F08FQF (ZHEEVD) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix.
If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the QL or QR algorithm.
SUBROUTINE F08FQF ( |
JOB, UPLO, N, A, LDA, W, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO) |
INTEGER |
N, LDA, LWORK, LRWORK, IWORK(max(1,LIWORK)), LIWORK, INFO |
REAL (KIND=nag_wp) |
W(*), RWORK(max(1,LRWORK)) |
COMPLEX (KIND=nag_wp) |
A(LDA,*), WORK(max(1,LWORK)) |
CHARACTER(1) |
JOB, UPLO |
|
F08FQF (ZHEEVD) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix
A.
In other words, it can compute the spectral factorization of
A as
where
Λ is a real diagonal matrix whose diagonal elements are the eigenvalues
λi, and
Z is the (complex) unitary matrix whose columns are the eigenvectors
zi. Thus
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
The computed eigenvalues and eigenvectors are exact for a nearby matrix
A+E, where
and
ε is the
machine precision. See Section 4.7 of
Anderson et al. (1999) for further details.
The real analogue of this routine is
F08FCF (DSYEVD).
This example computes all the eigenvalues and eigenvectors of the Hermitian matrix
A, where
The example program for F08FQF (ZHEEVD) illustrates the computation of error bounds for the eigenvalues and eigenvectors.