NAG Toolbox: nag_opt_lsq_uncon_quasi_deriv_comp (e04gb)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{m}$ – int64int32nag_int scalar

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

Constraint: $1\le \mathbf{n}\le \mathbf{m}$.

2:     $\mathrm{lsqlin}$ – function handle or string containing name of m-file

lsqlin enables you to specify whether the linear minimizations (i.e., minimizations of $F\left(x^{\left(k\right)}+{\alpha }^{\left(k\right)}{p}^{\left(k\right)}\right)$ with respect to ${\alpha }^{\left(k\right)}$) are to be performed by a function which just requires the evaluation of the $f_i\left(x\right)$ (nag_opt_lsq_uncon_quasi_deriv_comp_lsqlin_fun (e04fcv)), or by a function which also requires the first derivatives of the $f_i\left(x\right)$ (nag_opt_lsq_uncon_quasi_deriv_comp_lsqlin_deriv (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 nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) should be called with string nag_opt_lsq_uncon_quasi_deriv_comp_lsqlin_deriv (e04hev)>. However, if the evaluation of the derivatives takes more than about $4$ times as long as the evaluation of the residuals, then nag_opt_lsq_uncon_quasi_deriv_comp_lsqlin_fun (e04fcv) will usually be preferable. If in doubt, use as it is slightly more robust.

3:     $\mathrm{lsqfun}$ – function handle or string containing name of m-file

lsqfun must calculate the vector of values $f_i\left(x\right)$ and Jacobian matrix of first derivatives $\frac{\partial f_i}{\partial x_j}$ 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 nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) to terminate immediately.)

[iflag, fvec, fjac, user] = lsqfun(iflag, m, n, xc, ldfjac, user)

Input Parameters

1:     $\mathrm{iflag}$ – int64int32nag_int scalar

Will be set to $0$, $1$ or $2$.

$\mathbf{iflag}=0$

Indicates that only the residuals need to be evaluated

$\mathbf{iflag}=1$

Indicates that only the Jacobian matrix needs to be evaluated

$\mathbf{iflag}=2$

Indicates that both the residuals and the Jacobian matrix must be calculated.

If nag_opt_lsq_uncon_quasi_deriv_comp_lsqlin_deriv (e04hev) is used as nag_opt_lsq_uncon_quasi_deriv_comp (e04gb)'s lsqlin, lsqfun will always be called with iflag set to $2$.

2:     $\mathrm{m}$ – int64int32nag_int scalar

$m$, the number of residuals.

3:     $\mathrm{n}$ – int64int32nag_int scalar

$n$, the number of variables.

4:     $\mathrm{xc}\left(\mathbf{n}\right)$ – double array

The point $x$ at which the values of the $f_i$ and the $\frac{\partial f_i}{\partial x_j}$ are required.

5:     $\mathrm{ldfjac}$ – int64int32nag_int scalar

The first dimension of the array fjac.

6:     $\mathrm{user}$ – Any MATLAB object

lsqfun is called from nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) with the object supplied to nag_opt_lsq_uncon_quasi_deriv_comp (e04gb).

Output Parameters

1:     $\mathrm{iflag}$ – int64int32nag_int scalar

If it is not possible to evaluate the $f_i\left(x\right)$ 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 nag_opt_lsq_uncon_quasi_deriv_comp (e04gb). nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) will then terminate immediately, with ifail set to your setting of iflag.

2:     $\mathrm{fvec}\left(\mathbf{m}\right)$ – double array

Unless $\mathbf{iflag}=1$ on entry, or iflag is reset to a negative number, then $\mathbf{fvec}\left(i\right)$ must contain the value of $f_{\mathit{i}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$.

3:     $\mathrm{fjac}\left(\mathit{ldfjac},\mathbf{n}\right)$ – double array

Unless $\mathbf{iflag}=0$ on entry, or iflag is reset to a negative number, then $\mathbf{fjac}\left(\mathit{i},\mathit{j}\right)$ must contain the value of $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,n$.

4:     $\mathrm{user}$ – Any MATLAB object

Note:  lsqfun should be tested separately before being used in conjunction with nag_opt_lsq_uncon_quasi_deriv_comp (e04gb).

4:     $\mathrm{lsqmon}$ – function handle or string containing name of m-file

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 NAG Toolbox string nag_opt_lsq_dummy_lsqmon (e04fdz) can be used as lsqmon.

[user] = lsqmon(m, n, xc, fvec, fjac, ldfjac, s, igrade, niter, nf, user)

Input Parameters

1:     $\mathrm{m}$ – int64int32nag_int scalar

$m$, the numbers of residuals.

2:     $\mathrm{n}$ – int64int32nag_int scalar

$n$, the numbers of variables.

3:     $\mathrm{xc}\left(\mathbf{n}\right)$ – double array

The coordinates of the current point $x$.

4:     $\mathrm{fvec}\left(\mathbf{m}\right)$ – double array

The values of the residuals $f_i$ at the current point $x$.

5:     $\mathrm{fjac}\left(\mathit{ldfjac},\mathbf{n}\right)$ – double array

$\mathbf{fjac}\left(\mathit{i},\mathit{j}\right)$ 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:     $\mathrm{ldfjac}$ – int64int32nag_int scalar

The first dimension of the array fjac.

7:     $\mathrm{s}\left(\mathbf{n}\right)$ – double array

The singular values of the current Jacobian matrix. Thus s may be useful as information about the structure of your problem.

8:     $\mathrm{igrade}$ – int64int32nag_int scalar

nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) 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:     $\mathrm{niter}$ – int64int32nag_int scalar

The number of iterations which have been performed in nag_opt_lsq_uncon_quasi_deriv_comp (e04gb).

10:   $\mathrm{nf}$ – int64int32nag_int scalar

The number of evaluations of the residuals. (If nag_opt_lsq_uncon_quasi_deriv_comp_lsqlin_deriv (e04hev) is used as lsqlin, nf is also the number of evaluations of the Jacobian matrix.)

11:   $\mathrm{user}$ – Any MATLAB object

lsqmon is called from nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) with the object supplied to nag_opt_lsq_uncon_quasi_deriv_comp (e04gb).

Output Parameters

1:     $\mathrm{user}$ – Any MATLAB object

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 Accuracy. It is usually helpful to also print xc, the gradient of the sum of squares, niter and nf.

5:     $\mathrm{maxcal}$ – int64int32nag_int scalar

Suggested value:

• $\mathbf{maxcal}=75\times n$ if nag_opt_lsq_uncon_quasi_deriv_comp_lsqlin_fun (e04fcv) is used as lsqlin,
• $\mathbf{maxcal}=50\times n$ if nag_opt_lsq_uncon_quasi_deriv_comp_lsqlin_deriv (e04hev) is used as lsqlin.

Enables you to limit the number of times that lsqfun is called by nag_opt_lsq_uncon_quasi_deriv_comp (e04gb). There will be an error exit (see Error Indicators and Warnings) after maxcal calls of lsqfun.

Constraint: $\mathbf{maxcal}\ge 1$.

6:     $\mathrm{eta}$ – double scalar

Suggested value:

• $\mathbf{eta}=0.9$ if $\mathbf{n}>1$ and nag_opt_lsq_uncon_quasi_deriv_comp_lsqlin_deriv (e04hev) is used as lsqlin,
• $\mathbf{eta}=0.5$ if $\mathbf{n}>1$ and nag_opt_lsq_uncon_quasi_deriv_comp_lsqlin_fun (e04fcv) is uses as lsqlin,
• $\mathbf{eta}=0.0$ if $\mathbf{n}=1$.

Every iteration of nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) involves a linear minimization (i.e., minimization of $F\left(x^{\left(k\right)}+{\alpha }^{\left(k\right)}{p}^{\left(k\right)}\right)$ with respect to ${\alpha }^{\left(k\right)}$). eta specifies how accurately these 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 nag_opt_lsq_uncon_quasi_deriv_comp (e04gb), 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.

Constraint: $0.0\le \mathbf{eta}<1.0$.

7:     $\mathrm{xtol}$ – double scalar

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

$$‖x_{\mathrm{sol}}-x_{\mathrm{true}}‖<\mathbf{xtol}\times \left(1.0+‖x_{\mathrm{true}}‖\right)\text{,}$$

where $‖y‖=\sqrt{{\displaystyle \sum _{j=1}^n}{y}_j^2}$. For example, if the elements of $x_{\mathrm{sol}}$ are not much larger than $1.0$ in modulus and if $\mathbf{xtol}=\text{1.0e−5}$, then $x_{\mathrm{sol}}$ is usually accurate to about five decimal places. (For further details see Accuracy.)

If $F\left(x\right)$ and the variables are scaled roughly as described in Further Comments 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 $10\epsilon$, nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) will use $10\epsilon$ instead of xtol, since $10\epsilon$ is probably the smallest reasonable setting.

Constraint: $\mathbf{xtol}\ge 0.0$.

8:     $\mathrm{x}\left(\mathbf{n}\right)$ – double array

$\mathbf{x}\left(\mathit{j}\right)$ must be set to a guess at the $\mathit{j}$th component of the position of the minimum, for $\mathit{j}=1,2,\dots ,n$.

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: For n, the dimension of the array x.

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

Constraint: $1\le \mathbf{n}\le \mathbf{m}$.

2:     $\mathrm{iprint}$ – int64int32nag_int scalar

Default: $1$

The frequency with which lsqmon is to be called.

$\mathbf{iprint}>0$

lsqmon is called once every iprint iterations and just before exit from nag_opt_lsq_uncon_quasi_deriv_comp (e04gb).

$\mathbf{iprint}=0$

lsqmon is just called at the final point.

$\mathbf{iprint}<0$

lsqmon is not called at all.

iprint should normally be set to a small positive number.

3:     $\mathrm{stepmx}$ – double scalar

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.)

nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) will ensure that, for each iteration,

$$\sum _{j=1}^n{\left(x_j^{\left(k\right)}-x_j^{\left(k-1\right)}\right)}^2\le {\left(\mathbf{stepmx}\right)}^2$$

where $k$ is the iteration number. Thus, if the problem has more than one solution, nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) is most likely to find the one nearest to the starting point. On difficult problems, a realistic choice can prevent the sequence of $x^{\left(k\right)}$ 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.

Constraint: $\mathbf{stepmx}\ge \mathbf{xtol}$.

4:     $\mathrm{user}$ – Any MATLAB object

user is not used by nag_opt_lsq_uncon_quasi_deriv_comp (e04gb), but is passed to lsqfun and lsqmon. Note that for large objects it may be more efficient to use a global variable which is accessible from the m-files than to use user.

Output Parameters

1:     $\mathrm{x}\left(\mathbf{n}\right)$ – double array

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.

2:     $\mathrm{fsumsq}$ – double scalar

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.

3:     $\mathrm{fvec}\left(\mathbf{m}\right)$ – double array

The value of the residual $f_{\mathit{i}}\left(x\right)$ at the final point given in x, for $\mathit{i}=1,2,\dots ,m$.

4:     $\mathrm{fjac}\left(\mathit{ldfjac},\mathbf{n}\right)$ – double array

The value of the first derivative $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ evaluated at the final point given in x, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,n$.

5:     $\mathrm{s}\left(\mathbf{n}\right)$ – 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.

6:     $\mathrm{v}\left(\mathit{ldv},\mathbf{n}\right)$ – double array

The matrix $V$ associated with the singular value decomposition

$$J=US{V}^{\mathrm{T}}$$

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 $J^{\mathrm{T}}J$.

7:     $\mathrm{niter}$ – int64int32nag_int scalar

The number of iterations which have been performed in nag_opt_lsq_uncon_quasi_deriv_comp (e04gb).

8:     $\mathrm{nf}$ – int64int32nag_int scalar

The number of times that the residuals have been evaluated (i.e., the number of calls of lsqfun). If nag_opt_lsq_uncon_quasi_deriv_comp_lsqlin_deriv (e04hev) is used as lsqlin, nf is also the number of times that the Jacobian matrix has been evaluated.

9:     $\mathrm{user}$ – Any MATLAB object

10:   $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  $\mathbf{ifail}<0$

A negative value of ifail indicates an exit from nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) because you have set iflag negative in lsqfun. The value of ifail will be the same as your setting of iflag.

   $\mathbf{ifail}=1$

On entry,	$\mathbf{n}<1$,
or	$\mathbf{m}<\mathbf{n}$,
or	$\mathbf{maxcal}<1$,
or	$\mathbf{eta}<0.0$,
or	$\mathbf{eta}\ge 1.0$,
or	$\mathbf{xtol}<0.0$,
or	$\mathbf{stepmx}<\mathbf{xtol}$,
or	$\mathit{ldfjac}<\mathbf{m}$,
or	$\mathit{ldv}<\mathbf{n}$,
or	$\mathit{liw}<1$,
or	$\mathit{lw}<7\times \mathbf{n}+\mathbf{m}\times \mathbf{n}+2\times \mathbf{m}+\mathbf{n}\times \mathbf{n}$ when $\mathbf{n}>1$,
or	$\mathit{lw}<9+3\times \mathbf{m}$ when $\mathbf{n}=1$.

When this exit occurs, no values will have been assigned to fsumsq, or to the elements of fvec, fjac, s or v.

   $\mathbf{ifail}=2$

There have been maxcal calls of 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.

W  $\mathbf{ifail}=3$

The conditions for a minimum have not all been satisfied, but a lower point could not be found. This could be because xtol has been set so small that rounding errors in the evaluation of the residuals and derivatives make attainment of the convergence conditions impossible. See Accuracy for further information.

   $\mathbf{ifail}=4$

The method for computing the singular value decomposition of the Jacobian matrix has failed to converge in a reasonable number of sub-iterations. It may be worth applying nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) again starting with an initial approximation which is not too close to the point at which the failure occurred.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: e04gb_example

function e04gb_example


fprintf('e04gb example results\n\n');

global y t;

% Model fitting data
m = int64(15);
y = [ 0.14, 0.18, 0.22, 0.25, 0.29,...
      0.32, 0.35, 0.39, 0.37, 0.58,...
      0.73, 0.96, 1.34, 2.10, 4.39];
t = [ 1.0, 15.0, 1.0;
      2.0, 14.0, 2.0;
      3.0, 13.0, 3.0;
      4.0, 12.0, 4.0;
      5.0, 11.0, 5.0;
      6.0, 10.0, 6.0;
      7.0,  9.0, 7.0;
      8.0,  8.0, 8.0;
      9.0,  7.0, 7.0;
     10.0,  6.0, 6.0;
     11.0,  5.0, 5.0;
     12.0,  4.0, 4.0;
     13.0,  3.0, 3.0;
     14.0,  2.0, 2.0;
     15.0,  1.0, 1.0];

% Input parameters
maxcal = int64(150);
eta    = 0.9;
xtol   = sqrt(x02aj());

n      = 3;
x      = [0.5; 1; 1.5];

[x, fsumsq, fvec, fjac, s, v, niter, nf, user, ifail] = ...
     e04gb(...
           m, 'e04hev', @lsqfun, @lsqmon, maxcal, eta, xtol, x);

fprintf('Best fit model parameters are:\n');
for i = 1:n
  fprintf('        x_%d = %10.3f\n',i,x(i));
end
fprintf('\nResiduals for observed data:\n');
fprintf('  %8.4f  %8.4f  %8.4f  %8.4f  %8.4f\n',fvec);
fprintf('\nSum of squares of residuals    = %7.4f\n',fsumsq);
fprintf('Number of iterations           = %7d\n',niter);
fprintf('Number of function evaluations = %7d\n',nf);



function [iflag, fvecc, fjacc, user] = lsqfun(iflag, m, n, xc, ljc, user)

  global y t;

  fvecc = zeros(m, 1);
  fjacc = zeros(ljc, n);

  for i = 1:double(m)
    denom = xc(2)*t(i,2) + xc(3)*t(i,3);
    fvecc(i) = xc(1) + t(i,1)/denom - y(i);
    if (iflag ~= 0)
      fjacc(i,1) = 1;
      dummy = -1/(denom*denom);
      fjacc(i,2) = t(i,1)*t(i,2)*dummy;
      fjacc(i,3) = t(i,1)*t(i,3)*dummy;
    end
  end

function [user] = lsqmon(m, n, xc, fvecc, fjacc, ljc, ...
                         s, igrade, niter, nf, user)

e04gb example results

Best fit model parameters are:
        x_1 =      0.082
        x_2 =      1.133
        x_3 =      2.344

Residuals for observed data:
   -0.0059   -0.0003    0.0003    0.0065   -0.0008
   -0.0013   -0.0045   -0.0200    0.0822   -0.0182
   -0.0148   -0.0147   -0.0112   -0.0042    0.0068

Sum of squares of residuals    =  0.0082
Number of iterations           =       6
Number of function evaluations =       7