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      <text>12.3.1.1 The Newton-Raphson Method</text>
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      <text><![CDATA[<h3>The Newton&ndash;Raphson Method</h3>
<p>We first remind the reader of some basic notation: If \(f(x)\) is a given function the value of \(x\) for which \(f(x)=0\) is called a <strong>root</strong> of the equation or <strong>zero</strong> of the function. We also distinguish between various
    types of roots: simple roots and multiple roots. The following figures illustrate some common examples.
</p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/12.3%20Types%20of%20roots.svg" alt="Three plots showing a simple root, a double root and a triple root." width="800" height="173" class="img-responsive atto_image_button_text-bottom"></figure>
<p>More precisely; a root \(x_0\) is said to be:</p>
<ul>
    <li>
        <p>a <strong>simple root</strong> if</p>
        \[f(x_0) = 0 \quad \text{and} \quad \left.\frac{\mathrm{d}f}{\mathrm{d}x}\right\rvert_{x_0} \neq 0.\]</li>
    <li>
        <p>a <strong>double root</strong> if</p>
        \[f(x_0) = 0, \quad \left.\frac{\mathrm{d}f}{\mathrm{d}x}\right\rvert_{x_0} = 0 \quad \text{and} \quad \left.\frac{\mathrm{d}^2f}{\mathrm{d}x^2}\right\rvert_{x_0} = 0,\]
        <p>and so on.</p>
    </li>
</ul>
<p>In this Section we shall concentrate on the location of simple roots of a given function \(f(x)\).</p>]]></text>
<file name="12.3 Types of roots.svg" path="/" encoding="base64">[encoded data, 77688 characters]</file>
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