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| <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Terminating the while loop · MultiPrecisionArrays.jl</title><meta name="title" content="Terminating the while loop · MultiPrecisionArrays.jl"/><meta property="og:title" content="Terminating the while loop · MultiPrecisionArrays.jl"/><meta property="twitter:title" content="Terminating the while loop · MultiPrecisionArrays.jl"/><meta name="description" content="Documentation for MultiPrecisionArrays.jl."/><meta property="og:description" content="Documentation for MultiPrecisionArrays.jl."/><meta property="twitter:description" content="Documentation for MultiPrecisionArrays.jl."/><script data-outdated-warner src="../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" 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data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../../">MultiPrecisionArrays.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><span class="tocitem">GMRES-IR</span><ul><li><a class="tocitem" href="../../Half_1/">Half Precision and GMRES-IR</a></li></ul></li><li><span class="tocitem">More than you want to know</span><ul><li class="is-active"><a class="tocitem" href>Terminating the while loop</a></li><li><a class="tocitem" href="../Interprecision_1/">Interprecision Transfers: Part I</a></li></ul></li><li><span class="tocitem">MPArray Constructors</span><ul><li><a class="tocitem" href="../../functions/MPArray/">MPArray: constructor</a></li></ul></li><li><span class="tocitem">Factorizations</span><ul><li><a class="tocitem" href="../../functions/hlu!/">hlu!: Get LU to perform reasonably well for Float16</a></li><li><a class="tocitem" href="../../functions/mplu!/">mplu!: Simple MPArray factorization</a></li><li><a class="tocitem" href="../../functions/mplu/">mplu: Combine MPArray construction and factorization</a></li><li><a class="tocitem" href="../../functions/mpglu!/">mpglu!: Factor a MPArray and set it up for GMRES with \</a></li><li><a class="tocitem" href="../../functions/mpglu/">mpglu: Combine MPArray construction and factorization</a></li></ul></li><li><span class="tocitem">Solvers</span><ul><li><a class="tocitem" href="../../functions/mpgeslir/">mpgeslir: IR solver</a></li><li><a class="tocitem" href="../../functions/mpgmir/">mpgmir: GMRES-IR solver</a></li></ul></li><li><span class="tocitem">References</span><ul><li><a class="tocitem" href="../../References/">References</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">More than you want to know</a></li><li class="is-active"><a href>Terminating the while loop</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Terminating the while loop</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/ctkelley/MultiPrecisionArrays.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/ctkelley/MultiPrecisionArrays.jl/blob/main/docs/src/Details/Termination.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Terminating-the-while-loop"><a class="docs-heading-anchor" href="#Terminating-the-while-loop">Terminating the while loop</a><a id="Terminating-the-while-loop-1"></a><a class="docs-heading-anchor-permalink" href="#Terminating-the-while-loop" title="Permalink"></a></h1><p>We terminate the loop when </p><p class="math-container">\[\| r \| < \tau \| b \|\]</p><p>where we use <span>$\tau = 10 * eps(TH)$</span>, where <span>$eps(TH)$</span> is high precision floating point machine epsilon. The problem with this criterion is that IR can stagnate, especially for ill-conditioned problems, before the termination criterion is attained. We detect stagnation by looking for a unacceptable decrease (or increase) in the residual norm. So we will terminate the iteration if</p><p class="math-container">\[\| r_{new} \| \ge .9 \| r_{old} \|\]</p><p>even if the small residual condition is not satisfied.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../Half_1/">« Half Precision and GMRES-IR</a><a class="docs-footer-nextpage" href="../Interprecision_1/">Interprecision Transfers: Part I »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Thursday 16 November 2023 19:27">Thursday 16 November 2023</span>. Using Julia version 1.9.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> | ||
| <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Terminating the while loop · MultiPrecisionArrays.jl</title><meta name="title" content="Terminating the while loop · MultiPrecisionArrays.jl"/><meta property="og:title" content="Terminating the while loop · MultiPrecisionArrays.jl"/><meta property="twitter:title" content="Terminating the while loop · MultiPrecisionArrays.jl"/><meta name="description" content="Documentation for MultiPrecisionArrays.jl."/><meta property="og:description" content="Documentation for MultiPrecisionArrays.jl."/><meta property="twitter:description" content="Documentation for MultiPrecisionArrays.jl."/><script data-outdated-warner src="../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../search_index.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../../">MultiPrecisionArrays.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><span class="tocitem">GMRES-IR</span><ul><li><a class="tocitem" href="../../Half_1/">Half Precision and GMRES-IR</a></li></ul></li><li><span class="tocitem">More than you want to know</span><ul><li class="is-active"><a class="tocitem" href>Terminating the while loop</a></li><li><a class="tocitem" href="../Interprecision_1/">Interprecision Transfers: Part I</a></li></ul></li><li><span class="tocitem">MPArray Constructors</span><ul><li><a class="tocitem" href="../../functions/MPArray/">MPArray: constructor</a></li></ul></li><li><span class="tocitem">Factorizations</span><ul><li><a class="tocitem" href="../../functions/hlu!/">hlu!: Get LU to perform reasonably well for Float16</a></li><li><a class="tocitem" href="../../functions/mplu!/">mplu!: Simple MPArray factorization</a></li><li><a class="tocitem" href="../../functions/mplu/">mplu: Combine MPArray construction and factorization</a></li><li><a class="tocitem" href="../../functions/mpglu!/">mpglu!: Factor a MPGArray and set it up for GMRES y allocating room</a></li><li><a class="tocitem" href="../../functions/mpglu/">mpglu: Combine MPGArray construction and factorization</a></li></ul></li><li><span class="tocitem">Solvers</span><ul><li><a class="tocitem" href="../../functions/mpgeslir/">mpgeslir: IR solver</a></li><li><a class="tocitem" href="../../functions/mpgmir/">mpgmir: GMRES-IR solver</a></li></ul></li><li><span class="tocitem">References</span><ul><li><a class="tocitem" href="../../References/">References</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">More than you want to know</a></li><li class="is-active"><a href>Terminating the while loop</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Terminating the while loop</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/ctkelley/MultiPrecisionArrays.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/ctkelley/MultiPrecisionArrays.jl/blob/main/docs/src/Details/Termination.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Terminating-the-while-loop"><a class="docs-heading-anchor" href="#Terminating-the-while-loop">Terminating the while loop</a><a id="Terminating-the-while-loop-1"></a><a class="docs-heading-anchor-permalink" href="#Terminating-the-while-loop" title="Permalink"></a></h1><p>We terminate the loop when </p><p class="math-container">\[\| r \| < \tau \| b \|\]</p><p>where we use <span>$\tau = 10 * eps(TH)$</span>, where <span>$eps(TH)$</span> is high precision floating point machine epsilon. The problem with this criterion is that IR can stagnate, especially for ill-conditioned problems, before the termination criterion is attained. We detect stagnation by looking for a unacceptable decrease (or increase) in the residual norm. So we will terminate the iteration if</p><p class="math-container">\[\| r_{new} \| \ge .9 \| r_{old} \|\]</p><p>even if the small residual condition is not satisfied.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../Half_1/">« Half Precision and GMRES-IR</a><a class="docs-footer-nextpage" href="../Interprecision_1/">Interprecision Transfers: Part I »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.1.2 on <span class="colophon-date" title="Sunday 19 November 2023 17:02">Sunday 19 November 2023</span>. Using Julia version 1.9.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> |
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