NAG Library Routine Document
E04GZF
1 Purpose
E04GZF is an easytouse modified Gauss–Newton algorithm for finding an unconstrained minimum of a sum of squares of $m$ nonlinear functions in $n$ variables $\left(m\ge n\right)$. First 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).
2 Specification
INTEGER 
M, N, LW, IUSER(*), IFAIL 
REAL (KIND=nag_wp) 
X(N), FSUMSQ, W(LW), RUSER(*) 
EXTERNAL 
LSFUN2 

3 Description
E04GZF is similar to the subroutine LSFDN2 in the NPL Algorithms Library. It is applicable to problems of the form
where
$x={\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)}^{\mathrm{T}}$ and
$m\ge n$. (The functions
${f}_{i}\left(x\right)$ are often referred to as ‘residuals’.)
You must supply a subroutine to evaluate the residuals and their first derivatives at any point $x$.
Before attempting to minimize the sum of squares, the algorithm checks the subroutine for consistency. Then, 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 $F\left(x\right)$.
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 Parameters
 1: $\mathrm{M}$ – INTEGERInput
 2: $\mathrm{N}$ – INTEGERInput

On entry: the number $m$ of residuals, ${f}_{i}\left(x\right)$, and the number $n$ of variables, ${x}_{j}$.
Constraint:
$1\le {\mathbf{N}}\le {\mathbf{M}}$.
 3: $\mathrm{LSFUN2}$ – SUBROUTINE, supplied by the user.External Procedure

You must supply this routine to calculate the vector of values ${f}_{i}\left(x\right)$ and the Jacobian matrix of first derivatives $\frac{\partial {f}_{i}}{\partial {x}_{j}}$ at any point $x$. It should be tested separately before being used in conjunction with E04GZF.
The specification of
LSFUN2 is:
INTEGER 
M, N, LDFJAC, IUSER(*) 
REAL (KIND=nag_wp) 
XC(N), FVEC(M), FJAC(LDFJAC,N), RUSER(*) 

Important: the dimension declaration for
FJAC must contain the variable
LDFJAC, not an integer constant.
 1: $\mathrm{M}$ – INTEGERInput

On entry: $m$, the numbers of residuals.
 2: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the numbers of variables.
 3: $\mathrm{XC}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the point $x$ at which the values of the ${f}_{i}$ and the $\frac{\partial {f}_{i}}{\partial {x}_{j}}$ are required.
 4: $\mathrm{FVEC}\left({\mathbf{M}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: ${\mathbf{FVEC}}\left(i\right)$ must be set to the value of
${f}_{\mathit{i}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$.
 5: $\mathrm{FJAC}\left({\mathbf{LDFJAC}},{\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: ${\mathbf{FJAC}}\left(\mathit{i},\mathit{j}\right)$ must be set to the value of $\frac{\partial {f}_{\mathit{i}}}{\partial {x}_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,n$.
 6: $\mathrm{LDFJAC}$ – INTEGERInput

On entry: the first dimension of the array
FJAC as declared in the (sub)program from which E04GZF is called.
 7: $\mathrm{IUSER}\left(*\right)$ – INTEGER arrayUser Workspace
 8: $\mathrm{RUSER}\left(*\right)$ – REAL (KIND=nag_wp) arrayUser Workspace

LSFUN2 is called with the parameters
IUSER and
RUSER as supplied to E04GZF. You are free to use the arrays
IUSER and
RUSER to supply information to
LSFUN2 as an alternative to using COMMON global variables.
LSFUN2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which E04GZF is called. Parameters denoted as
Input must
not be changed by this procedure.
 4: $\mathrm{X}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry:
${\mathbf{X}}\left(\mathit{j}\right)$ must be set to a guess at the
$\mathit{j}$th component of the position of the minimum, for
$\mathit{j}=1,2,\dots ,n$. The routine checks the first derivatives calculated by
LSFUN2 at the starting point and so is more likely to detect any error in your routines if the initial
${\mathbf{X}}\left(j\right)$ are nonzero and mutually distinct.
On exit: the lowest point found during the calculations. Thus, if ${\mathbf{IFAIL}}={\mathbf{0}}$ on exit, ${\mathbf{X}}\left(j\right)$ is the $j$th component of the position of the minimum.
 5: $\mathrm{FSUMSQ}$ – REAL (KIND=nag_wp)Output

On exit: the value of the sum of squares,
$F\left(x\right)$, corresponding to the final point stored in
X.
 6: $\mathrm{W}\left({\mathbf{LW}}\right)$ – REAL (KIND=nag_wp) arrayWorkspace
 7: $\mathrm{LW}$ – INTEGERInput

On entry: the dimension of the array
W as declared in the (sub)program from which E04GZF is called.
Constraints:
 if ${\mathbf{N}}>1$, ${\mathbf{LW}}\ge 8\times {\mathbf{N}}+2\times {\mathbf{N}}\times {\mathbf{N}}+2\times {\mathbf{M}}\times {\mathbf{N}}+3\times {\mathbf{M}}$;
 if ${\mathbf{N}}=1$, ${\mathbf{LW}}\ge 11+5\times {\mathbf{M}}$.
 8: $\mathrm{IUSER}\left(*\right)$ – INTEGER arrayUser Workspace
 9: $\mathrm{RUSER}\left(*\right)$ – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and
RUSER are not used by E04GZF, but are passed directly to
LSFUN2 and may be used to pass information to this routine as an alternative to using COMMON global variables.
 10: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is
$1$.
When the value $\mathbf{1}\text{ or}1$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Note: E04GZF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

On entry,  ${\mathbf{N}}<1$, 
or  ${\mathbf{M}}<{\mathbf{N}}$, 
or  ${\mathbf{LW}}<8\times {\mathbf{N}}+2\times {\mathbf{N}}\times {\mathbf{N}}+2\times {\mathbf{M}}\times {\mathbf{N}}+3\times {\mathbf{M}}$, when ${\mathbf{N}}>1$, 
or  ${\mathbf{LW}}<11+5\times {\mathbf{M}}$, when ${\mathbf{N}}=1$. 
 ${\mathbf{IFAIL}}=2$

There have been
$50\times n$ calls of
LSFUN2, 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 E04GZF from the final point held in
X.
 ${\mathbf{IFAIL}}=3$
The final point does not satisfy the conditions for acceptance as a minimum, but no lower point could be found.
 ${\mathbf{IFAIL}}=4$
An auxiliary routine has been unable to complete a singular value decomposition in a reasonable number of subiterations.
 ${\mathbf{IFAIL}}=5$
 ${\mathbf{IFAIL}}=6$
 ${\mathbf{IFAIL}}=7$
 ${\mathbf{IFAIL}}=8$

There is some doubt about whether the point
X$x$ found by E04GZF is a minimum of
$F\left(x\right)$. The degree of confidence in the result decreases as
IFAIL increases. Thus, when
${\mathbf{IFAIL}}={\mathbf{5}}$, it is probable that the final
$x$ gives a good estimate of the position of a minimum, but when
${\mathbf{IFAIL}}={\mathbf{8}}$ it is very unlikely that the routine has found a minimum.
 ${\mathbf{IFAIL}}=9$
It is very likely that you have made an error in forming the derivatives
$\frac{\partial {f}_{i}}{\partial {x}_{j}}$ in
LSFUN2.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
If you are not satisfied with the result (e.g., because
IFAIL lies between
$3$ and
$8$), 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.
7 Accuracy
If the problem is reasonably well scaled and a successful exit is made, then, for a computer with a mantissa of $t$ decimals, one would expect to get about $t/21$ decimals accuracy in the components of $x$ and between $t1$ (if $F\left(x\right)$ is of order $1$ at the minimum) and $2t2$ (if $F\left(x\right)$ is close to zero at the minimum) decimals accuracy in $F\left(x\right)$.
8 Parallelism and Performance
E04GZF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
E04GZF 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 implementationspecific information.
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 E04GZF varies, but for
$m\gg n$ is approximately
$n\times {m}^{2}+\mathit{O}\left({n}^{3}\right)$. In addition, each iteration makes at least one call of
LSFUN2. So, unless the residuals and their derivatives can be evaluated very quickly, the run time will be dominated by the time spent in
LSFUN2.
Ideally, the problem should be scaled so that the minimum value of the sum of squares is in the range $\left(0,+1\right)$ 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 E04GZF will take less computer time.
When the sum of squares represents the goodnessoffit of a nonlinear model to observed data, elements of the variancecovariance 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.
10 Example
This example finds least squares estimates of
${x}_{1}$,
${x}_{2}$ and
${x}_{3}$ in the model
using the
$15$ sets of data given in the following table.
The program uses
$\left(0.5,1.0,1.5\right)$ as the initial guess at the position of the minimum.
10.1 Program Text
Program Text (e04gzfe.f90)
10.2 Program Data
Program Data (e04gzfe.d)
10.3 Program Results
Program Results (e04gzfe.r)