Error estimates for the summation of real numbers with application to floating-point summation

Error Bounds for Computer Arithmetics

“…Estimate (22) holds true for rounding upwards. We improve that by showing that under Assumption B, that is (26) for the ε k defined in (25), a linearized error bound is true for any directed or faithful rounding.…”

“…Assumption A, that is (15) is the key to the proof. In fact, the weaker assumption (20) in the following Theorem 4.1 suffices to prove more: in [22] it is shown that, for any order of evaluation, both the linearized error estimates (5) for the sum and (7) for the dot product remain true for an arithmetic according to the first standard model (14) with the additional assumption (20). The proof is more involved.…”

“…Theorem 4.1: [22] Let a binary tree T with root r be given. For a node j of T , denote the set of inner nodes of the subtree with root j by N j , and the set of its leaves by L j .…”

“…The restriction on n is necessary for any base β: for x 1 " 1 and x 2...n arbitrarily small positive numbers the sum increases each intermediate result to the next successor in F. Up to n ď β p´βp´1 summands the error is 2u, but passing the intermediate result β increases the error to βu, eventually spoiling the estimate (22).…”

“…Theorem 4.2: [22] Let a binary tree T with n leaves be given. To each leaf associate a real number x i and to each inner node associate a real number s k forming vectors x P R n and s P R n´1 .…”