NAG Library Function Document
nag_zero_nonlin_eqns_easy (c05qbc)
1 Purpose
nag_zero_nonlin_eqns_easy (c05qbc) is an easytouse function that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.
2 Specification
#include <nag.h> 
#include <nagc05.h> 
void 
nag_zero_nonlin_eqns_easy (
Integer n,
double x[],
double fvec[],
double xtol,
Nag_Comm *comm,
NagError *fail) 

3 Description
The system of equations is defined as:
nag_zero_nonlin_eqns_easy (c05qbc) is based on the MINPACK routine HYBRD1 (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 approximated by forward differences, but these are not used 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 userExternal Function

fcn must return the values of the functions
${f}_{i}$ at a point
$x$.
The specification of
fcn is:
void 
fcn (Integer n,
const double x[],
double fvec[],
Nag_Comm *comm, Integer *iflag)


 1:
$\mathbf{n}$ – IntegerInput

On entry: $n$, the number of equations.
 2:
$\mathbf{x}\left[{\mathbf{n}}\right]$ – const doubleInput

On entry: the components of the point $x$ at which the functions must be evaluated.
 3:
$\mathbf{fvec}\left[{\mathbf{n}}\right]$ – doubleOutput

On exit: the function values
${f}_{i}\left(x\right)$ (unless
iflag is set to a negative value by
fcn).
 4:
$\mathbf{comm}$ – 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 nag_zero_nonlin_eqns_easy (c05qbc) you may allocate memory and initialize these pointers with various quantities for use by
fcn when called from nag_zero_nonlin_eqns_easy (c05qbc) (see
Section 3.2.1.1 in the Essential Introduction).
 5:
$\mathbf{iflag}$ – Integer *Input/Output

On entry: ${\mathbf{iflag}}>0$.
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), then
iflag should be set to a negative integer.
 2:
$\mathbf{n}$ – IntegerInput

On entry: $n$, the number of equations.
Constraint:
${\mathbf{n}}>0$.
 3:
$\mathbf{x}\left[{\mathbf{n}}\right]$ – doubleInput/Output

On entry: an initial guess at the solution vector.
On exit: the final estimate of the solution vector.
 4:
$\mathbf{fvec}\left[{\mathbf{n}}\right]$ – doubleOutput

On exit: the function values at the final point returned in
x.
 5:
$\mathbf{xtol}$ – doubleInput

On entry: the accuracy in
x to which the solution is required.
Suggested value:
$\sqrt{\epsilon}$, where
$\epsilon $ is the
machine precision returned by
nag_machine_precision (X02AJC).
Constraint:
${\mathbf{xtol}}\ge 0.0$.
 6:
$\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see
Section 3.2.1.1 in the Essential Introduction).
 7:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.2.1.2 in the Essential Introduction 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.
An unexpected error has been triggered by this function. Please contact
NAG.
See
Section 3.6.6 in the Essential Introduction 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 nag_zero_nonlin_eqns_easy (c05qbc) 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 3.6.5 in the Essential Introduction 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
$200\times \left({\mathbf{n}}+1\right)$ 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, nag_zero_nonlin_eqns_easy (c05qbc) 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 nag_zero_nonlin_eqns_easy (c05qbc) 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 are reasonably well behaved. If this condition is not satisfied, then nag_zero_nonlin_eqns_easy (c05qbc) may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning nag_zero_nonlin_eqns_easy (c05qbc) with a lower value for
xtol.
8 Parallelism and Performance
nag_zero_nonlin_eqns_easy (c05qbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zero_nonlin_eqns_easy (c05qbc) 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 nag_zero_nonlin_eqns_easy (c05qbc). The total size of these arrays amounts to $n\times \left(3\times n+13\right)/2$ double elements.
The time required by nag_zero_nonlin_eqns_easy (c05qbc) 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_zero_nonlin_eqns_easy (c05qbc) to process each evaluation of the functions is approximately $11.5\times {n}^{2}$. The timing of nag_zero_nonlin_eqns_easy (c05qbc) 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
Program Text (c05qbce.c)
10.2 Program Data
None.
10.3 Program Results
Program Results (c05qbce.r)