It should be no surprise, then, that ode45 obtains a very accurate answer for the equation we solved before with Euler's method. So, in general, the smaller the number odeNN, the looser the solver's error tolerance is. Similarly, ode23 uses a 2nd-order and 3rd-order Runge-Kutta comparison. ode45 compares the results of a 4th-order Runge-Kutta method and a 5th-order Runge-Kutta method to determine the error. MATLAB's ODE solvers follow a naming convention that reveals information about which methods they use. The error estimate used to determine the step size is typically obtained by comparing the results of two different methods. This is another improvement over the fixed step sizes used above, since a solver that does more work per step is able to compensate by taking steps of varying size. Sophisticated ODE solvers, like the ones in MATLAB, also estimate the error in each step to determine how big the next step size should be. Multi-step algorithms, such as those employed by ode113 and ode15s, use the results of several past steps.

ode45, ode23, ode23s, ode23t, and ode23tb all employ single-step algorithms. Runge-Kutta algorithms are all single-step solvers, since each step only depends on the result of the previous step. Theoretically, this numerical solution technique is possible because of the connection between differential equations and integrals provided by the fundamental theorem of calculus: The final result is that the ODE solver returns a vector of time steps $t_0,t_1.,t_f$ as well as the corresponding solution at each time step $y_0,y_1.,y_f$. At the first such step, the initial conditions provide the necessary information that allows the integration to proceed. Starting with the initial conditions $y_0$, and a period of time over which the answer is to be obtained $(t_0,t_f)$, the solution is obtained iteratively by using the results of previous steps according to the solver's algorithm. Where $M(t,y)$ is referred to as the mass matrix. The ODE solvers in MATLAB all work on initial value problems of the form, To choose between the solvers, it's first necessary to understand why one solver might be better than another for a given problem. ODE23/ODE45 are optimized for a variable step, run faster with a variable step size, and clearly the results are more accurate.(note that ode15i is left out of this discussion because it solves its own class of initial value problems: fully implicit ODEs of the form $f(t,y,y') = 0$) Using a variable step ensures that a large step size is used for low frequencies and a small step size is used for high frequencies.

Using an algorithm that uses a fixed step size is dangerous since you may miss points where your signal's frequency is greater than the solver's frequency. These integration methods do not lend themselves to a fixed step size. The way that ODE23 and ODE45 utilize these methods is by selecting a point, taking the derivative of the function at that point, checking to see if the value is greater than or less than the tolerance, and altering the step size accordingly. ODE23 is based on the integration method, Runge Kutta23, and ODE45 is based on the integration method, Runge Kutta45.

ODE23 and ODE45 are MATLAB's ordinary differential equation solver functions.

Ode45 matlab