e04fcf is a comprehensive algorithm for finding an unconstrained minimum of a sum of squares of $m$ nonlinear functions in $n$ variables $(m\ge n)$. No 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).
where $x={({x}_{1},{x}_{2},\dots ,{x}_{n})}^{\mathrm{T}}$ and $m\ge n$. (The functions ${f}_{i}\left(x\right)$ are often referred to as ‘residuals’.)
You must supply lsqfun to calculate the values of the ${f}_{i}\left(x\right)$ at any point $x$.
From a starting point ${x}^{\left(1\right)}$ supplied by you, the routine generates a sequence of points ${x}^{\left(2\right)},{x}^{\left(3\right)},\dots $, which is intended to converge to a local minimum of $F\left(x\right)$. The sequence of points is given by
where the vector ${p}^{\left(k\right)}$ is a direction of search, and ${\alpha}^{\left(k\right)}$ is chosen such that $F({x}^{\left(k\right)}+{\alpha}^{\left(k\right)}{p}^{\left(k\right)})$ is approximately a minimum with respect to ${\alpha}^{\left(k\right)}$.
The vector ${p}^{\left(k\right)}$ used depends upon the reduction in the sum of squares obtained during the last iteration. If the sum of squares was sufficiently reduced, then ${p}^{\left(k\right)}$ is an approximation to the Gauss–Newton direction; otherwise additional function evaluations are made so as to enable ${p}^{\left(k\right)}$ to be a more accurate approximation to the Newton direction.
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.
4References
Gill P E and Murray W (1978) Algorithms for the solution of the nonlinear least squares problem SIAM J. Numer. Anal.15 977–992
5Arguments
1: $\mathbf{m}$ – IntegerInput
2: $\mathbf{n}$ – 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: $\mathbf{lsqfun}$ – Subroutine, supplied by the user.External Procedure
lsqfun must calculate the vector of values ${f}_{i}\left(x\right)$ at any point $x$. (However, if you do not wish to calculate the residuals at a particular $x$, there is the option of setting an argument to cause e04fcf to terminate immediately.)
On exit: if lsqfun resets iflag to some negative number, e04fcf will terminate immediately, with ifail set to your setting of iflag.
2: $\mathbf{m}$ – IntegerInput
On entry: $m$, the numbers of residuals.
3: $\mathbf{n}$ – IntegerInput
On entry: $n$, the numbers of variables.
4: $\mathbf{xc}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the point $x$ at which the values of the ${f}_{i}$ are required.
5: $\mathbf{fvec}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: unless iflag is reset to a negative number,
${\mathbf{fvec}}\left(\mathit{i}\right)$ must contain the value of ${f}_{\mathit{i}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$.
8: $\mathbf{w}\left({\mathbf{lw}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
9: $\mathbf{lw}$ – IntegerInput
lsqfun is called with these arguments as in the call to e04fcf, so you can pass quantities to lsqfun from the subroutine which calls e04fcf by using partitions of iw and w beyond those used as workspace by e04fcf. However, because of the danger of mistakes in partitioning, it is recommended that this facility be used very selectively, e.g., for stable applications packages which need to pass their own variable dimension workspace to lsqfun. It is recommended that the normal method for passing information from your subroutine to lsqfun should be via COMMON global variables. In any case, you must not changeliw, lw or the elements of iw and w used as workspace by e04fcf.
lsqfun must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04fcf 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 e04fcf. If your code inadvertently does return any NaNs or infinities, e04fcf is likely to produce unexpected results.
lsqfun should be tested separately before being used in conjunction with e04fcf.
4: $\mathbf{lsqmon}$ – Subroutine, supplied by the NAG Library or the user.External Procedure
If ${\mathbf{iprint}}\ge 0$, you must supply lsqmon which is suitable for monitoring the minimization process. lsqmon must not change the values of any of its arguments.
If ${\mathbf{iprint}}<0$, the dummy routine e04fdz can be used as lsqmon.
Important: the dimension declaration for fjac must contain the variable ldfjac, not an integer constant.
1: $\mathbf{m}$ – IntegerInput
On entry: $m$, the numbers of residuals.
2: $\mathbf{n}$ – IntegerInput
On entry: $n$, the numbers of variables.
3: $\mathbf{xc}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the coordinates of the current point $x$.
4: $\mathbf{fvec}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the values of the residuals ${f}_{i}$ at the current point $x$.
5: $\mathbf{fjac}({\mathbf{ldfjac}},{\mathbf{n}})$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{fjac}}(\mathit{i},\mathit{j})$ contains the value of $\frac{\partial {f}_{\mathit{i}}}{\partial {x}_{\mathit{j}}}$ at the current point $x$, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,n$.
6: $\mathbf{ldfjac}$ – IntegerInput
On entry: the first dimension of the array fjac as declared in the (sub)program from which e04fcf is called.
7: $\mathbf{s}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the singular values of the current approximation to the Jacobian matrix. Thus s may be useful as information about the structure of your problem.
8: $\mathbf{igrade}$ – IntegerInput
On entry: e04fcf 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: $\mathbf{niter}$ – IntegerInput
On entry: the number of iterations which have been performed in e04fcf.
10: $\mathbf{nf}$ – IntegerInput
On entry: the number of times that lsqfun has been called so far. (However, for intermediate calls of lsqmon, nf is calculated on the assumption that the latest linear search has been successful. If this is not the case, the $n$ evaluations allowed for approximating the Jacobian at the new point will not in fact have been made. nf will be accurate at the final call of lsqmon.)
13: $\mathbf{w}\left({\mathbf{lw}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
14: $\mathbf{lw}$ – IntegerInput
These arguments correspond to the arguments iw, liw, w and lw of e04fcf. They are included in lsqmon's argument list primarily for when e04fcf 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 e04fcf 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 $F\left(x\right)$ mentioned in Section 7. It is usually helpful to print xc, the estimated gradient of the sum of squares, niter and nf.
5: $\mathbf{iprint}$ – IntegerInput
On entry: the frequency with which lsqmon is to be called.
If ${\mathbf{iprint}}>0$, lsqmon is called once every iprint iterations and just before exit from e04fcf.
If ${\mathbf{iprint}}=0$, lsqmon is just called at the final point.
If ${\mathbf{iprint}}<0$, lsqmon is not called at all.
iprint should normally be set to a small positive number.
Suggested value:
${\mathbf{iprint}}=1$.
6: $\mathbf{maxcal}$ – IntegerInput
On entry: the limit you set on the number of times that lsqfun may be called by e04fcf. There will be an error exit (see Section 6) after maxcal calls of lsqfun.
Every iteration of e04fcf involves a linear minimization, i.e., minimization of $F({x}^{\left(k\right)}+{\alpha}^{\left(k\right)}{p}^{\left(k\right)})$ with respect to ${\alpha}^{\left(k\right)}$.
On entry: specifies how accurately the linear minimizations are to be performed. The minimum with respect to ${\alpha}^{\left(k\right)}$ will be located more accurately for small values of eta (say, $0.01$) than for large values (say, $0.9$). Although accurate linear minimizations will generally reduce the number of iterations performed by e04fcf, they will increase the number of calls of lsqfun made each iteration. On balance it is usually more efficient to perform a low accuracy minimization.
Suggested value:
${\mathbf{eta}}=0.5$ (${\mathbf{eta}}=0.0$ if ${\mathbf{n}}=1$).
Constraint:
$0.0\le {\mathbf{eta}}<1.0$.
8: $\mathbf{xtol}$ – Real (Kind=nag_wp)Input
On entry: the accuracy in $x$ to which the solution is required.
If ${x}_{\mathrm{true}}$ is the true value of $x$ at the minimum, then ${x}_{\mathrm{sol}}$, the estimated position before a normal exit, is such that
where $\Vert y\Vert =\sqrt{{\displaystyle \sum _{j=1}^{n}}{y}_{j}^{2}}$. For example, if the elements of ${x}_{\mathrm{sol}}$ are not much larger than $1.0$ in modulus and if ${\mathbf{xtol}}=\text{1.0E\u22125}$, then ${x}_{\mathrm{sol}}$ is usually accurate to about five decimal places. (For further details see Section 7.)
Suggested value:
if $F\left(x\right)$ and the variables are scaled roughly as described in Section 9 and $\epsilon $ is the machine precision, then a setting of order ${\mathbf{xtol}}=\sqrt{\epsilon}$ will usually be appropriate. If xtol is set to $0.0$ or some positive value less than $10\epsilon $, e04fcf will use $10\epsilon $ instead of xtol, since $10\epsilon $ is probably the smallest reasonable setting.
Constraint:
${\mathbf{xtol}}\ge 0.0$.
9: $\mathbf{stepmx}$ – 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.) e04fcf will ensure that, for each iteration,
where $k$ is the iteration number. Thus, if the problem has more than one solution, e04fcf is most likely to find the one nearest to the starting point. On difficult problems, a realistic choice can prevent the sequence ${x}^{\left(k\right)}$ entering a region where the problem is ill-behaved and can help avoid overflow in the evaluation of $F\left(x\right)$. However, an underestimate of stepmx can lead to inefficiency.
10: $\mathbf{x}\left({\mathbf{n}}\right)$ – 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 $\mathit{j}=1,2,\dots ,n$.
On exit: the final point ${x}^{\left(k\right)}$. Thus, if ${\mathbf{ifail}}={\mathbf{0}}$ on exit, ${\mathbf{x}}\left(j\right)$ is the $j$th component of the estimated position of the minimum.
11: $\mathbf{fsumsq}$ – Real (Kind=nag_wp)Output
On exit: the value of $F\left(x\right)$, the sum of squares of the residuals ${f}_{i}\left(x\right)$, at the final point given in x.
12: $\mathbf{fvec}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the value of the residual
${f}_{\mathit{i}}\left(x\right)$ at the final point given in x, for $\mathit{i}=1,2,\dots ,m$.
13: $\mathbf{fjac}({\mathbf{ldfjac}},{\mathbf{n}})$ – Real (Kind=nag_wp) arrayOutput
On exit: the estimate of the first derivative
$\frac{\partial {f}_{\mathit{i}}}{\partial {x}_{\mathit{j}}}$ at the final point given in x, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,n$.
14: $\mathbf{ldfjac}$ – IntegerInput
On entry: the first dimension of the array fjac as declared in the (sub)program from which e04fcf is called.
Constraint:
${\mathbf{ldfjac}}\ge {\mathbf{m}}$.
15: $\mathbf{s}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the singular values of the estimated Jacobian matrix at the final point. Thus s may be useful as information about the structure of your problem.
16: $\mathbf{v}({\mathbf{ldv}},{\mathbf{n}})$ – Real (Kind=nag_wp) arrayOutput
On exit: the matrix $V$ associated with the singular value decomposition
$$J=US{V}^{\mathrm{T}}$$
of the estimated 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 ${J}^{\mathrm{T}}J$.
17: $\mathbf{ldv}$ – IntegerInput
On entry: the first dimension of the array v as declared in the (sub)program from which e04fcf is called.
Constraint:
${\mathbf{ldv}}\ge {\mathbf{n}}$.
18: $\mathbf{niter}$ – IntegerOutput
On exit: the number of iterations which have been performed in e04fcf.
19: $\mathbf{nf}$ – IntegerOutput
On exit: the number of times that the residuals have been evaluated (i.e., number of calls of lsqfun).
On entry: the dimension of the array iw as declared in the (sub)program from which e04fcf is called.
Constraint:
${\mathbf{liw}}\ge 1$.
22: $\mathbf{w}\left({\mathbf{lw}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
23: $\mathbf{lw}$ – IntegerInput
On entry: the dimension of the array w as declared in the (sub)program from which e04fcf is called.
Constraints:
if ${\mathbf{n}}>1$, ${\mathbf{lw}}\ge 6\times {\mathbf{n}}+{\mathbf{m}}\times {\mathbf{n}}+2\times {\mathbf{m}}+{\mathbf{n}}\times ({\mathbf{n}}-1)/2$;
if ${\mathbf{n}}=1$, ${\mathbf{lw}}\ge 7+3\times {\mathbf{m}}$.
24: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $\mathrm{-1}$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ 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).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, 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 e04fcf may return useful information.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{eta}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: $0.0\le {\mathbf{eta}}<1.0$.
On entry, ${\mathbf{ldfjac}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldfjac}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{ldv}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldv}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{liw}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{liw}}\ge 1$.
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{maxcal}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{maxcal}}\ge 1$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{n}}=1$ and ${\mathbf{lw}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: if ${\mathbf{n}}=1$ then ${\mathbf{lw}}\ge 7+3\times {\mathbf{m}}$; that is, $\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{n}}>1$ and ${\mathbf{lw}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: if ${\mathbf{n}}>1$ then ${\mathbf{lw}}\ge 6\times {\mathbf{n}}+{\mathbf{m}}\times {\mathbf{n}}+2\times {\mathbf{m}}+{\mathbf{n}}\times ({\mathbf{n}}-1)/2$; that is, $\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{stepmx}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{xtol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{stepmx}}\ge {\mathbf{xtol}}$.
On entry, ${\mathbf{xtol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
${\mathbf{ifail}}=2$
There have been ${\mathbf{maxcal}}=\u27e8\mathit{\text{value}}\u27e9$ calls to lsqfun.
If steady reductions in the sum of squares, $F\left(x\right)$, 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 $F\left(x\right)$ has no minimum.
The error 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.
${\mathbf{ifail}}=3$
The conditions for a minimum have not all been satisfied, but a lower point could not be found. See Section 7 for further information.
The error 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.
${\mathbf{ifail}}=4$
Failure in computing SVD of estimated Jacobian matrix.
It may be worth applying e04fcf again starting with an initial approximation which is not too close to the point at which the failure occurred.
The error 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.
${\mathbf{ifail}}<0$
User requested termination by setting iflag negative in lsqfun.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
A successful exit (${\mathbf{ifail}}={\mathbf{0}}$) is made from e04fcf when (B1, B2 and B3) or B4 or B5 hold, where
and where $\Vert .\Vert $ and $\epsilon $ are as defined in Section 5, and ${F}^{\left(k\right)}$ and ${g}^{\left(k\right)}$ are the values of $F\left(x\right)$ and its vector of estimated first derivatives at ${x}^{\left(k\right)}$. If ${\mathbf{ifail}}={\mathbf{0}}$ then the vector in x on exit, ${x}_{\mathrm{sol}}$, is almost certainly an estimate of ${x}_{\mathrm{true}}$, the position of the minimum to the accuracy specified by xtol.
If ${\mathbf{ifail}}={\mathbf{3}}$, then ${x}_{\mathrm{sol}}$ may still be a good estimate of ${x}_{\mathrm{true}}$, but to verify this you should make the following checks. If
(a)the sequence $\left\{F\left({x}^{\left(k\right)}\right)\right\}$ converges to $F\left({x}_{\mathrm{sol}}\right)$ at a superlinear or a fast linear rate, and
(b)$g{\left({x}_{\mathrm{sol}}\right)}^{\mathrm{T}}g\left({x}_{\mathrm{sol}}\right)<10\epsilon $, where $\mathrm{T}$ denotes transpose, then it is almost certain that ${x}_{\mathrm{sol}}$ is a close approximation to the minimum. When (b) is true, then usually $F\left({x}_{\mathrm{sol}}\right)$ is a close approximation to $F\left({x}_{\mathrm{true}}\right)$. The values of $F\left({x}^{\left(k\right)}\right)$ can be calculated in lsqmon, and the vector $g\left({x}_{\mathrm{sol}}\right)$ can be calculated from the contents of fvec and fjac on exit from e04fcf.
Further suggestions about confirmation of a computed solution are given in the E04 Chapter Introduction.
8Parallelism and Performance
e04fcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
e04fcf 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.
9Further Comments
The number of iterations required depends on the number of variables, the number of residuals, the behaviour of $F\left(x\right)$, the accuracy demanded and the distance of the starting point from the solution. The number of multiplications performed per iteration of e04fcf 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 $n+1$ calls of lsqfun. So, unless the residuals 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, $F\left(x\right)$ and the corresponding values of the ${x}_{j}$ are each in the range $(\mathrm{-1},+1)$, and so that at points one unit away from the solution, $F\left(x\right)$ differs from its value at the solution by approximately one unit. This will usually imply that the Hessian matrix of $F\left(x\right)$ 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 e04fcf 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.
10Example
This example finds least squares estimates of ${x}_{1},{x}_{2}$ and ${x}_{3}$ in the model