NonlinearSystem

struct NonlinearSystem <: AbstractTimeIndependentSystem

A nonlinear system of equations.

Fields

• tag: A tag for the system. If two systems have the same tag, then they are structurally identical.

• eqs: Vector of equations defining the system.

• unknowns: Unknown variables.

• ps: Parameters.

• var_to_name: Array variables.

• observed: Observed equations.

• jac: Jacobian matrix. Note: this field will not be defined until calculate_jacobian is called on the system.

• name: The name of the system.

• description: A description of the system.

• systems: The internal systems. These are required to have unique names.

• defaults: The default values to use when initial conditions and/or parameters are not supplied in ODEProblem.

• guesses: The guesses to use as the initial conditions for the initialization system.

• initializesystem: The system for performing the initialization.

• initialization_eqs: Extra equations to be enforced during the initialization sequence.

• connector_type: Type of the system.

• parameter_dependencies: Topologically sorted parameter dependency equations, where all symbols are parameters and the LHS is a single parameter.

• metadata: Metadata for the system, to be used by downstream packages.

• gui_metadata: Metadata for MTK GUI.

• is_initializesystem: Whether this is an initialization system.

• tearing_state: Cache for intermediate tearing state.

• substitutions: Substitutions generated by tearing.

• namespacing: If false, then sys.x no longer performs namespacing.

• complete: If true, denotes the model will not be modified any further.

• index_cache: Cached data for fast symbolic indexing.

• parent: The hierarchical parent system before simplification.

• isscheduled

Examples

@variables x y z
@parameters σ ρ β

eqs = [0 ~ σ*(y-x),
       0 ~ x*(ρ-z)-y,
       0 ~ x*y - β*z]
@named ns = NonlinearSystem(eqs, [x,y,z],[σ,ρ,β])

structural_simplify(sys; ...)
structural_simplify(
    sys,
    io;
    additional_passes,
    simplify,
    split,
    allow_symbolic,
    allow_parameter,
    conservative,
    fully_determined,
    kwargs...
)

Structurally simplify algebraic equations in a system and compute the topological sort of the observed equations in sys.

Optional Arguments:

• optional argument io may take a tuple (inputs, outputs). This will convert all inputs to parameters and allow them to be unconnected, i.e., simplification will allow models where n_unknowns = n_equations - n_inputs.

Optional Keyword Arguments:

• When simplify=true, the simplify function will be applied during the tearing process.
• allow_symbolic=false, allow_parameter=true, and conservative=false limit the coefficient types during tearing. In particular, conservative=true limits tearing to only solve for trivial linear systems where the coefficient has the absolute value of $1$.
• fully_determined=true controls whether or not an error will be thrown if the number of equations don't match the number of inputs, outputs, and equations.
• sort_eqs=true controls whether equations are sorted lexicographically before simplification or not.

tearing(sys; simplify=false)

Tear the nonlinear equations in system. When simplify=true, we simplify the new residual equations after tearing. End users are encouraged to call structural_simplify instead, which calls this function internally.

calculate_jacobian(sys::AbstractSystem)

Calculate the Jacobian matrix of a system.

Returns a matrix of Num instances. The result from the first call will be cached in the system object.

generate_jacobian(sys::AbstractSystem, dvs = unknowns(sys), ps = parameters(sys),
                  expression = Val{true}; sparse = false, kwargs...)

Generates a function for the Jacobian matrix of a system. Extra arguments control the arguments to the internal build_function call.

SciMLBase.NonlinearFunction{iip}(sys::NonlinearSystem, dvs = unknowns(sys),
                                 ps = parameters(sys);
                                 version = nothing,
                                 jac = false,
                                 sparse = false,
                                 kwargs...) where {iip}

Create a NonlinearFunction from the NonlinearSystem. The arguments dvs and ps are used to set the order of the dependent variable and parameter vectors, respectively.

DiffEqBase.NonlinearProblem{iip}(sys::NonlinearSystem, u0map,
                                 parammap = DiffEqBase.NullParameters();
                                 jac = false, sparse = false,
                                 checkbounds = false,
                                 linenumbers = true, parallel = SerialForm(),
                                 kwargs...) where {iip}

Generates an NonlinearProblem from a NonlinearSystem and allows for automatically symbolically calculating numerical enhancements.

SciMLBase.NonlinearFunctionExpr{iip}(sys::NonlinearSystem, dvs = unknowns(sys),
                                     ps = parameters(sys);
                                     version = nothing,
                                     jac = false,
                                     sparse = false,
                                     kwargs...) where {iip}

Create a Julia expression for a NonlinearFunction from the NonlinearSystem. The arguments dvs and ps are used to set the order of the dependent variable and parameter vectors, respectively.

DiffEqBase.NonlinearProblemExpr{iip}(sys::NonlinearSystem, u0map,
                                     parammap = DiffEqBase.NullParameters();
                                     jac = false, sparse = false,
                                     checkbounds = false,
                                     linenumbers = true, parallel = SerialForm(),
                                     kwargs...) where {iip}

Generates a Julia expression for a NonlinearProblem from a NonlinearSystem and allows for automatically symbolically calculating numerical enhancements.