NAG CL Interface
c05qdc (sys_func_rcomm)
1
Purpose
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
void 
c05qdc (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) 

The function may be called by the names: c05qdc, nag_roots_sys_func_rcomm or nag_zero_nonlin_eqns_rcomm.
3
Description
The system of equations is defined as:
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:
$\mathbf{irevcm}$ – Integer *
Input/Output

On initial entry: must have the value $0$.
On intermediate exit:
specifies what action you must take before reentering
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 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 $2$.
Note: any values you return to c05qdc as part of the reverse communication procedure should not include floatingpoint NaN (Not a Number) or infinity values, since these are not handled by c05qdc. If your code inadvertently does return any NaNs or infinities, c05qdc is likely to produce unexpected results.

2:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of equations.
Constraint:
${\mathbf{n}}>0$.

3:
$\mathbf{x}\left[{\mathbf{n}}\right]$ – double
Input/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:
$\mathbf{fvec}\left[{\mathbf{n}}\right]$ – double
Input/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:
$\mathbf{xtol}$ – double
Input

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
X02AJC.
Constraint:
${\mathbf{xtol}}\ge 0.0$.

6:
$\mathbf{ml}$ – Integer
Input

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:
$\mathbf{mu}$ – Integer
Input

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:
$\mathbf{epsfcn}$ – double
Input

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
X02AJC) then
machine precision is used. Consequently a value of
$0.0$ will often be suitable.
Suggested value:
${\mathbf{epsfcn}}=0.0$.

9:
$\mathbf{scale\_mode}$ – Nag_ScaleType
Input

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:
$\mathbf{diag}\left[{\mathbf{n}}\right]$ – double
Input/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:
$\mathbf{factor}$ – double
Input

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:
$\mathbf{fjac}\left[{\mathbf{n}}\times {\mathbf{n}}\right]$ – double
Input/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:
$\mathbf{r}\left[{\mathbf{n}}\times \left({\mathbf{n}}+1\right)/2\right]$ – double
Input/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:
$\mathbf{qtf}\left[{\mathbf{n}}\right]$ – double
Input/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:
$\mathbf{iwsav}\left[17\right]$ – Integer
Communication Array

16:
$\mathbf{rwsav}\left[4\times {\mathbf{n}}+10\right]$ – double
Communication Array

The arrays
iwsav and
rwsav MUST NOT be altered between calls to
c05qdc.

17:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface 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, ${\mathbf{irevcm}}=\u2329\mathit{\text{value}}\u232a$.
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.
See
Section 7.5 in the Introduction to the NAG Library CL Interface 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.
The iteration is not making good progress, as measured by the improvement from the last $\u2329\mathit{\text{value}}\u232a$ Jacobian evaluations.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface 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_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
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
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
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ 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
c05qdc may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning
c05qdc with a lower value for
xtol.
8
Parallelism and Performance
c05qdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05qdc 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.
The time required by 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 c05qdc to process the evaluation of functions in the main program in each exit is approximately $11.5\times {n}^{2}$. The timing of 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.
10
Example
This example determines the values
${x}_{1},\dots ,{x}_{9}$ which satisfy the tridiagonal equations:
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results