# NAG Library Function Document

## 1Purpose

nag_zero_nonlin_eqns_expert (c05qcc) is a comprehensive function that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.

## 2Specification

 #include #include
void  nag_zero_nonlin_eqns_expert (
 void (*fcn)(Integer n, const double x[], double fvec[], Nag_Comm *comm, Integer *iflag),
Integer n, double x[], double fvec[], double xtol, Integer maxfev, Integer ml, Integer mu, double epsfcn, Nag_ScaleType scale_mode, double diag[], double factor, Integer nprint, Integer *nfev, double fjac[], double r[], double qtf[], Nag_Comm *comm, NagError *fail)

## 3Description

The system of equations is defined as:
 $fi x1,x2,…,xn = 0 , ​ i= 1, 2, …, n .$
nag_zero_nonlin_eqns_expert (c05qcc) is based on the MINPACK routine HYBRD (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).
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

## 5Arguments

1:    $\mathbf{fcn}$function, supplied by the userExternal Function
fcn must return the values of the functions ${f}_{i}$ at a point $x$, unless ${\mathbf{iflag}}=0$ on entry to nag_zero_nonlin_eqns_expert (c05qcc).
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:    $\mathbf{fvec}\left[{\mathbf{n}}\right]$doubleInput/Output
On entry: if ${\mathbf{iflag}}=0$, fvec contains the function values ${f}_{i}\left(x\right)$ and must not be changed.
On exit: if ${\mathbf{iflag}}>0$ on entry, fvec must contain the function values ${f}_{i}\left(x\right)$ (unless iflag is set to a negative value by fcn).
4:    $\mathbf{comm}$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_expert (c05qcc) you may allocate memory and initialize these pointers with various quantities for use by fcn when called from nag_zero_nonlin_eqns_expert (c05qcc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
5:    $\mathbf{iflag}$Integer *Input/Output
On entry: ${\mathbf{iflag}}\ge 0$.
${\mathbf{iflag}}=0$
x and fvec are available for printing (see nprint).
${\mathbf{iflag}}>0$
fvec must 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 floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_zero_nonlin_eqns_expert (c05qcc). If your code inadvertently does return any NaNs or infinities, nag_zero_nonlin_eqns_expert (c05qcc) is likely to produce unexpected results.
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:    $\mathbf{maxfev}$IntegerInput
On entry: the maximum number of calls to fcn with ${\mathbf{iflag}}\ne 0$. nag_zero_nonlin_eqns_expert (c05qcc) will exit with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_TOO_MANY_FEVALS, if, at the end of an iteration, the number of calls to fcn exceeds maxfev.
Suggested value: ${\mathbf{maxfev}}=200×\left({\mathbf{n}}+1\right)$.
Constraint: ${\mathbf{maxfev}}>0$.
7:    $\mathbf{ml}$IntegerInput
On entry: the number of subdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set ${\mathbf{ml}}={\mathbf{n}}-1$.)
Constraint: ${\mathbf{ml}}\ge 0$.
8:    $\mathbf{mu}$IntegerInput
On entry: the number of superdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set ${\mathbf{mu}}={\mathbf{n}}-1$.)
Constraint: ${\mathbf{mu}}\ge 0$.
9:    $\mathbf{epsfcn}$doubleInput
On entry: a rough estimate of the largest relative error in the functions. It is used in determining a suitable step for a forward difference approximation to the Jacobian. If epsfcn is less than machine precision (returned by nag_machine_precision (X02AJC)) then machine precision is used. Consequently a value of $0.0$ will often be suitable.
Suggested value: ${\mathbf{epsfcn}}=0.0$.
10:  $\mathbf{scale_mode}$Nag_ScaleTypeInput
On entry: indicates whether or not you have provided scaling factors in diag.
If ${\mathbf{scale_mode}}=\mathrm{Nag_ScaleProvided}$, the scaling must have been specified in diag.
Otherwise, if ${\mathbf{scale_mode}}=\mathrm{Nag_NoScaleProvided}$, the variables will be scaled internally.
Constraint: ${\mathbf{scale_mode}}=\mathrm{Nag_NoScaleProvided}$ or $\mathrm{Nag_ScaleProvided}$.
11:  $\mathbf{diag}\left[{\mathbf{n}}\right]$doubleInput/Output
On entry: if ${\mathbf{scale_mode}}=\mathrm{Nag_ScaleProvided}$, diag must contain multiplicative scale factors for the variables.
If ${\mathbf{scale_mode}}=\mathrm{Nag_NoScaleProvided}$, diag need not be set.
Constraint: if ${\mathbf{scale_mode}}=\mathrm{Nag_ScaleProvided}$, ${\mathbf{diag}}\left[\mathit{i}-1\right]>0.0$, for $\mathit{i}=1,2,\dots ,n$.
On exit: the scale factors actually used (computed internally if ${\mathbf{scale_mode}}=\mathrm{Nag_NoScaleProvided}$).
12:  $\mathbf{factor}$doubleInput
On entry: 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.)
Suggested value: ${\mathbf{factor}}=100.0$.
Constraint: ${\mathbf{factor}}>0.0$.
13:  $\mathbf{nprint}$IntegerInput
On entry: 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_zero_nonlin_eqns_expert (c05qcc).
14:  $\mathbf{nfev}$Integer *Output
On exit: the number of calls made to fcn with ${\mathbf{iflag}}>0$.
15:  $\mathbf{fjac}\left[{\mathbf{n}}×{\mathbf{n}}\right]$doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{fjac}}\left[\left(j-1\right)×{\mathbf{n}}+i-1\right]$.
On exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian.
16:  $\mathbf{r}\left[{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right]$doubleOutput
On exit: the upper triangular matrix $R$ produced by the $QR$ factorization of the final approximate Jacobian, stored row-wise.
17:  $\mathbf{qtf}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the vector ${Q}^{\mathrm{T}}f$.
18:  $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
19:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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_DIAG_ELEMENTS
On entry, ${\mathbf{scale_mode}}=\mathrm{Nag_ScaleProvided}$ and diag contained a non-positive element.
NE_INT
On entry, ${\mathbf{maxfev}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{maxfev}}>0$.
On entry, ${\mathbf{ml}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ml}}\ge 0$.
On entry, ${\mathbf{mu}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mu}}\ge 0$.
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.
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, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ iterations.
The iteration is not making good progress, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ Jacobian evaluations.
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, ${\mathbf{factor}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{factor}}>0.0$.
On entry, ${\mathbf{xtol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
NE_TOO_MANY_FEVALS
There have been at least maxfev calls to fcn: ${\mathbf{maxfev}}=〈\mathit{\text{value}}〉$. Consider restarting the calculation from the final 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. ${\mathbf{iflag}}=〈\mathit{\text{value}}〉$.

## 7Accuracy

If $\stackrel{^}{x}$ is the true solution and $D$ denotes the diagonal matrix whose entries are defined by the array diag, then nag_zero_nonlin_eqns_expert (c05qcc) 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_zero_nonlin_eqns_expert (c05qcc) 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_expert (c05qcc) may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning nag_zero_nonlin_eqns_expert (c05qcc) with a lower value for xtol.

## 8Parallelism and Performance

nag_zero_nonlin_eqns_expert (c05qcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zero_nonlin_eqns_expert (c05qcc) 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.

Local workspace arrays of fixed lengths are allocated internally by nag_zero_nonlin_eqns_expert (c05qcc). The total size of these arrays amounts to $4×n$ double elements.
The time required by nag_zero_nonlin_eqns_expert (c05qcc) 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_expert (c05qcc) to process each evaluation of the functions is approximately $11.5×{n}^{2}$. The timing of nag_zero_nonlin_eqns_expert (c05qcc) 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.
The number of function evaluations required to evaluate the Jacobian may be reduced if you can specify ml and mu accurately.

## 10Example

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

### 10.1Program Text

Program Text (c05qcce.c)

None.

### 10.3Program Results

Program Results (c05qcce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017