odeMassMatrix
Description
An odeMassMatrix
object represents the mass matrix for a system
of ordinary differential equations.
ode
objects can represent problems of the form , where is a mass matrix that can be full or sparse. The mass matrix encodes linear
combinations of derivatives on the left side of the equation.
Create an ode
object to
represent the ODE problem, and specify the odeMassMatrix
object as the value
of the MassMatrix
property to incorporate a mass matrix into the
problem.
Creation
Description
creates an
M
= odeMassMatrixodeMassMatrix
object with empty properties.
specifies one or more property values using name-value arguments.M
= odeMassMatrix(Name=Value
)
Properties
MassMatrix
— Mass matrix
matrix | function handle
Mass matrix, specified as a matrix or handle to a function that evaluates the mass matrix.
ode
objects can represent problems of the form , where is a mass matrix that can be full or sparse. The mass matrix encodes
linear combinations of derivatives on the left side of the equation.
When the mass matrix is nonsingular, the equation simplifies to and the ODE has a solution for any initial value. However, it is often more convenient and natural to express the model in terms of the mass matrix directly using , and avoiding the computation of the matrix inverse reduces the storage and execution time needed to solve the problem.
When the mass matrix is singular, the problem is a system of differential algebraic equations (DAEs). A DAE has a solution only when the initial values are consistent; that is, you must specify the initial slope using the
InitialSlope
property of theode
object such that the initial conditions are all consistent, . For more information, see Solve Differential Algebraic Equations (DAEs).
You can specify the value of the MassMatrix
property as:
A constant matrix with calculated values for .
A handle to a function that computes the matrix elements and that accepts two input arguments,
M = Mass(t,y)
. To give the function access to parameter values in theParameters
property, specify a third input argument in the function definition,M = Mass(t,y,p)
.
Example: M = odeMassMatrix(MassMatrix=@Mass)
specifies the
function Mass
that evaluates the mass matrix.
Example: M = odeMassMatrix(MassMatrix=[1 0; -2 1])
specifies a
constant mass matrix.
Data Types: single
| double
| function_handle
Complex Number Support: Yes
Singular
— Singular mass matrix toggle
"maybe"
(default) | "yes"
| "no"
Singular mass matrix toggle, specified as "maybe"
,
"yes"
, or "no"
. The default value of
"maybe"
causes the solver to test whether the problem is a
differential algebraic equation (DAE) by testing whether the mass matrix is singular.
Avoid this check by specifying "yes"
if you know the system is a DAE
with a singular mass matrix, or "no"
if it is not.
Example:
M = odeMassMatrix(MassMatrix=X,Singular="no")
StateDependence
— State dependence of mass matrix
"weak"
(default) | "none"
| "strong"
State dependence of the mass matrix, specified as "weak"
,
"strong"
, or "none"
.
For ODE problems of the form , set the
StateDependence
property to"none"
. This value ensures that the solver calls the mass matrix function with a single argument as inMass(t)
(or two arguments as inMass(t,p)
if you use theParameters
property of theode
object to pass parameters).If the mass matrix depends on
y
, then setStateDependence
to either"weak"
(default) or"strong"
. In both cases, the solver calls the mass matrix function with two arguments as inMass(t,y)
(or three arguments as inMass(t,y,p)
if you use theParameters
property of theode
object to pass parameters). However, a value of"weak"
results in implicit solvers using approximations when solving algebraic equations.
Example:
M = odeMassMatrix(MassMatrix=@Mass,StateDependence="none")
specifies
that the mass matrix function Mass
depends only on
t
.
SparsityPattern
— Mass matrix sparsity pattern
sparse matrix
Mass matrix sparsity pattern, specified as a sparse matrix. Use this property to
specify the sparsity pattern of the matrix . The sparse matrix S
has S(i,j) =
1
if, for any k
, the (i,k)
component
of depends on component j
of y
.
This property is similar to the MvPattern
option of
odeset
.
Note
The SparsityPattern
property is used by the
ode15s
, ode23t
, and
ode23tb
solvers when StateDependence
is
"strong"
.
Example:
M =
odeMassMatrix(MassMatrix=@Mass,StateDependence="strong",SparsityPattern=S)
Data Types: double
Examples
Solve System of DAEs with Mass Matrix
Consider this system of first-order equations.
The left side of the equations contain time derivatives, . However, because the derivative for does not appear in the system, the equations define a system of differential algebraic equations. Rewriting the system in the form shows a constant, singular mass matrix on the left side.
Solve the system of equations using the initial values [1 1 -2]
by creating an ode
object to represent the problem.
Specify the initial values in the
InitialValue
property.Specify the system of equations as an anonymous function in the
ODEFcn
property.Use an
odeMassMatrix
object to specify the constant, singular mass matrix in theMassMatrix
property.
F = ode;
F.InitialValue = [1 1 -2];
F.ODEFcn = @(t,y) [y(1)*y(3)-y(2);
y(1)-1;
y(1)+y(2)+y(3)];
F.MassMatrix = odeMassMatrix(MassMatrix=[1 0 0; 0 1 0; 0 0 0],Singular="yes");
Display the ode object. The SelectedSolver
property shows that the ode15s
solver was automatically chosen for this problem.
F
F = ode with properties: Problem definition ODEFcn: @(t,y)[y(1)*y(3)-y(2);y(1)-1;y(1)+y(2)+y(3)] InitialTime: 0 InitialValue: [1 1 -2] Jacobian: [] MassMatrix: [1x1 odeMassMatrix] EquationType: standard Solver properties AbsoluteTolerance: 1.0000e-06 RelativeTolerance: 1.0000e-03 Solver: auto SelectedSolver: ode15s Show all properties
Solve the system of equations over the time interval [0 10]
by using the solve
method. Plot all three solution components.
S = solve(F,0,10); plot(S.Time,S.Solution,"-o") legend("y_1","y_2","y_3",Location="southeast")
Version History
Introduced in R2023b
See Also
ode
| odeJacobian
| odeEvent
| ODEResults
| odeset
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