NAG Library Routine Document
F08UAF (DSBGV)
1 Purpose
F08UAF (DSBGV) computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form
where
A and
B are symmetric and banded, and
B is also positive definite.
2 Specification
SUBROUTINE F08UAF ( |
JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, INFO) |
INTEGER |
N, KA, KB, LDAB, LDBB, LDZ, INFO |
REAL (KIND=nag_wp) |
AB(LDAB,*), BB(LDBB,*), W(N), Z(LDZ,*), WORK(3*N) |
CHARACTER(1) |
JOBZ, UPLO |
|
The routine may be called by its
LAPACK
name dsbgv.
3 Description
The generalized symmetric-definite band problem
is first reduced to a standard band symmetric problem
where
C is a symmetric band matrix, using Wilkinson's modification to Crawford's algorithm (see
Crawford (1973) and
Wilkinson (1977)). The symmetric eigenvalue problem is then solved for the eigenvalues and the eigenvectors, if required, which are then backtransformed to the eigenvectors of the original problem.
The eigenvectors are normalized so that the matrix of eigenvectors,
Z, satisfies
where
Λ
is the diagonal matrix whose diagonal elements are the eigenvalues.
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
Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem
Comm. ACM 16 41–44
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1977) Some recent advances in numerical linear algebra
The State of the Art in Numerical Analysis (ed D A H Jacobs) Academic Press
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: UPLO – CHARACTER(1)Input
On entry: if
UPLO='U', the upper triangles of
A and
B are stored.
If UPLO='L', the lower triangles of A and B are stored.
Constraint:
UPLO='U' or 'L'.
- 3: N – INTEGERInput
On entry: n, the order of the matrices A and B.
Constraint:
N≥0.
- 4: KA – INTEGERInput
On entry: if
UPLO='U', the number of superdiagonals,
ka, of the matrix
A.
If UPLO='L', the number of subdiagonals, ka, of the matrix A.
Constraint:
KA≥0.
- 5: KB – INTEGERInput
On entry: if
UPLO='U', the number of superdiagonals,
kb, of the matrix
B.
If UPLO='L', the number of subdiagonals, kb, of the matrix B.
Constraint:
KA≥KB≥0.
- 6: AB(LDAB,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
AB
must be at least
max1,N.
On entry: the upper or lower triangle of the
n by
n symmetric band matrix
A.
The matrix is stored in rows
1 to
ka+1, more precisely,
- if UPLO='U', the elements of the upper triangle of A within the band must be stored with element Aij in ABka+1+i-jj for max1,j-ka≤i≤j;
- if UPLO='L', the elements of the lower triangle of A within the band must be stored with element Aij in AB1+i-jj for j≤i≤minn,j+ka.
On exit: the contents of
AB are overwritten.
- 7: LDAB – INTEGERInput
On entry: the first dimension of the array
AB as declared in the (sub)program from which F08UAF (DSBGV) is called.
Constraint:
LDAB≥KA+1.
- 8: BB(LDBB,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
BB
must be at least
max1,N.
On entry: the upper or lower triangle of the
n by
n symmetric band matrix
B.
The matrix is stored in rows
1 to
kb+1, more precisely,
- if UPLO='U', the elements of the upper triangle of B within the band must be stored with element Bij in BBkb+1+i-jj for max1,j-kb≤i≤j;
- if UPLO='L', the elements of the lower triangle of B within the band must be stored with element Bij in BB1+i-jj for j≤i≤minn,j+kb.
On exit: the factor
S from the split Cholesky factorization
B=STS, as returned by
F08UFF (DPBSTF).
- 9: LDBB – INTEGERInput
On entry: the first dimension of the array
BB as declared in the (sub)program from which F08UAF (DSBGV) is called.
Constraint:
LDBB≥KB+1.
- 10: W(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
- 11: Z(LDZ,*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
Z
must be at least
max1,N if
JOBZ='V', and at least
1 otherwise.
On exit: if
JOBZ='V',
Z contains the matrix
Z of eigenvectors, with the
ith column of
Z holding the eigenvector associated with
Wi. The eigenvectors are normalized so that
ZTBZ=I.
If
JOBZ='N',
Z is not referenced.
- 12: LDZ – INTEGERInput
On entry: the first dimension of the array
Z as declared in the (sub)program from which F08UAF (DSBGV) is called.
Constraints:
- if JOBZ='V', LDZ≥ max1,N ;
- otherwise LDZ≥1.
- 13: WORK(3×N) – REAL (KIND=nag_wp) arrayWorkspace
- 14: 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=i and i≤N, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
If
INFO=i and
i>N, if
INFO=N+i, for
1≤i≤N, then
F08UFF (DPBSTF)
returned
INFO=i:
B is not positive definite. The factorization of
B could not be completed and no eigenvalues or eigenvectors were computed.
7 Accuracy
If
B is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of
B differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of
B would suggest. See Section 4.10 of
Anderson et al. (1999) for details of the error bounds.
8 Further Comments
The total number of floating point operations is proportional to
n3
if JOBZ='V' and, assuming that
n≫ka
, is approximately proportional to
n2
ka
otherwise.
The complex analogue of this routine is
F08UNF (ZHBGV).
9 Example
This example finds all the eigenvalues of the generalized band symmetric eigenproblem
Az
=
λ
Bz
, where
9.1 Program Text
Program Text (f08uafe.f90)
9.2 Program Data
Program Data (f08uafe.d)
9.3 Program Results
Program Results (f08uafe.r)