NAG Toolbox: nag_roots_sys_deriv_rcomm (c05rd)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

On initial entry: must have the value $0$.

Constraint: $\mathbf{irevcm}=0$, $1$, $2$ or $3$.

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

On initial entry: an initial guess at the solution vector.

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

On initial entry: need not be set.

On intermediate re-entry: if $\mathbf{irevcm}\ne 2$, fvec must not be changed.

If $\mathbf{irevcm}=2$, fvec must be set to the values of the functions computed at the current point x.

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

On initial entry: need not be set.

On intermediate re-entry: if $\mathbf{irevcm}\ne 3$, fjac must not be changed.

If $\mathbf{irevcm}=3$, $\mathbf{fjac}\left(\mathit{i},\mathit{j}\right)$ must contain the value of $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,n$.

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

On initial entry: indicates whether or not you have provided scaling factors in diag.

If $\mathbf{mode}=2$ the scaling must have been supplied in diag.

Otherwise, if $\mathbf{mode}=1$, the variables will be scaled internally.

Constraint: $\mathbf{mode}=1$ or $2$.

6:     $\mathrm{diag}\left(\mathbf{n}\right)$ – double array

On initial entry: if $\mathbf{mode}=2$, diag must contain multiplicative scale factors for the variables.

If $\mathbf{mode}=1$, diag need not be set.

Constraint: if $\mathbf{mode}=2$, $\mathbf{diag}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,n$.

7:     $\mathrm{r}\left(\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$ – double array

On initial entry: need not be set.

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

On initial entry: need not be set.

9:     $\mathrm{iwsav}\left(17\right)$ – int64int32nag_int array

10:   $\mathrm{rwsav}\left(4\times \mathbf{n}+10\right)$ – double array

The arrays iwsav and rwsav must not be altered between calls to nag_roots_sys_deriv_rcomm (c05rd).

Optional Input Parameters

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

Default: the dimension of the arrays x, fvec, diag, qtf and the first dimension of the array fjac and the second dimension of the array fjac. (An error is raised if these dimensions are not equal.)

$n$, the number of equations.

Constraint: $\mathbf{n}>0$.

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

Suggested value: $\sqrt{\epsilon }$, where $\epsilon$ is the machine precision returned by nag_machine_precision (x02aj).

Default: $\sqrt{\mathit{machine\; precision}}$

On initial entry: the accuracy in x to which the solution is required.

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

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

Default: $100.0$

On initial entry: a quantity to be used in determining the initial step bound. In most cases, factor should lie between $0.1$ and $100.0$. (The step bound is $\mathbf{factor}\times {‖\mathbf{diag}\times \mathbf{x}‖}_2$ if this is nonzero; otherwise the bound is factor.)

Constraint: $\mathbf{factor}>0.0$.

Output Parameters

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

On intermediate exit: specifies what action you must take before re-entering nag_roots_sys_deriv_rcomm (c05rd) with irevcm unchanged. The value of irevcm should be interpreted as follows:

$\mathbf{irevcm}=1$

Indicates the start of a new iteration. No action is required by you, but x and fvec are available for printing.

$\mathbf{irevcm}=2$

Indicates that before re-entry to nag_roots_sys_deriv_rcomm (c05rd), fvec must contain the function values $f_i\left(x\right)$.

$\mathbf{irevcm}=3$

Indicates that before re-entry to nag_roots_sys_deriv_rcomm (c05rd), $\mathbf{fjac}\left(\mathit{i},\mathit{j}\right)$ must contain the value of $\frac{\partial f_{\mathit{i}}}{\partial x_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,n$.

On final exit: $\mathbf{irevcm}=0$, and the algorithm has terminated.

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

On intermediate exit: contains the current point.

On final exit: the final estimate of the solution vector.

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

On final exit: the function values at the final point, x.

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

On final exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian.

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

On intermediate exit: diag must not be changed.

On final exit: the scale factors actually used (computed internally if $\mathbf{mode}=1$).

6:     $\mathrm{r}\left(\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$ – double array

On intermediate exit: must not be changed.

On final exit: the upper triangular matrix $R$ produced by the $QR$ factorization of the final approximate Jacobian, stored row-wise.

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

On intermediate exit: must not be changed.

On final exit: the vector $Q^{\mathrm{T}}f$.

8:     $\mathrm{iwsav}\left(17\right)$ – int64int32nag_int array

9:     $\mathrm{rwsav}\left(4\times \mathbf{n}+10\right)$ – double array

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

On final exit: $\mathbf{ifail}=\mathbf{0}$ 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.

   $\mathbf{ifail}=2$

Constraint: $\mathbf{irevcm}=0$, $1$, $2$ or $3$.

W  $\mathbf{ifail}=3$

No further improvement in the solution is possible.

W  $\mathbf{ifail}=4$

The iteration is not making good progress, as measured by the improvement from the last $\_$ Jacobian evaluations. This failure exit may indicate that the system does not have a zero, or that the solution is very close to the origin (see Accuracy). Otherwise, rerunning nag_roots_sys_deriv_rcomm (c05rd) from a different starting point may avoid the region of difficulty.

W  $\mathbf{ifail}=5$

The iteration is not making good progress, as measured by the improvement from the last $\_$ iterations. This failure exit may indicate that the system does not have a zero, or that the solution is very close to the origin (see Accuracy). Otherwise, rerunning nag_roots_sys_deriv_rcomm (c05rd) from a different starting point may avoid the region of difficulty.

   $\mathbf{ifail}=11$

Constraint: $\mathbf{n}>0$.

   $\mathbf{ifail}=12$

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

   $\mathbf{ifail}=13$

Constraint: $\mathbf{mode}=1$ or $2$.

   $\mathbf{ifail}=14$

Constraint: $\mathbf{factor}>0.0$.

   $\mathbf{ifail}=15$

On entry, $\mathbf{mode}=2$ and diag contained a non-positive element.

   $\mathbf{ifail}=-99$

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: c05rd_example

function c05rd_example


fprintf('c05rd example results\n\n');

% The following starting values provide a rough solution.
x = -ones(9, 1);
diagnl = ones(9,1);
mode = int64(2);
irevcm = int64(0);
fjac = zeros(9, 9);
fvec = zeros(9, 1);
qtf = zeros(9, 1);
r = zeros(45, 1);
rwsav = zeros(46, 1);
iwsav = zeros(17, 1, 'int64');

icount = int64(0);
first = true;
while (irevcm ~= 0 || first )
  first = false;
  [irevcm, x, fvec, fjac, diag, r, qtf, iwsav, rwsav, ifail] = ...
      c05rd(irevcm, x, fvec, fjac, mode, diagnl, r, qtf, iwsav, rwsav);

  switch irevcm
    case {1}
      icount = icount + 1;
    case {2}
      % Evaluate functions at given point
      fvec(1:9) = (3.0-2.0.*x).*x + 1.0;
      fvec(2:9) = fvec(2:9) - x(1:8);
      fvec(1:8) = fvec(1:8) - 2.0.*x(2:9);
    case {3}
      % Evaluate Jacobian at current point
      fjac = zeros(9, 9);
      for k = 1:9
        fjac(k,k) = 3 - 4*x(k);
        if (k ~= 1)
          fjac(k,k-1) = -1;
        end
        if (k ~= 9)
          fjac(k,k+1) = -2;
        end
      end
  end
end

switch ifail
  case {0}
    fprintf('\nFinal 2-norm of the residuals after %d iterations = %12.4e\n',...
      icount, norm(fvec));
    fprintf('\nFinal approximate solution\n');
    disp(x);
  case {3, 4, 5}
    fprintf('\nApproximate solution\n');
    disp(x);
end

c05rd example results


Final 2-norm of the residuals after 11 iterations =   1.1926e-08

Final approximate solution
   -0.5707
   -0.6816
   -0.7017
   -0.7042
   -0.7014
   -0.6919
   -0.6658
   -0.5960
   -0.4164