https://dataleaf-642346.gitlab.io/categories/methods/Recent content in Methods on DataLeafHugoen-USTue, 18 Aug 2026 00:00:00 +0000https://dataleaf-642346.gitlab.io/posts/least-squares-three-ways/Tue, 18 Aug 2026 00:00:00 +0000https://dataleaf-642346.gitlab.io/posts/least-squares-three-ways/Polynomial fits make the difference visible: with twelve columns, the normal equations lose every digit while QR and the SVD keep eight or nine. One identity about condition numbers explains it, and also says when the fast method is good enough.https://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>