# NonlinearFunctions and Jacobian Types

The SciML ecosystem provides an extensive interface for declaring extra functions associated with the differential equation's data. In traditional libraries there is usually only one option: the Jacobian. However, we allow for a large array of pre-computed functions to speed up the calculations. This is offered via the NonlinearFunction types, which can be passed to the problems.

## Function Type Definitions

SciMLBase.NonlinearFunctionType

NonlinearFunction{iip,F,TMM,Ta,Tt,TJ,JVP,VJP,JP,SP,TW,TWt,TPJ,S,O,TCV} <: AbstractNonlinearFunction{iip}

A representation of an nonlinear system of equations f, defined by:

$$$0 = f(u,p)$$$

and all of its related functions, such as the Jacobian of f, its gradient with respect to time, and more. For all cases, u0 is the initial condition, p are the parameters, and t is the independent variable.

Constructor

NonlinearFunction{iip,recompile}(f;
analytic=nothing,
jac=nothing,
jvp=nothing,
vjp=nothing,
jac_prototype=nothing,
sparsity=jac_prototype,
paramjac = nothing,
syms = nothing,
indepsym = nothing,
colorvec = nothing)

Note that only the function f itself is required. This function should be given as f!(du,u,p) or du = f(u,p). See the section on iip for more details on in-place vs out-of-place handling.

All of the remaining functions are optional for improving or accelerating the usage of f. These include:

• analytic(u0,p): used to pass an analytical solution function for the analytical solution of the ODE. Generally only used for testing and development of the solvers.
• jac(J,u,p) or J=jac(u,p): returns $\frac{df}{du}$
• jvp(Jv,v,u,p) or Jv=jvp(v,u,p): returns the directional derivative$\frac{df}{du} v$
• vjp(Jv,v,u,p) or Jv=vjp(v,u,p): returns the adjoint derivative$\frac{df}{du}^\ast v$
• jac_prototype: a prototype matrix matching the type that matches the Jacobian. For example, if the Jacobian is tridiagonal, then an appropriately sized Tridiagonal matrix can be used as the prototype and integrators will specialize on this structure where possible. Non-structured sparsity patterns should use a SparseMatrixCSC with a correct sparsity pattern for the Jacobian. The default is nothing, which means a dense Jacobian.
• paramjac(pJ,u,p): returns the parameter Jacobian $\frac{df}{dp}$.
• syms: the symbol names for the elements of the equation. This should match u0 in size. For example, if u0 = [0.0,1.0] and syms = [:x, :y], this will apply a canonical naming to the values, allowing sol[:x] in the solution and automatically naming values in plots.
• indepsym: the canonical naming for the independent variable. Defaults to nothing, which internally uses t as the representation in any plots.
• colorvec: a color vector according to the SparseDiffTools.jl definition for the sparsity pattern of the jac_prototype. This specializes the Jacobian construction when using finite differences and automatic differentiation to be computed in an accelerated manner based on the sparsity pattern. Defaults to nothing, which means a color vector will be internally computed on demand when required. The cost of this operation is highly dependent on the sparsity pattern.

iip: In-Place vs Out-Of-Place

For more details on this argument, see the ODEFunction documentation.

recompile: Controlling Compilation and Specialization

For more details on this argument, see the ODEFunction documentation.

Fields

The fields of the NonlinearFunction type directly match the names of the inputs.