A Significance Rule for Multiple-Precision Arithmetic
Christopher B. Jones · ACM Transactions on Mathematical Software · 1984
Multiple-precision arithmetic overcomes the round-off error incurred in conventional floating-point arithmetic, at the cost of increased processing overhead.Significance arithmetic takes into account the Inexactness of the operands of a calculation, but can lead to loss of significant digits after a long series of operations.A new technique is described whmh alleviates the overhead of multiple-precision arithmetic by allowing nonsignificant digits to be discarded, while limiting the significance loss per operation to a controllable and acceptable rate.The technique is based on storing an inexact number as an interval, using a criterion of slgmficance to determine the precision with which the limits of the interval should be stored.A procedure referred to as a slgmficance rule uses this criterion to remove some of the nonsignificant digits from the limits of an interval prior to storage.A certain number of nonslgmficant digits are retained as guard digits.Calculations are performed using exact interval anthmetm and the significance-rule procedure is invoked after each operation to remove superfluous &glts.Round-off in the procedure causes a slight increase in the interval width on each operation.This results in a cumulative loss of significance at a rate related to the number of guard digits.