NAG Library Routine Document
C05NCF is a comprehensive routine that finds a solution of a system of nonlinear equations by a modification of the Powell hybrid method.
|SUBROUTINE C05NCF (
||FCN, N, X, FVEC, XTOL, MAXFEV, ML, MU, EPSFCN, DIAG, MODE, FACTOR, NPRINT, NFEV, FJAC, LDFJAC, R, LR, QTF, W, IFAIL)
||N, MAXFEV, ML, MU, MODE, NPRINT, NFEV, LDFJAC, LR, IFAIL
||X(N), FVEC(N), XTOL, EPSFCN, DIAG(N), FACTOR, FJAC(LDFJAC,N), R(N*(N+1)/2), QTF(N), W(1,1)
The system of equations is defined as:
C05NCF 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 rank-1 method of Broyden. At the starting point, the Jacobian is approximated by forward differences, but these are not used again until the rank-1 method fails to produce satisfactory progress. For more details see Powell (1970)
Moré J J, Garbow B S and Hillstrom K E (1980) User guide for MINPACK-1 Technical Report ANL-80-74 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
- 1: FCN – SUBROUTINE, supplied by the user.External Procedure
must return the values of the functions
at a point
The specification of FCN
- 1: N – INTEGERInput
On entry: , the number of equations.
- 2: X(N) – REAL (KIND=nag_wp) arrayInput
On entry: the components of the point at which the functions must be evaluated.
- 3: FVEC(N) – REAL (KIND=nag_wp) arrayInput/Output
contains the function values
and must not be changed.
on entry, FVEC
must contain the function values
is set to a negative value by FCN
- 4: IFLAG – INTEGERInput/Output
- X and FVEC are available for printing (see NPRINT below).
- FVEC must be updated.
: 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. This value will be returned through IFAIL
must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which C05NCF is called. Parameters denoted as Input
be changed by this procedure.
- 2: N – INTEGERInput
On entry: , the number of equations.
- 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
: the function values at the final point returned in X
- 5: XTOL – REAL (KIND=nag_wp)Input
: the accuracy in X
to which the solution is required.
is the machine precision
returned by X02AJF
- 6: MAXFEV – INTEGERInput
: the maximum number of calls to FCN
. C05NCF will exit with
, if, at the end of an iteration, the number of calls to FCN
- 7: ML – INTEGERInput
On entry: the number of subdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set .)
- 8: MU – INTEGERInput
On entry: the number of superdiagonals within the band of the Jacobian matrix. (If the Jacobian is not banded, or you are unsure, set .)
- 9: EPSFCN – REAL (KIND=nag_wp)Input
: a rough estimate 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
will often be suitable.
- 10: DIAG(N) – REAL (KIND=nag_wp) arrayInput/Output
must contain multiplicative scale factors for the variables.
need not be set.
if , , for .
On exit: the scale factors actually used (computed internally if ).
- 11: MODE – INTEGERInput
: indicates whether or not you have provided scaling factors in DIAG
the scaling must have been specified in DIAG
Otherwise, the variables will be scaled internally.
- 12: FACTOR – REAL (KIND=nag_wp)Input
: a quantity to be used in determining the initial step bound. In most cases, FACTOR
should lie between
. (The step bound is
if this is nonzero; otherwise the bound is FACTOR
- 13: NPRINT – INTEGERInput
: indicates whether (and how often) special calls to FCN
, with IFLAG
, are to be made for printing purposes.
- No calls are made.
- FCN is called at the beginning of the first iteration, every NPRINT iterations thereafter and immediately before the return from C05NCF.
- 14: NFEV – INTEGEROutput
: the number of calls made to FCN
- 15: FJAC(LDFJAC,N) – REAL (KIND=nag_wp) arrayOutput
On exit: the orthogonal matrix produced by the factorisation of the final approximate Jacobian.
- 16: LDFJAC – INTEGERInput
: the first dimension of the array FJAC
as declared in the (sub)program from which C05NCF is called.
- 17: R() – REAL (KIND=nag_wp) arrayOutput
On exit: the upper triangular matrix produced by the factorization of the final approximate Jacobian, stored row-wise.
- 18: LR – INTEGERDummy
This parameter is no longer accessed by C05NCF.
- 19: QTF(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the vector .
- 20: W(,) – REAL (KIND=nag_wp) arrayInput
This parameter is no longer accessed by C05NCF. Workspace is provided internally by dynamic allocation instead.
- 21: IFAIL – INTEGERInput/Output
must be set to
. 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
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
This indicates an exit from C05NCF because you have set IFLAG
negative in FCN
. The value of IFAIL
will be the same as your setting of IFLAG
|or|| and for some , for .|
There have been at least MAXFEV
evaluations of FCN
. Consider restarting the calculation from the final point held in X
No further improvement in the approximate solution X
is possible; XTOL
is too small.
The iteration is not making good progress, as measured by the improvement from the last five Jacobian evaluations.
The iteration is not making good progress, as measured by the improvement from the last ten iterations.
Internal memory allocation failed.
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 C05NCF from a different starting point may avoid the region of difficulty.
is the true solution and
denotes the diagonal matrix whose entries are defined by the array DIAG
, then C05NCF tries to ensure that
If this condition is satisfied with
, then the larger components of
significant decimal digits. There is a danger that the smaller components of
may have large relative errors, but the fast rate of convergence of C05NCF usually obviates this possibility.
is less than machine precision
and the above test is satisfied with the machine precision
in place of XTOL
, then the routine exits with
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 test assumes that the functions are reasonably well behaved. If this condition is not satisfied, then C05NCF may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning C05NCF with a lower value for XTOL
Local workspace arrays of fixed lengths are allocated internally by C05NCF. The total size of these arrays amounts to real elements.
The time required by C05NCF to solve a given problem depends on
, the behaviour of the functions, the accuracy requested and the starting point. The number of arithmetic operations executed by C05NCF to process each call of FCN
. Unless FCN
can be evaluated quickly, the timing of C05NCF will be strongly influenced by the time spent in FCN
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
This example determines the values
which satisfy the tridiagonal equations:
9.1 Program Text
Program Text (c05ncfe.f90)
9.2 Program Data
9.3 Program Results
Program Results (c05ncfe.r)