Error stability in finite mantissa floating point computers

Abstract

Floating point computing machines if operating perfectly depart from the ideal macazine in that the exponent is confined to a fixed number of places and that the mantissa is also confined to a fixed number of places° The former restriction leads to overflow and underflow problems and the latter leads to the generation of round off error. We are not concerned here with overflow or underflow problems. We shall be concerned only with the generation of round off error and its effect on the propagation of error. We assume that the round off error generated is that produced automatically by the machine truncating the true resultant of a normalized floa ting point ope ration. By relative error is meant the absolute error (exact resultant minus the machine resultant) divided by the machine resultant. Note that the absolute error referred to includes both the error in the resultant which propagated from the error in the operands as well as the new round off error generated in combining the operands arithmetically. Consider a machine using ~ place mantissas expressed to the base ~. If the resultant of combining two machine, umbers has exponent ~_