fA is computed using a spectral factorization of
A
where
D is the real diagonal matrix whose diagonal elements,
di, are the eigenvalues of
A,
Q is a unitary matrix whose columns are the eigenvectors of
A and
QH is the conjugate transpose of
Q.
fA is then given by
where
fD is the diagonal matrix whose
ith diagonal element is
fdi. See for example Section 4.5 of
Higham (2008).
fdi is assumed to be real.
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).
Provided that
fD can be computed accurately then the computed matrix function will be close to the exact matrix function. See Section 10.2 of
Higham (2008) for details and further discussion.
The cost of the algorithm is
On3 plus the cost of evaluating
fD.
If
λ^i is the
ith computed eigenvalue of
A, then the user-supplied subroutine
F will be asked to evaluate the function
f at
fλ^i, for
i=1,2,…,n.
For further information on matrix functions, see
Higham (2008).
F01EFF can be used to find the matrix function
fA for a real symmetric matrix
A.
This example finds the matrix cosine,
cosA, of the Hermitian matrix