c05 Chapter Contents
c05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zero_nonlin_eqns_deriv_expert (c05rcc)

## 1  Purpose

nag_zero_nonlin_eqns_deriv_expert (c05rcc) is a comprehensive 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

 #include #include
void  nag_zero_nonlin_eqns_deriv_expert (
 void (*fcn)(Integer n, const double x[], double fvec[], double fjac[], Nag_Comm *comm, Integer *iflag),
Integer n, double x[], double fvec[], double fjac[], double xtol, Integer maxfev, Nag_ScaleType scale_mode, double diag[], double factor, Integer nprint, Integer *nfev, Integer *njev, double r[], double qtf[], 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_deriv_expert (c05rcc) is based on the MINPACK routine HYBRJ (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 requested, but it is not asked for 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:     fcnfunction, supplied by the userExternal 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:     nIntegerInput
On entry: $n$, the number of equations.
2:     x[n]const doubleInput
On entry: the components of the point $x$ at which the functions or the Jacobian must be evaluated.
3:     fvec[n]doubleInput/Output
On entry: if ${\mathbf{iflag}}=0$ or $2$, 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:     fjac[${\mathbf{n}}×{\mathbf{n}}$]doubleInput/Output
On entry: if ${\mathbf{iflag}}=0$, ${\mathbf{fjac}}\left[\left(\mathit{j}-1\right)×{\mathbf{n}}+\mathit{i}-1\right]$ 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 $\mathit{j}=1,2,\dots ,n$. When ${\mathbf{iflag}}=0$ or $1$, fjac must not be changed.
On exit: if ${\mathbf{iflag}}=2$ on entry, ${\mathbf{fjac}}\left[\left(\mathit{j}-1\right)×{\mathbf{n}}+\mathit{i}-1\right]$ 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:     commNag_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_deriv_expert (c05rcc) you may allocate memory and initialize these pointers with various quantities for use by fcn when called from nag_zero_nonlin_eqns_deriv_expert (c05rcc) (see Section 3.2.1 in the Essential Introduction).
6:     iflagInteger *Input/Output
On entry: ${\mathbf{iflag}}=0$, $1$ or $2$.
${\mathbf{iflag}}=0$
x, fvec and fjac are available for printing (see nprint).
${\mathbf{iflag}}=1$
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), then iflag should be set to a negative integer value.
2:     nIntegerInput
On entry: $n$, the number of equations.
Constraint: ${\mathbf{n}}>0$.
3:     x[n]doubleInput/Output
On entry: an initial guess at the solution vector.
On exit: the final estimate of the solution vector.
4:     fvec[n]doubleOutput
On exit: the function values at the final point returned in x.
5:     fjac[${\mathbf{n}}×{\mathbf{n}}$]doubleOutput
On exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian, stored by columns.
6:     xtoldoubleInput
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: ${\mathbf{xtol}}\ge 0.0$.
7:     maxfevIntegerInput
On entry: the maximum number of calls to fcn with ${\mathbf{iflag}}\ne 0$. nag_zero_nonlin_eqns_deriv_expert (c05rcc) will exit with NE_TOO_MANY_FEVALS, if, at the end of an iteration, the number of calls to fcn exceeds maxfev.
Suggested value: ${\mathbf{maxfev}}=100×\left({\mathbf{n}}+1\right)$.
Constraint: ${\mathbf{maxfev}}>0$.
8:     scale_modeNag_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 $\mathrm{Nag_ScaleProvided}$.
9:     diag[n]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 ${\mathbf{scale_mode}}=\mathrm{Nag_NoScaleProvided}$).
10:   factordoubleInput
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: ${\mathbf{factor}}>0.0$.
11:   nprintIntegerInput
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_deriv_expert (c05rcc).
12:   nfevInteger *Output
On exit: the number of calls made to fcn to evaluate the functions.
13:   njevInteger *Output
On exit: the number of calls made to fcn to evaluate the Jacobian.
14:   r[${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$]doubleOutput
On exit: the upper triangular matrix $R$ produced by the $QR$ factorization of the final approximate Jacobian, stored row-wise.
15:   qtf[n]doubleOutput
On exit: the vector ${Q}^{\mathrm{T}}f$.
16:   commNag_Comm *Communication Structure
The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).
17:   failNagError *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.
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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
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.
NE_NO_IMPROVEMENT
The iteration is not making good progress, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ iterations. 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_deriv_expert (c05rcc) from a different starting point may avoid the region of difficulty.
The iteration is not making good progress, as measured by the improvement from the last $〈\mathit{\text{value}}〉$ Jacobian evaluations. 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_deriv_expert (c05rcc) from a different starting point may avoid the region of difficulty.
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: ${\mathbf{maxfev}}=〈\mathit{\text{value}}〉$.
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 and $D$ denotes the diagonal matrix whose entries are defined by the array diag, then nag_zero_nonlin_eqns_deriv_expert (c05rcc) 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_deriv_expert (c05rcc) 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 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 nag_zero_nonlin_eqns_deriv_expert (c05rcc) may incorrectly indicate convergence. The coding of the Jacobian can be checked using nag_check_derivs (c05zdc). If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning nag_zero_nonlin_eqns_deriv_expert (c05rcc) with a lower value for xtol.

Local workspace arrays of fixed lengths are allocated internally by nag_zero_nonlin_eqns_deriv_expert (c05rcc). The total size of these arrays amounts to $4×n$ double elements.
The time required by nag_zero_nonlin_eqns_deriv_expert (c05rcc) 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_deriv_expert (c05rcc) is approximately $11.5×{n}^{2}$ to process each evaluation of the functions and approximately $1.3×{n}^{3}$ to process each evaluation of the Jacobian. The timing of nag_zero_nonlin_eqns_deriv_expert (c05rcc) 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.

## 9  Example

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

### 9.1  Program Text

Program Text (c05rcce.c)

None.

### 9.3  Program Results

Program Results (c05rcce.r)