e04he {NAGFWrappers}R Documentation

e04he: Unconstrained minimum of a sum of squares, combined Gauss-Newton and modified Newton algorithm, using second derivatives (comprehensive)

Description

e04he is a comprehensive modified Gauss-Newton algorithm for finding an unconstrained minimum of a sum of squares of m nonlinear functions in n variables (m >= n). First and second derivatives are required.

The function is intended for functions which have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities).

Usage

e04he(m, lsqfun, lsqhes, lsqmon, maxcal, xtol, x,
      n = nrow(x),
      iprint = 1,
      eta = if (n==1) 0.0 else 0.5,
      stepmx = 100000.0)

Arguments

m

integer

lsqfun

function

lsqfun must calculate the vector of values f_i(x) and Jacobian matrix of first derivatives ( \partial f_i)/( \partial x_j) at any point x. (However, if you do not wish to calculate the residuals or first derivatives at a particular x, there is the option of setting a argument to cause e04he to terminate immediately.)

(IFLAG,FVEC,FJAC) = lsqfun(iflag,m,n,xc,ldfjac)

lsqhes

function

lsqhes must calculate the elements of the symmetric matrix

B(x) = ∑_i = 1^mf_i(x)G_i(x),

at any point x, where G_i(x) is the Hessian matrix of f_i(x). (As with lsqfun, there is the option of causing e04he to terminate immediately.)

(IFLAG,B) = lsqhes(iflag,m,n,fvec,xc,lb)

lsqmon

function

If iprint >= 0, you must supply lsqmon which is suitable for monitoring the minimization process. lsqmon must not change the values of any of its arguments.

() = lsqmon(m,n,xc,fvec,fjac,ldfjac,s,igrade,niter,nf)

maxcal

integer

This argument is present so as to enable you to limit the number of times that lsqfun is called by e04he. There will be an error exit (see the Errors section in Fortran library documentation) after maxcal calls of lsqfun.

xtol

double

The accuracy in x to which the solution is required.

x

double array

x[j]

must be set to a guess at the jth component of the position of the minimum for j=1 . . . n.

n

integer: default = nrow(x)

The number m of residuals, f_i(x), and the number n of variables, x_j.

iprint

integer: default = 1

Specifies the frequency with which lsqmon is to be called.

iprint > 0

: lsqmon is called once every iprint iterations and just before exit from e04he.

iprint = 0

: lsqmon is just called at the final point.

iprint < 0

: lsqmon is not called at all.

eta

double: default = if (n==1) 0.0 else 0.5

Every iteration of e04he involves a linear minimization (i.e., minimization of F(x^(k) + α^(k)p^(k)) with respect to α^(k)). eta must lie in the range 0.0 <= eta < 1.0, and specifies how accurately these linear minimizations are to be performed. The minimum with respect to α^(k) will be located more accurately for small values of eta (say, 0.01) than for large values (say, 0.9).

stepmx

double: default = 100000.0

An estimate of the Euclidean distance between the solution and the starting point supplied by you. (For maximum efficiency, a slight overestimate is preferable.)

Details

R interface to the NAG Fortran routine E04HEF.

Value

X

double array

The final point x^(k). Thus, if ifail =0 on exit, x[j] is the jth component of the estimated position of the minimum.

FSUMSQ

double

The value of F(x), the sum of squares of the residuals f_i(x), at the final point given in x.

FVEC

double array

The value of the residual f_i(x) at the final point given in x for i=1 . . . m.

FJAC

double array

The value of the first derivative ( \partial f_i)/( \partial x_j) evaluated at the final point given in x for j=1 . . . n for i=1 . . . m.

S

double array

The singular values of the Jacobian matrix at the final point. Thus s may be useful as information about the structure of your problem.

V

double array

The matrix V associated with the singular value decomposition

J = USV^T

of the Jacobian matrix at the final point, stored by columns. This matrix may be useful for statistical purposes, since it is the matrix of orthonormalized eigenvectors of J^TJ.

NITER

integer

The number of iterations which have been performed in e04he.

NF

integer

The number of times that the residuals and Jacobian matrix have been evaluated (i.e., number of calls of lsqfun).

IFAIL

integer

ifail =0

unless the function detects an error or a warning has been flagged (see the Errors section in Fortran library documentation).

Author(s)

NAG

References

http://www.nag.co.uk/numeric/FL/nagdoc_fl23/pdf/E04/e04hef.pdf

Examples


ifail <- 0
lsqfun = function(iflag, m, n, xc, ljc) {
    
    fvec <- as.matrix(mat.or.vec(m, 1))
    fjacc <- as.matrix(mat.or.vec(ljc, n))
    for (i in c(1:m)) {
        denom <- xc[2] %*% t[i, 2] + xc[3] %*% t[i, 3]
        
        fvec[i] <- xc[1] + t[i, 1]/denom - y[i]
        
        if (iflag != 0) {
            
            fjacc[i, 1] <- 1
            
            dummy <- -1/(denom %*% denom)
            
            fjacc[i, 2] <- t[i, 1] %*% t[i, 2] %*% dummy
            
            fjacc[i, 3] <- t[i, 1] %*% t[i, 3] %*% dummy
            
        }
    }
    list(IFLAG = as.integer(iflag), FVEC = as.matrix(fvec), FJAC = as.matrix(fjacc))
}
lsqhes = function(iflag, m, n, fvec, xc, lb) {
    
    b <- as.matrix(mat.or.vec(lb, 1))
    b[1] <- 0
    b[2] <- 0
    sum22 <- 0
    sum32 <- 0
    sum33 <- 0
    for (i in c(1:m)) {
        dummy <- 2 %*% t[i, 1]/(xc[2] %*% t[i, 2] + xc[3] %*% 
            t[i, 3])^3
        
        sum22 <- sum22 + fvec[i] %*% dummy %*% t[i, 2]^2
        
        sum32 <- sum32 + fvec[i] %*% dummy %*% t[i, 2] %*% t[i, 
            3]
        
        sum33 <- sum33 + fvec[i] %*% dummy %*% t[i, 3]^2
    }
    b[3] <- sum22
    b[4] <- 0
    b[5] <- sum32
    b[6] <- sum33
    list(IFLAG = as.integer(iflag), B = as.matrix(b))
}
lsqmon = function(m, n, xc, fvec, fjacc, ljc, s, igrade, 
    niter, nf) {
    
    list()
}

m <- 15

maxcal <- 150

xtol <- 1.05418557512311e-07

x <- matrix(c(0.5, 1, 1.5), nrow = 3, ncol = 1, byrow = TRUE)



iw <- matrix(c(0), nrow = 1, ncol = 1, byrow = TRUE)



w <- as.matrix(mat.or.vec(105, 1))

y <- matrix(c(0.14, 0.18, 0.22, 0.25, 0.29, 0.32, 
    0.35, 0.39, 0.37, 0.58, 0.73, 0.96, 1.34, 2.1, 4.39), nrow = 1, 
    ncol = 15, byrow = TRUE)



t <- matrix(c(1, 15, 1, 2, 14, 2, 3, 13, 3, 4, 12, 
    4, 5, 11, 5, 6, 10, 6, 7, 9, 7, 8, 8, 8, 9, 7, 7, 10, 6, 
    6, 11, 5, 5, 12, 4, 4, 13, 3, 3, 14, 2, 2, 15, 1, 1), nrow = 15, 
    ncol = 3, byrow = TRUE)



e04he(m, lsqfun, lsqhes, lsqmon, maxcal, xtol, x) 


[Package NAGFWrappers version 24.0 Index]