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https://github.com/junyuan-chen/nonlinearsystems.jl

Solve nonlinear systems of equations and nonlinear least squares in Julia
https://github.com/junyuan-chen/nonlinearsystems.jl

dogleg-method julia nonlinear-equations nonlinear-least-squares nonlinear-optimization optimization scientific-computing solver trust-region

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Solve nonlinear systems of equations and nonlinear least squares in Julia

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# NonlinearSystems.jl



*Solve nonlinear systems of equations and nonlinear least squares in Julia*



[![CI-stable][CI-stable-img]][CI-stable-url]

[![codecov][codecov-img]][codecov-url]

[![PkgEval][pkgeval-img]][pkgeval-url]

[![docs-stable][docs-stable-img]][docs-stable-url]

[![docs-dev][docs-dev-img]][docs-dev-url]



[CI-stable-img]: https://github.com/junyuan-chen/NonlinearSystems.jl/workflows/CI-stable/badge.svg

[CI-stable-url]: https://github.com/junyuan-chen/NonlinearSystems.jl/actions?query=workflow%3ACI-stable



[codecov-img]: https://codecov.io/gh/junyuan-chen/NonlinearSystems.jl/branch/main/graph/badge.svg

[codecov-url]: https://codecov.io/gh/junyuan-chen/NonlinearSystems.jl



[pkgeval-img]: https://juliaci.github.io/NanosoldierReports/pkgeval_badges/N/NonlinearSystems.svg

[pkgeval-url]: https://juliaci.github.io/NanosoldierReports/pkgeval_badges/N/NonlinearSystems.html



[docs-stable-img]: https://img.shields.io/badge/docs-stable-blue.svg

[docs-stable-url]: https://junyuan-chen.github.io/NonlinearSystems.jl/stable/



[docs-dev-img]: https://img.shields.io/badge/docs-dev-blue.svg

[docs-dev-url]: https://junyuan-chen.github.io/NonlinearSystems.jl/dev/



[NonlinearSystems.jl](https://github.com/junyuan-chen/NonlinearSystems.jl)

is a Julia package for solving nonlinear systems of equations and nonlinear least squares.

It renovates well-trusted solution algorithms with

highly performant and extensible implementation in native Julia language.



NonlinearSystems.jl places special emphasis on



- Low number of evaluations needed for updating the Jacobian matrix

- Flexibility of swapping the underlying linear solver based on array type and hardware

- Zero memory allocation incurred in iteration steps



At this moment, the only solution algorithm implemented

is a modified version of Powell's hybrid method

(a trust region method with dogleg).

Relations to existing packages are further discussed towards the end.



## Quick Start



NonlinearSystems.jl adopts the

[CommonSolve.jl](https://github.com/SciML/CommonSolve.jl) interface

and wraps a residual function as `OnceDifferentiable` defined in

[NLSolversBase.jl](https://github.com/JuliaNLSolvers/NLSolversBase.jl)

with an optionally user-provided Jacobian function.

The same interface is shared for

solving a root-finding problems and solving a least-squares problems.



```julia

using NonlinearSystems



# Residual function

function f!(F, x)

    F[1] = (x[1] + 3) * (x[2]^3 - 7) + 18

    F[2] = sin(x[2] * exp(x[1]) - 1)

    return F

end



# Jacobian function (optional)

function j!(J, x)

    J[1,1] = x[2]^3 - 7

    J[1,2] = 3 * x[2]^2 * (x[1] + 3)

    u = exp(x[1]) * cos(x[2] * exp(x[1]) - 1)

    J[2,1] = x[2] * u

    J[2,2] = u

    return J

end



# Initial value

x0 = [0.1, 1.2]



# Evaluate Jacobians via finite differencing methods from FiniteDiff.jl

solve(Hybrid{RootFinding}, f!, x0)



# Use user-specified Jacobian function and separate out the initialization step

s = init(Hybrid{LeastSquares}, f!, j!, x0)

solve!(s)

```



For more details, please see the [documentation][docs-stable-url].



## Related Packages



NonlinearSystems.jl addresses the following limitations that the related packages do not:



- [MINPACK.jl](https://github.com/sglyon/MINPACK.jl) and [GSL.jl](https://github.com/JuliaMath/GSL.jl)

  - No option for swapping the linear solver

  - Use of rank-1 update of the Jacobian matrix and factorization cannot be adjusted

  - Wrappers of C interface; no native Julia implementation

  - MINPACK.jl does not work on Apple Silicon

- [NLsolve.jl](https://github.com/JuliaNLSolvers/NLsolve.jl)

  - No option for swapping the linear solver

  - No rank-1 update of the Jacobian matrix and factorization

  - Iteration steps are not non-allocating

  - No support for nonlinear least squares

- [NonlinearSolve.jl](https://github.com/SciML/NonlinearSolve.jl)

  - No rank-1 update of the Jacobian matrix and factorization

  - Iteration steps are not non-allocating

  - No support for nonlinear least squares

- [LeastSquaresOptim.jl](https://github.com/matthieugomez/LeastSquaresOptim.jl)

  - Only solves nonlinear least squares

  - No rank-1 update of the Jacobian matrix and factorization

- [LsqFit.jl](https://github.com/JuliaNLSolvers/LsqFit.jl)

  - Only solves nonlinear least squares

  - Performance seems to be dominated by LeastSquaresOptim.jl



## Roadmap



At this moment, only trust-region methods are considered and

the linear problem involved in each iteration is only solved by dense matrix factorization.

The package is intended to be lightweight.



## References



**Moré, Jorge J., Danny C. Sorenson, Burton S. Garbow, and Kenneth E. Hillstrom.** 1984.

"The MINPACK Project."

In *Sources and Development of Mathematical Software*,

ed. Wayne R. Cowell, 88-111. New Jersey: Prentice-Hall.



**Nocedal, Jorge, and Stephen J. Wright.** 2006.

*Numerical Optimization.* 2nd ed. New York: Springer.



**Powell, Michael J. D.** 1970.

"A Hybrid Method for Nonlinear Equations."

In *Numerical Methods for Nonlinear Algebraic Equations*,

ed. Philip Rabinowitz, 87-114. London: Gordon and Breach.