c05 Chapter Contents
c05 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zero_nonlin_eqns_easy (c05qbc)

## 1  Purpose

nag_zero_nonlin_eqns_easy (c05qbc) is an easy-to-use function that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.

## 2  Specification

 #include #include
void  nag_zero_nonlin_eqns_easy (
 void (*fcn)(Integer n, const double x[], double fvec[], Nag_Comm *comm, Integer *iflag),
Integer n, double x[], double fvec[], double xtol, Nag_Comm *comm, NagError *fail)

## 3  Description

The system of equations is defined as:
 $fi x1,x2,…,xn = 0 , ​ i= 1, 2, …, n .$
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 rank-1 method of Broyden. At the starting point, the Jacobian is approximated by forward differences, but these are not used again until the rank-1 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 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

## 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.
userdouble *
iuserInteger *
pPointer
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 2.3.1.1 in How to Use the NAG Library and its Documentation).
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 2.3.1.1 in How to Use the NAG Library and its Documentation).
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
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 2.7.6 in How to Use the NAG Library and its Documentation 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 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
NE_TOO_MANY_FEVALS
There have been at least $200×\left({\mathbf{n}}+1\right)$ 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: ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.
NE_USER_STOP
iflag was set negative in fcn. ${\mathbf{iflag}}=〈\mathit{\text{value}}〉$.

## 7  Accuracy

If $\stackrel{^}{x}$ is the true solution, nag_zero_nonlin_eqns_easy (c05qbc) tries to ensure that
 $x-x^ 2 ≤ xtol × x^ 2 .$
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 implementation-specific information.

## 9  Further Comments

Local workspace arrays of fixed lengths are allocated internally by nag_zero_nonlin_eqns_easy (c05qbc). The total size of these arrays amounts to $n×\left(3×n+13\right)/2$ 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 $11.5×{n}^{2}$. 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:
 $3-2x1x1-2x2 = -1, -xi-1+3-2xixi-2xi+1 = -1, i=2,3,…,8 -x8+3-2x9x9 = -1.$

### 10.1  Program Text

Program Text (c05qbce.c)

None.

### 10.3  Program Results

Program Results (c05qbce.r)