Nonlinear Equation Solver with Modern Fortran.

A basic Newton-Raphson type nonlinear equation solver for dense systems with m functions of n input variables.

A work in progress.

Status

Features

• Is object-oriented.
• Works with square, under-determined, or over-determined systems.
• Can use different methods to solve the linear system:
  1. LAPACK routines (dgesv or dgels) for dense systems: If n=m, uses dgesv (LU decomposition). If n/=m, uses dgels (if m>n uses QR factorization, if m<n uses LQ factorization).
  2. lsqr -- a conjugate-gradient type method for solving sparse linear equations and sparse least-squares problems.
  3. lusol -- A sparse LU factorization for square and rectangular matrices, with Bartels-Golub-Reid updates for column replacement and other rank-1 modifications.
  4. lsmr -- a conjugate-gradient type method for solving sparse linear equations and sparse least-squares problems
  5. The user can also provide a custom linear solver.
• Has a Broyden update option (sparse and dense versions).
• Has various line search options.
  • use a specified constant step size (0,1]
  • backtracking linesearch method
  • exact linesearch method using fmin minimizer
  • evaluate function at specified fixed points

Compiling

• A Fortran Package Manager file is also included, so that the library and tests cases can be compiled with FPM. For example:

fpm build --profile release
fpm test --profile release

By default, the library is built with double precision (real64) real values. Explicitly specifying the real kind can be done using the following preprocessor flags:

Preprocessor flag	Kind	Number of bytes
REAL32	real(kind=real32)	4
REAL64	real(kind=real64)	8
REAL128	real(kind=real128)	16

For example, to build a single precision version of the library, use:

fpm build --profile release --flag "-DREAL32"

To use nlesolver within your fpm project, add the following to your fpm.toml file:

[dependencies]
nlesolver-fortran = { git="https://github.com/jacobwilliams/nlesolver-fortran.git" }

Or to use a specific version:

[dependencies]
nlesolver-fortran = { git="https://github.com/jacobwilliams/nlesolver-fortran.git", tag="1.1.0" }

Note that LAPACK is required to build. The fmin, lsqr, lusol, and lsmr libraries are also dependencies (which will be automatically downloaded by fpm).

Documentation

• The API documentation for the current master branch can be found here. This is generated by processing the source files with FORD.

License

• The NLESolver-Fortran source code and related files and documentation are distributed under a permissive free software license (BSD-3).

References

• C. G. Broyden, "A Class of Methods for Solving Nonlinear Simultaneous Equations", Math. Comp. 19 (1965), 577-593
• L. K. Schubert, "Modification of a Quasi-Newton Method for Nonlinear Equations with a Sparse Jacobian", Mathematics of Computation, Vol. 24, No. 109 (Jan., 1970), pp. 27-30.

See also

• MINPACK -- Modernized Minpack: for solving nonlinear equations and nonlinear least squares problems
• LUSOL Stanford University Systems Optimization Laboratory
• qr_mumps -- a software package for the solution of sparse, linear systems on multicore computers.