NAG Library Routine Document
F08FRF (ZHEEVR)
1 Purpose
F08FRF (ZHEEVR) computes selected eigenvalues and, optionally, eigenvectors of a complex n by n Hermitian matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
2 Specification
SUBROUTINE F08FRF ( |
JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO) |
INTEGER |
N, LDA, IL, IU, M, LDZ, ISUPPZ(*), LWORK, LRWORK, IWORK(max(1,LIWORK)), LIWORK, INFO |
REAL (KIND=nag_wp) |
VL, VU, ABSTOL, W(*), RWORK(max(1,LRWORK)) |
COMPLEX (KIND=nag_wp) |
A(LDA,*), Z(LDZ,*), WORK(max(1,LWORK)) |
CHARACTER(1) |
JOBZ, RANGE, UPLO |
|
The routine may be called by its
LAPACK
name zheevr.
3 Description
The Hermitian matrix is first reduced to a real tridiagonal matrix
T, using unitary similarity transformations. Then whenever possible, F08FRF (ZHEEVR) computes the eigenspectrum using Relatively Robust Representations. F08FRF (ZHEEVR) computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various ‘good’
LDLT representations (also known as Relatively Robust Representations). Gram–Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the
ith unreduced block of
T:
(a) |
compute
T
-
σi
I
=
Li
Di
LiT
, such that
Li
Di
LiT
is a relatively robust representation, |
(b) |
compute the eigenvalues, λj, of
Li
Di
LiT
to high relative accuracy by the dqds algorithm, |
(c) |
if there is a cluster of close eigenvalues, ‘choose’ σi close to the cluster, and go to (a), |
(d) |
given the approximate eigenvalue λj of
Li
Di
LiT
, compute the corresponding eigenvector by forming a rank-revealing twisted factorization. |
The desired accuracy of the output can be specified by the parameter
ABSTOL. For more details, see
Dhillon (1997) and
Parlett and Dhillon (2000).
4 References
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
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices
SIAM J. Numer. Anal. 27 762–791
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices
SIAM J. Sci. Statist. Comput. 11 873–912
Dhillon I (1997) A new
On2 algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem
Computer Science Division Technical Report No. UCB//CSD-97-971 UC Berkeley
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals
Linear Algebra Appl. 309 121–151
5 Parameters
- 1: JOBZ – CHARACTER(1)Input
On entry: indicates whether eigenvectors are computed.
- JOBZ='N'
- Only eigenvalues are computed.
- JOBZ='V'
- Eigenvalues and eigenvectors are computed.
Constraint:
JOBZ='N' or 'V'.
- 2: RANGE – CHARACTER(1)Input
On entry: if
RANGE='A', all eigenvalues will be found.
If RANGE='V', all eigenvalues in the half-open interval VL,VU will be found.
If
RANGE='I', the
ILth to
IUth eigenvalues will be found.
For
RANGE='V' or
'I' and
IU-IL<N-1,
F08JJF (DSTEBZ) and
F08JXF (ZSTEIN) are called.
Constraint:
RANGE='A', 'V' or 'I'.
- 3: UPLO – CHARACTER(1)Input
On entry: if
UPLO='U', the upper triangular part of
A is stored.
If UPLO='L', the lower triangular part of A is stored.
Constraint:
UPLO='U' or 'L'.
- 4: N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint:
N≥0.
- 5: A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the
n by
n Hermitian matrix
A.
- If UPLO='U', the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
- If UPLO='L', the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: the lower triangle (if
UPLO='L') or the upper triangle (if
UPLO='U') of
A, including the diagonal, is overwritten.
- 6: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08FRF (ZHEEVR) is called.
Constraint:
LDA≥max1,N.
- 7: VL – REAL (KIND=nag_wp)Input
- 8: VU – REAL (KIND=nag_wp)Input
On entry: if
RANGE='V', the lower and upper bounds of the interval to be searched for eigenvalues.
If
RANGE='A' or
'I',
VL and
VU are not referenced.
Constraint:
if RANGE='V', VL<VU.
- 9: IL – INTEGERInput
- 10: IU – INTEGERInput
On entry: if
RANGE='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If
RANGE='A' or
'V',
IL and
IU are not referenced.
Constraints:
- if RANGE='I' and N=0, IL=1 and IU=0;
- if RANGE='I' and N>0, 1≤ IL≤ IU≤ N .
- 11: ABSTOL – REAL (KIND=nag_wp)Input
On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval
a,b of width less than or equal to
where
ε is the
machine precision. If
ABSTOL is less than or equal to zero, then
ε T1 will be used in its place, where
T is the real tridiagonal matrix obtained by reducing
A to tridiagonal form. See
Demmel and Kahan (1990).
If high relative accuracy is important, set
ABSTOL to
X02AMF , although doing so does not currently guarantee that eigenvalues are computed to high relative accuracy. See
Barlow and Demmel (1990) for a discussion of which matrices can define their eigenvalues to high relative accuracy.
- 12: M – INTEGEROutput
On exit: the total number of eigenvalues found.
0≤M≤N.
If RANGE='A', M=N.
If RANGE='I', M=IU-IL+1.
- 13: W(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
W
must be at least
max1,N.
On exit: the first
M elements contain the selected eigenvalues in ascending order.
- 14: Z(LDZ,*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
Z
must be at least
max1,M if
JOBZ='V', and at least
1 otherwise.
On exit: if
JOBZ='V', the first
M columns of
Z contain the orthonormal eigenvectors of the matrix
A corresponding to the selected eigenvalues, with the
ith column of
Z holding the eigenvector associated with
Wi.
If
JOBZ='N',
Z is not referenced.
Note: you must ensure that at least
max1,M columns are supplied in the array
Z; if
RANGE='V', the exact value of
M is not known in advance and an upper bound of at least
N must be used.
- 15: LDZ – INTEGERInput
On entry: the first dimension of the array
Z as declared in the (sub)program from which F08FRF (ZHEEVR) is called.
Constraints:
- if JOBZ='V', LDZ≥ max1,N ;
- otherwise LDZ≥1.
- 16: ISUPPZ(*) – INTEGER arrayOutput
-
Note: the dimension of the array
ISUPPZ
must be at least
max1,2×M.
On exit: the support of the eigenvectors in
Z, i.e., the indices indicating the nonzero elements in
Z. The
ith eigenvector is nonzero only in elements
ISUPPZ2×i-1 through
ISUPPZ2×i. Implemented only for
RANGE='A' or
'I' and
IU-IL=N-1.
- 17: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0, the real part of
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 18: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08FRF (ZHEEVR) is called.
If
LWORK=-1, a workspace query is assumed; the routine only calculates the optimal sizes of the
WORK,
RWORK and
IWORK arrays, returns these values as the first entries of the
WORK,
RWORK and
IWORK arrays, and no error message related to
LWORK,
LRWORK or
LIWORK is issued.
Suggested value:
for optimal performance,
LWORK≥nb+1×N, where
nb is the largest optimal
block size for
F08FSF (ZHETRD) and for
F08FUF (ZUNMTR).
Constraint:
LWORK≥max1,2×N.
- 19: RWORK(max1,LRWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
RWORK1 returns the optimal (and minimal)
LRWORK.
- 20: LRWORK – INTEGERInput
On entry: the dimension of the array
RWORK as declared in the (sub)program from which F08FRF (ZHEEVR) is called.
If
LRWORK=-1, a workspace query is assumed; the routine only calculates the optimal sizes of the
WORK,
RWORK and
IWORK arrays, returns these values as the first entries of the
WORK,
RWORK and
IWORK arrays, and no error message related to
LWORK,
LRWORK or
LIWORK is issued.
Constraint:
LRWORK≥max1,24×N.
- 21: IWORK(max1,LIWORK) – INTEGER arrayWorkspace
On exit: if
INFO=0,
IWORK1 returns the optimal (and minimal)
LIWORK.
- 22: LIWORK – INTEGERInput
On entry: the dimension of the array
IWORK as declared in the (sub)program from which F08FRF (ZHEEVR) is called.
If
LIWORK=-1, a workspace query is assumed; the routine only calculates the optimal sizes of the
WORK,
RWORK and
IWORK arrays, returns these values as the first entries of the
WORK,
RWORK and
IWORK arrays, and no error message related to
LWORK,
LRWORK or
LIWORK is issued.
Constraint:
LIWORK≥ max1,10×N .
- 23: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO>0
F08FRF (ZHEEVR) failed to converge.
7 Accuracy
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.
8 Further Comments
The total number of floating point operations is proportional to n3.
The real analogue of this routine is
F08FDF (DSYEVR).
9 Example
This example finds the eigenvalues with indices in the range
2,3
, and the corresponding eigenvectors, of the Hermitian matrix
Information on required and provided workspace is also output.
9.1 Program Text
Program Text (f08frfe.f90)
9.2 Program Data
Program Data (f08frfe.d)
9.3 Program Results
Program Results (f08frfe.r)