- </style></head><body><div class="content"><h1>Numerical Differentiation Toolbox Documentation</h1><!--introduction--><p><img vspace="5" hspace="5" src="numerical_differentiation_toolbox.png" alt=""> </p><p>Copyright © 2021 Tamas Kis</p><!--/introduction--><h2>Contents</h2><div><ul><li><a href="#1">Functions</a></li><li><a href="#2">Written Documentation</a></li><li><a href="#3">Methology</a></li><li><a href="#4">Warnings/Disclaimers</a></li><li><a href="#5">Installation</a></li></ul></div><h2 id="1">Functions</h2><div><ul><li><a href="iderivative_doc.html"><b><tt>iderivative</tt></b></a> Derivative of a univariate, scalar or vector-valued function.</li><li><a href="ipartial_doc.html"><b><tt>ipartial</tt></b></a> Partial derivative of a multivariate, scalar or vector-valued function.</li><li><a href="igradient_doc.html"><b><tt>igradient</tt></b></a> Gradient of a multivariate, scalar-valued function.</li><li><a href="idirectional_doc.html"><b><tt>idirectional</tt></b></a> Directional derivative of a multivariate, scalar-valued function.</li><li><a href="ijacobian_doc.html"><b><tt>ijacobian</tt></b></a> Jacobian matrix of a multivariate, vector-valued function.</li><li><a href="ihessian_doc.html"><b><tt>ihessian</tt></b></a> Hessian matrix of a multivariate, scalar-valued function.</li></ul></div><p><b>NOTE:</b> Two other functions, <tt>iatan2</tt> and <tt>iabs</tt>, are also included with this toolbox. These functions are simply "complexified" versions of <tt>atan2</tt> and <tt>abs</tt> and are thus left undocumented.</p><h2 id="2">Written Documentation</h2><p>Click <a href="https://tamaskis.github.io/documentation/Numerical%20Differentiation%20Using%20the%20Complex-Step%20Approximation.pdf">here</a>.</p><h2 id="3">Methology</h2><p>All of these functions are implemented using the complex-step approximation of the derivative (this is why each function is prefixed with an "i"; it is meant to indicate the imaginary unit). A complete <a href="https://tamaskis.github.io/documentation/Numerical%20Differentiation%20Using%20the%20Complex-Step%20Approximation.pdf">written documentation</a> is available that details the implementation in depth.</p><h2 id="4">Warnings/Disclaimers</h2><div><ul><li>There are some special cases of functions where the complex-step approximation will not work directly; for example, trying to differentiate functions using MATLAB's <tt>atan2</tt> or <tt>abs</tt> would result in errors. We can "complexify" these functions to make them suitable for use with the complex-step approximation; in this toolbox, we have included <tt>iatan2</tt> and <tt>iabs</tt> in src/complexified. However, more complexified functions can be found (programmed in Fortran) at <a href="https://mdolab.engin.umich.edu/misc/files/complexify.f90">https://mdolab.engin.umich.edu/misc/files/complexify.f90</a>.</li><li>Most differentiable functions can be used with the various functions of this toolbox. Functions that result in errors (likely due to complexification issues) are summarized in Section 1.3.3 of the <a href="https://tamaskis.github.io/documentation/Numerical%20Differentiation%20Using%20the%20Complex-Step%20Approximation.pdf">written documentation</a>.</li><li>The functions in this toolbox cannot perform higher-order derivatives (this limitation is discussed in Section 1.3.1 of the <a href="https://tamaskis.github.io/documentation/Numerical%20Differentiation%20Using%20the%20Complex-Step%20Approximation.pdf">written documentation</a>).</li><li>All of the functions are (generally) accurate to double precision <b>with the the exception of</b> <a href="ihessian_doc.html"><tt>ihessian</tt></a>. An alternative that will perform better is the <tt>hessian</tt> function provided by the <a href="https://www.mathworks.com/matlabcentral/fileexchange/13490-adaptive-robust-numerical-differentiation?s_tid=srchtitle">Adaptive Robust Numerical Differentiation</a> toolbox.</li><li>No error estimates are provided.</li></ul></div><h2 id="5">Installation</h2><p>The toolbox can be downloaded from < File Exchange> or <a href="https://github.com/tamaskis/Numerical_Differentiation_Toolbox-MATLAB">GitHub</a>. The downloaded zip folder contains the following:</p><div><ul><li><b>docs</b> → Contains the HTML documentation. To open a copy of the HTML documentation locally on your computer (without need of an internet connection), open docs/index.html.</li><li><b>docsgeneration</b> → Contains all scripts/functions/images needed to produce the HTML documentation.</li><li><b>LICENSE</b> → Software license.</li><li><b>Numerical Differentiation Toolbox.mltbx</b> → Toolbox installer.</li><li><b>Numerical Differentiation Using the Complex-Step Approximation.pdf</b> → Written documentation.</li><li><b>README.md</b> → Markdown documentation for GitHub repository.</li><li><b>src</b> → Contains the actual functions/software.</li><li><b>test</b> → Contains scripts that perform unit tests on various functions.</li></ul></div><p><b>To install as a toolbox</b>, simply open "Numerical Differentiation Toolbox.mltbx" and MATLAB will automatically perform the installation. Afterwards, you will always have access to all functions of this toolbox. Alternatively, you can use individual functions by copying them from the "src" folder to whatever folder your other code is in.</p><p class="footer"><br><a href="https://www.mathworks.com/products/matlab/">Published with MATLAB® R2021a</a><br></p></div><!--
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