MATLAB: What interpolation algorithm does the ODE45 function in MATLAB use to evaluate the outputs at the time steps specified in the “tspan” input

dormanddormand-princeinterpolateMATLABodeprincetimespantimesteps

The syntax of the ODE45 function is as follows:

[T,Y] = solver(odefun,tspan,y0)

If I specify the "tspan" input with more than two elements, the solver uses interpolation to evaluate and return the output at each intermediate time steps. I would like to know what interpolation algorithm does the ODE45 function use in this process.

Best Answer

The ODE45 function uses the Dormand-Prince DOPRI5(4) method described in the following reference:

 Hairer, Norsett, Wanner, "Solving Ordinary Differential Equations", first edition, vol. I, p. 171 - 179

The interpolation uses slightly differenct coefficients than the implementation in the reference. For details of the implementation refer to the NTRP45 function which can be viewed using the following command:

type('private/ntrp45')

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MATLAB: Can I use the ode45 solver for the state update in a unscented Kalman filter

You cannot use the ode45 function directly in the above example code as it will output integrated function values for all the time points instead of a single time-step value as required here. However, you can implement single-step updates of higher order methods such as Runge-Kutta ( https://en.wikipedia.org/wiki/Runge%2525E2%252580%252593Kutta_methods ) instead of the Euler method for better accuracy.

ode45 function itself uses an explicit Runge-Kutta formula, the Dormand-Prince pair (refer to the documentation page https://www.mathworks.com/help/matlab/ref/ode45.html#bu0200e-1)

You can find more information on other ODE integration methods related to this here https://blogs.mathworks.com/cleve/2014/05/26/ordinary-differential-equation-solvers-ode23-and-ode45/

MATLAB: How to use a fixed step size with ODE23 and ODE45 in MATLAB

ODE23 and ODE45 are MATLAB's ordinary differential equation solver functions. ODE23 is based on the integration method, Runge Kutta23, and ODE45 is based on the integration method, Runge Kutta45. 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. These integration methods do not lend themselves to a fixed step size. 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. 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. ODE23/ODE45 are optimized for a variable step, run faster with a variable step size, and clearly the results are more accurate. If you wish to obtain only those values at a certain fixed increment, do the following:

- Use ODE23/ODE45 to solve the differential equation.

- Use INTERP1 to extract only the desired points.

For example:

% the fixed step vector for desired
% output:
t0 = 0:.01:10; 
[t,y] = ode23('filename',0,10);
y0 = interp1(t0,t,y);

Now, t0 and y0 are the outputs at a fixed interval.

Note that, as of MATLAB 5, you can also obtain solutions at specific time points by specifying tspan as a vector of the desired times. The time values must be in order, either increasing or decreasing.

For example:

tspan = 0:.01:10;
[t,y] = ode23('filename',tspan);

Best Answer

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MATLAB: Can I use the ode45 solver for the state update in a unscented Kalman filter

MATLAB: How to use a fixed step size with ODE23 and ODE45 in MATLAB

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