NAG Library Routine Document
C05QBF
1 Purpose
C05QBF is an easytouse routine that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.
2 Specification
INTEGER 
N, IUSER(*), IFAIL 
REAL (KIND=nag_wp) 
X(N), FVEC(N), XTOL, RUSER(*) 
EXTERNAL 
FCN 

3 Description
The system of equations is defined as:
C05QBF is based on the MINPACK routine HYBRD1 (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
 1: $\mathrm{FCN}$ – SUBROUTINE, supplied by the user.External Procedure

FCN must return the values of the functions
${f}_{i}$ at a point
$x$.
The specification of
FCN is:
INTEGER 
N, IUSER(*), IFLAG 
REAL (KIND=nag_wp) 
X(N), FVEC(N), RUSER(*) 

 1: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the number of equations.
 2: $\mathrm{X}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the components of the point $x$ at which the functions must be evaluated.
 3: $\mathrm{FVEC}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the function values
${f}_{i}\left(x\right)$ (unless
IFLAG is set to a negative value by
FCN).
 4: $\mathrm{IUSER}\left(*\right)$ – INTEGER arrayUser Workspace
 5: $\mathrm{RUSER}\left(*\right)$ – REAL (KIND=nag_wp) arrayUser Workspace

FCN is called with the arguments
IUSER and
RUSER as supplied to C05QBF. You should use the arrays
IUSER and
RUSER to supply information to
FCN.
 6: $\mathrm{IFLAG}$ – INTEGERInput/Output

On entry: ${\mathbf{IFLAG}}>0$.
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), then
IFLAG should be set to a negative integer.
FCN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which C05QBF is called. Arguments denoted as
Input must
not be changed by this procedure.
 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 entry: an initial guess at the solution vector.
On exit: the final estimate of the solution vector.
 4: $\mathrm{FVEC}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the function values at the final point returned in
X.
 5: $\mathrm{XTOL}$ – REAL (KIND=nag_wp)Input

On 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: $\mathrm{IUSER}\left(*\right)$ – INTEGER arrayUser Workspace
 7: $\mathrm{RUSER}\left(*\right)$ – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and
RUSER are not used by C05QBF, but are passed directly to
FCN and should be used to pass information to this routine.
 8: $\mathrm{IFAIL}$ – INTEGERInput/Output

On 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, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On 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:
 ${\mathbf{IFAIL}}=2$

There have been at least
$200\times \left({\mathbf{N}}+1\right)$ calls to
FCN. Consider restarting the calculation from the point held in
X.
 ${\mathbf{IFAIL}}=3$

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. 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 C05QBF from a different starting point may avoid the region of difficulty.
 ${\mathbf{IFAIL}}=5$

IFLAG was set negative in
FCN.
${\mathbf{IFLAG}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{IFAIL}}=11$

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: ${\mathbf{XTOL}}\ge 0.0$.
 ${\mathbf{IFAIL}}=99$
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.
7 Accuracy
If
$\hat{x}$ is the true solution, C05QBF tries to ensure that
If this condition is satisfied with
${\mathbf{XTOL}}={10}^{k}$, then the larger components of
$x$ have
$k$ significant decimal digits. There is a danger that the smaller components of
$x$ may have large relative errors, but the fast rate of convergence of C05QBF 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 C05QBF may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning C05QBF with a lower value for
XTOL.
8 Parallelism and Performance
C05QBF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
C05QBF 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.
Local workspace arrays of fixed lengths are allocated internally by C05QBF. The total size of these arrays amounts to $n\times \left(3\times n+13\right)/2$ real elements.
The time required by C05QBF 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 C05QBF to process each evaluation of the functions is approximately $11.5\times {n}^{2}$. The timing of C05QBF 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 (c05qbfe.f90)
10.2 Program Data
None.
10.3 Program Results
Program Results (c05qbfe.r)