Does the function "ode45" use matrix decomposition?

Hi all,

I wish to know the algorithms of ODE45 when it need to calculate multi-variable Runge Kutta.

Say, I want to solve ODE problem with muti variables such that y(x,t)'=Ay(x,t)

where y is the vector component of varibles, x is the vector component of corrdinates, A is the system matrix and t is time.

Does ODE45 solve this problem only by the explicit method without matrix decomposition?

Is there limitations for the usage of ODE45 if matrix A is singular or ill-conditioned?

Thanks for your help.

[Marco Feder]

Since ode45 is an explicit method, in the MatLab routine no linear system is solved, and hence no matrix decomposition will be done. Other implicit integrators for sure will use matrix decomposition (like LU) since Newton's-like method are used to find the solution at each time step.

[Sangwoo Kim]

Thank you, Marco.

Answer 1

[Christine Tobler]

ODE45 does not use a decomposition or solve a linear system with A. However, ODE15s and ODE23s, which are specialized for "stiff" problems, do compute the LU decomposition and solve a linear system with the matrix A. These methods will only work well if the matrix A is not singular or ill-conditioned, but they can work much better than ODE45 and other "non-stiff" problem solvers for these cases.

See the Choose an ODE Solver doc page for more explanations about the different ODE solvers.

[Sangwoo Kim]

Thanks for your kind explanation. Your answer was what I preassumed, and I needed some documented clarification. Thanks again.