NAG FL Interface
c05qsf (sparsys_​func_​easy)

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1 Purpose

c05qsf is an easy-to-use routine that finds a solution of a sparse system of nonlinear equations by a modification of the Powell hybrid method.

2 Specification

Fortran Interface
Subroutine c05qsf ( fcn, n, x, fvec, xtol, init, rcomm, lrcomm, icomm, licomm, iuser, ruser, ifail)
Integer, Intent (In) :: n, lrcomm, licomm
Integer, Intent (Inout) :: icomm(licomm), iuser(*), ifail
Real (Kind=nag_wp), Intent (In) :: xtol
Real (Kind=nag_wp), Intent (Inout) :: x(n), rcomm(lrcomm), ruser(*)
Real (Kind=nag_wp), Intent (Out) :: fvec(n)
Logical, Intent (In) :: init
External :: fcn
C Header Interface
#include <nag.h>
void  c05qsf_ (
void (NAG_CALL *fcn)(const Integer *n, const Integer *lindf, const Integer indf[], const double x[], double fvec[], Integer iuser[], double ruser[], Integer *iflag),
const Integer *n, double x[], double fvec[], const double *xtol, const logical *init, double rcomm[], const Integer *lrcomm, Integer icomm[], const Integer *licomm, Integer iuser[], double ruser[], Integer *ifail)
The routine may be called by the names c05qsf or nagf_roots_sparsys_func_easy.

3 Description

The system of equations is defined as:
fi (x1,x2,,xn) = 0 ,   ​ i= 1, 2, , n .  
c05qsf 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 sparse rank-1 method of Schubert (see Schubert (1970)). At the starting point, the sparsity pattern is determined and the Jacobian is approximated by forward differences, but these are not used again until the rank-1 method fails to produce satisfactory progress. Then, the sparsity structure is used to recompute an approximation to the Jacobian by forward differences with the least number of function evaluations. The subroutine you supply must be able to compute only the requested subset of the function values. The sparse Jacobian linear system is solved at each iteration with f11mef computing the Newton step. For more details see Powell (1970) and Broyden (1965).

4 References

Broyden C G (1965) A class of methods for solving nonlinear simultaneous equations Mathematics of Computation 19(92) 577–593
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
Schubert L K (1970) Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian Mathematics of Computation 24(109) 27–30

5 Arguments

1: fcn Subroutine, supplied by the user. External Procedure
fcn must return the values of the functions fi at a point x.
The specification of fcn is:
Fortran Interface
Subroutine fcn ( n, lindf, indf, x, fvec, iuser, ruser, iflag)
Integer, Intent (In) :: n, lindf, indf(lindf)
Integer, Intent (Inout) :: iuser(*), iflag
Real (Kind=nag_wp), Intent (In) :: x(n)
Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
Real (Kind=nag_wp), Intent (Out) :: fvec(n)
C Header Interface
void  fcn (const Integer *n, const Integer *lindf, const Integer indf[], const double x[], double fvec[], Integer iuser[], double ruser[], Integer *iflag)
1: n Integer Input
On entry: n, the number of equations.
2: lindf Integer Input
On entry: lindf specifies the number of indices i for which values of fi(x) must be computed.
3: indf(lindf) Integer array Input
On entry: indf specifies the indices i for which values of fi(x) must be computed. The indices are specified in strictly ascending order.
4: x(n) Real (Kind=nag_wp) array Input
On entry: the components of the point x at which the functions must be evaluated. x(i) contains the coordinate xi.
5: fvec(n) Real (Kind=nag_wp) array Output
On exit: fvec(i) must contain the function values fi(x), for all indices i in indf.
6: iuser(*) Integer array User Workspace
7: ruser(*) Real (Kind=nag_wp) array User Workspace
fcn is called with the arguments iuser and ruser as supplied to c05qsf. You should use the arrays iuser and ruser to supply information to fcn.
8: iflag Integer Input/Output
On entry: 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), 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 c05qsf 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 c05qsf. If your code inadvertently does return any NaNs or infinities, c05qsf 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. x(i) must contain the coordinate xi.
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. fvec(i) contains the function values fi.
5: 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 .
6: init Logical Input
On entry: init must be set to .TRUE. to indicate that this is the first time c05qsf is called for this specific problem. c05qsf then computes the dense Jacobian and detects and stores its sparsity pattern (in rcomm and icomm) before proceeding with the iterations. This is noticeably time consuming when n is large. If not enough storage has been provided for rcomm or icomm, c05qsf will fail. On exit with ifail=0, 2, 3 or 4, icomm(1) contains nnz, the number of nonzero entries found in the Jacobian. On subsequent calls, init can be set to .FALSE. if the problem has a Jacobian of the same sparsity pattern. In that case, the computation time required for the detection of the sparsity pattern will be smaller.
7: rcomm(lrcomm) Real (Kind=nag_wp) array Communication Array
rcomm must not be altered between successive calls to c05qsf.
8: lrcomm Integer Input
On entry: the dimension of the array rcomm as declared in the (sub)program from which c05qsf is called.
Constraint: lrcomm12+nnz where nnz is the number of nonzero entries in the Jacobian, as computed by c05qsf.
9: icomm(licomm) Integer array Communication Array
If ifail=0, 2, 3 or 4 on exit, icomm(1) contains nnz where nnz is the number of nonzero entries in the Jacobian.
icomm must not be altered between successive calls to c05qsf.
10: licomm Integer Input
On entry: the dimension of the array icomm as declared in the (sub)program from which c05qsf is called.
Constraint: licomm8×n+19+nnz where nnz is the number of nonzero entries in the Jacobian, as computed by c05qsf.
11: iuser(*) Integer array User Workspace
12: ruser(*) Real (Kind=nag_wp) array User Workspace
iuser and ruser are not used by c05qsf, but are passed directly to fcn and may be used to pass information to this routine.
13: 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 200×(n+1) calls to fcn. Consider setting init=.FALSE. and 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 c05qsf from a different starting point may avoid the region of difficulty. The condition number of the Jacobian is value.
Termination requested in fcn.
On entry, lrcomm=value.
Constraint: lrcommvalue.
On entry, licomm=value.
Constraint: licommvalue.
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, c05qsf 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 c05qsf 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 are reasonably well behaved. If this condition is not satisfied, then c05qsf may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning c05qsf with a lower value for xtol.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
c05qsf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05qsf 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 c05qsf. The total size of these arrays amounts to 8×n+2×q real elements and 10×n+2×q+5 integer elements where the integer q is bounded by 8×nnz and n2 and depends on the sparsity pattern of the Jacobian.
The time required by c05qsf 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 c05qsf to process each evaluation of the functions depends on the number of nonzero entries in the Jacobian. The timing of c05qsf is strongly influenced by the time spent evaluating the functions.
When init is .TRUE., the dense Jacobian is first evaluated and that will take time proportional to n2.
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.  
It then perturbs the equations by a small amount and solves the new system.

10.1 Program Text

Program Text (c05qsfe.f90)

10.2 Program Data


10.3 Program Results

Program Results (c05qsfe.r)