NAG Library Routine Document
C05RBF
1 Purpose
C05RBF is an easytouse routine 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
INTEGER 
N, IUSER(*), IFAIL 
REAL (KIND=nag_wp) 
X(N), FVEC(N), FJAC(N,N), XTOL, RUSER(*) 
EXTERNAL 
FCN 

3 Description
The system of equations is defined as:
C05RBF is based on the MINPACK routine HYBRJ1 (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 requested, but it is not asked for 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 Parameters
 1: FCN – SUBROUTINE, supplied by the user.External Procedure
Depending upon the value of
IFLAG,
FCN must either return the values of the functions
${f}_{i}$ at a point
$x$ or return the Jacobian at
$x$.
The specification of
FCN is:
INTEGER 
N, IUSER(*), IFLAG 
REAL (KIND=nag_wp) 
X(N), FVEC(N), FJAC(N,N), RUSER(*) 

 1: N – INTEGERInput
On entry: $n$, the number of equations.
 2: X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the components of the point $x$ at which the functions or the Jacobian must be evaluated.
 3: FVEC(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: if
${\mathbf{IFLAG}}=2$,
FVEC contains the function values
${f}_{i}\left(x\right)$ and must not be changed.
On exit: if
${\mathbf{IFLAG}}=1$ on entry,
FVEC must contain the function values
${f}_{i}\left(x\right)$ (unless
IFLAG is set to a negative value by
FCN).
 4: FJAC(N,N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: if
${\mathbf{IFLAG}}=1$,
FJAC contains the value of
$\frac{\partial {f}_{\mathit{i}}}{\partial {x}_{\mathit{j}}}$ at the point
$x$, for
$\mathit{i}=1,2,\dots ,n$ and
$\mathit{j}=1,2,\dots ,n$, and must not be changed.
On exit: if
${\mathbf{IFLAG}}=2$ on entry,
${\mathbf{FJAC}}\left(\mathit{i},\mathit{j}\right)$ 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
$\mathit{j}=1,2,\dots ,n$, (unless
IFLAG is set to a negative value by
FCN).
 5: IUSER($*$) – INTEGER arrayUser Workspace
 6: RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

FCN is called with the parameters
IUSER and
RUSER as supplied to C05RBF. You are free to use the arrays
IUSER and
RUSER to supply information to
FCN as an alternative to using COMMON global variables.
 7: IFLAG – INTEGERInput/Output
On entry:
${\mathbf{IFLAG}}=1$ or
$2$.
 ${\mathbf{IFLAG}}=1$
 FVEC is to be updated.
 ${\mathbf{IFLAG}}=2$
 FJAC is to be updated.
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 C05RBF is called. Parameters denoted as
Input must
not be changed by this procedure.
 2: N – INTEGERInput
On entry: $n$, the number of equations.
Constraint:
${\mathbf{N}}>0$.
 3: X(N) – 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: FVEC(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the function values at the final point returned in
X.
 5: FJAC(N,N) – REAL (KIND=nag_wp) arrayOutput
On exit: the orthogonal matrix $Q$ produced by the $QR$ factorization of the final approximate Jacobian.
 6: 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:
${\mathbf{XTOL}}\ge 0.0$.
 7: IUSER($*$) – INTEGER arrayUser Workspace
 8: RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and
RUSER are not used by C05RBF, but are passed directly to
FCN and may be used to pass information to this routine as an alternative to using COMMON global variables.
 9: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction 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 parameter, 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
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, 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
$100\times \left({\mathbf{N}}+1\right)$ evaluations of
FCN. Consider restarting the calculation from the final point held in
X.
 ${\mathbf{IFAIL}}=3$
No further improvement in the approximate solution
X is possible;
XTOL is too small.
 ${\mathbf{IFAIL}}=4$
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 C05RBF from a different starting point may avoid the region of difficulty.
 ${\mathbf{IFAIL}}=5$
You have set
IFLAG negative in
FCN.
 ${\mathbf{IFAIL}}=11$
On entry, ${\mathbf{N}}\le 0$.
 ${\mathbf{IFAIL}}=12$
On entry, ${\mathbf{XTOL}}<0.0$.
 ${\mathbf{IFAIL}}=999$

Internal memory allocation failed.
7 Accuracy
If
$\hat{x}$ is the true solution, C05RBF 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 C05RBF 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 and the Jacobian are coded consistently and that the functions are reasonably well behaved. If these conditions are not satisfied, then C05RBF may incorrectly indicate convergence. The coding of the Jacobian can be checked using
C05ZDF. If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning C05RBF with a lower value for
XTOL.
Local workspace arrays of fixed lengths are allocated internally by C05RBF. The total size of these arrays amounts to $n\times \left(n+13\right)/2$ real elements.
The time required by C05RBF 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 C05RBF 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 C05RBF 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.
9 Example
This example determines the values
${x}_{1},\dots ,{x}_{9}$ which satisfy the tridiagonal equations:
9.1 Program Text
Program Text (c05rbfe.f90)
9.2 Program Data
None.
9.3 Program Results
Program Results (c05rbfe.r)