GNU Octave 2.1.x 日本語マニュアル/微分方程式のソースを表示

== 23　微分方程式　==
Octaveは微分方程式を解くための組み込み関数を2つ持っています。
両方ともFortranで書かれた信頼性の高いODE(常微分方程式：Ordinary Differential Equation)ソルバーに基づいています。

=== 23.1 Ordinary Differential Equations ===
The function lsode can be used to solve ODEs of the form
dx
dt
= f(x; t)
using Hindmarsh’s ODE solver Lsode.

==== [x, istate, msg] lsode (fcn, x_0, t, t_crit) ====
[Loadable Function]

Solve the set of differential equations

<math>
dx
dt
= f(x; t)
</math>

with

x(t0) = x0

The solution is returned in the matrix x, with each row corresponding to an elementof the vector t. 

The first element of t should be t0 and should correspond to the initialstate of the system x 0, so that the first row of the output is x 0.

The first argument, fcn, is a string that names the function to call to compute the vector of right hand sides for the set of equations. The function must have the form

<math>
xdot = f (x, t)
</math>

in which xdot and x are vectors and t is a scalar.

If fcn is a two-element string array, the first element names the function f described above, and the second element names a function to compute the Jacobian of f.
The Jacobian function must have the form

<math>
jac = j (x, t)
</math>

in which jac is the matrix of partial derivatives

 J = @fi
 @xj
 =
 2
 66664
 @f1
 @x1
 @f1
 @x2
 ¢ ¢ ¢ @f1
 @xN
 @f2
 @x1
 @f2
 @x2
 ¢ ¢ ¢ @f2
 @xN ......
 . . .
 ...
 @f3
 @x1
 @f3
 @x2
 ¢ ¢ ¢ @f3
 @xN
 3
 77775

The second and third arguments specify the intial state of the system, x0, and the initial value of the independent variable t0.

The fourth argument is optional, and may be used to specify a set of times that the ODE 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 2 (consistent with the Fortran version of Lsode).

If the computation is not successful, istate will be something other than 2 and msg will contain additional information.

You can use the function lsode_options to set optional parameters for lsode.

==== lsode_options (opt, val) ====
[Loadable Function]

When called with two arguments, this function allows you set options parameters for
the function lsode. Given one argument, lsode_options returns the value of the
corresponding option. If no arguments are supplied, the names of all the available
options and their current values are displayed.

Options include

"absolute tolerance"

Absolute tolerance. May be either vector or scalar. If a vector, it must
match the dimension of the state vector.

"relative tolerance"

Relative tolerance parameter. Unlike the absolute tolerance, this parameter
may only be a scalar.

The local error test applied at each integration step is abs (local error in x(i)) <= rtol * abs (y(i)) + atol(i)

"integration method"

A string specifing the method of integration to use to solve the ODE
system. Valid values are

"adams"

"non-stiff"

No Jacobian used (even if it is available).

"bdf"

"stiff" Use stiff backward differentiation formula (BDF) method. If
a function to compute the Jacobian is not supplied, lsode
will compute a finite difference approximation of the Jacobian
matrix.

"initial step size"

The step size to be attempted on the first step (default is determined
automatically).

"maximum order"

Restrict the maximum order of the solution method. If using the Adams
method, this option must be between 1 and 12. Otherwise, it 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).

"minimum step size"

The minimum absolute step size allowed (default is 0).

"step limit"

Maximum number of steps allowed (default is 100000).

Here is an example of solving a set of three differential equations using lsode. Given the function

<source lang=matlab>
function xdot = f (x, t)
xdot = zeros (3,1);
xdot(1) = 77.27 * (x(2) - x(1)*x(2) + x(1) \
- 8.375e-06*x(1)^2);
xdot(2) = (x(3) - x(1)*x(2) - x(2)) / 77.27;
xdot(3) = 0.161*(x(1) - x(3));
endfunction
</source>

and the initial condition x0 = [ 4; 1.1; 4 ], the set of equations can be integrated using
the command

<source lang=matlab>
t = linspace (0, 500, 1000);
y = lsode ("f", x0, t);
</source>

If you try this, you will see that the value of the result changes dramatically between t
= 0 and 5, and again around t = 305. A more efficient set of output points might be

<source lang=matlab>
t = [0, logspace (-1, log10(303), 150), \
logspace (log10(304), log10(500), 150)];
</source>

See Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in
Scientific Computing, R. S. Stepleman, editor, (1983) for more information about the inner
workings of lsode.

=== 23.2 Differential-Algebraic Equations ===
The function daspk can be used to solve DAEs of the form

0 = f(x˙ ; x; t); x(t = 0) = x0; x˙ (t = 0) = x˙ 0

using Petzold’s DAE solver Daspk.

==== [x, xdot, istate, msg] = daspk (fcn, x_0, xdot_0, t, t_crit) ====
[Loadable Function]

Solve the set of differential-algebraic equations

0 = f(x; x˙ ; t)

with

x(t0) = x0; x˙ (t0) = x˙ 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 t0 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 that names the function 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, the first element names the function f described
above, and the second element names a function to compute the modified Jacobian

J = @f
@x
+ c
@f
@x˙

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.

==== daspk_options (opt, val) ====
[Loadable Function]

When called with two arguments, this function allows you set options parameters for
the function daspk. Given one argument, daspk_options returns the value of the
corresponding option. If no arguments are supplied, the names of all the available
options and their current values are displayed.
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 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 number of Newton iterations
allowed in all 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 occaisionally 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, t_out, istat, msg] = dasrt (fcn [, g], x_0,xdot_0, t [, t_crit]) ====
[Loadable Function]

Solve the set of differential-algebraic equations

0 = f(x; x˙ ; t)

with

x(t0) = x0; x˙ (t0) = x˙ 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 t0 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 that names the function 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, the first element names the function f described
above, and the second element names a function to compute the modified Jacobian

J = @f
@x
+ c
@f
@x˙

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 roundoff
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.

==== dasrt_options (opt, val) ====
[Loadable Function]

When called with two arguments, this function allows you set options parameters for
the function dasrt. Given one argument, dasrt_options returns the value of the
corresponding option. If no arguments are supplied, the names of all the available
options and their current values are displayed.
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 occaisionally 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.

[[Category:GNU Octave|2.1.x にほんこまにゆある]]
[[カテゴリ:微分方程式]]