The design of fixed-point iterations

Abstract

where the residual ~(~) tends to zero as v increases without limit° Such iterations are self-correcting. The ultimate accuracy of ~ will depend upon the rounding errors committed in determining f(~)~(~). These rounding errors may be minimized in a large class of problems by computinE [~kf(~)]~(~)2-k provided one ~nows a priori that If~)I~ ~k The previous iteration always provides a lower bound for k. In this manner it is possible to continue the iteration until the interval of uncertainty of the final ~ is minimal. For second-order iterative procedures in a binary computer with places to the right of the binary point, then one obtains