NAG Library Routine Document
F08JYF (ZSTEGR)
1 Purpose
F08JYF (ZSTEGR) computes all the eigenvalues and, optionally, all the eigenvectors of a real n by n symmetric tridiagonal matrix.
2 Specification
SUBROUTINE F08JYF ( |
JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) |
INTEGER |
N, IL, IU, M, LDZ, ISUPPZ(*), LWORK, IWORK(max(1,LIWORK)), LIWORK, INFO |
REAL (KIND=nag_wp) |
D(*), E(*), VL, VU, ABSTOL, W(*), WORK(max(1,LWORK)) |
COMPLEX (KIND=nag_wp) |
Z(LDZ,*) |
CHARACTER(1) |
JOBZ, RANGE |
|
The routine may be called by its
LAPACK
name zstegr.
3 Description
F08JYF (ZSTEGR) computes all the eigenvalues and, optionally, the eigenvectors, of a real symmetric tridiagonal matrix
T. That is, the routine computes the spectral factorization of
T given by
where
Λ is a diagonal matrix whose diagonal elements are the eigenvalues,
λi, of
T and
Z is an orthogonal matrix whose columns are the eigenvectors,
zi, of
T. Thus
The routine stores the real orthogonal matrix
Z in a complex array, so that it may also be used to compute all the eigenvalues and eigenvectors of a complex Hermitian matrix
A which has been reduced to tridiagonal form
T:
In this case, the matrix
Q must be explicitly applied to the output matrix
Z. The routines which must be called to perform the reduction to tridiagonal form and apply
Q are:
This routine uses the dqds and the Relatively Robust Representation algorithms to compute the eigenvalues and eigenvectors respectively; see for example
Parlett and Dhillon (2000) and
Dhillon and Parlett (2004) for further details. F08JYF (ZSTEGR) can usually compute all the eigenvalues and eigenvectors in
On2 floating point operations and so, for large matrices, is often considerably faster than the other symmetric tridiagonal routines in this chapter when all the eigenvectors are required, particularly so compared to those routines that are based on the
QR algorithm.
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
Dhillon I S and Parlett B N (2004) Orthogonal eigenvectors and relative gaps.
SIAM J. Appl. Math. 25 858–899
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: indicates which eigenvalues should be returned.
- RANGE='A'
- All eigenvalues will be found.
- RANGE='V'
- All eigenvalues in the half-open interval VL,VU will be found.
- RANGE='I'
- The ILth through IUth eigenvectors will be found.
Constraint:
RANGE='A', 'V' or 'I'.
- 3: N – INTEGERInput
On entry: n, the order of the matrix T.
Constraint:
N≥0.
- 4: D(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
D
must be at least
max1,N.
On entry: the n diagonal elements of the tridiagonal matrix T.
On exit:
D is overwritten.
- 5: E(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
E
must be at least
max1,N.
On entry: E1:N-1 contains the subdiagonal elements of the tridiagonal matrix T. EN need not be set.
On exit:
E is overwritten.
- 6: VL – REAL (KIND=nag_wp)Input
- 7: VU – REAL (KIND=nag_wp)Input
On entry: if
RANGE='V',
VL and
VU contain the lower and upper bounds respectively 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.
- 8: IL – INTEGERInput
- 9: IU – INTEGERInput
On entry: if
RANGE='I',
IL and
IU contains the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.
If
RANGE='A' or
'V',
IL and
IU are not referenced.
Constraints:
- if RANGE='I' and N>0, 1≤ IL≤ IU≤ N ;
- if RANGE='I' and N=0, IL=1 and IU=0.
- 10: ABSTOL – REAL (KIND=nag_wp)Input
On entry: in earlier versions, this argument was the absolute error tolerance for the eigenvalues/eigenvectors. It is now deprecated, and only included for backwards-compatibility.
- 11: 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.
- 12: W(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
W
must be at least
max1,N.
On exit: the eigenvalues in ascending order.
- 13: 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', then if
INFO=0, the columns of
Z contain the orthonormal eigenvectors of the matrix
T, 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.
- 14: LDZ – INTEGERInput
On entry: the first dimension of the array
Z as declared in the (sub)program from which F08JYF (ZSTEGR) is called.
Constraints:
- if JOBZ='V', LDZ≥ max1,N ;
- otherwise LDZ≥1.
- 15: 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.
- 16: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 returns the minimum
LWORK.
- 17: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08JYF (ZSTEGR) is called.
If
LWORK=-1, a workspace query is assumed; the routine only calculates the minimum sizes of the
WORK and
IWORK arrays, returns these values as the first entries of the
WORK and
IWORK arrays, and no error message related to
LWORK or
LIWORK is issued.
Constraint:
LWORK≥max1,18×N or LWORK=-1.
- 18: IWORK(max1,LIWORK) – INTEGER arrayWorkspace
On exit: if
INFO=0,
WORK1 returns the minimum
LIWORK.
- 19: LIWORK – INTEGERInput
On entry: the dimension of the array
IWORK as declared in the (sub)program from which F08JYF (ZSTEGR) is called.
If
LIWORK=-1, a workspace query is assumed; the routine only calculates the minimum sizes of the
WORK and
IWORK arrays, returns these values as the first entries of the
WORK and
IWORK arrays, and no error message related to
LWORK or
LIWORK is issued.
Constraint:
LIWORK≥max1,10×N or LIWORK=-1.
- 20: 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
If INFO=1, the dqds algorithm failed to converge, if INFO=2, inverse iteration failed to converge.
7 Accuracy
See the description for
ABSTOL. See also Section 4.7 of
Anderson et al. (1999) and
Barlow and Demmel (1990) for further details.
8 Further Comments
The total number of floating point operations required to compute all the eigenvalues and eigenvectors is approximately proportional to n2.
The real analogue of this routine is
F08JLF (DSTEGR).
9 Example
This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
ABSTOL is set to zero so that the default tolerance of
nε T1 is used.
9.1 Program Text
Program Text (f08jyfe.f90)
9.2 Program Data
Program Data (f08jyfe.d)
9.3 Program Results
Program Results (f08jyfe.r)