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{"documenter":{"julia_version":"1.12.5","generation_timestamp":"2026-02-18T17:19:59","documenter_version":"1.16.1"}}
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{"documenter":{"julia_version":"1.12.5","generation_timestamp":"2026-03-03T12:25:44","documenter_version":"1.17.0"}}

DifferentiationInterface/dev/api/index.html

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DifferentiationInterface/dev/assets/documenter.js

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DifferentiationInterface/dev/assets/warner.js

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closer.addEventListener("click", function () {
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document.body.removeChild(div);
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});
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const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE;
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var target_href =
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window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE;
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// try to stay on the same page when linking to the stable version
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// get the current page path relative to the version root
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var current_page = window.location.pathname;
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// resolve the documenterBaseURL to an absolute path
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// documenterBaseURL is a relative path (usually "."), so we need to resolve it
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var base_url_absolute = new URL(documenterBaseURL, window.location.href)
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.pathname;
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if (!base_url_absolute.endsWith("/")) {
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base_url_absolute = base_url_absolute + "/";
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}
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// extract the page path after the version directory
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// e.g., if we're on /stable/man/guide.html, we want "man/guide.html"
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var page_path = "";
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if (current_page.startsWith(base_url_absolute)) {
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page_path = current_page.substring(base_url_absolute.length);
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}
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// construct the target URL with the same page path
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var target_url = target_href;
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if (page_path && page_path !== "" && page_path !== "index.html") {
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// ensure target_href ends with a slash before appending page path
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if (!target_url.endsWith("/")) {
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target_url = target_url + "/";
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target_url = target_url + page_path;
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"This documentation is for an <strong>older version</strong> that may be missing recent changes.<br>";
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'<a href="' +
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href +
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'">Click here to go to the documentation for the latest stable release.</a>';
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// Create the link element with same-page navigation
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const link = document.createElement("a");
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link.href = target_url + current_hash;
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link.textContent =
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"Click here to go to the documentation for the latest stable release.";
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link.addEventListener("click", function (e) {
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fetch(target_url, { method: "HEAD" })
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.then(function (response) {
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window.location.href = target_url + current_hash;
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window.location.href = target_href;
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}
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})
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.catch(function (error) {
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window.location.href = target_href;
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div.appendChild(link);
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DifferentiationInterface/dev/dev/contributing/index.html

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DifferentiationInterface/dev/dev/internals/index.html

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DifferentiationInterface/dev/dev/math/index.html

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DifferentiationInterface/dev/explanation/advanced/index.html

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seed = 3
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coloring_algorithm = GreedyColoringAlgorithm(
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RandomOrder(StableRNG(seed), seed); postprocessing=true
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)</code></pre><h2 id="Batch-mode"><a class="docs-heading-anchor" href="#Batch-mode">Batch mode</a><a id="Batch-mode-1"></a><a class="docs-heading-anchor-permalink" href="#Batch-mode" title="Permalink"></a></h2><h3 id="Multiple-tangents"><a class="docs-heading-anchor" href="#Multiple-tangents">Multiple tangents</a><a id="Multiple-tangents-1"></a><a class="docs-heading-anchor-permalink" href="#Multiple-tangents" title="Permalink"></a></h3><p>The <a href="../../api/#DifferentiationInterface.jacobian"><code>jacobian</code></a> and <a href="../../api/#DifferentiationInterface.hessian"><code>hessian</code></a> operators compute matrices by repeatedly applying lower-level operators (<a href="../../api/#DifferentiationInterface.pushforward"><code>pushforward</code></a>, <a href="../../api/#DifferentiationInterface.pullback"><code>pullback</code></a> or <a href="../../api/#DifferentiationInterface.hvp"><code>hvp</code></a>) to a set of tangents. The tangents usually correspond to basis elements of the appropriate vector space. We could call the lower-level operator on each tangent separately, but some packages (<a href="https://github.com/JuliaDiff/ForwardDiff.jl">ForwardDiff.jl</a> and <a href="https://github.com/EnzymeAD/Enzyme.jl">Enzyme.jl</a>) have optimized implementations to handle multiple tangents at once.</p><p>This behavior is often called &quot;vector mode&quot; AD, but we call it &quot;batch mode&quot; to avoid confusion with Julia&#39;s <code>Vector</code> type. As a matter of fact, the optimal batch size <span>$B$</span> (number of simultaneous tangents) is usually very small, so tangents are passed within an <code>NTuple</code> and not a <code>Vector</code>. When the underlying vector space has dimension <span>$N$</span>, the operators <code>jacobian</code> and <code>hessian</code> process <span>$\lceil N / B \rceil$</span> batches of size <span>$B$</span> each.</p><h3 id="Optimal-batch-size"><a class="docs-heading-anchor" href="#Optimal-batch-size">Optimal batch size</a><a id="Optimal-batch-size-1"></a><a class="docs-heading-anchor-permalink" href="#Optimal-batch-size" title="Permalink"></a></h3><p>For every backend which does not support batch mode, the batch size is set to <span>$B = 1$</span>. But for <a href="https://sciml.github.io/ADTypes.jl/stable/#ADTypes.AutoForwardDiff"><code>AutoForwardDiff</code></a> and <a href="https://sciml.github.io/ADTypes.jl/stable/#ADTypes.AutoEnzyme"><code>AutoEnzyme</code></a>, more complicated rules apply. If the backend object has a pre-determined batch size <span>$B_0$</span>, then we always set <span>$B = B_0$</span>. In particular, this will throw errors when <span>$N &lt; B_0$</span>. On the other hand, without a pre-determined batch size, we apply backend-specific heuristics to pick <span>$B$</span> based on <span>$N$</span>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../backends/">« Backends</a><a class="docs-footer-nextpage" href="../../faq/limitations/">Limitations »</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="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.16.1 on <span class="colophon-date" title="Wednesday 18 February 2026 17:19">Wednesday 18 February 2026</span>. Using Julia version 1.12.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
9+
)</code></pre><h2 id="Batch-mode"><a class="docs-heading-anchor" href="#Batch-mode">Batch mode</a><a id="Batch-mode-1"></a><a class="docs-heading-anchor-permalink" href="#Batch-mode" title="Permalink"></a></h2><h3 id="Multiple-tangents"><a class="docs-heading-anchor" href="#Multiple-tangents">Multiple tangents</a><a id="Multiple-tangents-1"></a><a class="docs-heading-anchor-permalink" href="#Multiple-tangents" title="Permalink"></a></h3><p>The <a href="../../api/#DifferentiationInterface.jacobian"><code>jacobian</code></a> and <a href="../../api/#DifferentiationInterface.hessian"><code>hessian</code></a> operators compute matrices by repeatedly applying lower-level operators (<a href="../../api/#DifferentiationInterface.pushforward"><code>pushforward</code></a>, <a href="../../api/#DifferentiationInterface.pullback"><code>pullback</code></a> or <a href="../../api/#DifferentiationInterface.hvp"><code>hvp</code></a>) to a set of tangents. The tangents usually correspond to basis elements of the appropriate vector space. We could call the lower-level operator on each tangent separately, but some packages (<a href="https://github.com/JuliaDiff/ForwardDiff.jl">ForwardDiff.jl</a> and <a href="https://github.com/EnzymeAD/Enzyme.jl">Enzyme.jl</a>) have optimized implementations to handle multiple tangents at once.</p><p>This behavior is often called &quot;vector mode&quot; AD, but we call it &quot;batch mode&quot; to avoid confusion with Julia&#39;s <code>Vector</code> type. As a matter of fact, the optimal batch size <span>$B$</span> (number of simultaneous tangents) is usually very small, so tangents are passed within an <code>NTuple</code> and not a <code>Vector</code>. When the underlying vector space has dimension <span>$N$</span>, the operators <code>jacobian</code> and <code>hessian</code> process <span>$\lceil N / B \rceil$</span> batches of size <span>$B$</span> each.</p><h3 id="Optimal-batch-size"><a class="docs-heading-anchor" href="#Optimal-batch-size">Optimal batch size</a><a id="Optimal-batch-size-1"></a><a class="docs-heading-anchor-permalink" href="#Optimal-batch-size" title="Permalink"></a></h3><p>For every backend which does not support batch mode, the batch size is set to <span>$B = 1$</span>. But for <a href="https://sciml.github.io/ADTypes.jl/stable/#ADTypes.AutoForwardDiff"><code>AutoForwardDiff</code></a> and <a href="https://sciml.github.io/ADTypes.jl/stable/#ADTypes.AutoEnzyme"><code>AutoEnzyme</code></a>, more complicated rules apply. If the backend object has a pre-determined batch size <span>$B_0$</span>, then we always set <span>$B = B_0$</span>. In particular, this will throw errors when <span>$N &lt; B_0$</span>. On the other hand, without a pre-determined batch size, we apply backend-specific heuristics to pick <span>$B$</span> based on <span>$N$</span>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../backends/">« Backends</a><a class="docs-footer-nextpage" href="../../faq/limitations/">Limitations »</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="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.17.0 on <span class="colophon-date" title="Tuesday 3 March 2026 12:25">Tuesday 3 March 2026</span>. Using Julia version 1.12.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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