NAG CL Interface
c05rdc (sys_deriv_rcomm)
1
Purpose
c05rdc is a comprehensive reverse communication function 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
void |
c05rdc (Integer *irevcm,
Integer n,
double x[],
double fvec[],
double fjac[],
double xtol,
Nag_ScaleType scale_mode,
double diag[],
double factor,
double r[],
double qtf[],
Integer iwsav[],
double rwsav[],
NagError *fail) |
|
The function may be called by the names: c05rdc, nag_roots_sys_deriv_rcomm or nag_zero_nonlin_eqns_deriv_rcomm.
3
Description
The system of equations is defined as:
c05rdc is based on the MINPACK routine HYBRJ (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. 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
Note: this function uses
reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument
irevcm. Between intermediate exits and re-entries,
all arguments other than fvec and fjac must remain unchanged.
-
1:
– Integer *
Input/Output
-
On initial entry: must have the value .
On intermediate exit:
specifies what action you must take before re-entering
c05rdc with
irevcm unchanged. The value of
irevcm should be interpreted as follows:
- Indicates the start of a new iteration. No action is required by you, but x and fvec are available for printing.
- Indicates that before re-entry to c05rdc, fvec must contain the function values .
- Indicates that before re-entry to c05rdc,
must contain the value of at the point , for and .
On final exit: and the algorithm has terminated.
Constraint:
, , or .
Note: any values you return to c05rdc as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by c05rdc. If your code inadvertently does return any NaNs or infinities, c05rdc is likely to produce unexpected results.
-
2:
– Integer
Input
-
On entry: , the number of equations.
Constraint:
.
-
3:
– double
Input/Output
-
On initial entry: an initial guess at the solution vector.
On intermediate exit:
contains the current point.
On final exit: the final estimate of the solution vector.
-
4:
– double
Input/Output
-
On initial entry: need not be set.
On intermediate re-entry: if
,
fvec must not be changed.
If
,
fvec must be set to the values of the functions computed at the current point
x.
On final exit: the function values at the final point,
x.
-
5:
– double
Input/Output
-
Note: the th element of the matrix is stored in .
On initial entry: need not be set.
On intermediate re-entry: if
,
fjac must not be changed.
If ,
must contain the value of at the point , for and .
On final exit: the orthogonal matrix produced by the factorization of the final approximate Jacobian, stored by columns.
-
6:
– double
Input
-
On initial entry: the accuracy in
x to which the solution is required.
Suggested value:
, where
is the
machine precision returned by
X02AJC.
Constraint:
.
-
7:
– Nag_ScaleType
Input
-
On initial entry: indicates whether or not you have provided scaling factors in
diag.
If
, the scaling must have been supplied in
diag.
Otherwise, if , the variables will be scaled internally.
Constraint:
or .
-
8:
– double
Input/Output
-
On initial entry: if
,
diag must contain multiplicative scale factors for the variables.
If
,
diag need not be set.
Constraint:
if ,, for .
On intermediate exit:
diag must not be changed.
On final exit: the scale factors actually used (computed internally if ).
-
9:
– double
Input
-
On initial entry: a quantity to be used in determining the initial step bound. In most cases,
factor should lie between
and
. (The step bound is
if this is nonzero; otherwise the bound is
factor.)
Suggested value:
.
Constraint:
.
-
10:
– double
Input/Output
-
On initial entry: need not be set.
On intermediate exit:
must not be changed.
On final exit: the upper triangular matrix produced by the factorization of the final approximate Jacobian, stored row-wise.
-
11:
– double
Input/Output
-
On initial entry: need not be set.
On intermediate exit:
must not be changed.
On final exit: the vector .
-
12:
– Integer
Communication Array
-
13:
– double
Communication Array
-
The arrays
iwsav and
rwsav MUST NOT be altered between calls to
c05rdc.
-
14:
– 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_DIAG_ELEMENTS
-
On entry,
and
diag contained a non-positive element.
- NE_INT
-
On entry, .
Constraint: , , or .
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, as measured by the improvement from the last
iterations. 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
c05rdc from a different starting point may avoid the region of difficulty.
The iteration is not making good progress, as measured by the improvement from the last
Jacobian evaluations. 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
c05rdc from a different starting point may avoid the region of difficulty.
- 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: .
On entry, .
Constraint: .
- NE_TOO_SMALL
-
No further improvement in the solution is possible.
xtol is too small:
.
7
Accuracy
If
is the true solution and
denotes the diagonal matrix whose entries are defined by the array
diag, then
c05rdc 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
c05rdc 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 and the Jacobian are coded consistently and that the functions are reasonably well behaved. If these conditions are not satisfied, then
c05rdc may incorrectly indicate convergence. The coding of the Jacobian can be checked using
c05zdc. If the Jacobian is coded correctly, then the validity of the answer can be checked by rerunning
c05rdc with a lower value for
xtol.
8
Parallelism and Performance
c05rdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
c05rdc 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.
The time required by c05rdc 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 c05rdc is approximately to process each evaluation of the functions and approximately to process each evaluation of the Jacobian. The timing of c05rdc 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
which satisfy the tridiagonal equations:
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results