NAG Library Routine Document
E04FYF is an easy-to-use algorithm for finding an unconstrained minimum of a sum of squares of nonlinear functions in variables . No derivatives are required.
It is intended for functions which are continuous and which have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities).
||M, N, LW, IUSER(*), IFAIL
||X(N), FSUMSQ, W(LW), RUSER(*)
E04FYF is essentially identical to the subroutine LSNDN1 in the NPL Algorithms Library. It is applicable to problems of the form
. (The functions
are often referred to as ‘residuals’.)
You must supply a subroutine to evaluate functions at any point .
From a starting point supplied by you, a sequence of points is generated which is intended to converge to a local minimum of the sum of squares. These points are generated using estimates of the curvature of .
Gill P E and Murray W (1978) Algorithms for the solution of the nonlinear least-squares problem SIAM J. Numer. Anal. 15 977–992
- 1: M – INTEGERInput
- 2: N – INTEGERInput
On entry: the number of residuals, , and the number of variables, .
- 3: LSFUN1 – SUBROUTINE, supplied by the user.External Procedure
You must supply this routine to calculate the vector of values
at any point
. It should be tested separately before being used in conjunction with E04FYF (see the E04 Chapter Introduction
The specification of LSFUN1
||M, N, IUSER(*)
||XC(N), FVEC(M), RUSER(*)
- 1: M – INTEGERInput
On entry: , the numbers of residuals.
- 2: N – INTEGERInput
On entry: , the numbers of variables.
- 3: XC(N) – REAL (KIND=nag_wp) arrayInput
On entry: the point at which the values of the are required.
- 4: FVEC(M) – REAL (KIND=nag_wp) arrayOutput
On exit: must contain the value of at the point , for .
- 5: IUSER() – INTEGER arrayUser Workspace
- 6: RUSER() – REAL (KIND=nag_wp) arrayUser Workspace
is called with the parameters IUSER
as supplied to E04FYF. You are free to use the arrays IUSER
to supply information to LSFUN1
as an alternative to using COMMON global variables.
must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which E04FYF is called. Parameters denoted as Input
be changed by this procedure.
- 4: X(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: must be set to a guess at the th component of the position of the minimum, for .
On exit: the lowest point found during the calculations. Thus, if on exit, is the th component of the position of the minimum.
- 5: FSUMSQ – REAL (KIND=nag_wp)Output
: the value of the sum of squares,
, corresponding to the final point stored in X
- 6: W(LW) – REAL (KIND=nag_wp) arrayWorkspace
- 7: LW – INTEGERInput
: the dimension of the array W
as declared in the (sub)program from which E04FYF is called.
- if , ;
- if , .
- 8: IUSER() – INTEGER arrayUser Workspace
- 9: RUSER() – REAL (KIND=nag_wp) arrayUser Workspace
are not used by E04FYF, but are passed directly to LSFUN1
and may be used to pass information to this routine as an alternative to using COMMON global variables.
- 10: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
on exit, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Note: E04FYF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
|or||, when ,|
|or||, when .|
There have been
calls of LSFUN1
, yet the algorithm does not seem to have converged. This may be due to an awkward function or to a poor starting point, so it is worth restarting E04FYF from the final point held in X
The final point does not satisfy the conditions for acceptance as a minimum, but no lower point could be found.
An auxiliary routine has been unable to complete a singular value decomposition in a reasonable number of sub-iterations.
There is some doubt about whether the point
found by E04FYF is a minimum of
. The degree of confidence in the result decreases as IFAIL
increases. Thus, when
it is probable that the final
gives a good estimate of the position of a minimum, but when
it is very unlikely that the routine has found a minimum.
If you are not satisfied with the result (e.g., because IFAIL
), it is worth restarting the calculations from a different starting point (not the point at which the failure occurred) in order to avoid the region which caused the failure. Repeated failure may indicate some defect in the formulation of the problem.
If the problem is reasonably well scaled and a successful exit is made, then, for a computer with a mantissa of decimals, one would expect to get about decimals accuracy in the components of and between (if is of order at the minimum) and (if is close to zero at the minimum) decimals accuracy in .
The number of iterations required depends on the number of variables, the number of residuals and their behaviour, and the distance of the starting point from the solution. The number of multiplications performed per iteration of E04FYF varies, but for
. In addition, each iteration makes at least
calls of LSFUN1
. So, unless the residuals can be evaluated very quickly, the run time will be dominated by the time spent in LSFUN1
Ideally, the problem should be scaled so that the minimum value of the sum of squares is in the range , and so that at points a unit distance away from the solution the sum of squares is approximately a unit value greater than at the minimum. It is unlikely that you will be able to follow these recommendations very closely, but it is worth trying (by guesswork), as sensible scaling will reduce the difficulty of the minimization problem, so that E04FYF will take less computer time.
When the sum of squares represents the goodness-of-fit of a nonlinear model to observed data, elements of the variance-covariance matrix of the estimated regression coefficients can be computed by a subsequent call to E04YCF
, using information returned in segments of the workspace array W
. See E04YCF
for further details.
This example finds least squares estimates of
in the model
sets of data given in the following table.
The program uses
as the initial guess at the position of the minimum.
9.1 Program Text
Program Text (e04fyfe.f90)
9.2 Program Data
Program Data (e04fyfe.d)
9.3 Program Results
Program Results (e04fyfe.r)