Automatic rounding error estimation and its applications. Solution of linear equations, Newton's and Mcauley's methods

Numerical computations are usually performed by floating arithmetic. Since the operations are performed by a finite number of digits, inevitably a rounding error is produced which is difficult mathematically to handle. This is because associative and distributive laws do not in general apply to the rounding operation. This paper describes a method which estimates the rounding error without using an analytical method. The number is represented by describing simultaneously the approximate value and the error. In parallel to the result of operation, the rounding error produced in that operation is determined and then summed for each operation. When a series of operations is completed, the range of existence for the result can be determined from the result of computation and the sum of errors. From the viewpoint that an unreliable number should not be given as output, the upper and lower limits of the range of existence for the result are given as output, indicating the extent to which the result of computation can be relied upon. It is shown that the proposed method is applicable to the sweeping-out method, Gauss-Seidel's method, Newton's method, and McAuley's method.