Description Details References

Solve a system of nonlinear equations using a Broyden or a Newton method with a choice of global strategies such as line search and trust region. There are options for using a numerical or user supplied Jacobian, for specifying a banded numerical Jacobian and for allowing a singular or ill-conditioned Jacobian.

The nleqslv package provides two algorithms for solving (dense) nonlinear systems of equations. The methods provided are

a Broyden Secant method where the matrix of derivatives is updated after each major iteration using the Broyden rank 1 update.

a full Newton method where the Jacobian matrix of derivatives is recalculated at each iteration

Both methods utilize global strategies such as line search or trust region methods whenever the standard Newton/Broyden step does not lead to a point closer to a root of the equation system. Both methods can also be used without a norm reducing global strategy. Line search may be either cubic, quadratic or geometric. The trust region methods are either the double dogleg method, the Powell single dogleg method or a Levenberg-Marquardt type method.

There is a facility for specifying that the Jacobian is banded; this can significantly speedup the calculation of a numerical Jacobian when the number of sub- and super diagonals is small compared to the size of the system of equations. For example the Jacobian of a tridiagonal system can be calculated with only three evaluations of the function.

The package provides an option to attempt to solve the system of equations when the Jacobian is singular or ill-conditioned using an approximation to the Moore-Penrose pseudoinverse of the Jacobian.

The algorithms provided in this package are derived from Dennis and Schnabel (1996). The code is written in Fortran 77 and Fortran 95 and uses Lapack and BLAS routines as provided by the R system.

Dennis, J.E. Jr and Schnabel, R.B. (1996), *Numerical Methods for Unconstrained Optimization
and Nonlinear Equations*, Siam.

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