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<!DOCTYPE html
PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html><head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
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This HTML was auto-generated from MATLAB code.
To make changes, update the MATLAB code and republish this document.
--><title>AB4</title><meta name="generator" content="MATLAB 9.12"><link rel="schema.DC" href="http://purl.org/dc/elements/1.1/"><meta name="DC.date" content="2022-09-18"><meta name="DC.source" content="AB4_doc.m"><style type="text/css">
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</style></head><body><div class="content"><h1><tt>AB4</tt></h1><!--introduction--><p>Propagates the state vector forward one time step using the Adams-Bashforth 4th-order method.</p><p><a href="index.html">Back to IVP Solver Toolbox Contents</a>.</p><!--/introduction--><h2>Contents</h2><div><ul><li><a href="#1">Syntax</a></li><li><a href="#2">Description</a></li><li><a href="#3">Input/Output Parameters</a></li><li><a href="#4">See also</a></li></ul></div><h2 id="1">Syntax</h2><pre class="language-matlab">F = AB4(f,t,F,h)
</pre><h2 id="2">Description</h2><p><tt>F = AB4(f,t,y,h)</tt> updates the <img src="AB4_doc_eq03148538246847756930.png" alt="$\mathbf{F}$" style="width:8px;height:8px;"> matrix, <tt>F</tt>, for the next sample time, given the <img src="AB4_doc_eq03148538246847756930.png" alt="$\mathbf{F}$" style="width:8px;height:8px;"> matrix at the current time <tt>t</tt>, the function <tt>f(t,y)</tt> defining the ODE <img src="AB4_doc_eq11546780198861005830.png" alt="$d\mathbf{y}/dt=\mathbf{f}(t,\mathbf{y})$" style="width:71px;height:11px;">, and the step size <tt>h</tt>.</p><h2 id="3">Input/Output Parameters</h2><p>
<table border=1>
<tr>
<td></td>
<td style="text-align:center"><b>Variable</b></td>
<td style="text-align:center"><b>Symbol</b></td>
<td style="text-align:center"><b>Description</b></td>
<td style="text-align:center"><b>Format</b></td>
</tr>
<tr>
<td rowspan="4" style="text-align:center"><b>Input</b></td>
<td style="text-align:center"><TT>f</TT></td>
<td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{f}(t,\mathbf{y})" title="" /></td>
<td>multivariate, vector-valued function (<img
src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{f}:\mathbb{R}\times\mathbb{R}^{p}\rightarrow\mathbb{R}^{p}"
title="" />) defining the ordinary differential equation <img src="https://latex.codecogs.com/svg.latex?\inline&space;d\mathbf{y}/dt=\mathbf{f}(t,\mathbf{y})" title="" />
<BR> - inputs to <TT>f</TT> are the current time (<TT>t</TT>, 1×1 double) and the current state vector (<TT>y</TT>, p×1 double)
<BR> - output of <TT>f</TT> is the state vector derivative (<TT>dydt</TT>, p×1 double) at the current time/state</td>
<td style="text-align:center">1×1<BR>function_handle</td>
</tr>
<tr>
<td style="text-align:center"><TT>t</TT></td>
<td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;t_{n}" title="" /></td>
<td>current sample time</td>
<td style="text-align:center">1×1<BR>double</td>
</tr>
<tr>
<td style="text-align:center"><TT>F</TT></td>
<td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{F}" title="" /></td>
<td><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{F}" title="" /> matrix (see Section 3.5.2 of the <a href="https://tamaskis.github.io/documentation/Solving_Initial_Value_Problems_for_ODEs.pdf">technical documentation</a>) for the current sample time</td>
<td style="text-align:center">p×5<BR>double</td>
</tr>
<tr>
<td style="text-align:center"><TT>h</TT></td>
<td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;h" title="h" /></td>
<td>step size</td>
<td style="text-align:center">1×1<BR>double</td>
</tr>
<tr>
<td rowspan="1" style="text-align:center"><b>Output</b></td>
<td style="text-align:center"><TT>F</TT></td>
<td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{F}" title="" /></td>
<td><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{F}" title="" /> matrix (see Section 3.5.2 of the <a href="https://tamaskis.github.io/documentation/Solving_Initial_Value_Problems_for_ODEs.pdf">technical documentation</a>) for the next sample time</td>
<td style="text-align:center">p×5<BR>double</td>
</tr>
</table>
</p><h2 id="4">See also</h2><p><a href="AB2_doc.html"><tt>AB2</tt></a> | <a href="AB3_doc.html"><tt>AB3</tt></a> | <a href="AB5_doc.html"><tt>AB5</tt></a> | <a href="AB6_doc.html"><tt>AB6</tt></a> | <a href="AB7_doc.html"><tt>AB7</tt></a> | <a href="AB8_doc.html"><tt>AB8</tt></a></p><p class="footer"><br><a href="https://www.mathworks.com/products/matlab/">Published with MATLAB® R2022a</a><br></p></div><!--
##### SOURCE BEGIN #####
%% |AB4|
% Propagates the state vector forward one time step using the
% Adams-Bashforth 4th-order method.
%
% <index.html Back to IVP Solver Toolbox Contents>.
%% Syntax
% F = AB4(f,t,F,h)
%% Description
% |F = AB4(f,t,y,h)| updates the $\mathbf{F}$ matrix, |F|, for the next
% sample time, given the $\mathbf{F}$ matrix at the current time |t|, the
% function |f(t,y)| defining the ODE
% $d\mathbf{y}/dt=\mathbf{f}(t,\mathbf{y})$, and the step size |h|.
%% Input/Output Parameters
% <html>
% <table border=1>
% <tr>
% <td></td>
% <td style="text-align:center"><b>Variable</b></td>
% <td style="text-align:center"><b>Symbol</b></td>
% <td style="text-align:center"><b>Description</b></td>
% <td style="text-align:center"><b>Format</b></td>
% </tr>
% <tr>
% <td rowspan="4" style="text-align:center"><b>Input</b></td>
% <td style="text-align:center"><TT>f</TT></td>
% <td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{f}(t,\mathbf{y})" title="" /></td>
% <td>multivariate, vector-valued function (<img
% src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{f}:\mathbb{R}\times\mathbb{R}^{p}\rightarrow\mathbb{R}^{p}"
% title="" />) defining the ordinary differential equation <img src="https://latex.codecogs.com/svg.latex?\inline&space;d\mathbf{y}/dt=\mathbf{f}(t,\mathbf{y})" title="" />
% <BR> - inputs to <TT>f</TT> are the current time (<TT>t</TT>, 1×1 double) and the current state vector (<TT>y</TT>, p×1 double)
% <BR> - output of <TT>f</TT> is the state vector derivative (<TT>dydt</TT>, p×1 double) at the current time/state</td>
% <td style="text-align:center">1×1<BR>function_handle</td>
% </tr>
% <tr>
% <td style="text-align:center"><TT>t</TT></td>
% <td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;t_{n}" title="" /></td>
% <td>current sample time</td>
% <td style="text-align:center">1×1<BR>double</td>
% </tr>
% <tr>
% <td style="text-align:center"><TT>F</TT></td>
% <td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{F}" title="" /></td>
% <td><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{F}" title="" /> matrix (see Section 3.5.2 of the <a href="https://tamaskis.github.io/documentation/Solving_Initial_Value_Problems_for_ODEs.pdf">technical documentation</a>) for the current sample time</td>
% <td style="text-align:center">p×5<BR>double</td>
% </tr>
% <tr>
% <td style="text-align:center"><TT>h</TT></td>
% <td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;h" title="h" /></td>
% <td>step size</td>
% <td style="text-align:center">1×1<BR>double</td>
% </tr>
% <tr>
% <td rowspan="1" style="text-align:center"><b>Output</b></td>
% <td style="text-align:center"><TT>F</TT></td>
% <td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{F}" title="" /></td>
% <td><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{F}" title="" /> matrix (see Section 3.5.2 of the <a href="https://tamaskis.github.io/documentation/Solving_Initial_Value_Problems_for_ODEs.pdf">technical documentation</a>) for the next sample time</td>
% <td style="text-align:center">p×5<BR>double</td>
% </tr>
% </table>
% </html>
%% See also
% <AB2_doc.html |AB2|> |
% <AB3_doc.html |AB3|> |
% <AB5_doc.html |AB5|> |
% <AB6_doc.html |AB6|> |
% <AB7_doc.html |AB7|> |
% <AB8_doc.html |AB8|>
##### SOURCE END #####
--></body></html>