BIT 18 (1978), 42-51 ITERATIVE METHODS FOR INTERVAL INCLUSION OF FIXED POINTS OLE CAPRANI and KAJ MADSEN Abstract. The paper discusses a technique for handling numerical, iterative processes that combines the efficiency of ordinary floating-point iterations with the accuracy control that may be obtained by iterations in interval arithmetic. As illustration the technique is used for the solution of fixed point problems in one and several variables. 1. Introduction. It is well known (see, e.g., Moore [1]) that calculations in ordinary floating- point arithmetic do not normally produce any information about the accuracy of the results obtained; in order to obtain such information it has been proposed (e.g., Moore [1]) to employ interval arithmetic. However, if used exclusively, interval arithmetic is very costly and further produces very pessimistic error bounds. It is the purpose of the present study to show how a combined use of floating-point and interval arithmetic in solving one type of problems, viz. iterative determination of fixed points, can lead to effective processes for finding solutions with known, narrow error bounds. Our approach follows the advice of Wilkinson [2]: "In general it is the best in algebraic computations to leave the use of interval arithmetic as late as possible so that it effectively becomes an a posteriori weapon". In order to illustrate the simplicity of the idea we start with an example. EXAMPLE 1 (Nickel [3]). Find the fixed point of the real function f(x)= (1--X2)/(3+X2). A 'standard technique for the solution of this problem is the iteration process, process (i): c.hoose Xo arbitrarily, and compute x~+l =f(x) for j--0, 1,2 .... end of process (i). The sequence {xj} obtained from this process converges to the fixed point because If'(x)l < 1. If we want the fixed point with an absolute accuracy of, say 10 -9, a common technique is to carry out the iteration by means of floating-point arithmetic and Received May 25, 1977. Revised December 2, 1977.