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Accelerated Sequential Residual Method for Nonlinear Systems

This is dfsaneacc, an R package for solving large-scale nonlinear systems of equations using derivative-free accelerated sequential residual method.

This repository includes all source files related to the publication Birgin, Gardenghi, Marcondes and Martínez (2021), namely:

How to install and use the package in R

How to install

Download dfsaneacc_1.0.tar.gz inside the package folder. Then, unpack this and type in your terminal

R CMD INSTALL dfsaneacc

How to use dfsaneacc

You must call the routine

dfsaneacc (x, evalr, nhlim=6, epsf=1.e-06, maxit=Inf, iprint=-1, ...)

where:

This routine returns:

Details

Convergence of the algorithm is declared when the L2-norm is less or equal than epsf. The default value for epsf is 1.0e-6.

The algorithm employs the function evalr to compute the value of the nonlinear system at a given point x. The function evalr must have the form evalr (x, ...).

The function has four output levels, based on the value of the input parameter iprint: iprint=-1 no output is generated, iprint=0 means basic information at every iteration, iprint=1 adds additional information related to the backtracking strategy, and iprint=2 adds information related to the computation of the acceleration step. Its default value is iprint=-1.

Example

To solve the Exponential Function 2 (see La Cruz and Raydan, 2003):

n <- 3
x0 <- rep(1/n^2, n)

expfun2 <- function(x) {
    n <- length(x)
    f <- rep(NA, n)
    f[1] <- exp(x[1]) - 1.0
    f[2:n] <- (2:n)/10.0 * (exp(x[2:n]) + x[1:n-1] - 1)
    f
}

ret <- dfsaneacc(x=x0, evalr=expfun2, nhlim=6, epsf=1.0e-6*sqrt(n), iprint=0)
ret

The CUTEst interface

This interface consists in a set of R routines to evaluate the objective function of the nonlinear system problems from CUTEst testset.

It has an initialization routine that aims to generate a dynamic library R-problem-forcutest.so to be loaded in R using dyn.load routine. This initialization routine tries to generate ELFUN.f, EXTER.f, GROUP.f, and RANGE.f Fortran source files using sifdecoder from CUTEst and then to compile them together with cutest.c interface to generate the dynamic library.

All this machinery was tested in an Ubuntu Linux with gfortran and gcc. If your environment is different, you can decode your problem manually using sifdecoder, compile the .f files with cutest.c interface using your compiler and generate a dynamic library called R-problem-forcutest.so. Then, you can call cutest_init with optional logical variables sifdecode and compile set to FALSE.

See further details in the next section.

How to use the interface

The interface consists in five routines:

This interface works only with the 70 problems from the set of square nonlinear systems of equations.

For example, to solve BOOTH problem using this interface:

cutest_init("BOOTH")

n <- cutest_getn()
x0 <- cutest_getx0()
ret <- dfsaneacc(x=x0, evalr=cutest_evalr, nhlim=6,
                 epsf=1.0e-6*sqrt(n),iprint=0)

cutest_end()

You can find this example in cutest/example.R file.

About the tests from the publication

Requirements

To reproduce the results, one must have:

Source included

The folder paper-experiments includes all necessary code to reproduce the results from (Birgin, Gardenghi, Marcondes, Martínez 2021), namely:

The script runall-cutest.sh compiles and runs one of the three above in all the 70 nonlinear systems from CUTEst testset. Its usage is:

./runall-cutest.sh [algorithm]
[algorithm] must be 1, 2, or 3:
   1  to run Accelerated DF-SANE in R
   2  to run Accelerated DF-SANE in Fortran
   3  to run NITSOL

References

E. G. Birgin, J. L. Gardenghi, D. S. Marcondes, J. M. Martínez (2021), Accelerated derivative-free spectral residual method for nonlinear systems of equations, Submitted.

E. G. Birgin, J. M. Martínez (2021), Secant acceleration of sequential residual methods for solving large-scale nonlinear systems of equations. Technical Report arXiv:2012.13251v1.

W. La Cruz, and M. Raydan (2003), Nonmonotone spectral methods for large-scale nonlinear systems, Optimization Methods and Software, 18, 583-599. DOI 10.1080/10556780310001610493

W. La Cruz, J. M. Martínez, and M. Raydan (2006), Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, 75, 1429-1448. DOI 10.1090/S0025-5718-06-01840-0