Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.
The routine may be called by its
LAPACK
name dsyev.
3 Description
The symmetric matrix A is first reduced to tridiagonal form, using orthogonal similarity transformations, and then the QR algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.
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: 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 triangular part of A is stored.
If UPLO='L', the lower triangular part of A is stored.
Constraint:
UPLO='U' or 'L'.
3: N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint:
N≥0.
4: A(LDA,*) – REAL (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 symmetric 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', then A contains the orthonormal eigenvectors of the matrix A.
If JOBZ='N', then on exit the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is overwritten.
5: LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08FAF (DSYEV) is called.
7: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if INFO=0, 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 F08FAF (DSYEV) 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+2×N, where nb is the optimal block size for F08FEF (DSYTRD).
Constraint:
LWORK≥max1,3×N-1.
9: 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, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
7 Accuracy
The computed eigenvalues and eigenvectors are exact for a nearby matrix A+E, where
E2=OεA2,
and ε is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.
8 Further Comments
The total number of floating point operations is proportional to n3.