Shin'ichi Oishi

Accurate Sum and Dot Product

Algorithms for summation and dot product of floating-point numbers are presented which are fast in terms of measured computing time. We show that the computed results are as accurate as if computed in twice or K-fold working precision,

Accurate Floating-Point Summation Part I: Faithful Rounding

K\ge 3

Fast verification of solutions of matrix equations

. For twice the working precision our algorithms for summation and dot product are some 40% faster than the corresponding XBLAS routines while sharing similar error estimates. Our algorithms are widely applicable because they require only addition, subtraction, and multiplication of floating-point numbers in the same working precision as the given data. Higher precision is unnecessary, algorithms are straight loops without branch, and no access to mantissa or exponent is necessary.

Accurate Floating-Point Summation Part II: Sign,

Given a vector of floating-point numbers with exact sum

K

s

-Fold Faithful and Rounding to Nearest

, we present an algorithm for calculating a faithful rounding of

Self-synchronization of coupled oscillators with hysteretic responses

s

Numerical verification of existence and inclusion of solutions for nonlinear operator equations

, i.e., the result is one of the immediate floating-point neighbors of

VERIFIED EIGENVALUE EVALUATION FOR THE LAPLACIAN OVER POLYGONAL DOMAINS OF ARBITRARY SHAPE

s

Verified Computations for Hyperbolic 3-Manifolds

. If the sum