NAG Library Function Document
nag_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 <nag.h> 
#include <nagc05.h> 
void 
nag_zero_nonlin_eqns_deriv_expert (
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:
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 rank1 method of Broyden. At the starting point, the Jacobian is requested, but it is not asked for 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

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:
$\mathbf{n}$ – 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 or the Jacobian must be evaluated.
 3:
$\mathbf{fvec}\left[{\mathbf{n}}\right]$ – 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:
$\mathbf{fjac}\left[{\mathbf{n}}\times {\mathbf{n}}\right]$ – doubleInput/Output

Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{fjac}}\left[\left(j1\right)\times {\mathbf{n}}+i1\right]$.
On entry: if
${\mathbf{iflag}}=0$,
${\mathbf{fjac}}\left[\left(\mathit{j}1\right)\times {\mathbf{n}}+\mathit{i}1\right]$ 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,
${\mathbf{fjac}}\left[\left(\mathit{j}1\right)\times {\mathbf{n}}+\mathit{i}1\right]$ 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:
$\mathbf{comm}$ – 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_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 3.3.1.1 in How to Use the NAG Library and its Documentation).
 6:
$\mathbf{iflag}$ – Integer *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),
iflag should be set to a negative integer value.
Note: fcn should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
nag_zero_nonlin_eqns_deriv_expert (c05rcc). If your code inadvertently
does return any NaNs or infinities,
nag_zero_nonlin_eqns_deriv_expert (c05rcc) 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:
$\mathbf{fjac}\left[{\mathbf{n}}\times {\mathbf{n}}\right]$ – doubleOutput

Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{fjac}}\left[\left(j1\right)\times {\mathbf{n}}+i1\right]$.
On exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian, stored by columns.
 6:
$\mathbf{xtol}$ – 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:
${\mathbf{xtol}}\ge 0.0$.
 7:
$\mathbf{maxfev}$ – IntegerInput

On entry: the maximum number of calls to
fcn with
${\mathbf{iflag}}\ne 0$.
nag_zero_nonlin_eqns_deriv_expert (c05rcc) 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:
${\mathbf{maxfev}}=100\times \left({\mathbf{n}}+1\right)$.
Constraint:
${\mathbf{maxfev}}>0$.
 8:
$\mathbf{scale\_mode}$ – 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 $\mathrm{Nag\_ScaleProvided}$.
 9:
$\mathbf{diag}\left[{\mathbf{n}}\right]$ – 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 ${\mathbf{scale\_mode}}=\mathrm{Nag\_NoScaleProvided}$).
 10:
$\mathbf{factor}$ – 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
${\mathbf{factor}}\times {\Vert {\mathbf{diag}}\times {\mathbf{x}}\Vert}_{2}$ if this is nonzero; otherwise the bound is
factor.)
Suggested value:
${\mathbf{factor}}=100.0$.
Constraint:
${\mathbf{factor}}>0.0$.
 11:
$\mathbf{nprint}$ – IntegerInput

On entry: indicates whether (and how often) special calls to
fcn, with
iflag set to
$0$, are to be made for printing purposes.
 ${\mathbf{nprint}}\le 0$
 No calls are made.
 ${\mathbf{nprint}}>0$
 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:
$\mathbf{nfev}$ – Integer *Output

On exit: the number of calls made to
fcn to evaluate the functions.
 13:
$\mathbf{njev}$ – Integer *Output

On exit: the number of calls made to
fcn to evaluate the Jacobian.
 14:
$\mathbf{r}\left[{\mathbf{n}}\times \left({\mathbf{n}}+1\right)/2\right]$ – doubleOutput

On exit: the upper triangular matrix $R$ produced by the $QR$ factorization of the final approximate Jacobian, stored rowwise.
 15:
$\mathbf{qtf}\left[{\mathbf{n}}\right]$ – doubleOutput

On exit: the vector ${Q}^{\mathrm{T}}f$.
 16:
$\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
 17:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 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.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_DIAG_ELEMENTS

On entry,
${\mathbf{scale\_mode}}=\mathrm{Nag\_ScaleProvided}$ and
diag contained a nonpositive element.
 NE_INT

On entry, ${\mathbf{maxfev}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{maxfev}}>0$.
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.
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
$\u2329\mathit{\text{value}}\u232a$ 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
$\u2329\mathit{\text{value}}\u232a$ 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_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, ${\mathbf{factor}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{factor}}>0.0$.
On entry, ${\mathbf{xtol}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
 NE_TOO_MANY_FEVALS

There have been at least
maxfev calls to
fcn:
${\mathbf{maxfev}}=\u2329\mathit{\text{value}}\u232a$. 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:
${\mathbf{xtol}}=\u2329\mathit{\text{value}}\u232a$.
 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 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
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
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ 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 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.
8
Parallelism and Performance
nag_zero_nonlin_eqns_deriv_expert (c05rcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zero_nonlin_eqns_deriv_expert (c05rcc) 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_deriv_expert (c05rcc). The total size of these arrays amounts to $4\times n$ 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 $11.5\times {n}^{2}$ to process each evaluation of the functions and approximately $1.3\times {n}^{3}$ 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.
10
Example
This example determines the values
${x}_{1},\dots ,{x}_{9}$ which satisfy the tridiagonal equations:
10.1
Program Text
Program Text (c05rcce.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (c05rcce.r)