INTEGRALS WITH A KERNEL IN THE SOLUTION OF NONLINEAR EQUATIONS IN N DIMENSIONS

We consider iterations for solving the nonlinear equation F(x) the N dimensional Banach space, 1 ^ N ^ + «>, which use "integral information with a kernel". This information consists of the "standard information" F V j ; ( x d ), j « 0,1,...,s and the integral J g(t) F(x d + ty d )dt where s ^ 1, x, is an approximation to the solution and y, depends on the standard in- u a formation. We show there exists an iteration with order 2s + 1 + 6„ , and N,l prove its optimality.


INTRODUCTION
We want to approximate the simple solution a of the nonlinear equation where x^ is a close approximation to or.
In previous papers we investigated another kind of information, namely the integral information Since the maximal order of iterations using the standard information is equal to s + 1, the use of the integral increases the maximal order by 2 -6.
In this paper we consider more general kind of integral information, namely integral information with a kernel. Note that if g(t) s 1 then «Jr. =51 . The question is how the -I , S -I , S maximal order of iteration depends on g.
In Section 2 we define the iteration I*% which uses Ut^-for optimally -l,s -1,8 J chosen y d (see Section 4) and is of order min(s+1-hn, 2s+1+8 N j) (see Section 3 and Corollary 1 in Section 4), where m is an integer depending on g (defined in Section 2) and 8.. is the Kronecker delta. In Section 4 we prove the iteration I -is maximal. Furthermore we show there exists a polynomial -1, s g = g(t) independent on F such that m « s+6 M «• Since for such g the order is equal to 2s+1+6 «I > the value of the integral with a kernel, which is represented by the vector of size N, increases the maximal order by s+8^ g In Section 5 we show that for N sufficiently large the iteration 1^ ^ has smaller complexity index than any interpolatory iteration IQ ^, which uses the information 51^, k ^ 1, under some assumptions on the cost of computing the value of function, its derivatives, and the integral. In   (with a criterion of its selection, e.g., the nearest zero to x^) , where w is given as follows.
Note that to find a good approximation of h in numerical practice it is d possible to perform a few Newton steps on the equation (2.9).
We see that for m ~ 0 I 8 is equal "to the well known interpolatory -1, s iteration I rt which uses the standard information !Jl and is of order 0,s s s+1. Hence we assume that m ^ 1.
One can verify that the polynomial w satisfies the following interpolatory conditions. For N = 1, 3. CONVERGENCE OF THE ITERATION I 8 . -l,s If the function F is sufficiently smooth in the neighborhood of the zero a, then from (2.10), (2.11), (2.12) and due to the special form of y d given by (2.8) we have , 1 r (s+2) w .s+2 , -..I ,,s+3. , From the Brouwer fix point theorem for N < + co or the Schauder fix point theorem for N « + » (see Ortega and Rheinboldt [70], p.164), from the definition (2.9) of I 8 .
and ( For this purpose we define the order of iteration and the order of information as in Wozniakowski [75b]. Wozniakowski [75a] proved that the order of information is equal to the maximal order of convergence. We shall use this property to show I is -1 ,s maximal.
We now prove the theorem about order of information 51 -• -1, s Theorem 2 Let 91 -be the integral information with a kernel -l,s  Hence, (4.6) and (4.7) prove (4.4) for N = 1.
If there exist F € % F(«) » 0 and fx,}, lim x, = a such that a , a a _ iiv«ii  where (3. is close to the solution (but not equal), and x N-2 has the order of convergence greater than s+1, i.e., greater than the order of used information, which is a contradiction.
Finally, from (4.11) and (4.10) follows the inequality (4.4) for m > s, g which means that p(51 . ) ^ 2s+1. This proves Case II and also the first part -1, s of Theorem 2.
We shall prove the second part of Theorem 2. We want to show that for Note that the order of information and at the same time order of iteration