odeJacobian
Description
An odeJacobian
object represents the Jacobian matrix for a system
of ordinary differential equations. The Jacobian is a matrix of partial derivatives of the
functions in the system of differential equations.
Create an ode
object to
represent the ODE problem, and specify an odeJacobian
object as the value of
the Jacobian
property to incorporate a Jacobian matrix or its sparsity
pattern into the problem.
Creation
Description
creates an
J
= odeJacobianodeJacobian
object with empty properties.
specifies one or more property values using name-value arguments. For example, J
= odeJacobian(Name=Value
)J
= odeJacobian(Jacobian=[0 1; -2 1])
specifies a constant Jacobian
matrix.
Properties
Jacobian
— Jacobian matrix
matrix | cell array | function handle
Jacobian matrix, specified as a matrix, cell array, or handle to a function that evaluates the Jacobian. The Jacobian is a matrix of partial derivatives of the functions that define the system of differential equations.
For stiff ODE solvers (ode15s
, ode23s
,
ode23t
, and ode23tb
), providing information
about the Jacobian matrix is critical for reliability and efficiency. If you do not
provide the Jacobian, then the ODE solver approximates it numerically using finite
differences.
For large systems of equations where it is not feasible to provide the entire
analytic Jacobian, use the SparsityPattern
property to pass in the
sparsity pattern of the Jacobian matrix. The solver uses the sparsity pattern to
calculate a sparse Jacobian.
You can specify the Jacobian
property as:
A constant matrix with calculated values for .
When
EquationType
is"fullyimplicit"
, a two-element cell array with calculated values for the constant Jacobian with respect toy
in the first element andyp
in the second element. If you specify one of the elements as[]
, the ODE solver approximates the corresponding Jacobian numerically while taking the provided values in the other element into account. (since R2024b)A handle to a function that computes the matrix elements and that accepts two input arguments,
dfdy = Fjac(t,y)
. To give the function access to parameter values in theParameters
property, specify a third input argument in the function definition,dfdy = Fjac(t,y,p)
.When
EquationType
is"fullyimplicit"
, a handle to a function that computes the matrix elements and that accepts three input arguments,[dfdy,dfdp] = Fjac(t,y,yp)
. To give the function access to parameter values in theParameters
property, specify a fourth input argument in the function definition,[dfdy,dfdp] = Fjac(t,y,yp,p)
. (since R2024b)
Example: J = odeJacobian(Jacobian=@Fjac)
specifies the function
Fjac
that evaluates the Jacobian matrix.
Example: J = odeJacobian(Jacobian=[0 1; -2 1])
specifies a
constant Jacobian matrix.
Data Types: single
| double
| cell
| function_handle
SparsityPattern
— Jacobian sparsity pattern
sparse matrix | cell array
Jacobian sparsity pattern, specified as a sparse matrix or cell array. The sparse
matrix contains 1
s where there might be nonzero entries in the
Jacobian. The ODE solver uses the sparsity pattern to generate a sparse Jacobian matrix
numerically. Use this property to improve execution time when the ODE system is large,
sparse, and you cannot provide an analytic Jacobian.
If the EquationType
property of the ode
object
is "fullyimplicit"
, specify a constant sparsity pattern as a cell
array where the first element is the sparsity pattern for the Jacobian with respect to
y
and the second element is the sparsity pattern for the Jacobian
with respect to yp
. If you specify one of the elements of the cell
array as []
, the ODE solver approximates the corresponding Jacobian
numerically while taking the provided values in the other element into account.
Note
If you specify a Jacobian matrix using the Jacobian
property,
then the solver ignores the SparsityPattern
property.
Example: J = odeJacobian(SparsityPattern=S)
specifies the Jacobian
sparsity pattern using sparse matrix S
.
Data Types: double
| cell
(since R2024b)
Examples
Specify Jacobian Matrix for Stiff ODE
The Van der Pol oscillator equation is a second-order differential equation. The equation includes a parameter , and the equation becomes stiff when the value of is large.
Using the substitutions and produces a system of two first-order equations.
The Jacobian matrix for these equations is the matrix of partial derivatives of each equation with respect to both and .
Solve the Van der Pol oscillator using and initial values of [2; 0]
by creating an ode
object to represent the problem.
Store the value of in the
Parameters
property.Specify the initial values in the
InitialValue
property.Specify the system of equations in the
ODEFcn
property, specifying three input arguments so that the value for is passed to the function.Specify a function that calculates the Jacobian matrix in the
Jacobian
property, specifying three input arguments so that the value for is passed to the function.
F = ode; F.Parameters = 1000; F.InitialValue = [2; 0]; F.ODEFcn = @(t,y,p) [y(2); p(1)*(1-y(1)^2)*y(2)-y(1)]; F.Jacobian = @(t,y,p) [0 1; -2*p(1)*y(1)*y(2)-1 p(1)*(1-y(1)^2)];
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,p)[y(2);p(1)*(1-y(1)^2)*y(2)-y(1)] Parameters: 1000 InitialTime: 0 InitialValue: [2x1 double] Jacobian: [1x1 odeJacobian] 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 3000]
by using the solve
method. Plot the first solution component.
S = solve(F,0,3000);
plot(S.Time,S.Solution(1,:),"-o")
Version History
Introduced in R2023bR2024b: Specify Jacobian for implicit ODEs
The Jacobian
property can accept functions of the form
dfdy = Fjac(t,y,yp)
for implicit ODEs. Specify the ode
object property EquationType
as "fullyimplicit"
to
use the three-argument form for Jacobian
.
If EquationType
is "fullyimplicit"
, the
Jacobian
and SparsityPattern
properties also
accept a two-element cell array, where the first element corresponds to the Jacobian with
respect to y
and the second element corresponds to the Jacobian with
respect to yp
.
See Also
ode
| odeMassMatrix
| odeEvent
| ODEResults
| odeset
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