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<!DOCTYPE html>
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<a href="index.html">CMDSolver Docs</a>
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<div class="doc-header-right"><div class="lang-switch"><button class="lang-btn" data-lang="de">DE</button><button class="lang-btn" data-lang="en">EN</button></div><span class="doc-badge solver">SolverType.BROYDEN</span></div></div>
<div class="hero">
<div class="hero-sub"><span data-lang="de">Solver 5 von 7</span><span data-lang="en">Solver 5 of 7</span></div>
<h1 data-lang="de">Broyden<br>Quasi-Newton</h1><h1 data-lang="en">Broyden<br>Quasi-Newton</h1>
<p class="tagline" data-lang="de">Quasi-Newton-Verfahren mit Rang-1-Update der Jacobi-Approximation. Günstiger pro Iteration als vollständiger Jacobi-Neuaufbau — besonders vorteilhaft bei Parameterstudien.</p>
<p class="tagline" data-lang="en">Quasi-Newton method with rank-1 update of the Jacobian approximation. Cheaper per iteration than full Jacobian rebuild — especially advantageous for parameter studies.</p>
<div class="hero-stats">
<div class="stat"><span class="stat-val">25</span><span class="stat-lbl" data-lang="de">Iter (SimpleSystem)</span><span class="stat-lbl" data-lang="en">Iter (SimpleSystem)</span></div>
<div class="stat"><span class="stat-val">8.02e-08</span><span class="stat-lbl">fsum</span></div>
<div class="stat"><span class="stat-val">~50ms</span><span class="stat-lbl" data-lang="de">Laufzeit</span><span class="stat-lbl" data-lang="en">Runtime</span></div>
<div class="stat"><span class="stat-val">Rang-1</span><span class="stat-lbl" data-lang="de">Update</span><span class="stat-lbl" data-lang="en">Update</span></div>
</div>
</div>
<nav>
<a href="#theorie" data-lang="de">Theorie</a><a href="#theorie" data-lang="en">Theory</a>
<a href="#pseudocode" data-lang="de">Pseudocode</a><a href="#pseudocode" data-lang="en">Pseudocode</a>
<a href="#dateien" data-lang="de">Dateien</a><a href="#dateien" data-lang="en">Files</a>
<a href="#parameter" data-lang="de">Parameter</a><a href="#parameter" data-lang="en">Parameters</a>
<a href="#verhalten" data-lang="de">Verhalten</a><a href="#verhalten" data-lang="en">Behavior</a>
<a href="#grenzen" data-lang="de">Grenzen</a><a href="#grenzen" data-lang="en">Limits</a>
</nav>
<main>
<section id="theorie">
<h2 data-lang="de">Theoretischer Kontext</h2><h2 data-lang="en">Theoretical Context</h2>
<h3 data-lang="de">Grundidee — allgemein verständlich</h3><h3 data-lang="en">Core idea — in plain terms</h3>
<p data-lang="de">
Bei Newton-Verfahren wird in jeder Iteration die vollständige Jacobi-Matrix
neu berechnet — das ist teuer. Broyden fragt: "Können wir die Jacobi aus dem
vorherigen Schritt <em>anpassen</em> statt sie neu zu berechnen?"
</p><p data-lang="en">
In Newton methods the full Jacobian is recomputed every iteration — that's
expensive. Broyden asks: "Can we <em>adjust</em> the Jacobian from the previous
step instead of rebuilding it?"
</p>
<p data-lang="de">
Die Antwort: Ja — durch einen <strong>Rang-1-Update</strong>. Man berechnet
wie sich F zwischen zwei Punkten verändert hat und korrigiert die
Jacobi-Approximation minimal so dass sie mit dieser Beobachtung konsistent ist.
Das kostet O(n²) statt O(n³) für einen vollständigen Neuaufbau.
</p><p data-lang="en">
The answer: yes — via a <strong>rank-1 update</strong>. We compute how F
changed between two points and minimally correct the Jacobian approximation
so it remains consistent with that observation. That costs O(n²) instead
of O(n³) for a full rebuild.
</p>
<div class="math-block">
<div class="eq"><span class="hi" data-lang="de">Broyden-Update (Rang-1):</span><span class="hi" data-lang="en">Broyden update (rank-1):</span></div>
<div class="eq"> Jₖ₊₁ = Jₖ + (Δf − Jₖ·Δx) · ΔxᵀT / ||Δx||²</div>
<div class="eq"></div>
<div class="eq"><span class="hi" data-lang="de">wobei:</span><span class="hi" data-lang="en">where:</span></div>
<div class="eq"> Δx = xₖ₊₁ − xₖ <span class="cmt" data-lang="de">// Schritt</span><span class="cmt" data-lang="en">// step</span></div>
<div class="eq"> Δf = F(xₖ₊₁) − F(xₖ) <span class="cmt" data-lang="de">// Residuumsänderung</span><span class="cmt" data-lang="en">// residual change</span></div>
<div class="eq"></div>
<div class="eq"><span class="hi" data-lang="de">Sekanten-Bedingung:</span><span class="hi" data-lang="en">Secant condition:</span> Jₖ₊₁ · Δx = Δf</div>
<div class="eq"><span class="cmt" data-lang="de">// Die neue Jacobi ist konsistent mit dem letzten Schritt</span><span class="cmt" data-lang="en">// The new Jacobian is consistent with the last step</span></div>
</div>
<h3 data-lang="de">Reset-Mechanismus</h3><h3 data-lang="en">Reset mechanism</h3>
<p data-lang="de">
Broyden akkumuliert Approximationsfehler über die Iterationen. CMDSolver
implementiert daher einen automatischen Reset: Nach <code>resetInterval</code>
Iterationen oder wenn die Konditionszahl κ einen Schwellwert überschreitet,
wird die Jacobi vollständig neu berechnet. Das verhindert Divergenz durch
akkumulierte Fehler.
</p><p data-lang="en">
Broyden accumulates approximation error over iterations. CMDSolver therefore
implements an automatic reset: after <code>resetInterval</code> iterations,
or when the condition number κ exceeds a threshold, the Jacobian is fully
rebuilt. This prevents divergence due to accumulated error.
</p>
</section>
<section id="pseudocode">
<h2 data-lang="de">Pseudocode</h2><h2 data-lang="en">Pseudocode</h2>
<pre><span class="kw">function</span> <span class="fn">broyden_solve</span>(F, x₀, tol, maxIter, resetInterval):
x ← InitGuessApplier(x₀)
J ← buildJacobian(F, x) <span class="cmt" data-lang="de">// vollständiger Aufbau beim Start</span><span class="cmt" data-lang="en">// full build at start</span>
Jinv ← inverse(J)
<span class="kw">for</span> k = 0 <span class="kw">to</span> maxIter:
f ← F(x)
<span class="cmt" data-lang="de">// ── Newton-Schritt mit approximierter Jacobi ──</span><span class="cmt" data-lang="en">// ── Newton step with approximated Jacobian ──</span>
Δx ← −Jinv · f
<span class="cmt" data-lang="de">// ── Armijo-Liniensuche ──</span><span class="cmt" data-lang="en">// ── Armijo line search ──</span>
α ← armijo_search(x, Δx, F)
x_new ← x + α · Δx
f_new ← F(x_new)
<span class="cmt" data-lang="de">// ── Broyden Rang-1-Update ──</span><span class="cmt" data-lang="en">// ── Broyden rank-1 update ──</span>
<span class="kw">if</span> k % resetInterval != 0 <span class="kw">and</span> κ(J) < κ_reset:
Δf ← f_new − f
J ← J + (Δf − J·Δx) · Δxᵀ / ||Δx||² <span class="cmt" data-lang="de">// O(n²)</span><span class="cmt" data-lang="en">// O(n²)</span>
Jinv ← update_inverse(Jinv, Δx, Δf)
<span class="kw">else</span>:
J ← buildJacobian(F, x_new) <span class="cmt" data-lang="de">// vollständiger Reset</span><span class="cmt" data-lang="en">// full reset</span>
Jinv ← inverse(J)
x ← x_new
x ← PhysicalProjector(x)
fsum ← Σ|Fᵢ(x)|
<span class="kw">if</span> fsum < tol: <span class="kw">return</span> CONVERGED, x, k, fsum
<span class="kw">return</span> NOT_CONVERGED, x, maxIter, fsum</pre>
</section>
<section id="dateien">
<h2 data-lang="de">Wichtige Dateien</h2><h2 data-lang="en">Key Files</h2>
<ul class="file-list">
<li>
<span class="file-icon" data-lang="de">KERN</span><span class="file-icon" data-lang="en">CORE</span>
<div class="file-desc">
<span data-lang="de"><strong>BroydenSolver.java</strong>
Implementiert Rang-1-Update, Reset-Mechanismus, Armijo-Liniensuche
und Checkpoint-System für Warm-Start bei Parameterstudien.
<br><code>apps/eqnParser/solver/BroydenSolver.java</code></span><span data-lang="en"><strong>BroydenSolver.java</strong>
Implements rank-1 update, reset mechanism, Armijo line search and
checkpoint system for warm-start in parameter studies.
<br><code>apps/eqnParser/solver/BroydenSolver.java</code></span>
</div>
</li>
<li>
<span class="file-icon" data-lang="de">BASIS</span><span class="file-icon" data-lang="en">BASE</span>
<div class="file-desc">
<span data-lang="de"><strong>AbstractSolver.java</strong>
Basisklasse — Jacobi-Aufbau, Iteration, Konvergenzprüfung.
<br><code>apps/eqnParser/solver/AbstractSolver.java</code></span><span data-lang="en"><strong>AbstractSolver.java</strong>
Base class — Jacobian build, iteration, convergence check.
<br><code>apps/eqnParser/solver/AbstractSolver.java</code></span>
</div>
</li>
</ul>
</section>
<section id="parameter">
<h2 data-lang="de">Parameter</h2><h2 data-lang="en">Parameters</h2>
<pre>java CASSolver -I:model.cas -S:BROYDEN</pre>
<div class="tbl-wrap">
<table>
<thead><tr><th>Parameter</th><th>Default</th><th data-lang="de">Beschreibung</th><th data-lang="en">Description</th></tr></thead>
<tbody>
<tr><td><code>tolerance</code></td><td><code>1e-7</code></td><td data-lang="de">Konvergenzschwelle</td><td data-lang="en">Convergence threshold</td></tr>
<tr><td><code>maxIter</code></td><td><code>100</code></td><td data-lang="de">Maximale Iterationen</td><td data-lang="en">Maximum iterations</td></tr>
<tr><td><code>broydenResetInterval</code></td><td><code>25</code></td><td data-lang="de">Vollständiger Jacobi-Neuaufbau alle n Iterationen</td><td data-lang="en">Full Jacobian rebuild every n iterations</td></tr>
<tr><td><code>broydenKappaReset</code></td><td><code>1e10</code></td><td data-lang="de">κ-Schwellwert für erzwungenen Reset</td><td data-lang="en">κ threshold for forced reset</td></tr>
</tbody>
</table>
</div>
</section>
<section id="verhalten">
<h2 data-lang="de">Konvergenzverhalten</h2><h2 data-lang="en">Convergence Behavior</h2>
<div class="tbl-wrap">
<table>
<thead><tr><th data-lang="de">Situation</th><th data-lang="en">Situation</th><th data-lang="de">Verhalten</th><th data-lang="en">Behavior</th><th data-lang="de">Empfehlung</th><th data-lang="en">Recommendation</th></tr></thead>
<tbody>
<tr><td data-lang="de">Parameterstudien (viele ähnliche Punkte)</td><td data-lang="en">Parameter studies (many similar points)</td><td><span class="badge b-ok" data-lang="de">✅ Sehr effizient</span><span class="badge b-ok" data-lang="en">✅ Very efficient</span></td><td data-lang="de">Jacobi vom Vorpunkt als Start</td><td data-lang="en">Jacobian from previous point as start</td></tr>
<tr><td data-lang="de">Gut konditioniertes System</td><td data-lang="en">Well-conditioned system</td><td><span class="badge b-warn" data-lang="de">⚠️ Mehr Iterationen als Newton</span><span class="badge b-warn" data-lang="en">⚠️ More iterations than Newton</span></td><td data-lang="de">Superlinear statt quadratisch</td><td data-lang="en">Superlinear instead of quadratic</td></tr>
<tr><td data-lang="de">Schlecht konditioniert</td><td data-lang="en">Ill-conditioned</td><td><span class="badge b-err" data-lang="de">❌ Reset nötig</span><span class="badge b-err" data-lang="en">❌ Reset required</span></td><td data-lang="de">→ BROYDEN_SPARSE oder LM</td><td data-lang="en">→ BROYDEN_SPARSE or LM</td></tr>
</tbody>
</table>
</div>
</section>
<section id="grenzen">
<h2 data-lang="de">Bekannte Grenzen</h2><h2 data-lang="en">Known Limitations</h2>
<div class="note amber" data-lang="de">
<strong>Akkumulierte Fehler:</strong> Die Jacobi-Approximation degradiert über
viele Iterationen. Der Reset-Mechanismus begrenzt das, kann es aber nicht
vollständig vermeiden. Bei sehr nichtlinearen Systemen häufiger Reset empfohlen.
</div><div class="note amber" data-lang="en">
<strong>Accumulated error:</strong> the Jacobian approximation degrades over
many iterations. The reset mechanism limits this but cannot fully prevent
it. For strongly nonlinear systems, more frequent resets are recommended.
</div>
<div class="note amber" data-lang="de">
<strong>Superlineare Konvergenz:</strong> Broyden konvergiert superlinear statt
quadratisch — mehr Iterationen als Newton bei gleicher Genauigkeit.
</div><div class="note amber" data-lang="en">
<strong>Superlinear convergence:</strong> Broyden converges superlinearly,
not quadratically — more iterations than Newton at the same accuracy.
</div>
</section>
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