F08QYF (ZTRSNA) estimates condition numbers for specified eigenvalues and/or right eigenvectors of a complex upper triangular matrix.
SUBROUTINE F08QYF ( |
JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK, INFO) |
INTEGER |
N, LDT, LDVL, LDVR, MM, M, LDWORK, INFO |
REAL (KIND=nag_wp) |
S(*), SEP(*), RWORK(*) |
COMPLEX (KIND=nag_wp) |
T(LDT,*), VL(LDVL,*), VR(LDVR,*), WORK(LDWORK,*) |
LOGICAL |
SELECT(*) |
CHARACTER(1) |
JOB, HOWMNY |
|
F08QYF (ZTRSNA) estimates condition numbers for specified eigenvalues and/or right eigenvectors of a complex upper triangular matrix T. These are the same as the condition numbers of the eigenvalues and right eigenvectors of an original matrix A=ZTZH (with unitary Z), from which T may have been derived.
F08QYF (ZTRSNA) computes the reciprocal of the condition number of an eigenvalue
λi as
where
u and
v are the right and left eigenvectors of
T, respectively, corresponding to
λi. This reciprocal condition number always lies between zero (i.e., ill-conditioned) and one (i.e., well-conditioned).
An approximate error estimate for a computed eigenvalue
λi is then given by
where
ε is the
machine precision.
To estimate the reciprocal of the condition number of the right eigenvector corresponding to
λi, the routine first calls
F08QTF (ZTREXC) to reorder the eigenvalues so that
λi is in the leading position:
The reciprocal condition number of the eigenvector is then estimated as
sepi, the smallest singular value of the matrix
T22-λiI. This number ranges from zero (i.e., ill-conditioned) to very large (i.e., well-conditioned).
An approximate error estimate for a computed right eigenvector
u corresponding to
λi is then given by
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
The real analogue of this routine is
F08QLF (DTRSNA).
This example computes approximate error estimates for all the eigenvalues and right eigenvectors of the matrix
T, where