========================================= Solving of IVPs of DAEs by Taylormethods ========================================= The integration methods solve initial value problems (IVPs) of DAEs like .. math:: f(x',x,t) = 0, t \in [t_0,T] \Pi (x(t_0)-\alpha) = 0 with a given initial guess :math:`\alpha` and implicitely defined matrix :math:`\Pi`. The matrix :math:`\Pi` underlines that, in opposit to the ODE case, we can not prescribe initial values for all of the components in the DAE case. The user has to provide the function *f* and an initial guess :math:`\alpha` only. The used integration methods are described in [4]: D. Estévez Schwarz and R. Lamour *Projected Explicit and Implicit Taylor Series Methods for DAEs* Preprint 2019-08, Fachbereich Mathematik, Humboldt-Universität zu Berlin. Further, the diagnosis of results obtained with other integration methods is also possible. ------------------------------------------- The integration modules ------------------------------------------- Three integration modules are offered .. function:: Taylor_integration_adaptiveStep(self,t0,T,alpha,max_iterations=20,tol=1e-6,hmin=1e-8,K_start=2,disp=False,RobustIntegration=True,ftol=1e-8,hstart=-1.) with stepsize control (Attention: The stepsize control works (up to now) on the basis of the explicit Taylormethod (method=0)), .. function:: Taylor_integration_timeList(self,ti_list,x0,ftol=1e-14,max_iterations=20) integrating at given timpoints (e.g. to compare the result with one from another integration method) and, .. function:: Taylor_integration_constant_h(self,t0,x0,T,NumSteps,ftol=1e-14,max_iterations=20) to integrate with constant stepsize. ------------------------------------------- The diagnosis modules ------------------------------------------- The diagnosis module .. function:: check_solutionpath(self,x,a,b,time_points=100) computes the index and a condition number along an analytically given function *x* at constant *time_points* of the interval [a,b]. The module .. function:: check_solution_along_path(self, path,ftol=1e-10) computes the index and a condition number in given points (e.g. discrete results from another integration method), which are red from a file *path*. -------------------------------------------- An example main program -------------------------------------------- To use the modules you have to prepare your DAE (see :doc:`How to represent the DAE `) and create an instance of the class TaylorIntegration(TaylorDAE) using the constructor .. function:: TaylorIntegration(example,K,tol=None,info_output='abc',method=x) The values for *'abc'* and *x* are described in the `detailed documentation `_. We give here an example to integrate the pendulum problem. .. literalinclude:: ../../AD_examples/Pendulum/PendulumIntegration.py A detailed main program to use all the different modules can be found at the end of the module DAE_Taylor_integration.