NAG Library Routine Document
F08SAF (DSYGV)
1 Purpose
F08SAF (DSYGV) computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form
where
and
are symmetric and
is also positive definite.
2 Specification
SUBROUTINE F08SAF ( |
ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, INFO) |
INTEGER |
ITYPE, N, LDA, LDB, LWORK, INFO |
REAL (KIND=nag_wp) |
A(LDA,*), B(LDB,*), W(N), WORK(max(1,LWORK)) |
CHARACTER(1) |
JOBZ, UPLO |
|
The routine may be called by its
LAPACK
name dsygv.
3 Description
F08SAF (DSYGV) first performs a Cholesky factorization of the matrix
as
, when
or
, when
. 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
, the eigenvectors are normalized so that the matrix of eigenvectors,
, satisfies
where
is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem
we correspondingly have
and for
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.
- .
- .
- .
Constraint:
, or .
- 2: JOBZ – CHARACTER(1)Input
On entry: indicates whether eigenvectors are computed.
- Only eigenvalues are computed.
- Eigenvalues and eigenvectors are computed.
Constraint:
or .
- 3: UPLO – CHARACTER(1)Input
On entry: if
, the upper triangles of
and
are stored.
If , the lower triangles of and are stored.
Constraint:
or .
- 4: N – INTEGERInput
On entry: , the order of the matrices and .
Constraint:
.
- 5: A(LDA,) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
.
On entry: the
by
symmetric matrix
.
- If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
- If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
On exit: if
,
A contains the matrix
of eigenvectors. The eigenvectors are normalized as follows:
- if or , ;
- if , .
If
, the upper triangle (if
) or the lower triangle (if
) 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 F08SAF (DSYGV) is called.
Constraint:
.
- 7: B(LDB,) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
.
On entry: the
by
symmetric positive definite matrix
.
- If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
- If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
On exit: if
, the part of
B containing the matrix is overwritten by the triangular factor
or
from the Cholesky factorization
or
.
- 8: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F08SAF (DSYGV) is called.
Constraint:
.
- 9: W(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
- 10: WORK() – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
,
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 F08SAF (DSYGV) is called.
If
, 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,
, where
is the optimal
block size for
F08FEF (DSYTRD).
Constraint:
.
- 12: INFO – INTEGEROutput
On exit:
unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If
,
F08FAF (DSYEV)
failed to converge;
off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
F07FDF (DPOTRF)
returned an error code; i.e., if
, for
, then the leading minor of order
of
is not positive definite. The factorization of
could not be completed and no eigenvalues or eigenvectors were computed.
7 Accuracy
If
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
differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of
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.
The total number of floating point operations is proportional to .
The complex analogue of this routine is
F08SNF (ZHEGV).
9 Example
This example finds all the eigenvalues and eigenvectors of the generalized symmetric eigenproblem
, where
together with and estimate of the condition number of
, and approximate error bounds for the computed eigenvalues and eigenvectors.
The example program for
F08SCF (DSYGVD) illustrates solving a generalized symmetric eigenproblem of the form
.
9.1 Program Text
Program Text (f08safe.f90)
9.2 Program Data
Program Data (f08safe.d)
9.3 Program Results
Program Results (f08safe.r)