NAG Library Function Document
nag_zero_nonlin_eqns_deriv_rcomm (c05rdc)
1 Purpose
nag_zero_nonlin_eqns_deriv_rcomm (c05rdc) is a comprehensive reverse communication 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_rcomm (Integer *irevcm,
Integer n,
double x[],
double fvec[],
double fjac[],
double xtol,
Nag_ScaleType scale_mode,
double diag[],
double factor,
double r[],
double qtf[],
Integer iwsav[],
double rwsav[],
NagError *fail) 

3 Description
The system of equations is defined as:
nag_zero_nonlin_eqns_deriv_rcomm (c05rdc) 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. 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 and fjac 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_deriv_rcomm (c05rdc)
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_deriv_rcomm (c05rdc), fvec must contain the function values ${f}_{i}\left(x\right)$.
 ${\mathbf{irevcm}}=3$
 Indicates that before reentry to nag_zero_nonlin_eqns_deriv_rcomm (c05rdc),
${\mathbf{fjac}}\left[\left(\mathit{j}1\right)\times {\mathbf{n}}+\mathit{i}1\right]$ 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 $\mathit{j}=1,2,\dots ,n$.
On final exit: ${\mathbf{irevcm}}=0$, and the algorithm has terminated.
Constraint:
${\mathbf{irevcm}}=0$, $1$, $2$ or $3$.
 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
${\mathbf{irevcm}}\ne 2$,
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:
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 reentry: if
${\mathbf{irevcm}}\ne 3$,
fjac must not be changed.
If ${\mathbf{irevcm}}=3$,
${\mathbf{fjac}}\left[\left(\mathit{j}1\right)\times {\mathbf{n}}+\mathit{i}1\right]$ 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 $\mathit{j}=1,2,\dots ,n$.
On final exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian, stored by columns.
 6:
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:
${\mathbf{xtol}}\ge 0.0$.
 7:
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 $\mathrm{Nag\_ScaleProvided}$.
 8:
diag[n] – doubleInput/Output
On initial 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 intermediate exit:
diag must not be changed.
On final exit: the scale factors actually used (computed internally if ${\mathbf{scale\_mode}}=\mathrm{Nag\_NoScaleProvided}$).
 9:
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:
${\mathbf{factor}}>0.0$.
 10:
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.
 11:
qtf[n] – doubleInput/Output
On initial entry: need not be set.
On intermediate exit:
must not be changed.
On final exit: the vector ${Q}^{\mathrm{T}}f$.
 12:
iwsav[$17$] – IntegerCommunication Array
 13:
rwsav[$4\times {\mathbf{n}}+10$] – doubleCommunication Array
The arrays
iwsav and
rwsav MUST NOT be altered between calls to nag_zero_nonlin_eqns_deriv_rcomm (c05rdc).
 14:
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 $\u27e8\mathit{\text{value}}\u27e9$ 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, ${\mathbf{irevcm}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{irevcm}}=0$, $1$, $2$ or $3$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
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
$\u27e8\mathit{\text{value}}\u27e9$ 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_rcomm (c05rdc) 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
$\u27e8\mathit{\text{value}}\u27e9$ 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_rcomm (c05rdc) from a different starting point may avoid the region of difficulty.
 NE_REAL

On entry, ${\mathbf{factor}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{factor}}>0.0$.
On entry, ${\mathbf{xtol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{xtol}}\ge 0.0$.
 NE_TOO_SMALL

No further improvement in the solution is possible.
xtol is too small:
${\mathbf{xtol}}=\u27e8\mathit{\text{value}}\u27e9$.
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_rcomm (c05rdc) 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_rcomm (c05rdc) 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_rcomm (c05rdc) 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_rcomm (c05rdc) with a lower value for
xtol.
8 Parallelism and Performance
nag_zero_nonlin_eqns_deriv_rcomm (c05rdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zero_nonlin_eqns_deriv_rcomm (c05rdc) 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
Users' Note for your implementation for any additional implementationspecific information.
The time required by nag_zero_nonlin_eqns_deriv_rcomm (c05rdc) 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_rcomm (c05rdc) 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_rcomm (c05rdc) 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 (c05rdce.c)
10.2 Program Data
None.
10.3 Program Results
Program Results (c05rdce.r)