NAG Library Routine Document

F08SCF (DSYGVD)

INTEGER	ITYPE, N, LDA, LDB, LWORK, IWORK(max(1,LIWORK)), LIWORK, 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 dsygvd.

3 Description

F08SCF (DSYGVD) first performs a Cholesky factorization of the matrix B as B=UTU , when UPLO='U' or B=LLT , when UPLO='L'. The generalized problem is then reduced to a standard symmetric eigenvalue problem

Cx=λx ,

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

ZT A Z = Λ and ZT B Z = I ,

where Λ is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem A B z = λ z we correspondingly have

Z-1 A Z-T = Λ and ZT B Z = I ,

and for B A z = λ z we have

ZT A Z = Λ and ZT B-1 Z = I .

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.

2: JOBZ – CHARACTER(1)Input

On entry: if JOBZ='N', compute eigenvalues only.

If JOBZ='V', compute eigenvalues and eigenvectors.

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,*) – 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 if INFO=0, A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:

if ITYPE=1 or 2, ZTBZ=I;
if ITYPE=3, ZTB-1Z=I.

If JOBZ='N', the upper triangle (if UPLO='U') or the lower triangle (if UPLO='L') of A, including the diagonal, is destroyed.

6: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08SCF (DSYGVD) is called.

Constraint: LDA≥max1,N.

7: B(LDB,*) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array B must be at least max1,N.

On entry: the symmetric matrix B:

if UPLO='U', the leading n by n upper triangular part of B contains the upper triangular part of the matrix B;
if UPLO='L', the leading n by n lower triangular part of B contains the lower triangular part of the matrix B.

On exit: if INFO≤N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B=UTU or B=LLT.

8: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F08SCF (DSYGVD) is called.

Constraint: LDB≥max1,N.

9: W(N) – REAL (KIND=nag_wp) arrayOutput

On exit: if INFO=0, the eigenvalues in ascending order.

10: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace

On exit: if INFO=0, 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 F08SCF (DSYGVD) is called.

If LWORK=-1, a workspace query is assumed; the routine only calculates the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued.

Suggested value: for optimal performance, LWORK should usually be larger than the minimum, try increasing by nb×N, where nb is the optimal block size.

Constraints:

if N≤1, LWORK≥1;
if JOBZ='N' and N>1, LWORK≥2×N+1;
if JOBZ='V' and N>1, LWORK≥1+6×N+2×N2.

12: IWORK(max1,LIWORK) – INTEGER arrayWorkspace

On exit: if INFO=0, IWORK1 returns the minimum LIWORK.

13: LIWORK – INTEGERInput

On entry: the dimension of the array IWORK as declared in the (sub)program from which F08SCF (DSYGVD) is called.

If LIWORK=-1, a workspace query is assumed; the routine only calculates the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued.

Constraints:

if N≤1, LIWORK≥1;
if JOBZ='N' and N>1, LIWORK≥1;
if JOBZ='V' and N>1, LIWORK≥3+5×N.

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

F07FDF (DPOTRF) or F08FCF (DSYEVD) returned an error code:

≤N	if INFO=i, F08FCF (DSYEVD) failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
>N	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.