NAG Toolbox: nag_opt_lsq_uncon_mod_deriv2_comp (e04he)

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:     $\mathrm{lsqfun}$ – 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_mod_deriv2_comp (e04he) to terminate immediately.)

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

Input Parameters

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

To lsqfun, iflag will be set to $2$.

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

$m$, the numbers of residuals.

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

$n$, the numbers 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_mod_deriv2_comp (e04he) with the object supplied to nag_opt_lsq_uncon_mod_deriv2_comp (e04he).

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 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_mod_deriv2_comp (e04he). nag_opt_lsq_uncon_mod_deriv2_comp (e04he) will then terminate immediately, with ifail set to your setting of iflag.

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

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 $\mathit{i}=1,2,\dots ,m$.

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

Unless iflag is reset to a negative number, $\mathbf{fjac}\left(\mathit{i},\mathit{j}\right)$ 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_mod_deriv2_comp (e04he).

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

lsqhes must calculate the elements of the symmetric matrix

$$B\left(x\right)=\sum _{i=1}^m{f}_i\left(x\right)G_i\left(x\right)\text{,}$$

at any point $x$, where $G_i\left(x\right)$ is the Hessian matrix of $f_i\left(x\right)$. (As with lsqfun, there is the option of causing nag_opt_lsq_uncon_mod_deriv2_comp (e04he) to terminate immediately.)

[iflag, b, user] = lsqhes(iflag, m, n, fvec, xc, lb, user)

Input Parameters

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

Is set to a non-negative number.

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

$m$, the numbers of residuals.

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

$n$, the numbers of variables.

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

The value of the residual $f_{\mathit{i}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,m$, so that the values of the $f_{\mathit{i}}$ can be used in the calculation of the elements of b.

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

The point $x$ at which the elements of b are to be evaluated.

6:     $\mathrm{lb}$ – int64int32nag_int scalar

The length of the array b.

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

lsqhes is called from nag_opt_lsq_uncon_mod_deriv2_comp (e04he) with the object supplied to nag_opt_lsq_uncon_mod_deriv2_comp (e04he).

Output Parameters

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

If lsqhes resets iflag to some negative number, nag_opt_lsq_uncon_mod_deriv2_comp (e04he) will terminate immediately, with ifail set to your setting of iflag.

2:     $\mathrm{b}\left(\mathbf{lb}\right)$ – double array

Unless iflag is reset to a negative number, b must contain the lower triangle of the matrix $\mathbf{b}\left(x\right)$, evaluated at the point $x$, stored by rows. (The upper triangle is not required because the matrix is symmetric.) More precisely, $\mathbf{b}\left(\mathit{j}\left(\mathit{j}-1\right)/2+\mathit{k}\right)$ must contain ${\displaystyle \sum _{\mathit{i}=1}^m}{f}_{\mathit{i}}\frac{{\partial }^2{f}_{\mathit{i}}}{\partial x_{\mathit{j}}\partial x_{\mathit{k}}}$ evaluated at the point $x$, for $\mathit{j}=1,2,\dots ,n$ and $\mathit{k}=1,2,\dots ,\mathit{j}$.

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

Note:  lsqhes should be tested separately before being used in conjunction with nag_opt_lsq_uncon_mod_deriv2_comp (e04he).

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 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. (If $\mathbf{iprint}>0$, lsqmon is called at the initial point before the singular values have been calculated, so the elements of s are set to zero for the first call of lsqmon.)

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

nag_opt_lsq_uncon_mod_deriv2_comp (e04he) 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_mod_deriv2_comp (e04he).

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

The number of times that lsqfun has been called so far. Thus nf gives the number of evaluations of the residuals and the Jacobian matrix.

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

lsqmon is called from nag_opt_lsq_uncon_mod_deriv2_comp (e04he) with the object supplied to nag_opt_lsq_uncon_mod_deriv2_comp (e04he).

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:     $\mathrm{xtol}$ – 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_mod_deriv2_comp (e04he) will use $10\epsilon$ instead of xtol, since $10\epsilon$ is probably the smallest reasonable setting.

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

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

Specifies 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_mod_deriv2_comp (e04he).

$\mathbf{iprint}=0$

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:     $\mathrm{maxcal}$ – int64int32nag_int scalar

Default: $\mathbf{maxcal}=50\times n$

This argument is present so as to enable you to limit the number of times that lsqfun is called by nag_opt_lsq_uncon_mod_deriv2_comp (e04he). There will be an error exit (see Error Indicators and Warnings) after maxcal calls of lsqfun.

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

4:     $\mathrm{eta}$ – double scalar

Suggested value: $\mathbf{eta}=0.5$ ($\mathbf{eta}=0.0$ if $\mathbf{n}=1$).

Default:

• if $\mathbf{n}=1$, $0.0$;
• otherwise $0.5$.

Every iteration of nag_opt_lsq_uncon_mod_deriv2_comp (e04he) 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 must lie in the range $0.0\le \mathbf{eta}<1.0$, and 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_mod_deriv2_comp (e04he), 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.

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

5:     $\mathrm{stepmx}$ – 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_mod_deriv2_comp (e04he) will ensure that, for each iteration

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

where $k$ is the iteration number. Thus, if the problem has more than one solution, nag_opt_lsq_uncon_mod_deriv2_comp (e04he) 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: $\mathbf{stepmx}\ge \mathbf{xtol}$.

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

user is not used by nag_opt_lsq_uncon_mod_deriv2_comp (e04he), but is passed to lsqfun, lsqhes 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_mod_deriv2_comp (e04he).

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

The number of times that the residuals and Jacobian matrix have been evaluated (i.e., number of calls of lsqfun).

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_mod_deriv2_comp (e04he) because you have set iflag negative in lsqfun or lsqhes. 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.

   $\mathbf{ifail}=4$

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_mod_deriv2_comp (e04he) 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: e04he_example

function e04he_example


fprintf('e04he 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];

% Initial guess
n = 3;
x = [0.5;  1;  1.5];

xtol = sqrt(x02aj());

[x, fsumsq, fvec, fjac, s, v, niter, nf, user, ifail] = ...
     e04he(...
           m, @lsqfun, @lsqhes, @lsqmon, 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: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 [iflag, b, user] = lsqhes(iflag, m, n, fvecc, xc, lb, user)

  global y t;

  b = zeros(lb, 1);

  sum22 = 0;
  sum32 = 0;
  sum33 = 0;
  for i = 1:double(m)
     dummy = 2*t(i,1)/(xc(2)*t(i,2)+xc(3)*t(i,3))^3;
     sum22 = sum22 + fvecc(i)*dummy*t(i,2)^2;
     sum32 = sum32 + fvecc(i)*dummy*t(i,2)*t(i,3);
     sum33 = sum33 + fvecc(i)*dummy*t(i,3)^2;
  end
  b(3) = sum22;
  b(5) = sum32;
  b(6) = sum33;


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

e04he 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           =       5
Number of function evaluations =       6