NAG Library Routine Document
F01JCF computes an estimate of the absolute condition number of a matrix function at a real by matrix in the -norm, using analytical derivatives of you have supplied.
|SUBROUTINE F01JCF (
||N, A, LDA, F, IUSER, RUSER, IFLAG, CONDA, NORMA, NORMFA, IFAIL)
||N, LDA, IUSER(*), IFLAG, IFAIL
||A(LDA,*), RUSER(*), CONDA, NORMA, NORMFA
The absolute condition number of
is given by the norm of the Fréchet derivative of
, which is defined by
The Fréchet derivative in the direction
is linear in
and can therefore be written as
operator stacks the columns of a matrix into one vector, so that
. F01JCF computes an estimate
. The relative condition number can then be computed via
The algorithm used to find
is detailed in Section 3.4 of Higham (2008)
, and the derivatives of
, are returned by subroutine F
which, given an integer
at a number of (generally complex) points
. For any
on the real line,
must also be real. F01JCF is therefore appropriate for routines that can be evaluated on the complex plane and whose derivatives, of arbitrary order, can also be evaluated on the complex plane.
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
- 1: N – INTEGERInput
On entry: , the order of the matrix .
- 2: A(LDA,) – REAL (KIND=nag_wp) arrayInput/Output
the second dimension of the array A
must be at least
On entry: the by matrix .
On exit: the by matrix, .
- 3: LDA – INTEGERInput
: the first dimension of the array A
as declared in the (sub)program from which F01JCF is called.
- 4: F – SUBROUTINE, supplied by the user.External Procedure
Given an integer
, the subroutine F
at a number of points
The specification of F
||M, IFLAG, NZ, IUSER(*)
- 1: M – INTEGERInput
: the order,
, of the derivative required.
If , should be returned. For , should be returned.
- 2: IFLAG – INTEGERInput/Output
will be zero.
should either be unchanged from its entry value of zero, or may be set nonzero to indicate that there is a problem in evaluating the function
; for instance
may not be defined. If IFLAG
is returned as nonzero then F01JCF will terminate the computation, with
- 3: NZ – INTEGERInput
On entry: , the number of function or derivative values required.
- 4: Z(NZ) – COMPLEX (KIND=nag_wp) arrayInput
On entry: the points at which the function is to be evaluated.
- 5: FZ(NZ) – COMPLEX (KIND=nag_wp) arrayOutput
On exit: the function or derivative values.
should return the value , for . If lies on the real line, then so must .
- 6: IUSER() – INTEGER arrayUser Workspace
- 7: RUSER() – REAL (KIND=nag_wp) arrayUser Workspace
is called with the parameters IUSER
as supplied to F01JCF. You are free to use the arrays IUSER
to supply information to F
as an alternative to using COMMON global variables.
must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which F01JCF is called. Parameters denoted as Input
be changed by this procedure.
- 5: IUSER() – INTEGER arrayUser Workspace
- 6: RUSER() – REAL (KIND=nag_wp) arrayUser Workspace
are not used by F01JCF, but are passed directly to F
and may be used to pass information to this routine as an alternative to using COMMON global variables.
- 7: IFLAG – INTEGEROutput
, unless IFLAG
has been set nonzero inside F
, in which case IFLAG
will be the value set and IFAIL
will be set to
- 8: CONDA – REAL (KIND=nag_wp)Output
On exit: an estimate of the absolute condition number of at .
- 9: NORMA – REAL (KIND=nag_wp)Output
On exit: the -norm of .
- 10: NORMFA – REAL (KIND=nag_wp)Output
On exit: the -norm of .
- 11: 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:
An internal error occurred when estimating the norm of the Fréchet derivative of
. Please contact NAG
An internal error occurred when evaluating the matrix function
. You can investigate further by calling F01EMF
with the matrix
and the function
has been set nonzero by the user-supplied subroutine.
On entry, .
Input argument number is invalid.
On entry, parameter LDA
Allocation of memory failed.
F01JCF uses the norm estimation routine F04YDF
to estimate a quantity
. For further details on the accuracy of norm estimation, see the documentation for F04YDF
of real allocatable memory is required by the routine, in addition to the memory used by the underlying matrix function routine F01EMF
F01JCF returns the matrix function
. This is computed using F01EMF
. If only
is required, without an estimate of the condition number, then it is far more efficient to use F01EMF
The complex analogue of this routine is F01KCF
This example estimates the absolute and relative condition numbers of the matrix function
9.1 Program Text
Program Text (f01jcfe.f90)
9.2 Program Data
Program Data (f01jcfe.d)
9.3 Program Results
Program Results (f01jcfe.r)