NAG Library Function Document
nag_zero_nonlin_eqns_rcomm (c05qdc)
1 Purpose
nag_zero_nonlin_eqns_rcomm (c05qdc) is a comprehensive reverse communication function that 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_rcomm (Integer *irevcm,
Integer n,
double x[],
double fvec[],
double xtol,
Integer ml,
Integer mu,
double epsfcn,
Nag_ScaleType scale_mode,
double diag[],
double factor,
double fjac[],
double r[],
double qtf[],
Integer iwsav[],
double rwsav[],
NagError *fail) 

3 Description
The system of equations is defined as:
nag_zero_nonlin_eqns_rcomm (c05qdc) 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 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
Note: this function uses
reverse communication. Its use involves an initial entry, intermediate exits and reentries, and a final exit, as indicated by the argument
irevcm. Between intermediate exits and reentries,
all arguments other than fvec must remain unchanged.
 1:
irevcm – Integer *Input/Output

On initial entry: must have the value $0$.
On intermediate exit:
specifies what action you must take before reentering nag_zero_nonlin_eqns_rcomm (c05qdc) with
irevcm unchanged. The value of
irevcm should be interpreted as follows:
 ${\mathbf{irevcm}}=1$
 Indicates the start of a new iteration. No action is required by you, but x and fvec are available for printing.
 ${\mathbf{irevcm}}=2$
 Indicates that before reentry to nag_zero_nonlin_eqns_rcomm (c05qdc), fvec must contain the function values ${f}_{i}\left(x\right)$.
On final exit: ${\mathbf{irevcm}}=0$, and the algorithm has terminated.
Constraint:
${\mathbf{irevcm}}=0$, $1$ or $2$.
 2:
n – IntegerInput

On entry:
$n$, the number of equations.
Constraint:
${\mathbf{n}}>0$.
 3:
x[n] – doubleInput/Output

On initial entry: an initial guess at the solution vector.
On intermediate exit:
contains the current point.
On final exit: the final estimate of the solution vector.
 4:
fvec[n] – doubleInput/Output

On initial entry: need not be set.
On intermediate reentry: if
${\mathbf{irevcm}}=1$,
fvec must not be changed.
If
${\mathbf{irevcm}}=2$,
fvec must be set to the values of the functions computed at the current point
x.
On final exit: the function values at the final point,
x.
 5:
xtol – doubleInput

On initial 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:
ml – IntegerInput

On initial 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:
${\mathbf{ml}}\ge 0$.
 7:
mu – IntegerInput

On initial 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:
${\mathbf{mu}}\ge 0$.
 8:
epsfcn – doubleInput

On initial entry: the order 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:
${\mathbf{epsfcn}}=0.0$.
 9:
scale_mode – Nag_ScaleTypeInput

On initial entry: indicates whether or not you have provided scaling factors in
diag.
If
${\mathbf{scale\_mode}}=\mathrm{Nag\_ScaleProvided}$ the scaling must have been supplied in
diag.
Otherwise, if ${\mathbf{scale\_mode}}=\mathrm{Nag\_NoScaleProvided}$, the variables will be scaled internally.
Constraint:
${\mathbf{scale\_mode}}=\mathrm{Nag\_NoScaleProvided}$ or $\mathrm{Nag\_ScaleProvided}$.
 10:
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
${\mathbf{scale\_mode}}=\mathrm{Nag\_NoScaleProvided}$).
 11:
factor – doubleInput

On initial 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:
${\mathbf{factor}}>0.0$.
 12:
fjac[${\mathbf{n}}\times {\mathbf{n}}$] – 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 initial entry: need not be set.
On intermediate exit:
must not be changed.
On final exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian.
 13:
r[${\mathbf{n}}\times \left({\mathbf{n}}+1\right)/2$] – doubleInput/Output

On initial entry: need not be set.
On intermediate exit:
must not be changed.
On final exit: the upper triangular matrix $R$ produced by the $QR$ factorization of the final approximate Jacobian, stored rowwise.
 14:
qtf[n] – doubleInput/Output

On initial entry: need not be set.
On intermediate exit:
must not be changed.
On final exit: the vector ${Q}^{\mathrm{T}}f$.
 15:
iwsav[$17$] – IntegerCommunication Array
 16:
rwsav[$4\times {\mathbf{n}}+10$] – doubleCommunication Array

The arrays
iwsav and
rwsav MUST NOT be altered between calls to nag_zero_nonlin_eqns_rcomm (c05qdc).
 17:
fail – NagError *Input/Output

The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
 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, ${\mathbf{irevcm}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{irevcm}}=0$, $1$ or $2$.
On entry, ${\mathbf{ml}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ml}}\ge 0$.
On entry, ${\mathbf{mu}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{mu}}\ge 0$.
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.
 NE_NO_IMPROVEMENT
The iteration is not making good progress,
as measured by the improvement from the last $\u2329\mathit{\text{value}}\u232a$ iterations.
The iteration is not making good progress,
as measured by the improvement from the last $\u2329\mathit{\text{value}}\u232a$ Jacobian evaluations.
 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_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 and
$D$ denotes the diagonal matrix whose entries are defined by the array
diag, then nag_zero_nonlin_eqns_rcomm (c05qdc) 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_rcomm (c05qdc) 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
${\mathbf{fail}}\mathbf{.}\mathbf{code}=\text{}$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_rcomm (c05qdc) may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning nag_zero_nonlin_eqns_rcomm (c05qdc) with a lower value for
xtol.
The time required by nag_zero_nonlin_eqns_rcomm (c05qdc) 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_rcomm (c05qdc) to process the evaluation of functions in the main program in each exit is approximately $11.5\times {n}^{2}$. The timing of nag_zero_nonlin_eqns_rcomm (c05qdc) 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.
9 Example
This example determines the values
${x}_{1},\dots ,{x}_{9}$ which satisfy the tridiagonal equations:
9.1 Program Text
Program Text (c05qdce.c)
9.2 Program Data
None.
9.3 Program Results
Program Results (c05qdce.r)