F01KCF computes an estimate of the absolute condition number of a matrix function f at a complex n by n matrix A in the 1-norm, using analytical derivatives of f you have supplied.
The absolute condition number of
f at
A,
condabsf,A is given by the norm of the Fréchet derivative of
f,
LA,E, which is defined by
The Fréchet derivative in the direction
E,
LX,E is linear in
E and can therefore be written as
where the
vec operator stacks the columns of a matrix into one vector, so that
KX is
n2×n2. F01KCF computes an estimate
γ such that
γ
≤
KX
1
, where
KX
1
∈
n-1
LX
1
,
n
LX
1
. The relative condition number can then be computed via
The algorithm used to find
γ is detailed in Section 3.4 of
Higham (2008).
The function
f, and the derivatives of
f, are returned by subroutine
F which, given an integer
m, evaluates
fmzi at a number of points
zi, for
i=1,2,…,nz, on the complex plane. F01KCF 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
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
F01KCF uses the norm estimation routine
F04ZDF to estimate a quantity
γ, where
γ
≤
KX
1
and
KX
1
∈
n-1
LX
1
,
n
LX
1
. For further details on the accuracy of norm estimation, see the documentation for
F04ZDF.
Approximately
6n2 of complex allocatable memory is required by the routine, in addition to the memory used by the underlying matrix function routine
F01FMF.
F01KCF returns the matrix function
fA. This is computed using
F01FMF. If only
fA is required, without an estimate of the condition number, then it is far more efficient to use
F01FMF directly.
The real analogue of this routine is
F01JCF.
This example estimates the absolute and relative condition numbers of the matrix function
e3A where