dfsane {BB}R Documentation

Solving Large-Scale Nonlinear System of Equations

Description

Derivative-Free Spectral Approach for solving nonlinear systems of equations

Usage

  dfsane(par, fn, method=2, control=list(), ...) 
 

Arguments

fn a function that takes a real vector as argument and returns a real vector of same length (see details).
par A real vector argument to fn, indicating the initial guess for the root of the nonlinear system.
method An integer (1, 2, or 3) specifying which Barzilai-Borwein steplength to use. The default is 2. See *Details*.
control A list of control parameters. See *Details*.
... Additional arguments passed to fn.

Details

The function dfsane is another algorithm for implementing non-monotone spectral residual method for finding a root of nonlinear systems, by working without gradient information. It stands for "derivative-free spectral approach for nonlinear equations". It differs from the function sane in that sane requires an approximation of a directional derivative at every iteration of the merit function F(x)^t F(x), where F is specified by the argument fn.

R adaptation, with significant modifications, by Ravi Varadhan, Johns Hopkins University (March 25, 2008), from the original FORTRAN code of La Cruz, Martinez, and Raydan (2006) available at http://kuainasi.ciens.ucv.ve/ccct/mraydan/mraydan.html.

A major modification in our R adaptation of the original FORTRAN code is the availability of 3 different options for Barzilai-Borwein (BB) steplengths: method = 1 is the BB steplength used in LaCruz, Martinez and Raydan (2006); method = 2 is equivalent to the other steplength proposed in Barzilai and Borwein's (1988) original paper. Finally, method = 3, is a new steplength, which is equivalent to that first proposed in Varadhan and Roland (2008) for accelerating the EM algorithm. In fact, Varadhan and Roland (2008) considered 3 similar steplength schemes in their EM acceleration work. Here, we have chosen method = 2 as the "default" method, since it generally performs better than others. However, it is not universally better and in some cases, one of the other two steplength schemes perform better. Therefore, if the default does not perform well in a particular nonlinear system, we recommend trying the other two steplength schemes.

Argument control is a list specifing any changes to default values of algorithm control parameters. Note that the names of these must be specified completely. Partial matching does not work. The list items are as follows:

M
A positive integer, typically between 5-20, that controls the monotonicity of the algorithm. M=1 would enforce strict monotonicity in the reduction of L2-norm of F, whereas larger values allow for more non-monotonicity. Global convergence under non-monotonicity is ensured by enforcing the Grippo-Lampariello-Lucidi condition (Grippo et al. 1986) in a non-monotone line-search algorithm. Values of M between 5 to 20 are generally good. The default is M = 10.

maxit
The maximum number of iterations. The default is maxit = 1500.
tol
The absolute convergence tolerance on the residual L2-norm of F. Convergence is declared when sum(F(x)^2) / npar < tol^2. Default is tol = 1.e-07.
trace
A logical variable (TRUE/FALSE). If TRUE, information on the progress of solving the system is produced. Default is trace = TRUE.
triter
An integer that controls the frequency of tracing when trace=TRUE. Default is triter=10, which means that the L2-norm of fn is printed at every 10-th iteration.

Value

A list with the following components:

par The best set of parameters that solves the nonlinear system.
residual L2-norm of the function at convergence, divided by sqrt(npar), where "npar" is the number of parameters.
fn.reduction Reduction in the L2-norm of the function from the initial L2-norm.
feval Number of times fn was evaluated.
iter Number of iterations taken by the algorithm.
convergence An integer code indicating type of convergence. 0 indicates successful convergence, in which case the resid is smaller than tol. Error codes are 1 indicates that the iteration limit maxit has been reached. 2 indicates failure due to an error in function evaluation. 3 indicates failure due to exceeding 100 steplength reductions in line-search.
message A text message explaining which termination criterion was used.

References

J Barzilai, and JM Borwein (1988), Two-point step size gradient methods, IMA J Numerical Analysis, 8, 141-148.

L Grippo, F Lampariello, and S Lucidi (1986), A nonmonotone line search technique for Newton's method, SIAM J on Numerical Analysis, 23, 707-716.

W LaCruz, JM Martinez, and M Raydan (2006), Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, 75, 1429-1448 (see http://kuainasi.ciens.ucv.ve/ccct/mraydan/mraydan.html).

R Varadhan and C Roland (2008), Simple and globally-convergent methods for accelerating the convergence of any EM algorithm, Scandinavian J Statistics, doi: 10.1111/j.1467-9469.2007.00585.x.

R Varadhan and PD Gilbert (2008), BB: An R package of Barzilai-Borwein spectral methods for solving and optimizing large-scale nonlinear systems, Unpublished.

See Also

sane, spg, grad

Examples

  trigexp <- function(x) {
# Test function No. 12 in the Appendix of LaCruz and Raydan (2003)
    n <- length(x)
    F <- rep(NA, n)
    F[1] <- 3*x[1]^2 + 2*x[2] - 5 + sin(x[1] - x[2]) * sin(x[1] + x[2])
    tn1 <- 2:(n-1)
    F[tn1] <- -x[tn1-1] * exp(x[tn1-1] - x[tn1]) + x[tn1] * ( 4 + 3*x[tn1]^2) +
        2 * x[tn1 + 1] + sin(x[tn1] - x[tn1 + 1]) * sin(x[tn1] + x[tn1 + 1]) - 8 
    F[n] <- -x[n-1] * exp(x[n-1] - x[n]) + 4*x[n] - 3
    F
    }

    p0 <- rnorm(1000)
    dfsane(par=p0, fn=trigexp)  # default is method=2
    dfsane(par=p0, fn=trigexp, method=1)
    dfsane(par=p0, fn=trigexp, method=3)
    dfsane(par=p0, fn=trigexp, control=list(triter=5, M=5))

[Package BB version 2008.8-1 Index]