NAG FL Interface
c05rbf (sys_​deriv_​easy)

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1 Purpose

c05rbf is an easy-to-use 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

Fortran Interface
Subroutine c05rbf ( fcn, n, x, fvec, fjac, xtol, iuser, ruser, ifail)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: iuser(*), ifail
Real (Kind=nag_wp), Intent (In) :: xtol
Real (Kind=nag_wp), Intent (Inout) :: x(n), ruser(*)
Real (Kind=nag_wp), Intent (Out) :: fvec(n), fjac(n,n)
External :: fcn
C Header Interface
#include <nag.h>
void  c05rbf_ (
void (NAG_CALL *fcn)(const Integer *n, const double x[], double fvec[], double fjac[], Integer iuser[], double ruser[], Integer *iflag),
const Integer *n, double x[], double fvec[], double fjac[], const double *xtol, Integer iuser[], double ruser[], Integer *ifail)
The routine may be called by the names c05rbf or nagf_roots_sys_deriv_easy.

3 Description

The system of equations is defined as:
fi (x1,x2,,xn) = 0 ,   i= 1, 2, , n .  
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 rank-1 method of Broyden. At the starting point, the Jacobian is requested, but it is not asked for again until the rank-1 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 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

5 Arguments

1: fcn Subroutine, supplied by the user. External Procedure
Depending upon the value of iflag, fcn must either return the values of the functions fi at a point x or return the Jacobian at x.
The specification of fcn is:
Fortran Interface
Subroutine fcn ( n, x, fvec, fjac, iuser, ruser, iflag)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: iuser(*), iflag
Real (Kind=nag_wp), Intent (In) :: x(n)
Real (Kind=nag_wp), Intent (Inout) :: fvec(n), fjac(n,n), ruser(*)
C Header Interface
void  fcn (const Integer *n, const double x[], double fvec[], double fjac[], Integer iuser[], double ruser[], Integer *iflag)
1: n Integer Input
On entry: n, the number of equations.
2: x(n) Real (Kind=nag_wp) array Input
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) array Input/Output
On entry: if iflag=2 , fvec contains the function values fi(x) and must not be changed.
On exit: if iflag=1 on entry, fvec must contain the function values fi(x) (unless iflag is set to a negative value by fcn).
4: fjac(n,n) Real (Kind=nag_wp) array Input/Output
On entry: if iflag=1 , fjac contains the value of fi xj at the point x, for i=1,2,,n and j=1,2,,n, and must not be changed.
On exit: if iflag=2 on entry, fjac(i,j) must contain the value of fi xj at the point x, for i=1,2,,n and j=1,2,,n, (unless iflag is set to a negative value by fcn).
5: iuser(*) Integer array User Workspace
6: ruser(*) Real (Kind=nag_wp) array User Workspace
fcn is called with the arguments iuser and ruser as supplied to c05rbf. You should use the arrays iuser and ruser to supply information to fcn.
7: iflag Integer Input/Output
On entry: iflag=1 or 2.
fvec is to be updated.
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), 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. Arguments denoted as Input must not be changed by this procedure.
Note: fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by c05rbf. If your code inadvertently does return any NaNs or infinities, c05rbf is likely to produce unexpected results.
2: n Integer Input
On entry: n, the number of equations.
Constraint: n>0 .
3: x(n) Real (Kind=nag_wp) array Input/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) array Output
On exit: the function values at the final point returned in x.
5: fjac(n,n) Real (Kind=nag_wp) array Output
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: ε, where ε is the machine precision returned by x02ajf.
Constraint: xtol0.0 .
7: iuser(*) Integer array User Workspace
8: ruser(*) Real (Kind=nag_wp) array User 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.
9: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry 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:
There have been at least 100×(n+1) calls to fcn. Consider restarting the calculation from the point held in x.
No further improvement in the solution is possible. xtol is too small: xtol=value.
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.
iflag was set negative in fcn. iflag=value.
On entry, n=value.
Constraint: n>0.
On entry, xtol=value.
Constraint: xtol0.0.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

If x^ is the true solution, c05rbf tries to ensure that
x-x^2 xtol × x^2 .  
If this condition is satisfied with 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 ifail=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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
c05rbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05rbf 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 implementation-specific information.

9 Further Comments

Local workspace arrays of fixed lengths are allocated internally by c05rbf. The total size of these arrays amounts to n×(n+13)/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×n2 to process each evaluation of the functions and approximately 1.3×n3 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.

10 Example

This example determines the values x1 , , x9 which satisfy the tridiagonal equations:
(3-2x1)x1-2x2 = −1, -xi-1+(3-2xi)xi-2xi+1 = −1,  i=2,3,,8 -x8+(3-2x9)x9 = −1.  

10.1 Program Text

Program Text (c05rbfe.f90)

10.2 Program Data


10.3 Program Results

Program Results (c05rbfe.r)