski [74]) 9ls = {f(x0),...,f (s) (x0)} where s ^ 1 and x^ is an approximation to the solution a.
The maximal order of convergence of such methods is equal to s + 1 (Traub [64], Wozniakowski [73]). We raise the question how other types of information can be used in iterative processes and what is the maximal order of convergence for this information.
This paper deals with integral information which consists of the standard information 9tg and additionally the value of an integral. Thus (1.2) S 1 s -{f(x0),...,f (s V0), J f(t) dtl where y^ is a complex number defined in Section 3.
In Section 2 we define an interpolatory -integral method I which uses integral information -I , s to estimate a and in Section 7 we prove its order for s > 1 is maximal. Sections 4, 5 and 6 contain theorems about the convergence of I , -1 ,s Wozniakowski [74] defined for the generalized information 9t an order of information pOJt) and proved it is equal to the maximal order of convergence. In Section 7 we prove that for s ^ 1 and for suitable x o chosen yQ, p( <^_ 1 g) = s + 3. Since pOft ) = s + 1, the additional information given by f f(t)dt increases the order of information by two. For systems of nonlinear equations similar results can be proved and will be reported to a future paper.

DEFINITION OF A LOWER LIMIT OF THE INTEGRAL
We want to define y.^ to maximize the order of I 1 g. Setting x = a in (2.12) we have (3.1) -w.(cy) = R(a) •* 1 -^ sufficiently close to a simple zero a.
Let us assume for a moment that w. has a zero x We see that the order of iteration depends mainly on R(a) . Therefore we shall choose y^^ to minimize As G^(a) and G^(cy) are in general unknown we want to minimize One can verify that the minimal value of (3.3) is for yi equal to y As we do not know a we have to replace it by an approximation to <y, z^ which depends only on the stan- It can be proved that one can drop | z -x^| in the denominator without the change of the order.
Finally, y is defined by Hence, from (3.6) and (2.12) we get where w^ is the interpolatory polynomial defined in Section 2.

THE CONVERGENCE OF THE ITERATIVE METHOD I -FOR s ;> 1 -1 ,s
In the previous section we have seen that the order of iteration mainly depends on R(a). From imal order of I., . for . * 1 it suffices to define approximation z± using Hence, to assure the max Newton method  ich completes the proof of Theorem 1.
In general, B is not equal zero (see point (iii) which means that s+3 is the order of the interpolatory -integral method I for s ^ 1, (Traub [64], Wozniakowski [74]). Note that iterative meth--1 , s (s) ods which use only the standard information f(x.),...,f (x.) have orders at most s+1. Additional in- x. Note that we cannot now define z. by the Newton method as we do not know the value of the first derivative. Let be given now by the secant method, In this case, the interpolatory -integral method 1^ Q is a one-point method with memory ( (iv) lim X i+1 = B p+1 where p = 1 + Ji .

X. -OF
The proof of this theorem is omitted since it is similar to the proof of Theorem 1. From (iv) follows that 1 + Ji is the order of the interpolator -integral method I_1)Q.
Next let us assume that where {x^} is an arbitrary sequence converging to a. Let wi be an interpolatory polynomial of degree at most s + 1 defined as follows: From (7.4) and (7.6) it follows that for any the order of information p = p(5l_1 g) exists. Let us assume that p > s+3. Let e > 0 be a number such that p -e > s + 3 + e. For f and [f 1 given by (7.5) we get from (7. It is easy to verify that from (7.4) and (7.6) it follows P(^1>s) = s + 3, which completes the proof of Theorem 3. M From Theorems 1 and 3 we get Corollary 1 The interpolatory -integral method I -is maximal, iQe., -1 , s