NAG Library Routine Document
F08JEF (DSTEQR)
1 Purpose
F08JEF (DSTEQR) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix, or of a real symmetric matrix which has been reduced to tridiagonal form.
2 Specification
| INTEGER |
N, LDZ, INFO |
| REAL (KIND=nag_wp) |
D(*), E(*), Z(LDZ,*), WORK(*) |
| CHARACTER(1) |
COMPZ |
|
The routine may be called by its
LAPACK
name dsteqr.
3 Description
F08JEF (DSTEQR) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix
T.
In other words, it can compute the spectral factorization of
T as
where
Λ is a diagonal matrix whose diagonal elements are the eigenvalues
λi, and
Z is the orthogonal matrix whose columns are the eigenvectors
zi. Thus
The routine may also be used to compute all the eigenvalues and eigenvectors of a real symmetric matrix
A which has been reduced to tridiagonal form
T:
In this case, the matrix
Q must be formed explicitly and passed to F08JEF (DSTEQR), which must be called with
COMPZ='V'. The routines which must be called to perform the reduction to tridiagonal form and form
Q are:
F08JEF (DSTEQR) uses the implicitly shifted
QR algorithm, switching between the
QR and
QL variants in order to handle graded matrices effectively (see
Greenbaum and Dongarra (1980)). The eigenvectors are normalized so that
zi2=1, but are determined only to within a factor
±1.
If only the eigenvalues of
T are required, it is more efficient to call
F08JFF (DSTERF) instead. If
T is positive definite, small eigenvalues can be computed more accurately by
F08JGF (DPTEQR).
4 References
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Greenbaum A and Dongarra J J (1980) Experiments with QR/QL methods for the symmetric triangular eigenproblem
LAPACK Working Note No. 17 (Technical Report CS-89-92) University of Tennessee, Knoxville
Parlett B N (1998)
The Symmetric Eigenvalue Problem SIAM, Philadelphia
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 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 T.
On exit: the
n eigenvalues in ascending order, unless
INFO>0 (in which case see
Section 6).
- 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 off-diagonal elements of the tridiagonal matrix T.
On exit:
E is overwritten.
- 5: Z(LDZ,*) – REAL (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 if
COMPZ='N'.
On entry: if
COMPZ='V',
Z must contain the
orthogonal
matrix
Q from the reduction to tridiagonal form.
If
COMPZ='I',
Z need not be set.
On exit: if
COMPZ='I' or
'V', the
n required orthonormal eigenvectors stored as columns of
Z; the
ith column corresponds to the
ith eigenvalue, where
i=1,2,…,n, unless
INFO>0.
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 F08JEF (DSTEQR) is called.
Constraints:
- if COMPZ='V' or 'I', LDZ≥max1,N;
- if COMPZ='N', LDZ≥1.
- 7: WORK(*) – REAL (KIND=nag_wp) arrayWorkspace
-
Note: the dimension of the array
WORK
must be at least
max1,2×N-1 if
COMPZ='V' or
'I' and at least
1 if
COMPZ='N'.
If
COMPZ='N',
WORK is not referenced.
- 8: 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 has failed to find all the eigenvalues after a total of
30×N iterations. In this case,
D and
E contain on exit the diagonal and off-diagonal elements, respectively, of a tridiagonal matrix
similar to
T. If
INFO=i, then
i off-diagonal elements have not converged to zero.
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.
8 Further Comments
The total number of floating point operations is typically about 24n2 if COMPZ='N' and about 7n3 if COMPZ='V' or 'I', but depends on how rapidly the algorithm converges. When COMPZ='N', the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when COMPZ='V' or 'I' can be vectorized and on some machines may be performed much faster.
The complex analogue of this routine is
F08JSF (ZSTEQR).
9 Example
This example computes all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
T, where
See also the examples for
F08FFF (DORGTR),
F08GFF (DOPGTR) or
F08HEF (DSBTRD), which illustrate the use of this routine to compute the eigenvalues and eigenvectors of a full or band symmetric matrix.
9.1 Program Text
Program Text (f08jefe.f90)
9.2 Program Data
Program Data (f08jefe.d)
9.3 Program Results
Program Results (f08jefe.r)