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main/algebra/fibonacci-numbers.html

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@@ -6772,7 +6772,7 @@
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<ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
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Last update:
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<span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="April 22, 2025 01:19:59">April 22, 2025</span>&emsp;
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<span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="April 22, 2025 03:58:03">April 22, 2025</span>&emsp;
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@@ -6929,7 +6929,17 @@ <h3 id="matrix-form">Matrix form<a class="headerlink" href="#matrix-form" title=
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F_{n}
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\end{pmatrix}
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$$</div>
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<p>where <span class="arithmatex">$F_1 = 1, F_0 = 0$</span>.</p>
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<p>where <span class="arithmatex">$F_1 = 1, F_0 = 0$</span>.
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In fact, since </p>
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<div class="arithmatex">$$
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\begin{pmatrix} 1 &amp; 1 \\ 1 &amp; 0 \end{pmatrix}
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= \begin{pmatrix} F_2 &amp; F_1 \\ F_1 &amp; F_0 \end{pmatrix}
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$$</div>
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<p>we can use the matrix directly:</p>
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<div class="arithmatex">$$
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\begin{pmatrix} 1 &amp; 1 \\ 1 &amp; 0 \end{pmatrix}^n
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= \begin{pmatrix} F_{n+1} &amp; F_n \\ F_n &amp; F_{n-1} \end{pmatrix}
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$$</div>
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<p>Thus, in order to find <span class="arithmatex">$F_n$</span> in <span class="arithmatex">$O(\log n)$</span> time, we must raise the matrix to n. (See <a href="binary-exp.html">Binary exponentiation</a>)</p>
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<div class="highlight"><pre><span></span><code><span class="k">struct</span><span class="w"> </span><span class="nc">matrix</span><span class="w"> </span><span class="p">{</span>
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<span class="w"> </span><span class="kt">long</span><span class="w"> </span><span class="kt">long</span><span class="w"> </span><span class="n">mat</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">];</span>
@@ -7034,7 +7044,7 @@ <h2 id="practice-problems">Practice Problems<a class="headerlink" href="#practic
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<ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
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<span class="contributors-text">Contributors:</span>
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