NAG Library Function Document
nag_zero_nonlin_eqns_1 (c05tbc)
1 Purpose
nag_zero_nonlin_eqns_1 (c05tbc) 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_1 (Integer n,
double x[],
double fvec[],
double xtol,
Nag_User *comm,
NagError *fail) 

3 Description
The system of equations is defined as:
nag_zero_nonlin_eqns_1 (c05tbc) is based upon 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. Under reasonable conditions this guarantees global convergence for starting points far from the solution and a fast rate of convergence. 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:
n – IntegerInput

On entry: $n$, the number of equations.
Constraint:
${\mathbf{n}}>0$.
 2:
x[n] – doubleInput/Output

On entry: an initial guess at the solution vector.
On exit: the final estimate of the solution vector.
 3:
fvec[n] – doubleOutput

On exit: the function values at the final point,
x.
 4:
f – function, supplied by the userExternal Function

f must return the values of the
${f}_{i}$ at a point
$x$.
The specification of
f is:
 1:
n – IntegerInput

On entry: $n$, the number of equations.
 2:
x[n] – const doubleInput

On entry: the components of the point $x$ at which the functions must be evaluated.
 3:
fvec[n] – doubleOutput

On exit: the function values
${f}_{i}\left(x\right)$ (unless
userflag is set to a negative value by
f).
 4:
userflag – Integer *Input/Output

On entry: ${\mathbf{userflag}}>0$.
On exit: in general,
userflag should not be reset by
f. If, however, you wish to terminate execution (perhaps because some illegal point
x has been reached), then
userflag should be set to a negative integer. This value will be returned through
${\mathbf{fail}}\mathbf{.}\mathbf{errnum}$.
 5:
comm – Nag_User *

Pointer to a structure of type Nag_User with the following member:
 p – Pointer

On entry/exit: the pointer
$\mathbf{comm}\mathbf{\to}\mathbf{p}$ should be cast to the required type, e.g.,
struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type. (See the argument
comm below.)
 5:
xtol – doubleInput

On entry: the accuracy in
x to which the solution is required.
Suggested value:
the square root of the machine precision.
Constraint:
${\mathbf{xtol}}\ge 0.0$.
 6:
comm – Nag_User *

Pointer to a structure of type Nag_User with the following member:
 p – Pointer

On entry/exit: the pointer
$\mathbf{comm}\mathbf{\to}\mathbf{p}$, of type Pointer, allows you to communicate information to and from
f(). You must declare an object of the required type, e.g., a structure, and its address assigned to the pointer
$\mathbf{comm}\mathbf{\to}\mathbf{p}$ by means of a cast to Pointer in the calling program, e.g.,
comm.p = (Pointer)&s. The type pointer will be
void * with a C compiler that defines
void * and
char * otherwise.
 7:
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.
 NE_INT_ARG_LE
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}>0$.
 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_1 (c05tbc) from a different starting point may avoid the region of difficulty.
 NE_REAL_ARG_LT
On entry,
xtol must not be less than 0.0:
${\mathbf{xtol}}=\u2329\mathit{\text{value}}\u232a$.
 NE_TOO_MANY_FUNC_EVAL
There have been at least 200*
$\left({\mathbf{n}}+1\right)$ evaluations of
f().
Consider restarting the calculation from the point held in
x.
 NE_USER_STOP
User requested termination, user flag value $\text{}=\u2329\mathit{\text{value}}\u232a$.
 NE_XTOL_TOO_SMALL
No further improvement in the solution is possible.
xtol is too small:
${\mathbf{xtol}}=\u2329\mathit{\text{value}}\u232a$.
7 Accuracy
If
$\hat{x}$ is the true solution, nag_zero_nonlin_eqns_1 (c05tbc) 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_1 (c05tbc) usually avoids 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
NE_XTOL_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 test assumes that the functions are reasonably well behaved. If this condition is not satisfied, then nag_zero_nonlin_eqns_1 (c05tbc) may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning nag_zero_nonlin_eqns_1 (c05tbc) with a tighter tolerance.
The time required by nag_zero_nonlin_eqns_1 (c05tbc) 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_1 (c05tbc) to process each call of
f is about
$11.5\times {n}^{2}$. Unless
f can be evaluated quickly, the timing of nag_zero_nonlin_eqns_1 (c05tbc) will be strongly influenced by the time spent in
f.
Ideally the problem should be scaled so that, at the solution, the function values are of comparable magnitude.
9 Example
This example determines the values
${x}_{1},\dots ,{x}_{9}$ which satisfy the tridiagonal equations:
9.1 Program Text
Program Text (c05tbce.c)
9.2 Program Data
None.
9.3 Program Results
Program Results (c05tbce.r)