Scope of the program¶
For general DAEs the index and a set of consistent initial values have to be determined. The module consistentValuesIndex computes both as described in
[1] D. Estévez Schwarz and R. Lamour A new approach for computing consistent initial values and Taylor coefficients for DAEs using projector based constrained optimization Numerical Algorithms 2017, DOI 10.1007/s11075-017-0379-9
[2] D. Estévez Schwarz and R. Lamour: A new projector based decoupling of linear DAEs for monitoring singularities. Numerical Algorithms, 73 (2), pp 535–565 (2016).
and
[3] D. Estévez Schwarz and R. Lamour Consistent initialization for higher-index DAEs using a projector based minimum-norm specification Preprint 2016-01, Fachbereich Mathematik, Humboldt-Universität zu Berlin.
To compute consistent initial values we solve the minimization problem
![\text{min } ||P(t_0)(x-\alpha)||_2, \text{s.t.
} g^{[k]}(x^{(k)},x^{(k-1)},\ldots,x,t_0) = 0,](_images/math/f46aed7dc32078cc0d5b424a0b8bf0c474c6d2c7.png)
with user given 
(see [1]). In terms of [3] the
minimization problem is equivalent to a classical
initial value problem formulation
![\Pi(x-\alpha) = 0,
g^{[k]}(x^{(k)},x^{(k-1)},\ldots,x,t_0) = 0,](_images/math/e09c1ea50509b1e6cda8dc5896a46cab58bdbb33.png)
where 
describes an appropriate projector and the
rank of coincides with the degree of freedom of the
DAE.
Additional user-given requirements belonging to single components and/or relations between components can be realized using a function u. In this case we solve the minimization problem
![\text{min } ||P(t_0)(x-\alpha)||_2,
\text{s.t. } g^{[k]}(x^{(k)},x^{(k-1)},\ldots,x,t_0)
= 0,
u(x) = 0.](_images/math/4f514d247464ab9f21f02e25a6436a1d226cbd5b.png)
Therefore, in terms of [3] the minimization problem is equivalent to a classical initial value problem formulation
![\Pi_u(x-\alpha) = 0,
g^{[k]}(x^{(k)},x^{(k-1)},\ldots,x,t_0) = 0,
u(x) = 0,](_images/math/826a433c1df1122767b1a1dd5d21bd8f1a84c571.png)
where 
describes an appropriate projector and
the subscript u underlines that is influenced by u.
The determination of the index in the computed consistent values bases on [1], [2].
To start the module the user has to provide , an initial guess for the consistent
initial value, and a class containing the DAE. Finally,
a main program has to be written.
Hints for coding a DAE and the main program are given in
this documentation. Take also account of the parameter
description of the module consistentValuesIndex,
which has to be called in the main programm.
InitDAE has been conceived for academic purposes. It delivers precise information thanks to algorithmic differentiation. For examples of moderate size (we tested DAEs of dimension up to 600) the run time was acceptable.