NAG CL Interface
c05qsc (sparsys_func_easy)
1
Purpose
c05qsc is an easy-to-use function that finds a solution of a sparse system of nonlinear equations by a modification of the Powell hybrid method.
2
Specification
void |
c05qsc (
Integer n,
double x[],
double fvec[],
double xtol,
Nag_Boolean init,
double rcomm[],
Integer lrcomm,
Integer icomm[],
Integer licomm,
Nag_Comm *comm,
NagError *fail) |
|
The function may be called by the names: c05qsc, nag_roots_sparsys_func_easy or nag_zero_sparse_nonlin_eqns_easy.
3
Description
The system of equations is defined as:
c05qsc 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 function 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
f11mec 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:
– function, supplied by the user
External Function
-
fcn must return the values of the functions
at a point
.
The specification of
fcn is:
void |
fcn (Integer n,
Integer lindf,
const Integer indf[],
const double x[],
double fvec[],
Nag_Comm *comm, Integer *iflag)
|
|
-
1:
– Integer
Input
-
On entry: , the number of equations.
-
2:
– Integer
Input
-
On entry:
lindf specifies the number of indices
for which values of
must be computed.
-
3:
– const Integer
Input
-
On entry:
indf specifies the indices
for which values of
must be computed. The indices are specified in strictly ascending order.
-
4:
– const double
Input
-
On entry: the components of the point at which the functions must be evaluated. contains the coordinate .
-
5:
– double
Output
-
On exit:
must contain the function values
, for all indices
in
indf.
-
6:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
fcn.
- user – double *
- iuser – Integer *
- p – Pointer
The type Pointer will be
void *. Before calling
c05qsc you may allocate memory and initialize these pointers with various quantities for use by
fcn when called from
c05qsc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
-
7:
– Integer *
Input/Output
-
On entry: .
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.
Note: fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
c05qsc. If your code inadvertently
does return any NaNs or infinities,
c05qsc is likely to produce unexpected results.
-
2:
– Integer
Input
-
On entry: , the number of equations.
Constraint:
.
-
3:
– double
Input/Output
-
On entry: an initial guess at the solution vector. must contain the coordinate .
On exit: the final estimate of the solution vector.
-
4:
– double
Output
-
On exit: the function values at the final point returned in
x.
contains the function values
.
-
5:
– double
Input
-
On entry: the accuracy in
x to which the solution is required.
Suggested value:
, where
is the
machine precision returned by
X02AJC.
Constraint:
.
-
6:
– Nag_Boolean
Input
-
On entry:
init must be set to Nag_TRUE to indicate that this is the first time
c05qsc is called for this specific problem.
c05qsc 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,
c05qsc will fail. On exit with
NE_NOERROR,
NE_NO_IMPROVEMENT,
NE_TOO_MANY_FEVALS or
NE_TOO_SMALL,
contains
, the number of nonzero entries found in the Jacobian. On subsequent calls,
init can be set to Nag_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:
– double
Communication Array
-
rcomm MUST NOT be altered between successive calls to
c05qsc.
-
8:
– Integer
Input
-
On entry: the dimension of the array
rcomm.
Constraint:
where is the number of nonzero entries in the Jacobian, as computed by c05qsc.
-
9:
– Integer
Communication Array
-
If
NE_NOERROR,
NE_NO_IMPROVEMENT,
NE_TOO_MANY_FEVALS or
NE_TOO_SMALL on exit,
contains
where
is the number of nonzero entries in the Jacobian.
icomm MUST NOT be altered between successive calls to
c05qsc.
-
10:
– Integer
Input
-
On entry: the dimension of the array
icomm.
Constraint:
where is the number of nonzero entries in the Jacobian, as computed by c05qsc.
-
11:
– Nag_Comm *
-
The NAG communication argument (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
-
12:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_IMPROVEMENT
-
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
c05qsc from a different starting point may avoid the region of difficulty. The condition number of the Jacobian is
.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
- NE_REAL
-
On entry, .
Constraint: .
- NE_TOO_MANY_FEVALS
-
There have been at least
calls to
fcn. Consider setting
and restarting the calculation from the point held in
x.
- NE_TOO_SMALL
-
No further improvement in the solution is possible.
xtol is too small:
.
- NE_USER_STOP
-
iflag was set negative in
fcn.
.
7
Accuracy
If
is the true solution,
c05qsc tries to ensure that
If this condition is satisfied with
, then the larger components of
have
significant decimal digits. There is a danger that the smaller components of
may have large relative errors, but the fast rate of convergence of
c05qsc 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 function exits with
NE_TOO_SMALL.
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
c05qsc may incorrectly indicate convergence. The validity of the answer can be checked, for example, by rerunning
c05qsc with a lower value for
xtol.
8
Parallelism and Performance
c05qsc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05qsc 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 function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
Local workspace arrays of fixed lengths are allocated internally by c05qsc. The total size of these arrays amounts to double elements and integer elements where the integer is bounded by and and depends on the sparsity pattern of the Jacobian.
The time required by c05qsc 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 c05qsc to process each evaluation of the functions depends on the number of nonzero entries in the Jacobian. The timing of c05qsc is strongly influenced by the time spent evaluating the functions.
When
init is Nag_TRUE, the dense Jacobian is first evaluated and that will take time proportional to
.
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
which satisfy the tridiagonal equations:
It then perturbs the equations by a small amount and solves the new system.
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results