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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_roots_sys_deriv_expert (c05rc)

Purpose

nag_roots_sys_deriv_expert (c05rc) is a comprehensive function that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method. You must provide the Jacobian.

Syntax

[x, fvec, fjac, diag, nfev, njev, r, qtf, user, ifail] = c05rc(fcn, x, mode, diag, nprint, 'n', n, 'xtol', xtol, 'maxfev', maxfev, 'factor', factor, 'user', user)
[x, fvec, fjac, diag, nfev, njev, r, qtf, user, ifail] = nag_roots_sys_deriv_expert(fcn, x, mode, diag, nprint, 'n', n, 'xtol', xtol, 'maxfev', maxfev, 'factor', factor, 'user', user)

Description

The system of equations is defined as:
 $fi x1,x2,…,xn = 0 , ​ i= 1, 2, …, n .$
nag_roots_sys_deriv_expert (c05rc) is based on the MINPACK routine HYBRJ (see Moré et al. (1980)). It chooses the correction at each step as a convex combination of the Newton and scaled gradient directions. The Jacobian is updated by the rank-1 method of Broyden. At the starting point, the Jacobian is requested, but it is not asked for again until the rank-1 method fails to produce satisfactory progress. For more details see Powell (1970).

References

Moré J J, Garbow B S and Hillstrom K E (1980) User guide for MINPACK-1 Technical Report ANL-80-74 Argonne National Laboratory
Powell M J D (1970) A hybrid method for nonlinear algebraic equations Numerical Methods for Nonlinear Algebraic Equations (ed P Rabinowitz) Gordon and Breach

Parameters

Compulsory Input Parameters

1:     $\mathrm{fcn}$ – function handle or string containing name of m-file
Depending upon the value of iflag, fcn must either return the values of the functions ${f}_{i}$ at a point $x$ or return the Jacobian at $x$.
[fvec, fjac, user, iflag] = fcn(n, x, fvec, fjac, user, iflag)

Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
$n$, the number of equations.
2:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
The components of the point $x$ at which the functions or the Jacobian must be evaluated.
3:     $\mathrm{fvec}\left({\mathbf{n}}\right)$ – double array
If ${\mathbf{iflag}}=0$ or $2$, fvec contains the function values ${f}_{i}\left(x\right)$ and must not be changed.
4:     $\mathrm{fjac}\left({\mathbf{n}},{\mathbf{n}}\right)$ – double array
If ${\mathbf{iflag}}=0$, ${\mathbf{fjac}}\left(\mathit{i},\mathit{j}\right)$ contains the value of $\frac{\partial {f}_{\mathit{i}}}{\partial {x}_{\mathit{j}}}$ at the point $x$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,n$. When ${\mathbf{iflag}}=0$ or $1$, fjac must not be changed.
5:     $\mathrm{user}$ – Any MATLAB object
fcn is called from nag_roots_sys_deriv_expert (c05rc) with the object supplied to nag_roots_sys_deriv_expert (c05rc).
6:     $\mathrm{iflag}$int64int32nag_int scalar
${\mathbf{iflag}}=0$, $1$ or $2$.
${\mathbf{iflag}}=0$
x, fvec and fjac are available for printing (see nprint).
${\mathbf{iflag}}=1$
fvec is to be updated.
${\mathbf{iflag}}=2$
fjac is to be updated.

Output Parameters

1:     $\mathrm{fvec}\left({\mathbf{n}}\right)$ – double array
If ${\mathbf{iflag}}=1$ on entry, fvec must contain the function values ${f}_{i}\left(x\right)$ (unless iflag is set to a negative value by fcn).
2:     $\mathrm{fjac}\left({\mathbf{n}},{\mathbf{n}}\right)$ – double array
If ${\mathbf{iflag}}=2$ on entry, ${\mathbf{fjac}}\left(\mathit{i},\mathit{j}\right)$ 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 $\mathit{j}=1,2,\dots ,n$, (unless iflag is set to a negative value by fcn).
3:     $\mathrm{user}$ – Any MATLAB object
4:     $\mathrm{iflag}$int64int32nag_int scalar
In general, iflag should not be reset by fcn. If, however, you wish to terminate execution (perhaps because some illegal point x has been reached), then iflag should be set to a negative integer value.
2:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
An initial guess at the solution vector.
3:     $\mathrm{mode}$int64int32nag_int scalar
Indicates whether or not you have provided scaling factors in diag.
If ${\mathbf{mode}}=2$ the scaling must have been specified in diag.
Otherwise, if ${\mathbf{mode}}=1$, the variables will be scaled internally.
Constraint: ${\mathbf{mode}}=1$ or $2$.
4:     $\mathrm{diag}\left({\mathbf{n}}\right)$ – double array
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 $\mathit{i}=1,2,\dots ,n$.
5:     $\mathrm{nprint}$int64int32nag_int scalar
Indicates whether (and how often) special calls to fcn, with iflag set to $0$, are to be made for printing purposes.
${\mathbf{nprint}}\le 0$
${\mathbf{nprint}}>0$
fcn is called at the beginning of the first iteration, every nprint iterations thereafter and immediately before the return from nag_roots_sys_deriv_expert (c05rc).

Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the arrays x, diag. (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: $\sqrt{\mathbit{machine precision}}$
The accuracy in x to which the solution is required.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
3:     $\mathrm{maxfev}$int64int32nag_int scalar
Default: $100×\left({\mathbf{n}}+1\right)$
The maximum number of calls to fcn with ${\mathbf{iflag}}\ne 0$. nag_roots_sys_deriv_expert (c05rc) will exit with ${\mathbf{ifail}}={\mathbf{2}}$, if, at the end of an iteration, the number of calls to fcn exceeds maxfev.
Constraint: ${\mathbf{maxfev}}>0$.
4:     $\mathrm{factor}$ – double scalar
Default: $100.0$
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 ${\mathbf{factor}}×{‖{\mathbf{diag}}×{\mathbf{x}}‖}_{2}$ if this is nonzero; otherwise the bound is factor.)
Constraint: ${\mathbf{factor}}>0.0$.
5:     $\mathrm{user}$ – Any MATLAB object
user is not used by nag_roots_sys_deriv_expert (c05rc), but is passed to fcn. 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 estimate of the solution vector.
2:     $\mathrm{fvec}\left({\mathbf{n}}\right)$ – double array
The function values at the final point returned in x.
3:     $\mathrm{fjac}\left({\mathbf{n}},{\mathbf{n}}\right)$ – double array
The orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian.
4:     $\mathrm{diag}\left({\mathbf{n}}\right)$ – double array
The scale factors actually used (computed internally if ${\mathbf{mode}}=1$).
5:     $\mathrm{nfev}$int64int32nag_int scalar
The number of calls made to fcn to evaluate the functions.
6:     $\mathrm{njev}$int64int32nag_int scalar
The number of calls made to fcn to evaluate the Jacobian.
7:     $\mathrm{r}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – double array
The upper triangular matrix $R$ produced by the $QR$ factorization of the final approximate Jacobian, stored row-wise.
8:     $\mathrm{qtf}\left({\mathbf{n}}\right)$ – double array
The vector ${Q}^{\mathrm{T}}f$.
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

Errors or warnings detected by the function:

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

W  ${\mathbf{ifail}}=2$
There have been at least maxfev calls to fcn.
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_expert (c05rc) 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_expert (c05rc) from a different starting point may avoid the region of difficulty.
W  ${\mathbf{ifail}}=6$
iflag was set negative in fcn.
${\mathbf{ifail}}=11$
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.
${\mathbf{ifail}}=18$
Constraint: ${\mathbf{maxfev}}>0$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Accuracy

If $\stackrel{^}{x}$ is the true solution and $D$ denotes the diagonal matrix whose entries are defined by the array diag, then nag_roots_sys_deriv_expert (c05rc) tries to ensure that
 $D x-x^ 2 ≤ xtol × D x^ 2 .$
If this condition is satisfied with ${\mathbf{xtol}}={10}^{-k}$, then the larger components of $Dx$ have $k$ significant decimal digits. There is a danger that the smaller components of $Dx$ may have large relative errors, but the fast rate of convergence of nag_roots_sys_deriv_expert (c05rc) usually obviates this possibility.
If xtol is less than machine precision and the above test is satisfied with the machine precision in place of xtol, then the function exits with ${\mathbf{ifail}}={\mathbf{3}}$.
Note:  this convergence test is based purely on relative error, and may not indicate convergence if the solution is very close to the origin.
The convergence test assumes that the functions and the Jacobian are coded consistently and that the functions are reasonably well behaved. If these conditions are not satisfied, then nag_roots_sys_deriv_expert (c05rc) may incorrectly indicate convergence. The coding of the Jacobian can be checked using nag_roots_sys_deriv_check (c05zd). If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning nag_roots_sys_deriv_expert (c05rc) with a lower value for xtol.

Local workspace arrays of fixed lengths are allocated internally by nag_roots_sys_deriv_expert (c05rc). The total size of these arrays amounts to $4×n$ double elements.
The time required by nag_roots_sys_deriv_expert (c05rc) to solve a given problem depends on $n$, the behaviour of the functions, the accuracy requested and the starting point. The number of arithmetic operations executed by nag_roots_sys_deriv_expert (c05rc) is approximately $11.5×{n}^{2}$ to process each evaluation of the functions and approximately $1.3×{n}^{3}$ to process each evaluation of the Jacobian. The timing of nag_roots_sys_deriv_expert (c05rc) is strongly influenced by the time spent evaluating the functions.
Ideally the problem should be scaled so that, at the solution, the function values are of comparable magnitude.

Example

This example determines the values ${x}_{1},\dots ,{x}_{9}$ which satisfy the tridiagonal equations:
 $3-2x1x1-2x2 = -1, -xi-1+3-2xixi-2xi+1 = -1, i=2,3,…,8 -x8+3-2x9x9 = -1.$
```function c05rc_example

fprintf('c05rc example results\n\n');

% The following starting values provide a rough solution.
x = -ones(9, 1);
diag = ones(9, 1);
mode = int64(2);
nprint = int64(0);
[xOut, fvec, fjac, diag, nfev, njev, r, qtf, user, ifail] = ...
c05rc(@fcn, x, mode, diag, nprint);

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

function [fvec, fjac, user, iflag] = fcn(n, x, fvec, fjac, user, iflag)
coeff = [-1, 3, -2, -2, -1];
nd = double(n); % Can't use 64 bit integers in loops
if (iflag == 0)
% We could print fvec and fjac if we wished
elseif (iflag == 1)
fvec(1:nd) = (coeff(2)+coeff(3)*x(1:nd)).*x(1:nd) - coeff(5);
fvec(2:nd) = fvec(2:nd) + coeff(1)*x(1:(nd-1));
fvec(1:(nd-1)) = fvec(1:(nd-1)) + coeff(4)*x(2:nd);
else
fjac = zeros(nd, nd);

fjac(1,1) = coeff(2) + 2*coeff(3)*x(1);
fjac(1,2) = coeff(4);
for k = 2:nd-1
fjac(k,k-1) = coeff(1);
fjac(k,k) = coeff(2) + 2*coeff(3)*x(k);
fjac(k,k+1) = coeff(4);
end
fjac(nd,nd-1) = coeff(1);
fjac(nd,nd) = coeff(2) + 2*coeff(3)*x(nd);
end
```
```c05rc example results

Final 2-norm of the residuals =   1.1926e-08

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

```