integer

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 e04gd to terminate immediately.)

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

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)

integer

Enables you to limit the number of times that lsqfun is called by e04gd. There will be an error exit (see the Errors section in Fortran library documentation) after maxcal evaluations of the residuals (i.e., calls of lsqfun with iflag set to 2). It should be borne in mind that, in addition to the calls of lsqfun which are limited directly by maxcal, there will be calls of lsqfun (with iflag set to 1) to evaluate only first derivatives.

double

The accuracy in x to which the solution is required.

double array

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

integer: default = nrow(x)

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

integer: default = 1

The frequency with which lsqmon is to be called.

: lsqmon is called once every iprint iterations and just before exit from e04gd.

: lsqmon is just called at the final point.

: lsqmon is not called at all.

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

Every iteration of e04gd involves a linear minimization, i.e., minimization of F(x^(k) + α^(k)p^(k)) with respect to α^(k). eta 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).

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.) e04gd will ensure that, for each iteration,

∑_j = 1^n(x_j^(k) - x_j^(k - 1))^2 <= (stepmx)^2

where k is the iteration number. Thus, if the problem has more than one solution, e04gd is most likely to find the one nearest to the starting point. On difficult problems, a realistic choice can prevent the sequence of x^(k) entering a region where the problem is ill-behaved and can help avoid overflow in the evaluation of F(x). However, an underestimate of stepmx can lead to inefficiency.

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.

double

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

double array

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

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.

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.

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.

integer

The number of iterations which have been performed in e04gd.

integer

The number of times that the residuals have been evaluated (i.e., number of calls of lsqfun with iflag set to 2).

integer

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