Modular Arithmetic for Linear Algebra Computations in the Real Field

The aim of this work is to decrease the bit precision required in computations without affecting the precision of the answer, whether this is computed exactly or within some tolerance. By precision we understand the number of bits in the binary representation of the values involved in the computation, hence a smaller precision requirement leads to a smaller complexity. We achieve this by combining the customary numerical techniques of rounding the least significant bits with the algebraic technique of reduction modulo an integer, which we extend to the reduction modulo a positive number. In particular, we show that, if the sum of several numbers has small magnitude, relative to the magnitude of the summands, then the precision used in the computation of this sum can be decreased without affecting the precision of the answer. Furthermore, if the magnitude of the inner product of two vectors is small and if one of them is filled with “short” binary numbers, then again we may decrease the precision of the computation. The method is applied to the iterative improvement algorithm for a linear system of equations whose coefficients are represented by “short” binary numbers, as well as to the solution of PDEs by means of multigrid methods. Some results of numerical experiments are presented to demonstrate the power of the method.