NAG Library Routine Document
C05QDF
1 Purpose
C05QDF is a comprehensive reverse communication routine that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.
2 Specification
SUBROUTINE C05QDF ( 
IREVCM, N, X, FVEC, XTOL, ML, MU, EPSFCN, MODE, DIAG, FACTOR, FJAC, R, QTF, IWSAV, RWSAV, IFAIL) 
INTEGER 
IREVCM, N, ML, MU, MODE, IWSAV(17), IFAIL 
REAL (KIND=nag_wp) 
X(N), FVEC(N), XTOL, EPSFCN, DIAG(N), FACTOR, FJAC(N,N), R(N*(N+1)/2), QTF(N), RWSAV(4*N+10) 

3 Description
The system of equations is defined as:
C05QDF 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 routine 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: $\mathrm{IREVCM}$ – INTEGERInput/Output

On initial entry: must have the value $0$.
On intermediate exit:
specifies what action you must take before reentering C05QDF 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 C05QDF, 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: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the number of equations.
Constraint:
${\mathbf{N}}>0$.
 3: $\mathrm{X}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayInput/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: $\mathrm{FVEC}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayInput/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: $\mathrm{XTOL}$ – REAL (KIND=nag_wp)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
X02AJF.
Constraint:
${\mathbf{XTOL}}\ge 0.0$.
 6: $\mathrm{ML}$ – 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: $\mathrm{MU}$ – 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: $\mathrm{EPSFCN}$ – REAL (KIND=nag_wp)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
X02AJF) then
machine precision is used. Consequently a value of
$0.0$ will often be suitable.
Suggested value:
${\mathbf{EPSFCN}}=0.0$.
 9: $\mathrm{MODE}$ – INTEGERInput

On initial entry: indicates whether or not you have provided scaling factors in
DIAG.
If
${\mathbf{MODE}}=2$ the scaling must have been supplied in
DIAG.
Otherwise, if ${\mathbf{MODE}}=1$, the variables will be scaled internally.
Constraint:
${\mathbf{MODE}}=1$ or $2$.
 10: $\mathrm{DIAG}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry: if
${\mathbf{MODE}}=2$,
DIAG must contain multiplicative scale factors for the variables.
If
${\mathbf{MODE}}=1$,
DIAG need not be set.
Constraint:
if ${\mathbf{MODE}}=2$, ${\mathbf{DIAG}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,n$.
On exit: the scale factors actually used (computed internally if ${\mathbf{MODE}}=1$).
 11: $\mathrm{FACTOR}$ – REAL (KIND=nag_wp)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: $\mathrm{FJAC}\left({\mathbf{N}},{\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayInput/Output

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: $\mathrm{R}\left({\mathbf{N}}\times \left({\mathbf{N}}+1\right)/2\right)$ – REAL (KIND=nag_wp) arrayInput/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: $\mathrm{QTF}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayInput/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: $\mathrm{IWSAV}\left(17\right)$ – INTEGER arrayCommunication Array
 16: $\mathrm{RWSAV}\left(4\times {\mathbf{N}}+10\right)$ – REAL (KIND=nag_wp) arrayCommunication Array

The arrays
IWSAV and
RWSAV must not be altered between calls to C05QDF.
 17: $\mathrm{IFAIL}$ – INTEGERInput/Output

On initial entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if
${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is
$1$.
When the value $\mathbf{1}\text{ or}1$ is used it is essential to test the value of IFAIL on exit.
On final exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=2$

On entry, ${\mathbf{IREVCM}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{IREVCM}}=0$, $1$ or $2$.
 ${\mathbf{IFAIL}}=3$

No further improvement in the solution is possible.
XTOL is too small:
${\mathbf{XTOL}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{IFAIL}}=4$

The iteration is not making good progress, as measured by the improvement from the last $\u2329\mathit{\text{value}}\u232a$ Jacobian evaluations.
 ${\mathbf{IFAIL}}=5$

The iteration is not making good progress, as measured by the improvement from the last $\u2329\mathit{\text{value}}\u232a$ iterations.
 ${\mathbf{IFAIL}}=11$

On entry, ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{N}}>0$.
 ${\mathbf{IFAIL}}=12$

On entry, ${\mathbf{XTOL}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{XTOL}}\ge 0.0$.
 ${\mathbf{IFAIL}}=13$

On entry, ${\mathbf{MODE}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{MODE}}=1$ or $2$.
 ${\mathbf{IFAIL}}=14$

On entry, ${\mathbf{FACTOR}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{FACTOR}}>0.0$.
 ${\mathbf{IFAIL}}=15$

On entry,
${\mathbf{MODE}}=2$ and
DIAG contained a nonpositive element.
 ${\mathbf{IFAIL}}=16$

On entry, ${\mathbf{ML}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ML}}\ge 0$.
 ${\mathbf{IFAIL}}=17$

On entry, ${\mathbf{MU}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{MU}}\ge 0$.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
A value of
${\mathbf{IFAIL}}={\mathbf{4}}$ or
${\mathbf{5}}$ 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 C05QDF from a different starting point may avoid the region of difficulty.
7 Accuracy
If
$\hat{x}$ is the true solution and
$D$ denotes the diagonal matrix whose entries are defined by the array
DIAG, then C05QDF 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 C05QDF 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 routine exits with
${\mathbf{IFAIL}}={\mathbf{3}}$.
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 C05QDF may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning C05QDF with a lower value for
XTOL.
8 Parallelism and Performance
C05QDF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
C05QDF 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 routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The time required by C05QDF 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 C05QDF to process the evaluation of functions in the main program in each exit is approximately $11.5\times {n}^{2}$. The timing of C05QDF 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
Program Text (c05qdfe.f90)
10.2 Program Data
None.
10.3 Program Results
Program Results (c05qdfe.r)