NAG Library Routine Document
F08JVF (ZSTEDC)
1 Purpose
F08JVF (ZSTEDC) computes all the eigenvalues and, optionally, all the eigenvectors of a real n by n symmetric tridiagonal matrix, or of a complex full or banded Hermitian matrix which has been reduced to tridiagonal form.
2 Specification
SUBROUTINE F08JVF ( |
COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO) |
INTEGER |
N, LDZ, LWORK, LRWORK, IWORK(max(1,LIWORK)), LIWORK, INFO |
REAL (KIND=nag_wp) |
D(*), E(*), RWORK(max(1,LRWORK)) |
COMPLEX (KIND=nag_wp) |
Z(LDZ,*), WORK(max(1,LWORK)) |
CHARACTER(1) |
COMPZ |
|
The routine may be called by its
LAPACK
name zstedc.
3 Description
F08JVF (ZSTEDC) 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 may also be used to compute all the eigenvalues and eigenvectors of a complex full, or banded, Hermitian matrix
A which has been reduced to real tridiagonal form
T as
where
Q is unitary. The spectral factorization of
A is then given by
In this case
Q must be formed explicitly and passed to F08JVF (ZSTEDC) in the array
Z, and the routine called with
COMPZ='V'. Routines which may be called to form
T and
Q are
When only eigenvalues are required then this routine calls
F08JFF (DSTERF) to compute the eigenvalues of the tridiagonal matrix
T, but when eigenvectors of
T are also required and the matrix is not too small, then a divide and conquer method is used, which can be much faster than
F08JSF (ZSTEQR), although more storage is required.
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
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: COMPZ – CHARACTER(1)Input
On entry: indicates whether the eigenvectors are to be computed.
- COMPZ='N'
- Only the eigenvalues are computed (and the array Z is not referenced).
- COMPZ='I'
- The eigenvalues and eigenvectors of T are computed (and the array Z is initialized by the routine).
- COMPZ='V'
- The eigenvalues and eigenvectors of A are computed (and the array Z must contain the matrix Q on entry).
Constraint:
COMPZ='N', 'V' or 'I'.
- 2: N – INTEGERInput
On entry: n, the order of the symmetric tridiagonal matrix T.
Constraint:
N≥0.
- 3: D(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
D
must be at least
max1,N.
On entry: the diagonal elements of the tridiagonal matrix.
On exit: if INFO=0, the eigenvalues in ascending order.
- 4: E(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
E
must be at least
max1,N-1.
On entry: the subdiagonal elements of the tridiagonal matrix.
On exit:
E is overwritten.
- 5: Z(LDZ,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
Z
must be at least
max1,N if
COMPZ='V' or
'I', and at least
1 otherwise.
On entry: if
COMPZ='V',
Z must contain the unitary matrix
Q used in the reduction to tridiagonal form.
On exit: if
COMPZ='V',
Z contains the orthonormal eigenvectors of the original Hermitian matrix
A, and if
COMPZ='I',
Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix
T.
If
COMPZ='N',
Z is not referenced.
- 6: LDZ – INTEGERInput
On entry: the first dimension of the array
Z as declared in the (sub)program from which F08JVF (ZSTEDC) is called.
Constraints:
- if COMPZ='V' or 'I', LDZ≥ max1,N ;
- otherwise LDZ≥1.
- 7: 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.
- 8: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08JVF (ZSTEDC) 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.
Constraints:
if
LWORK≠-1,
- if COMPZ='N' or 'I' or N≤1, LWORK must be at least 1;
- if COMPZ='V' and N>1, LWORK must be at least N2.
Note: that for
COMPZ='V', then if
N is less than or equal to the minimum divide size, usually
25, then
LWORK need only be
1.
- 9: RWORK(max1,LRWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
RWORK1 returns the optimal
LRWORK.
- 10: LRWORK – INTEGERInput
On entry: the dimension of the array
RWORK as declared in the (sub)program from which F08JVF (ZSTEDC) 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.
Constraints:
if
LRWORK≠-1,
- if COMPZ='N' or N≤1, LRWORK must be at least 1;
- if COMPZ='V' and N>1, LRWORK must be at least 1+3×N+2×N×lgN+4×N2, where lgN=smallest integer k such that 2k≥N;
- if COMPZ='I' and N>1, LRWORK must be at least 1+4×N+2×N2.
Note: that for
COMPZ='I' or
'V' then if
N is less than or equal to the minimum divide size, usually
25, then
LRWORK need only be
max1,2×N-1 .
- 11: IWORK(max1,LIWORK) – INTEGER arrayWorkspace
On exit: if
INFO=0,
IWORK1 returns the optimal
LIWORK.
- 12: LIWORK – INTEGERInput
On entry: the dimension of the array
IWORK as declared in the (sub)program from which F08JVF (ZSTEDC) 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.
Constraints:
if
LIWORK≠-1,
- if COMPZ='N' or N≤1, LIWORK must be at least 1;
- if COMPZ='V' or N>1, LIWORK must be at least 6+6×N+5×N×lgN;
- if COMPZ='I' or N>1, LIWORK must be at least 3+5×N.
- 13: 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
The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/N+1 through INFO mod N+1.
7 Accuracy
The computed eigenvalues and eigenvectors are exact for a nearby matrix
T+E, where
and
ε is the
machine precision.
If
λi is an exact eigenvalue and
λ~i is the corresponding computed value, then
where
cn is a modestly increasing function of
n.
If
zi is the corresponding exact eigenvector, and
z~i is the corresponding computed eigenvector, then the angle
θz~i,zi between them is bounded as follows:
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.
See Section 4.7 of
Anderson et al. (1999) for further details. See also
F08FLF (DDISNA).
8 Further Comments
If only eigenvalues are required, the total number of floating point operations is approximately proportional to
n2. When eigenvectors are required the number of operations is bounded above by approximately the same number of operations as
F08JSF (ZSTEQR), but for large matrices F08JVF (ZSTEDC) is usually much faster.
The real analogue of this routine is
F08JHF (DSTEDC).
9 Example
This example finds all the eigenvalues and eigenvectors of the Hermitian band matrix
A is first reduced to tridiagonal form by a call to
F08HSF (ZHBTRD).
9.1 Program Text
Program Text (f08jvfe.f90)
9.2 Program Data
Program Data (f08jvfe.d)
9.3 Program Results
Program Results (f08jvfe.r)