IMPROVED ERROR BOUNDS FOR INNER PRODUCTS IN FLOATING-POINT ARITHMETIC

Given two floating-point vectors x, y of dimension n and assuming rounding to nearest, we show that if no underflow or overflow occurs, any evaluation order for an inner product returns a floating-point number (r) over cap such that vertical bar(r) over cap - x(T)y vertical bar <= nu vertical bar x vertical bar(T)vertical bar y vertical bar with u the unit roundoff. This result, which holds for any radix and with no restriction on n, can be seen as a generalization of a similar bound given in [S. M. Rump, BIT, 52 (2012), pp. 201-220] for recursive summation in radix 2, namely, vertical bar(r) over cap - x(T)e vertical bar <= (n - 1)u vertical bar x vertical bar(T)e with e = [1, 1, ... , 1](T). As a direct consequence, the error bound for the floating-point approximation (C) over cap of classical matrix multiplication with inner dimension n simplifies to vertical bar(C) over cap - AB vertical bar <= nu vertical bar A parallel to B vertical bar.