I want to solve the following Matrix Riccati ODE.
My Matrix Riccati ODE
After searching on the Internet, I found that there is something called continuous-time algebraic Riccati equation (CARE) https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.solve_continuous_are.html#scipy.linalg.solve_continuous_are
However, I am not majored in math and not familiar with advanced ode, so I wanna ask that are these two the same thing or not? If not, how to solve the above Matrix Riccati ODE in Python? Thanks in advance. Any comments are welcomed.
You can use the modified Davison-Maki method to solve your Riccati equation.
See https://arxiv.org/pdf/1910.13362.pdf (Algorithm 2). The method is easy to implement.
Related
I was doing a problem and came across two differential equation. When I try to solve the equation using sympy , I was only able to solve one of them. Can someone tell me if it's possible.
This is the code,
Code here I tried to solve
I am trying to figure out the differences between Optimize.fsolve and SymPys solve function? **Is there a difference in computation or are the constraints as to which method to use?**Is Scipy fsolve more accurate than Numpy solve function?
The obvious difference I found is that Scipy.Optimize.Fsolve can solve non linear equations given a guess value, whereas SymPys solve function can be used to solve equations and expressions that contion symbolic math variables.
Any reference literature or website which you can share will be much appreciated.
Link to Complicated Equation
I'm trying to find Ts for a given time. I have all necessary variables in the above equation, except for Ts. I realized that I can integrate with respect to t, which makes things a little easier. From there, how do I go about solving for Ts?
This is not a root-finding problem. This is a differential equation, which you can solve using e.g. solve_ivp from scipy.integrate. You will need the initial value for T(t=0).
I'm trying to set up a fast numerical solver in Python for a differential problem of the form:
where r is some constant.
I want to integrate A over some time period, t of interest. However, this is complicated by the fact that the dA/dt equation includes another variable B, which itself is described by an ODE dB/dt. B is actually a vector, but I've simplified the expression to try and highlight my problems more clearly.
I currently have a solution using a manual Euler method: ie compute dB/dt (then use B = B_previous + dB/dt * dt) and manually step along using a fixed time step size dt. However, this is slow and unreliable. I imagine it would be far better to use the built-in ODE solvers in Numpy, but I'm not sure this is possible given the coupled nature of the problem I'm trying to solve?
Is this possible using Numpy odeint or solve_ivp please? And if so, can anyone suggest any pointers please! Thanks.
What you have is a coupled differential equation which are standard to solve using Runge kutta, Eulers, and many other methods. You can use this example to guide you in writting your python code:
https://scipy-cookbook.readthedocs.io/items/CoupledSpringMassSystem.html
Keep in mind that that not all equations can be solved with ODEINT. If your ODE is a "stiff" ODE then you will have to choose your algorithm precisely. The definition of a stiff ODE is not completely defined but usually they arise if you have large or non-integral powers of your dependent variable in your ODE.
The first step in solving a coupled ODE though is to use standard methods. If they don't work then look into something else.
I'm a Python beginner. I'm trying to switch some programs that I have in matlab.
I need solve a stiff ode equation, whose inputs are all matrices. In matlab I use
[ttT,uT] = ode23s('SST',t,fT);
For most things you do in Matlab, you can do them with the NumPy module in Python. It can be found here.
You might also find the related module SciPy useful as well.
PyDSTool might also be of relevance to you. It's a wrapper around the Radau solver.
Then you might like to try matplotlib for plotting. It works quite like Matlab's plotting thing.
The following links might help, too:
http://www.ews.uiuc.edu/~mrgates2/ode/
http://wiki.python.org/moin/NumericAndScientific?action=show&redirect=SciPy
Integrate stiff ODEs with Python
If you show me the differential equations I can help you a little more, but in general, a good way to solve a stiff ODE system is through the next sentence:
solution = scipy.integrate.solve_ivp(function, [t_0, t_f], y0, method='BDF', first_step =0.0001, dense_output=True)
where your function has to be defined previously in this way: function(t,variable,parameters)
t_0 = initial value for time
t_f = final value for time
y0 = value of your variables at t_0
For stiff ODE system, I suggest use the method 'BDF' (is typically used in the solve of microkinetic systems in reactors, where the importance of the time could change a lot)
for more infomation about the options of the code: https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html