F01KAF 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, where f is either the exponential, logarithm, sine, cosine, hyperbolic sine (sinh) or hyperbolic cosine (cosh). The evaluation of the matrix function, fA, is also returned.
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. F01KAF 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).
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).
F01KAF 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 routines
F01FCF,
F01FJF or
F01FKF.
F01KAF returns the matrix function
fA. This is computed using
F01FCF if
FUN='EXP',
F01FJF if
FUN='LOG' and
F01FKF otherwise. If only
fA is required, without an estimate of the condition number, then it is far more efficient to use
F01FCF,
F01FJF or
F01FKF directly.
F01JAF can be used to find the condition number of the exponential, logarithm, sine, cosine, sinh or cosh at a real matrix.
This example estimates the absolute and relative condition numbers of the matrix sinh function for