Cash–Karp method

<p>In <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a>, the Cash–Karp method is a method for solving <a href="/facts/Ordinary_differential_equations/ygKGQ8kD">ordinary differential equations</a> (ODEs). It was proposed by Professor Jeff R. Cash  from <a href="/facts/Imperial_College_London/SoXBRyax">Imperial College London</a> and Alan H. Karp from <a href="/facts/IBM/QszA7nwd">IBM</a> Scientific Center. The method is a member of the <a href="/facts/Runge%25E2%2580%2593Kutta/YRjbyHpY">Runge–Kutta</a> family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth order) solution. This error estimate is very convenient for <a href="/facts/Adaptive_stepsize/XcUzXp8m">adaptive stepsize</a> integration algorithms. Other similar integration methods are <a href="/facts/Fehlberg/6m1kDqb7">Fehlberg</a> (RKF) and <a href="/facts/Dormand%25E2%2580%2593Prince_method/B39udhzj">Dormand–Prince</a> (RKDP).
</p><p>The <a href="/facts/Butcher_tableau/YRjbyHpY">Butcher tableau</a> is:
</p>
<table><tbody><tr><td></td><td>0</td></tr><tr><td></td><td>1/5</td><td>1/5</td></tr><tr><td></td><td>3/10</td><td>3/40</td><td>9/40</td></tr><tr><td></td><td>3/5</td><td>3/10</td><td>−9/10</td><td>6/5</td></tr><tr><td></td><td>1</td><td>−11/54</td><td>5/2</td><td>−70/27</td><td>35/27</td></tr><tr><td></td><td>7/8</td><td>1631/55296</td><td>175/512</td><td>575/13824</td><td>44275/110592</td><td>253/4096</td><td></td></tr><tr><td></td><td></td><td>37/378</td><td>0</td><td>250/621</td><td>125/594</td><td>0</td><td>512/1771</td></tr><tr><td></td><td></td><td>2825/27648</td><td>0</td><td>18575/48384</td><td>13525/55296</td><td>277/14336</td><td>1/4</td></tr></tbody></table>
<p>The first row of <i>b</i> coefficients gives the fifth-order accurate solution, and the second row gives the fourth-order solution.
</p>

Cash–Karp method open-in-new

Cash–Karp method