NAG CL Interface
c05rbc (sys_deriv_easy)
1
Purpose
c05rbc is an easytouse function that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method. You must provide the Jacobian.
2
Specification
void 
c05rbc (
Integer n,
double x[],
double fvec[],
double fjac[],
double xtol,
Nag_Comm *comm,
NagError *fail) 

The function may be called by the names: c05rbc, nag_roots_sys_deriv_easy or nag_zero_nonlin_eqns_deriv_easy.
3
Description
The system of equations is defined as:
c05rbc is based on the MINPACK routine HYBRJ1 (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 rank1 method of Broyden. At the starting point, the Jacobian is requested, but it is not asked for again until the rank1 method fails to produce satisfactory progress. For more details see
Powell (1970).
4
References
Moré J J, Garbow B S and Hillstrom K E (1980) User guide for MINPACK1 Technical Report ANL8074 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
5
Arguments

1:
$\mathbf{fcn}$ – function, supplied by the user
External Function

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
$x$.
The specification of
fcn is:
void 
fcn (Integer n,
const double x[],
double fvec[],
double fjac[],
Nag_Comm *comm, Integer *iflag)



1:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of equations.

2:
$\mathbf{x}\left[{\mathbf{n}}\right]$ – const double
Input

On entry: the components of the point $x$ at which the functions or the Jacobian must be evaluated.

3:
$\mathbf{fvec}\left[{\mathbf{n}}\right]$ – double
Input/Output

On entry: if
${\mathbf{iflag}}=2$,
fvec contains the function values
${f}_{i}\left(x\right)$ and must not be changed.
On exit: 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).

4:
$\mathbf{fjac}\left[{\mathbf{n}}\times {\mathbf{n}}\right]$ – double
Input/Output

Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{fjac}}\left[\left(j1\right)\times {\mathbf{n}}+i1\right]$.
On entry: if
${\mathbf{iflag}}=1$,
fjac 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
$\mathit{j}=1,2,\dots ,n$, and must not be changed.
On exit: if
${\mathbf{iflag}}=2$ on entry,
${\mathbf{fjac}}\left[\left(\mathit{j}1\right)\times {\mathbf{n}}+\mathit{i}1\right]$ 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).

5:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
fcn.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
c05rbc you may allocate memory and initialize these pointers with various quantities for use by
fcn when called from
c05rbc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).

6:
$\mathbf{iflag}$ – Integer *
Input/Output

On entry:
${\mathbf{iflag}}=1$ or
$2$.
 ${\mathbf{iflag}}=1$
 fvec is to be updated.
 ${\mathbf{iflag}}=2$
 fjac is to be updated.
On exit: 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),
iflag should be set to a negative integer.
Note: fcn should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
c05rbc. If your code inadvertently
does return any NaNs or infinities,
c05rbc is likely to produce unexpected results.

2:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of equations.
Constraint:
${\mathbf{n}}>0$.

3:
$\mathbf{x}\left[{\mathbf{n}}\right]$ – double
Input/Output

On entry: an initial guess at the solution vector.
On exit: the final estimate of the solution vector.

4:
$\mathbf{fvec}\left[{\mathbf{n}}\right]$ – double
Output

On exit: the function values at the final point returned in
x.

5:
$\mathbf{fjac}\left[{\mathbf{n}}\times {\mathbf{n}}\right]$ – double
Output

Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{fjac}}\left[\left(j1\right)\times {\mathbf{n}}+i1\right]$.
On exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian, stored by columns.

6:
$\mathbf{xtol}$ – double
Input

On entry: the accuracy in
x to which the solution is required.
Suggested value:
$\sqrt{\epsilon}$, where
$\epsilon $ is the
machine precision returned by
X02AJC.
Constraint:
${\mathbf{xtol}}\ge 0.0$.

7:
$\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).

8:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}>0$.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_NO_IMPROVEMENT

The iteration is not making good progress. This failure exit may indicate that the system does not have a zero, or that the solution is very close to the origin (see
Section 7). Otherwise, rerunning
c05rbc from a different starting point may avoid the region of difficulty.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_REAL

On entry, ${\mathbf{xtol}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
 NE_TOO_MANY_FEVALS

There have been at least
$100\times \left({\mathbf{n}}+1\right)$ calls to
fcn. Consider restarting the calculation from the point held in
x.
 NE_TOO_SMALL

No further improvement in the solution is possible.
xtol is too small:
${\mathbf{xtol}}=\u2329\mathit{\text{value}}\u232a$.
 NE_USER_STOP

iflag was set negative in
fcn.
${\mathbf{iflag}}=\u2329\mathit{\text{value}}\u232a$.
7
Accuracy
If
$\hat{x}$ is the true solution,
c05rbc tries to ensure that
If this condition is satisfied with
${\mathbf{xtol}}={10}^{k}$, then the larger components of
$x$ have
$k$ significant decimal digits. There is a danger that the smaller components of
$x$ may have large relative errors, but the fast rate of convergence of
c05rbc 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
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_TOO_SMALL.
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
c05rbc may incorrectly indicate convergence. The coding of the Jacobian can be checked using
c05zdc. If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning
c05rbc with a lower value for
xtol.
8
Parallelism and Performance
c05rbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05rbc 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 function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
Local workspace arrays of fixed lengths are allocated internally by c05rbc. The total size of these arrays amounts to $n\times \left(n+13\right)/2$ double elements.
The time required by c05rbc 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 c05rbc is approximately $11.5\times {n}^{2}$ to process each evaluation of the functions and approximately $1.3\times {n}^{3}$ to process each evaluation of the Jacobian. The timing of c05rbc 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.
10
Example
This example determines the values
${x}_{1},\dots ,{x}_{9}$ which satisfy the tridiagonal equations:
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results