AlgorithmAlgorithm: AI-adaptedProof: Not applicablePipeline-generated

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Kahan compensated summation with range-failure status

Definition

For a finite list of $n \geq 0$ real floating-point inputs $x_{1}, \dots, x_{n}$ , initialize $s_{0} = c_{0} = 0$ . For $i = 1, \dots, n$ compute $y_{i} = fl (x_{i} - c_{i - 1})$ , $s_{i} = fl (s_{i - 1} + y_{i})$ , $d_{i} = fl (s_{i} - s_{i - 1})$ , and $c_{i} = fl (d_{i} - y_{i})$ , in that order. Return $s_{n}$ , or the first range-failure and its operation index.

Remarks

The separate Finite arithmetic schedules terminate, with early failure counted separately applies to the four ordered rounded operations at each input index. It gives finite execution with at most $4 n$ attempted rounded operations and exactly $4 n$ on success, including zero for $n = 0$ . The successful error identity is proved in Kahan and Neumaier summation satisfy explicit finite data-dependent error bounds.

Depends on

Used by

Kahan and Neumaier summation satisfy explicit finite data-dependent error bounds Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Fundamentals of Numerical Computation, floating-point arithmetic (standard reference, not scraped)
Higham, The Accuracy of Floating Point Summation (standard reference, not scraped)
Ogita, Rump, and Oishi, Accurate Sum and Dot Product (standard reference, not scraped)