NAG Library Routine Document
F08SNF (ZHEGV)
1 Purpose
F08SNF (ZHEGV) computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form
where
A and
B are Hermitian and
B is also positive definite.
2 Specification
SUBROUTINE F08SNF ( |
ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, RWORK, INFO) |
INTEGER |
ITYPE, N, LDA, LDB, LWORK, INFO |
REAL (KIND=nag_wp) |
W(N), RWORK(3*N-2) |
COMPLEX (KIND=nag_wp) |
A(LDA,*), B(LDB,*), WORK(max(1,LWORK)) |
CHARACTER(1) |
JOBZ, UPLO |
|
The routine may be called by its
LAPACK
name zhegv.
3 Description
F08SNF (ZHEGV) first performs a Cholesky factorization of the matrix
B as
B=UHU
, when
UPLO='U' or
B=LLH
, when
UPLO='L'. The generalized problem is then reduced to a standard symmetric eigenvalue problem
which is solved for the eigenvalues and, optionally, the eigenvectors; the eigenvectors are then backtransformed to give the eigenvectors of the original problem.
For the problem
Az=λBz
, the eigenvectors are normalized so that the matrix of eigenvectors,
z, satisfies
where
Λ
is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem
A
B
z
=
λ
z
we correspondingly have
and for
B
A
z
=
λ
z
we have
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: ITYPE – INTEGERInput
On entry: specifies the problem type to be solved.
- ITYPE=1
- Az=λBz.
- ITYPE=2
- ABz=λz.
- ITYPE=3
- BAz=λz.
Constraint:
ITYPE=1, 2 or 3.
- 2: 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'.
- 3: 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'.
- 4: N – INTEGERInput
On entry: n, the order of the matrices A and B.
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: if
JOBZ='V',
A contains the matrix
Z of eigenvectors. The eigenvectors are normalized as follows:
- if ITYPE=1 or 2, ZHBZ=I;
- if ITYPE=3, ZHB-1Z=I.
If
JOBZ='N', the upper triangle (if
UPLO='U') or the lower triangle (if
UPLO='L') 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 F08SNF (ZHEGV) is called.
Constraint:
LDA≥max1,N.
- 7: B(LDB,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
max1,N.
On entry: the
n by
n Hermitian positive definite matrix
B.
- If UPLO='U', the upper triangular part of B must be stored and the elements of the array below the diagonal are not referenced.
- If UPLO='L', the lower triangular part of B must be stored and the elements of the array above the diagonal are not referenced.
On exit: if
0≤INFO≤N, the part of
B containing the matrix is overwritten by the triangular factor
U or
L from the Cholesky factorization
B=UHU or
B=LLH.
- 8: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F08SNF (ZHEGV) is called.
Constraint:
LDB≥max1,N.
- 9: W(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
- 10: 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.
- 11: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08SNF (ZHEGV) is called.
If
LWORK=-1, a workspace query is assumed; the routine only calculates the optimal size of the
WORK array, returns this value as the first entry of the
WORK array, and no error message related to
LWORK is issued.
Suggested value:
for optimal performance,
LWORK≥nb+1×N, where
nb is the optimal
block size for
F08FSF (ZHETRD).
Constraint:
LWORK≥max1,2×N.
- 12: RWORK(3×N-2) – REAL (KIND=nag_wp) arrayWorkspace
- 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=1 to N
If
INFO=i,
F08FNF (ZHEEV)
failed to converge;
i i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
- INFO>N
F07FRF (ZPOTRF)
returned an error code; i.e., if
INFO=N+i, for
1≤i≤N, then the leading minor of order
i of
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.
The example program below illustrates the computation of approximate error bounds.
8 Further Comments
The total number of floating point operations is proportional to n3.
The real analogue of this routine is
F08SAF (DSYGV).
9 Example
This example finds all the eigenvalues and eigenvectors of the generalized Hermitian eigenproblem
Az=λBz, where
and
together with and estimate of the condition number of
B, and approximate error bounds for the computed eigenvalues and eigenvectors.
The example program for
F08SQF (ZHEGVD) illustrates solving a generalized Hermitian eigenproblem of the form
ABz=λz.
9.1 Program Text
Program Text (f08snfe.f90)
9.2 Program Data
Program Data (f08snfe.d)
9.3 Program Results
Program Results (f08snfe.r)