NAG FL Interface
e04gbf (lsq_uncon_quasi_deriv_comp)
1
Purpose
e04gbf is a comprehensive quasi-Newton algorithm for finding an unconstrained minimum of a sum of squares of nonlinear functions in variables . First derivatives are required. The routine is intended for functions which have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities).
2
Specification
Fortran Interface
Subroutine e04gbf ( |
m, n, lsqlin, lsqfun, lsqmon, iprint, maxcal, eta, xtol, stepmx, x, fsumsq, fvec, fjac, ldfjac, s, v, ldv, niter, nf, iw, liw, w, lw, ifail) |
Integer, Intent (In) |
:: |
m, n, iprint, maxcal, ldfjac, ldv, liw, lw |
Integer, Intent (Inout) |
:: |
iw(liw), ifail |
Integer, Intent (Out) |
:: |
niter, nf |
Real (Kind=nag_wp), Intent (In) |
:: |
eta, xtol, stepmx |
Real (Kind=nag_wp), Intent (Inout) |
:: |
x(n), fjac(ldfjac,n), v(ldv,n), w(lw) |
Real (Kind=nag_wp), Intent (Out) |
:: |
fsumsq, fvec(m), s(n) |
External |
:: |
lsqlin, lsqfun, lsqmon |
|
C Header Interface
#include <nag.h>
void |
e04gbf_ (const Integer *m, const Integer *n, void (NAG_CALL *lsqlin)(Integer *selct), void (NAG_CALL *lsqfun)(Integer *iflag, const Integer *m, const Integer *n, const double xc[], double fvec[], double fjac[], const Integer *ldfjac, Integer iw[], const Integer *liw, double w[], const Integer *lw), void (NAG_CALL *lsqmon)(const Integer *m, const Integer *n, const double xc[], const double fvec[], const double fjac[], const Integer *ldfjac, const double s[], const Integer *igrade, const Integer *niter, const Integer *nf, Integer iw[], const Integer *liw, double w[], const Integer *lw), const Integer *iprint, const Integer *maxcal, const double *eta, const double *xtol, const double *stepmx, double x[], double *fsumsq, double fvec[], double fjac[], const Integer *ldfjac, double s[], double v[], const Integer *ldv, Integer *niter, Integer *nf, Integer iw[], const Integer *liw, double w[], const Integer *lw, Integer *ifail) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
e04gbf_ (const Integer &m, const Integer &n, void (NAG_CALL *lsqlin)(Integer &selct), void (NAG_CALL *lsqfun)(Integer &iflag, const Integer &m, const Integer &n, const double xc[], double fvec[], double fjac[], const Integer &ldfjac, Integer iw[], const Integer &liw, double w[], const Integer &lw), void (NAG_CALL *lsqmon)(const Integer &m, const Integer &n, const double xc[], const double fvec[], const double fjac[], const Integer &ldfjac, const double s[], const Integer &igrade, const Integer &niter, const Integer &nf, Integer iw[], const Integer &liw, double w[], const Integer &lw), const Integer &iprint, const Integer &maxcal, const double &eta, const double &xtol, const double &stepmx, double x[], double &fsumsq, double fvec[], double fjac[], const Integer &ldfjac, double s[], double v[], const Integer &ldv, Integer &niter, Integer &nf, Integer iw[], const Integer &liw, double w[], const Integer &lw, Integer &ifail) |
}
|
The routine may be called by the names e04gbf or nagf_opt_lsq_uncon_quasi_deriv_comp.
3
Description
e04gbf is essentially identical to the subroutine LSQFDQ in the NPL Algorithms Library. It is applicable to problems of the form:
where
and
. (The functions
are often referred to as ‘residuals’.)
You must supply a subroutine to calculate the values of the and their first derivatives at any point .
From a starting point
supplied by you, the routine generates a sequence of points
, which is intended to converge to a local minimum of
. The sequence of points is given by
where the vector
is a direction of search, and
is chosen such that
is approximately a minimum with respect to
.
The vector used depends upon the reduction in the sum of squares obtained during the last iteration. If the sum of squares was sufficiently reduced, then is the Gauss–Newton direction; otherwise the second derivatives of the are taken into account using a quasi-Newton updating scheme.
The method is designed to ensure that steady progress is made whatever the starting point, and to have the rapid ultimate convergence of Newton's method.
4
References
Gill P E and Murray W (1978) Algorithms for the solution of the nonlinear least squares problem SIAM J. Numer. Anal. 15 977–992
5
Arguments
-
1:
– Integer
Input
-
2:
– Integer
Input
-
On entry: the number of residuals, , and the number of variables, .
Constraint:
.
-
3:
– Subroutine, supplied by the NAG Library.
External Procedure
-
lsqlin enables you to specify whether the linear minimizations (i.e., minimizations of
with respect to
) are to be performed by a routine which just requires the evaluation of the
(
e04fcv), or by a routine which also requires the first derivatives of the
(
e04hev).
It will often be possible to evaluate the first derivatives of the residuals in about the same amount of computer time that is required for the evaluation of the residuals themselves – if this is so, then
e04gbf should be called with routine
e04hev as the argument
lsqlin. However, if the evaluation of the derivatives takes more than about four times as long as the evaluation of the residuals, then
e04fcv will usually be preferable. If in doubt, use
e04hev as it is slightly more robust.
Whichever subroutine is used, must be declared as EXTERNAL in the subroutine from which e04gbf is called.
-
4:
– Subroutine, supplied by the user.
External Procedure
-
lsqfun must calculate the vector of values
and Jacobian matrix of first derivatives
at any point
. (However, if you do not wish to calculate the residuals or first derivatives at a particular
, there is the option of setting an argument to cause
e04gbf to terminate immediately.)
The specification of
lsqfun is:
Fortran Interface
Subroutine lsqfun ( |
iflag, m, n, xc, fvec, fjac, ldfjac, iw, liw, w, lw) |
Integer, Intent (In) |
:: |
m, n, ldfjac, liw, lw |
Integer, Intent (Inout) |
:: |
iflag, iw(liw) |
Real (Kind=nag_wp), Intent (In) |
:: |
xc(n) |
Real (Kind=nag_wp), Intent (Inout) |
:: |
fjac(ldfjac,n), w(lw) |
Real (Kind=nag_wp), Intent (Out) |
:: |
fvec(m) |
|
C Header Interface
void |
lsqfun_ (Integer *iflag, const Integer *m, const Integer *n, const double xc[], double fvec[], double fjac[], const Integer *ldfjac, Integer iw[], const Integer *liw, double w[], const Integer *lw) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
lsqfun_ (Integer &iflag, const Integer &m, const Integer &n, const double xc[], double fvec[], double fjac[], const Integer &ldfjac, Integer iw[], const Integer &liw, double w[], const Integer &lw) |
}
|
Important: the dimension declaration for
fjac must contain the variable
ldfjac, not an integer constant.
-
1:
– Integer
Input/Output
-
On entry: will be set to
,
or
.
- Indicates that only the residuals need to be evaluated
- Indicates that only the Jacobian matrix needs to be evaluated
- Indicates that both the residuals and the Jacobian matrix must be calculated.
If
e04hev is used as
e04gbf's
lsqlin,
lsqfun will always be called with
iflag set to
.
On exit: if it is not possible to evaluate the
or their first derivatives at the point given in
xc (or if it is wished to stop the calculations for any other reason), you should reset
iflag to some negative number and return control to
e04gbf.
e04gbf will then terminate immediately, with
ifail set to your setting of
iflag.
-
2:
– Integer
Input
-
On entry: , the number of residuals.
-
3:
– Integer
Input
-
On entry: , the number of variables.
-
4:
– Real (Kind=nag_wp) array
Input
-
On entry: the point at which the values of the and the are required.
-
5:
– Real (Kind=nag_wp) array
Output
-
On exit: unless
on entry, or
iflag is reset to a negative number,
must contain the value of
at the point
, for
.
-
6:
– Real (Kind=nag_wp) array
Output
-
On exit: unless
on entry, or
iflag is reset to a negative number,
must contain the value of
at the point
, for
and
.
-
7:
– Integer
Input
-
On entry: the first dimension of the array
fjac, set to
by
e04gbf.
-
8:
– Integer array
Workspace
-
9:
– Integer
Input
-
10:
– Real (Kind=nag_wp) array
Workspace
-
11:
– Integer
Input
-
lsqfun is called with
e04gbf's arguments
iw,
liw,
w,
lw as these arguments. They are present so that, when other library routines require the solution of a minimization subproblem, constants needed for the evaluation of residuals can be passed through
iw and
w. Similarly, you could pass quantities to
lsqfun from the segment which calls
e04gbf by using partitions of
iw and
w beyond those used as workspace by
e04gbf. However, because of the danger of mistakes in partitioning, it is
recommended that you should pass information to
lsqfun via COMMON global variables and
not use iw or w at all. In any case you
must not change the elements of
iw and
w used as workspace by
e04gbf.
lsqfun must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which
e04gbf is called. Arguments denoted as
Input must
not be changed by this procedure.
Note: lsqfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
e04gbf. If your code inadvertently
does return any NaNs or infinities,
e04gbf is likely to produce unexpected results.
lsqfun should be tested separately before being used in conjunction with
e04gbf.
-
5:
– Subroutine, supplied by the NAG Library or the user.
External Procedure
-
If
, you must supply
lsqmon which is suitable for monitoring the minimization process.
lsqmon must not change the values of any of its arguments.
If
, the NAG Library dummy routine
e04fdz can be used as
lsqmon.
The specification of
lsqmon is:
Fortran Interface
Subroutine lsqmon ( |
m, n, xc, fvec, fjac, ldfjac, s, igrade, niter, nf, iw, liw, w, lw) |
Integer, Intent (In) |
:: |
m, n, ldfjac, igrade, niter, nf, liw, lw |
Integer, Intent (Inout) |
:: |
iw(liw) |
Real (Kind=nag_wp), Intent (In) |
:: |
xc(n), fvec(m), fjac(ldfjac,n), s(n) |
Real (Kind=nag_wp), Intent (Inout) |
:: |
w(lw) |
|
C Header Interface
void |
lsqmon_ (const Integer *m, const Integer *n, const double xc[], const double fvec[], const double fjac[], const Integer *ldfjac, const double s[], const Integer *igrade, const Integer *niter, const Integer *nf, Integer iw[], const Integer *liw, double w[], const Integer *lw) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
lsqmon_ (const Integer &m, const Integer &n, const double xc[], const double fvec[], const double fjac[], const Integer &ldfjac, const double s[], const Integer &igrade, const Integer &niter, const Integer &nf, Integer iw[], const Integer &liw, double w[], const Integer &lw) |
}
|
Important: the dimension declaration for
fjac must contain the variable
ldfjac, not an integer constant.
-
1:
– Integer
Input
-
On entry: , the numbers of residuals.
-
2:
– Integer
Input
-
On entry: , the numbers of variables.
-
3:
– Real (Kind=nag_wp) array
Input
-
On entry: the coordinates of the current point .
-
4:
– Real (Kind=nag_wp) array
Input
-
On entry: the values of the residuals at the current point .
-
5:
– Real (Kind=nag_wp) array
Input
-
On entry: contains the value of at the current point , for and .
-
6:
– Integer
Input
-
On entry: the first dimension of the array
fjac as declared in the (sub)program from which
e04gbf is called.
-
7:
– Real (Kind=nag_wp) array
Input
-
On entry: the singular values of the current Jacobian matrix. Thus
s may be useful as information about the structure of your problem.
-
8:
– Integer
Input
-
On entry:
e04gbf estimates the dimension of the subspace for which the Jacobian matrix can be used as a valid approximation to the curvature (see
Gill and Murray (1978)). This estimate is called the grade of the Jacobian matrix, and
igrade gives its current value.
-
9:
– Integer
Input
-
On entry: the number of iterations which have been performed in e04gbf.
-
10:
– Integer
Input
-
On entry: the number of evaluations of the residuals. (If
e04hev is used as
lsqlin,
nf is also the number of evaluations of the Jacobian matrix.)
-
11:
– Integer array
Workspace
-
12:
– Integer
Input
-
13:
– Real (Kind=nag_wp) array
Workspace
-
14:
– Integer
Input
-
As in
lsqfun, these arguments correspond to the arguments
iw,
liw,
w,
lw of
e04gbf. They are included in
lsqmon's argument list primarily for when
e04gbf is called by other library routines.
lsqmon must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which
e04gbf is called. Arguments denoted as
Input must
not be changed by this procedure.
Note: you should normally print the sum of squares of residuals, so as to be able to examine the sequence of values of
mentioned in
Section 7. It is usually helpful to also print
xc, the gradient of the sum of squares,
niter and
nf.
-
6:
– Integer
Input
-
On entry: the frequency with which
lsqmon is to be called.
- lsqmon is called once every iprint iterations and just before exit from e04gbf.
- lsqmon is just called at the final point.
- lsqmon is not called at all.
iprint should normally be set to a small positive number.
Suggested value:
.
-
7:
– Integer
Input
-
On entry: enables you to limit the number of times that
lsqfun is called by
e04gbf. There will be an error exit (see
Section 6) after
maxcal calls of
lsqfun.
Suggested values:
- if e04fcv is used as lsqlin,
- if e04hev is used as lsqlin.
Constraint:
.
-
8:
– Real (Kind=nag_wp)
Input
-
On entry: every iteration of
e04gbf involves a linear minimization (i.e., minimization of
with respect to
).
eta specifies how accurately these linear minimizations are to be performed. The minimum with respect to
will be located more accurately for small values of
eta (say,
) than for large values (say,
).
Although accurate linear minimizations will generally reduce the number of iterations performed by
e04gbf, they will increase the number of calls of
lsqfun made every iteration. On balance it is usually more efficient to perform a low accuracy minimization.
Suggested values:
- if and e04hev is used as lsqlin,
- if and e04fcv is uses as lsqlin,
- if .
Constraint:
.
-
9:
– Real (Kind=nag_wp)
Input
-
On entry: the accuracy in
to which the solution is required.
If
is the true value of
at the minimum, then
, the estimated position before a normal exit, is such that
where
. For example, if the elements of
are not much larger than
in modulus and if
, then
is usually accurate to about five decimal places. (For further details see
Section 7.)
If
and the variables are scaled roughly as described in
Section 9 and
is the
machine precision, then a setting of order
will usually be appropriate. If
xtol is set to
or some positive value less than
,
e04gbf will use
instead of
xtol, since
is probably the smallest reasonable setting.
Constraint:
.
-
10:
– Real (Kind=nag_wp)
Input
-
On entry: an estimate of the Euclidean distance between the solution and the starting point supplied by you. (For maximum efficiency, a slight overestimate is preferable.)
e04gbf will ensure that, for each iteration,
where
is the iteration number. Thus, if the problem has more than one solution,
e04gbf is most likely to find the one nearest to the starting point. On difficult problems, a realistic choice can prevent the sequence of
entering a region where the problem is ill-behaved and can help avoid overflow in the evaluation of
. However, an underestimate of
stepmx can lead to inefficiency.
Suggested value:
.
Constraint:
.
-
11:
– Real (Kind=nag_wp) array
Input/Output
-
On entry: must be set to a guess at the th component of the position of the minimum, for .
On exit: the final point . Thus, if on exit, is the th component of the estimated position of the minimum.
-
12:
– Real (Kind=nag_wp)
Output
-
On exit: the value of
, the sum of squares of the residuals
, at the final point given in
x.
-
13:
– Real (Kind=nag_wp) array
Output
-
On exit: the value of the residual
at the final point given in
x, for
.
-
14:
– Real (Kind=nag_wp) array
Output
-
On exit: the value of the first derivative
evaluated at the final point given in
x, for
and
.
-
15:
– Integer
Input
-
On entry: the first dimension of the array
fjac as declared in the (sub)program from which
e04gbf is called.
Constraint:
.
-
16:
– Real (Kind=nag_wp) array
Output
-
On exit: the singular values of the Jacobian matrix at the final point. Thus
s may be useful as information about the structure of your problem.
-
17:
– Real (Kind=nag_wp) array
Output
-
On exit: the matrix
associated with the singular value decomposition
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
.
-
18:
– Integer
Input
-
On entry: the first dimension of the array
v as declared in the (sub)program from which
e04gbf is called.
Constraint:
.
-
19:
– Integer
Output
-
On exit: the number of iterations which have been performed in e04gbf.
-
20:
– Integer
Output
-
On exit: the number of times that the residuals have been evaluated (i.e., the number of calls of
lsqfun). If
e04hev is used as
lsqlin,
nf is also the number of times that the Jacobian matrix has been evaluated.
-
21:
– Integer array
Workspace
-
22:
– Integer
Input
-
On entry: the dimension of the array
iw as declared in the (sub)program from which
e04gbf is called.
Constraint:
.
-
23:
– Real (Kind=nag_wp) array
Workspace
-
24:
– Integer
Input
-
On entry: the dimension of the array
w as declared in the (sub)program from which
e04gbf is called.
Constraints:
- if , ;
- if , .
-
25:
– Integer
Input/Output
-
On entry:
ifail must be set to
,
. If you are unfamiliar with this argument you should refer to
Section 4 in the Introduction to the NAG Library FL Interface 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 arguments 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.
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
or
, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
Note: in some cases e04gbf may return useful information.
-
On entry, .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, .
Constraint: .
On entry, and .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, and .
Constraint: if then ; that is, .
On entry, and .
Constraint: if then ; that is, .
On entry, and .
Constraint: .
On entry, .
Constraint: .
-
There have been
calls to
lsqfun.
If steady reductions in the sum of squares,
, were monitored up to the point where this exit occurred, then the exit probably occurred simply because
maxcal was set too small, so the calculations should be restarted from the final point held in
x. This exit may also indicate that
has no minimum.
-
The conditions for a minimum have not all been satisfied, but a lower point could not be found. See
Section 7 for further information.
-
Failure in computing SVD of Jacobian matrix.
It may be worth applying e04gbf again starting with an initial approximation which is not too close to the point at which the failure occurred.
-
User requested termination by setting
iflag negative in
lsqfun.
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See
Section 9 in the Introduction to the NAG Library FL Interface for further information.
The values
,
or
may also be caused by mistakes in
lsqfun, by the formulation of the problem or by an awkward function. If there are no such mistakes 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.
7
Accuracy
A successful exit (
) is made from
e04gbf when (B1, B2 and B3) or B4 or B5 hold, where
and where
and
are as defined in
xtol, and
and
are the values of
and its vector of first derivatives at
.
If
, then the vector in
x on exit,
, is almost certainly an estimate of
, the position of the minimum to the accuracy specified by
xtol.
If
, then
may still be a good estimate of
, but to verify this you should make the following checks. If
-
(a)the sequence converges to at a superlinear or a fast linear rate, and
-
(b)where denotes transpose, then it is almost certain that is a close approximation to the minimum.
When
(b) is true, then usually
is a close approximation to
. The values of
can be calculated in
lsqmon, and the vector
can be calculated from the contents of
fvec and
fjac on exit from
e04gbf.
Further suggestions about confirmation of a computed solution are given in the
E04 Chapter Introduction.
8
Parallelism and Performance
e04gbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
e04gbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The number of iterations required depends on the number of variables, the number of residuals, the behaviour of
, the accuracy demanded and the distance of the starting point from the solution. The number of multiplications performed per iteration of
e04gbf varies, but for
is approximately
. In addition, each iteration makes at least one call of
lsqfun. So, unless the residuals and their derivatives can be evaluated very quickly, the run time will be dominated by the time spent in
lsqfun.
Ideally, the problem should be scaled so that, at the solution, and the corresponding values of the are each in the range , and so that at points one unit away from the solution differs from its value at the solution by approximately one unit. This will usually imply that the Hessian matrix of at the solution is well-conditioned. 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 e04gbf 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 the arrays
s and
v. See
e04ycf for further details.
10
Example
This example finds least squares estimates of
,
and
in the model
using the
sets of data given in the following table.
Before calling
e04gbf, the program calls
e04yaf to check
lsqfun. It uses
as the initial guess at the position of the minimum.
10.1
Program Text
10.2
Program Data
10.3
Program Results