**trapezoidal_fixed**,
a MATLAB code which
solves one or more ordinary differential equations (ODE)
using the (implicit) trapezoidal method, using a fixed-point method
to handle the implicit system.

Unless the right hand side of the ODE is linear in the dependent variable, each trapezoidal step requires the solution of an implicit nonlinear equation. Such equations can be approximately solved using methods such as fixed point iteration, or an implicit equation solver like fsolve().

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

**trapezoidal_fixed** is available in
a MATLAB version and
an Octave version and
a Python version and
an R version.

backward_euler, a MATLAB code which solves one or more ordinary differential equations (ODE) using the backward Euler method.

backward_euler_fixed, a MATLAB code which solves one or more ordinary differential equations (ODE) using the (implicit) backward Euler method, using a fixed point iteration for the implicit equation.

cauchy_method, a MATLAB code which solves one or more ordinary differential equations (ODE) using the Cauchy method.

euler, a MATLAB code which solves one or more ordinary differential equations (ODE) using the forward Euler method.

leapfrog, a MATLAB code which uses the leapfrog method to solve a second order ordinary differential equation (ODE) of the form y''=f(t,y).

midpoint, a MATLAB code which solves one or more ordinary differential equations (ODE) using the (implicit) midpoint method.

midpoint_explicit, a MATLAB code which solves one or more ordinary differential equations (ODE) using the (explicit) midpoint method, also called the modified Euler method.

midpoint_fixed, a MATLAB code which solves one or more ordinary differential equations (ODE) using the (implicit) midpoint method, using a simple fixed-point iteration to solve the implicit equation.

rk12, a MATLAB code which implements Runge-Kutta solvers of orders 1 and 2 for a system of ordinary differential equations (ODE).

rk23, a MATLAB code which implements Runge-Kutta ODE solvers of orders 2 and 3.

rk34, a MATLAB code which implements Runge-Kutta ODE solvers of orders 3 and 4.

rk4, a MATLAB code which applies the fourth order Runge-Kutta (RK) algorithm to estimate the solution of an ordinary differential equation (ODE).

rk45, a MATLAB code which implements Runge-Kutta ODE solvers of orders 4 and 5.

rkf45, a MATLAB code which implements the Runge-Kutta-Fehlberg ODE solver.

theta_method, a MATLAB code which solves one or more ordinary differential equations (ODE) using the theta method.

trapezoidal, a MATLAB code which solves one or more ordinary differential equations (ODE) using the (implicit) trapezoidal method, and MATLAB's fsolve() function to handle the implicit equation.

trapezoidal_explicit, a MATLAB code which solves one or more ordinary differential equations (ODE) using the (explicit) trapezoidal method.

velocity_verlet, a MATLAB code which uses a version of the velocity Verlet method to solve a secord order ordinary differential equation (ODE) of the form y''=f(t,y).

- trapezoidal_fixed.m, a version of the trapezoidal method that solves the implicit trapezoidal step equation using a fixed point iteration.