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L_23h
[INTERNAL]::
dae_reduce_index(dict:SX dae, dict opts) -> (dict:SX , dict OUTPUT) dae_reduce_index(dict:MX dae, dict opts) -> (dict:MX , dict OUTPUT)Reduce index.
Index reduction leads to a new set of variables and equations.
In the process, a set of constraints (algebraic equations or derivatives) a.k.a invariants is constructed that are invariant to the problem: whenever an initial point satisfies these constraints, the boundary-value-problem outcome will keep satisfying those constraints automatically, even though they are not part of the reduced DAE.
For any practical numerical integration method, there will be numerical drift away from satisfaction of those constraints. In other words, you will see the value of invariants slowly moving away from original zero.
A classic mitigation technique is Baumgarte stabilization: you add these invariants to the reduced DAE as a correction term that acts in a way to make small (numerical) perturbations to the invariants decay to the origin as a dampened linear system.
in which a certain set of constraints (algebraic equations or derivatives) has been dropped in favour of
dae: Expression dictionary describing the DAE
Each value must be a dense column vector.
keys: x_impl: symbol for implicit differential states
dx_impl: symbol for implicit differential state derivatives
z: symbol for algebraic variables
alg: expression for algebraic equations
t: symbol for time
p: symbol for parameters
opts: Option dictionary
'baumgarte_pole': double Poles (inverse time constants) of the Baumgarte invariant correction term. Must be <0 to dampen out perturbations 0 (default) amounts to no correction. Corresponds to -gamma of equation (1.5) in Ascher, Uri M., Hongsheng Chin, and Sebastian Reich. "Stabilization of DAEs and invariant manifolds." Numerische Mathematik 67.2 (1994): 131-149.
stats: Statistics
Expression dictionary describing the reduced DAE
In addition the fields allowed in the input DAE, the following keys occur:
x: symbol for explicit differential states
ode: expression for right-hand-side of explicit differential states
I: expression for invariants
.......
::
dae_reduce_index(dict:SX dae, dict opts)[INTERNAL] Reduce index.
Index reduction leads to a new set of variables and equations.
In the process, a set of constraints (algebraic equations or derivatives) a.k.a invariants is constructed that are invariant to the problem: whenever an initial point satisfies these constraints, the boundary-value-problem outcome will keep satisfying those constraints automatically, even though they are not part of the reduced DAE.
For any practical numerical integration method, there will be numerical drift away from satisfaction of those constraints. In other words, you will see the value of invariants slowly moving away from original zero.
A classic mitigation technique is Baumgarte stabilization: you add these invariants to the reduced DAE as a correction term that acts in a way to make small (numerical) perturbations to the invariants decay to the origin as a dampened linear system.
in which a certain set of constraints (algebraic equations or derivatives) has been dropped in favour of
dae: Expression dictionary describing the DAE
Each value must be a dense column vector.
keys: x_impl: symbol for implicit differential states
dx_impl: symbol for implicit differential state derivatives
z: symbol for algebraic variables
alg: expression for algebraic equations
t: symbol for time
p: symbol for parameters
opts: Option dictionary
'baumgarte_pole': double Poles (inverse time constants) of the Baumgarte invariant correction term. Must be <0 to dampen out perturbations 0 (default) amounts to no correction. Corresponds to -gamma of equation (1.5) in Ascher, Uri M., Hongsheng Chin, and Sebastian Reich. "Stabilization of DAEs and invariant manifolds." Numerische Mathematik 67.2 (1994): 131-149.
stats: Statistics
Expression dictionary describing the reduced DAE
In addition the fields allowed in the input DAE, the following keys occur:
x: symbol for explicit differential states
ode: expression for right-hand-side of explicit differential states
I: expression for invariants
.............
.......
::
dae_reduce_index(dict:MX dae, dict opts)[INTERNAL] Reduce index.
Index reduction leads to a new set of variables and equations.
In the process, a set of constraints (algebraic equations or derivatives) a.k.a invariants is constructed that are invariant to the problem: whenever an initial point satisfies these constraints, the boundary-value-problem outcome will keep satisfying those constraints automatically, even though they are not part of the reduced DAE.
For any practical numerical integration method, there will be numerical drift away from satisfaction of those constraints. In other words, you will see the value of invariants slowly moving away from original zero.
A classic mitigation technique is Baumgarte stabilization: you add these invariants to the reduced DAE as a correction term that acts in a way to make small (numerical) perturbations to the invariants decay to the origin as a dampened linear system.
in which a certain set of constraints (algebraic equations or derivatives) has been dropped in favour of
dae: Expression dictionary describing the DAE
Each value must be a dense column vector.
keys: x_impl: symbol for implicit differential states
dx_impl: symbol for implicit differential state derivatives
z: symbol for algebraic variables
alg: expression for algebraic equations
t: symbol for time
p: symbol for parameters
opts: Option dictionary
'baumgarte_pole': double Poles (inverse time constants) of the Baumgarte invariant correction term. Must be <0 to dampen out perturbations 0 (default) amounts to no correction. Corresponds to -gamma of equation (1.5) in Ascher, Uri M., Hongsheng Chin, and Sebastian Reich. "Stabilization of DAEs and invariant manifolds." Numerische Mathematik 67.2 (1994): 131-149.
stats: Statistics
Expression dictionary describing the reduced DAE
In addition the fields allowed in the input DAE, the following keys occur:
x: symbol for explicit differential states
ode: expression for right-hand-side of explicit differential states
I: expression for invariants
.............
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