https://dataleaf-642346.gitlab.io/tags/python/Recent content in Python on DataLeafHugoen-USTue, 03 Jun 2025 00:00:00 +0000https://dataleaf-642346.gitlab.io/posts/compensated-summation/Tue, 03 Jun 2025 00:00:00 +0000https://dataleaf-642346.gitlab.io/posts/compensated-summation/<p>A floating-point sum looks like the most innocent computation in a program. It is computed one addition at a time, <span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>s</mi><mo>^</mo></mover><mi>k</mi></msub><mo>=</mo><mi mathvariant="normal">fl</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mover accent="true"><mi>s</mi><mo>^</mo></mover><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat s_k = \operatorname{fl}(\hat s_{k-1} + x_k)</annotation></semantics></math></span>, and every addition rounds its result to the nearest representable number. Each rounding is tiny, but a long sum performs millions of them, and their errors do not have to cancel.</p> <div class="dataleaf-heading"><h2 id="how-large-can-the-error-get">How large can the error get?</h2> <a class="dataleaf-heading-anchor" href="#how-large-can-the-error-get" aria-label="Permalink to How large can the error get?" data-heading-anchor data-heading-copy="Copy link to How large can the error get?" data-heading-copied="Copied link to How large can the error get?.">#</a> </div> <p>For recursive summation, the classic bound is<sup id="fnref:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup></p>