# Nonlinear System Solvers

solve(prob::NonlinearProblem,alg;kwargs)

Solves for $f(u)=0$ in the problem defined by prob using the algorithm alg. If no algorithm is given, a default algorithm will be chosen.

NewtonRaphson is a good choice for most problems. It is non-allocating on static arrays and thus really well-optimized for small systems, while for large systems it can make use of sparsity patterns for sparse automatic differentiation and sparse linear solving of very large systems. That said, as a classic Newton method, its stability region can be smaller than other methods. NLSolveJL's :trust_region method can be a good choice for high stability, along with CMINPACK.

For a system which is very non-stiff (i.e., the condition number of the Jacobian is small, or the eigenvalues of the Jacobian are within a few orders of magnitude), then NLSolveJL's :anderson can be a good choice.

## Full List of Methods

### NonlinearSolve.jl

These are the core solvers.

• NewtonRaphson(;autodiff=true,chunk_size=12,diff_type=Val{:forward},linsolve=DEFAULT_LINSOLVE): A Newton-Raphson method with swappable nonlinear solvers and autodiff methods for high performance on large and sparse systems. When used on objects like static arrays, this method is non-allocating.

### SciMLNLSolve.jl

This is a wrapper package for importing solvers from other packages into this interface. Note that these solvers do not come by default, and thus one needs to install the package before using these solvers:

]add SciMLNLSolve
using SciMLNLSolve
• CMINPACK(): A wrapper for using the classic MINPACK method through MINPACK.jl
• NLSolveJL(): A wrapper for NLsolve.jl
NLSolveJL(;
method=:trust_region,
autodiff=:central,
store_trace=false,
extended_trace=false,
linesearch=LineSearches.Static(),
linsolve=(x, A, b) -> copyto!(x, A\b),
factor = one(Float64),
autoscale=true,
m=10,
beta=one(Float64),
show_trace=false,
)

Choices for methods in NLSolveJL:

• :fixedpoint: Fixed-point iteration
• :anderson: Anderson-accelerated fixed-point iteration
• :newton: Classical Newton method with an optional line search
• :trust_region: Trust region Newton method (the default choice)

### Sundials.jl

This is a wrapper package for the SUNDIALS C library, specifically the KINSOL nonlinear solver included in that ecosystem. Note that these solvers do not come by default, and thus one needs to install the package before using these solvers:

]add Sundials
using Sundials
• KINSOL: The KINSOL method of the SUNDIALS C library
KINSOL(;
linear_solver = :Dense,
jac_upper = 0,
jac_lower = 0,
userdata = nothing,
)

The choices for the linear solver are:

• :Dense: A dense linear solver
• :Band: A solver specialized for banded Jacobians. If used, you must set the position of the upper and lower non-zero diagonals via jac_upper and jac_lower.
• :LapackDense: A version of the dense linear solver that uses the Julia-provided OpenBLAS-linked LAPACK for multithreaded operations. This will be faster than :Dense on larger systems but has noticeable overhead on smaller (<100 ODE) systems.
• :LapackBand: A version of the banded linear solver that uses the Julia-provided OpenBLAS-linked LAPACK for multithreaded operations. This will be faster than :Band on larger systems but has noticeable overhead on smaller (<100 ODE) systems.
• :Diagonal: This method is specialized for diagonal Jacobians.
• :GMRES: A GMRES method. Recommended first choice Krylov method.
• :BCG: A biconjugate gradient method
• :PCG: A preconditioned conjugate gradient method. Only for symmetric linear systems.
• :TFQMR: A TFQMR method.
• :KLU: A sparse factorization method. Requires that the user specify a Jacobian. The Jacobian must be set as a sparse matrix in the ODEProblem type.