(octave.info)Differential-Algebraic Equations
24.2 Differential-Algebraic Equations
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The function ‘daspk’ can be used to solve DAEs of the form
0 = f (x-dot, x, t), x(t=0) = x_0, x-dot(t=0) = x-dot_0
where x-dot is the derivative of x. The equation is solved using
Petzold’s DAE solver DASPK.
-- : [X, XDOT, ISTATE, MSG] = daspk (FCN, X_0, XDOT_0, T, T_CRIT)
Solve a set of differential-algebraic equations.
‘daspk’ solves the set of equations
0 = f (x, xdot, t)
with
x(t_0) = x_0, xdot(t_0) = xdot_0
The solution is returned in the matrices X and XDOT, with each row
in the result matrices corresponding to one of the elements in the
vector T. The first element of T should be t_0 and correspond to
the initial state of the system X_0 and its derivative XDOT_0, so
that the first row of the output X is X_0 and the first row of the
output XDOT is XDOT_0.
The first argument, FCN, is a string, inline, or function handle
that names the function f to call to compute the vector of
residuals for the set of equations. It must have the form
RES = f (X, XDOT, T)
in which X, XDOT, and RES are vectors, and T is a scalar.
If FCN is a two-element string array or a two-element cell array of
strings, inline functions, or function handles, the first element
names the function f described above, and the second element names
a function to compute the modified Jacobian
df df
jac = -- + c ------
dx d xdot
The modified Jacobian function must have the form
JAC = j (X, XDOT, T, C)
The second and third arguments to ‘daspk’ specify the initial
condition of the states and their derivatives, and the fourth
argument specifies a vector of output times at which the solution
is desired, including the time corresponding to the initial
condition.
The set of initial states and derivatives are not strictly required
to be consistent. If they are not consistent, you must use the
‘daspk_options’ function to provide additional information so that
‘daspk’ can compute a consistent starting point.
The fifth argument is optional, and may be used to specify a set of
times that the DAE solver should not integrate past. It is useful
for avoiding difficulties with singularities and points where there
is a discontinuity in the derivative.
After a successful computation, the value of ISTATE will be greater
than zero (consistent with the Fortran version of DASPK).
If the computation is not successful, the value of ISTATE will be
less than zero and MSG will contain additional information.
You can use the function ‘daspk_options’ to set optional parameters
for ‘daspk’.
See also: Note: dassl.
-- : daspk_options ()
-- : val = daspk_options (OPT)
-- : daspk_options (OPT, VAL)
Query or set options for the function ‘daspk’.
When called with no arguments, the names of all available options
and their current values are displayed.
Given one argument, return the value of the option OPT.
When called with two arguments, ‘daspk_options’ sets the option OPT
to value VAL.
Options include
"absolute tolerance"
Absolute tolerance. May be either vector or scalar. If a
vector, it must match the dimension of the state vector, and
the relative tolerance must also be a vector of the same
length.
"relative tolerance"
Relative tolerance. May be either vector or scalar. If a
vector, it must match the dimension of the state vector, and
the absolute tolerance must also be a vector of the same
length.
The local error test applied at each integration step is
abs (local error in x(i))
<= rtol(i) * abs (Y(i)) + atol(i)
"compute consistent initial condition"
Denoting the differential variables in the state vector by
‘Y_d’ and the algebraic variables by ‘Y_a’, ‘ddaspk’ can solve
one of two initialization problems:
1. Given Y_d, calculate Y_a and Y’_d
2. Given Y’, calculate Y.
In either case, initial values for the given components are
input, and initial guesses for the unknown components must
also be provided as input. Set this option to 1 to solve the
first problem, or 2 to solve the second (the default is 0, so
you must provide a set of initial conditions that are
consistent).
If this option is set to a nonzero value, you must also set
the "algebraic variables" option to declare which variables in
the problem are algebraic.
"use initial condition heuristics"
Set to a nonzero value to use the initial condition heuristics
options described below.
"initial condition heuristics"
A vector of the following parameters that can be used to
control the initial condition calculation.
‘MXNIT’
Maximum number of Newton iterations (default is 5).
‘MXNJ’
Maximum number of Jacobian evaluations (default is 6).
‘MXNH’
Maximum number of values of the artificial stepsize
parameter to be tried if the "compute consistent initial
condition" option has been set to 1 (default is 5).
Note that the maximum total number of Newton iterations
allowed is ‘MXNIT*MXNJ*MXNH’ if the "compute consistent
initial condition" option has been set to 1 and
‘MXNIT*MXNJ’ if it is set to 2.
‘LSOFF’
Set to a nonzero value to disable the linesearch
algorithm (default is 0).
‘STPTOL’
Minimum scaled step in linesearch algorithm (default is
eps^(2/3)).
‘EPINIT’
Swing factor in the Newton iteration convergence test.
The test is applied to the residual vector, premultiplied
by the approximate Jacobian. For convergence, the
weighted RMS norm of this vector (scaled by the error
weights) must be less than ‘EPINIT*EPCON’, where ‘EPCON’
= 0.33 is the analogous test constant used in the time
steps. The default is ‘EPINIT’ = 0.01.
"print initial condition info"
Set this option to a nonzero value to display detailed
information about the initial condition calculation (default
is 0).
"exclude algebraic variables from error test"
Set to a nonzero value to exclude algebraic variables from the
error test. You must also set the "algebraic variables"
option to declare which variables in the problem are algebraic
(default is 0).
"algebraic variables"
A vector of the same length as the state vector. A nonzero
element indicates that the corresponding element of the state
vector is an algebraic variable (i.e., its derivative does not
appear explicitly in the equation set).
This option is required by the "compute consistent initial
condition" and "exclude algebraic variables from error test"
options.
"enforce inequality constraints"
Set to one of the following values to enforce the inequality
constraints specified by the "inequality constraint types"
option (default is 0).
1. To have constraint checking only in the initial condition
calculation.
2. To enforce constraint checking during the integration.
3. To enforce both options 1 and 2.
"inequality constraint types"
A vector of the same length as the state specifying the type
of inequality constraint. Each element of the vector
corresponds to an element of the state and should be assigned
one of the following codes
-2
Less than zero.
-1
Less than or equal to zero.
0
Not constrained.
1
Greater than or equal to zero.
2
Greater than zero.
This option only has an effect if the "enforce inequality
constraints" option is nonzero.
"initial step size"
Differential-algebraic problems may occasionally suffer from
severe scaling difficulties on the first step. If you know a
great deal about the scaling of your problem, you can help to
alleviate this problem by specifying an initial stepsize
(default is computed automatically).
"maximum order"
Restrict the maximum order of the solution method. This
option must be between 1 and 5, inclusive (default is 5).
"maximum step size"
Setting the maximum stepsize will avoid passing over very
large regions (default is not specified).
Octave also includes DASSL, an earlier version of DASPK, and DASRT,
which can be used to solve DAEs with constraints (stopping conditions).
-- : [X, XDOT, ISTATE, MSG] = dassl (FCN, X_0, XDOT_0, T, T_CRIT)
Solve a set of differential-algebraic equations.
‘dassl’ solves the set of equations
0 = f (x, xdot, t)
with
x(t_0) = x_0, xdot(t_0) = xdot_0
The solution is returned in the matrices X and XDOT, with each row
in the result matrices corresponding to one of the elements in the
vector T. The first element of T should be t_0 and correspond to
the initial state of the system X_0 and its derivative XDOT_0, so
that the first row of the output X is X_0 and the first row of the
output XDOT is XDOT_0.
The first argument, FCN, is a string, inline, or function handle
that names the function f to call to compute the vector of
residuals for the set of equations. It must have the form
RES = f (X, XDOT, T)
in which X, XDOT, and RES are vectors, and T is a scalar.
If FCN is a two-element string array or a two-element cell array of
strings, inline functions, or function handles, the first element
names the function f described above, and the second element names
a function to compute the modified Jacobian
df df
jac = -- + c ------
dx d xdot
The modified Jacobian function must have the form
JAC = j (X, XDOT, T, C)
The second and third arguments to ‘dassl’ specify the initial
condition of the states and their derivatives, and the fourth
argument specifies a vector of output times at which the solution
is desired, including the time corresponding to the initial
condition.
The set of initial states and derivatives are not strictly required
to be consistent. In practice, however, DASSL is not very good at
determining a consistent set for you, so it is best if you ensure
that the initial values result in the function evaluating to zero.
The fifth argument is optional, and may be used to specify a set of
times that the DAE solver should not integrate past. It is useful
for avoiding difficulties with singularities and points where there
is a discontinuity in the derivative.
After a successful computation, the value of ISTATE will be greater
than zero (consistent with the Fortran version of DASSL).
If the computation is not successful, the value of ISTATE will be
less than zero and MSG will contain additional information.
You can use the function ‘dassl_options’ to set optional parameters
for ‘dassl’.
See also: Note: daspk, Note: dasrt, Note:
lsode.
-- : dassl_options ()
-- : val = dassl_options (OPT)
-- : dassl_options (OPT, VAL)
Query or set options for the function ‘dassl’.
When called with no arguments, the names of all available options
and their current values are displayed.
Given one argument, return the value of the option OPT.
When called with two arguments, ‘dassl_options’ sets the option OPT
to value VAL.
Options include
"absolute tolerance"
Absolute tolerance. May be either vector or scalar. If a
vector, it must match the dimension of the state vector, and
the relative tolerance must also be a vector of the same
length.
"relative tolerance"
Relative tolerance. May be either vector or scalar. If a
vector, it must match the dimension of the state vector, and
the absolute tolerance must also be a vector of the same
length.
The local error test applied at each integration step is
abs (local error in x(i))
<= rtol(i) * abs (Y(i)) + atol(i)
"compute consistent initial condition"
If nonzero, ‘dassl’ will attempt to compute a consistent set
of initial conditions. This is generally not reliable, so it
is best to provide a consistent set and leave this option set
to zero.
"enforce nonnegativity constraints"
If you know that the solutions to your equations will always
be non-negative, it may help to set this parameter to a
nonzero value. However, it is probably best to try leaving
this option set to zero first, and only setting it to a
nonzero value if that doesn’t work very well.
"initial step size"
Differential-algebraic problems may occasionally suffer from
severe scaling difficulties on the first step. If you know a
great deal about the scaling of your problem, you can help to
alleviate this problem by specifying an initial stepsize.
"maximum order"
Restrict the maximum order of the solution method. This
option must be between 1 and 5, inclusive.
"maximum step size"
Setting the maximum stepsize will avoid passing over very
large regions (default is not specified).
"step limit"
Maximum number of integration steps to attempt on a single
call to the underlying Fortran code.
-- : [X, XDOT, T_OUT, ISTAT, MSG] = dasrt (FCN, G, X_0, XDOT_0, T)
-- : ... = dasrt (FCN, G, X_0, XDOT_0, T, T_CRIT)
-- : ... = dasrt (FCN, X_0, XDOT_0, T)
-- : ... = dasrt (FCN, X_0, XDOT_0, T, T_CRIT)
Solve a set of differential-algebraic equations.
‘dasrt’ solves the set of equations
0 = f (x, xdot, t)
with
x(t_0) = x_0, xdot(t_0) = xdot_0
with functional stopping criteria (root solving).
The solution is returned in the matrices X and XDOT, with each row
in the result matrices corresponding to one of the elements in the
vector T_OUT. The first element of T should be t_0 and correspond
to the initial state of the system X_0 and its derivative XDOT_0,
so that the first row of the output X is X_0 and the first row of
the output XDOT is XDOT_0.
The vector T provides an upper limit on the length of the
integration. If the stopping condition is met, the vector T_OUT
will be shorter than T, and the final element of T_OUT will be the
point at which the stopping condition was met, and may not
correspond to any element of the vector T.
The first argument, FCN, is a string, inline, or function handle
that names the function f to call to compute the vector of
residuals for the set of equations. It must have the form
RES = f (X, XDOT, T)
in which X, XDOT, and RES are vectors, and T is a scalar.
If FCN is a two-element string array or a two-element cell array of
strings, inline functions, or function handles, the first element
names the function f described above, and the second element names
a function to compute the modified Jacobian
df df
jac = -- + c ------
dx d xdot
The modified Jacobian function must have the form
JAC = j (X, XDOT, T, C)
The optional second argument names a function that defines the
constraint functions whose roots are desired during the
integration. This function must have the form
G_OUT = g (X, T)
and return a vector of the constraint function values. If the
value of any of the constraint functions changes sign, DASRT will
attempt to stop the integration at the point of the sign change.
If the name of the constraint function is omitted, ‘dasrt’ solves
the same problem as ‘daspk’ or ‘dassl’.
Note that because of numerical errors in the constraint functions
due to round-off and integration error, DASRT may return false
roots, or return the same root at two or more nearly equal values
of T. If such false roots are suspected, the user should consider
smaller error tolerances or higher precision in the evaluation of
the constraint functions.
If a root of some constraint function defines the end of the
problem, the input to DASRT should nevertheless allow integration
to a point slightly past that root, so that DASRT can locate the
root by interpolation.
The third and fourth arguments to ‘dasrt’ specify the initial
condition of the states and their derivatives, and the fourth
argument specifies a vector of output times at which the solution
is desired, including the time corresponding to the initial
condition.
The set of initial states and derivatives are not strictly required
to be consistent. In practice, however, DASSL is not very good at
determining a consistent set for you, so it is best if you ensure
that the initial values result in the function evaluating to zero.
The sixth argument is optional, and may be used to specify a set of
times that the DAE solver should not integrate past. It is useful
for avoiding difficulties with singularities and points where there
is a discontinuity in the derivative.
After a successful computation, the value of ISTATE will be greater
than zero (consistent with the Fortran version of DASSL).
If the computation is not successful, the value of ISTATE will be
less than zero and MSG will contain additional information.
You can use the function ‘dasrt_options’ to set optional parameters
for ‘dasrt’.
See also: Note: dasrt_options, *note daspk:
XREFdaspk, Note: dasrt, Note: lsode.
-- : dasrt_options ()
-- : val = dasrt_options (OPT)
-- : dasrt_options (OPT, VAL)
Query or set options for the function ‘dasrt’.
When called with no arguments, the names of all available options
and their current values are displayed.
Given one argument, return the value of the option OPT.
When called with two arguments, ‘dasrt_options’ sets the option OPT
to value VAL.
Options include
"absolute tolerance"
Absolute tolerance. May be either vector or scalar. If a
vector, it must match the dimension of the state vector, and
the relative tolerance must also be a vector of the same
length.
"relative tolerance"
Relative tolerance. May be either vector or scalar. If a
vector, it must match the dimension of the state vector, and
the absolute tolerance must also be a vector of the same
length.
The local error test applied at each integration step is
abs (local error in x(i)) <= ...
rtol(i) * abs (Y(i)) + atol(i)
"initial step size"
Differential-algebraic problems may occasionally suffer from
severe scaling difficulties on the first step. If you know a
great deal about the scaling of your problem, you can help to
alleviate this problem by specifying an initial stepsize.
"maximum order"
Restrict the maximum order of the solution method. This
option must be between 1 and 5, inclusive.
"maximum step size"
Setting the maximum stepsize will avoid passing over very
large regions.
"step limit"
Maximum number of integration steps to attempt on a single
call to the underlying Fortran code.
See K. E. Brenan, et al., ‘Numerical Solution of Initial-Value
Problems in Differential-Algebraic Equations’, North-Holland (1989) for
more information about the implementation of DASSL.
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