NAG Library Routine Document

F08UPF (ZHBGVX)

INTEGER	N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, IWORK(5N), JFAIL(), INFO
REAL (KIND=nag_wp)	VL, VU, ABSTOL, W(N), RWORK(7*N)
COMPLEX (KIND=nag_wp)	AB(LDAB,), BB(LDBB,), Q(LDQ,), Z(LDZ,), WORK(N)
CHARACTER(1)	JOBZ, RANGE, UPLO

The routine may be called by its LAPACK name zhbgvx.

3 Description

The generalized Hermitian-definite band problem

Az = λ Bz

is first reduced to a standard band Hermitian problem

Cx = λx ,

where C is a Hermitian band matrix, using Wilkinson's modification to Crawford's algorithm (see Crawford (1973) and Wilkinson (1977)). The Hermitian eigenvalue problem is then solved for the required eigenvalues and eigenvectors, and the eigenvectors are then backtransformed to the eigenvectors of the original problem.

The eigenvectors are normalized so that

zH A z = λ and zH B z = 1 .

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

Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912

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

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

Constraint: JOBZ='N' or 'V'.

2: RANGE – CHARACTER(1)Input

On entry: if RANGE='A', all eigenvalues will be found.

If RANGE='V', all eigenvalues in the half-open interval VL,VU will be found.

If RANGE='I', the ILth to IUth eigenvalues will be found.

Constraint: RANGE='A', 'V' or 'I'.

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: 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.

6: 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.

7: AB(LDAB,*) – COMPLEX (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 Hermitian 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 destroyed.

8: LDAB – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F08UPF (ZHBGVX) is called.

Constraint: LDAB≥KA+1.

9: BB(LDBB,*) – COMPLEX (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 Hermitian 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=SHS, as returned by F08UTF (ZPBSTF).

10: LDBB – INTEGERInput

On entry: the first dimension of the array BB as declared in the (sub)program from which F08UPF (ZHBGVX) is called.

Constraint: LDBB≥KB+1.

11: Q(LDQ,*) – COMPLEX (KIND=nag_wp) arrayOutput

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

On exit: if JOBZ='V', the n by n matrix used in the reduction of Az=λBz to standard form, i.e., Cx=λx, and subsequently C to tridiagonal form.

If JOBZ='N', Q is not referenced.

12: LDQ – INTEGERInput

On entry: the first dimension of the array Q as declared in the (sub)program from which F08UPF (ZHBGVX) is called.

Constraints:

if JOBZ='N', LDQ≥1;
if JOBZ='V', LDQ≥max1,N.

13: VL – REAL (KIND=nag_wp)Input

14: VU – REAL (KIND=nag_wp)Input

On entry: if RANGE='V', the lower and upper bounds of the interval to be searched for eigenvalues.

If RANGE='A' or 'I', VL and VU are not referenced.

Constraint: if RANGE='V', VL<VU.

15: IL – INTEGERInput

16: IU – INTEGERInput

On entry: if RANGE='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned.

If RANGE='A' or 'V', IL and IU are not referenced.

Constraints:

if RANGE='I' and N=0, IL=1 and IU=0;
if RANGE='I' and N>0, 1≤IL≤IU≤N.

17: ABSTOL – REAL (KIND=nag_wp)Input

On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval a,b of width less than or equal to

ABSTOL+ε maxa,b ,

where ε is the machine precision. If ABSTOL is less than or equal to zero, then ε T1 will be used in its place, where T is the tridiagonal matrix obtained by reducing C to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2 × X02AMF , not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2 × X02AMF . See Demmel and Kahan (1990).

18: M – INTEGEROutput

On exit: the total number of eigenvalues found. 0≤M≤N.

If RANGE='A', M=N.

If RANGE='I', M=IU-IL+1.

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

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

20: Z(LDZ,*) – COMPLEX (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', then if INFO=0, 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 ZHBZ=I.

If JOBZ='N', Z is not referenced.

21: LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08UPF (ZHBGVX) is called.

Constraints:

if JOBZ='V', LDZ≥max1,N;
otherwise LDZ≥1.

22: WORK(N) – COMPLEX (KIND=nag_wp) arrayWorkspace

23: RWORK(7×N) – REAL (KIND=nag_wp) arrayWorkspace

24: IWORK(5×N) – INTEGER arrayWorkspace

25: JFAIL(*) – INTEGER arrayOutput

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

On exit: if JOBZ='V', then if INFO=0, the first M elements of JFAIL are zero.

If INFO>0, JFAIL contains the indices of the eigenvectors that failed to converge.

If JOBZ='N', JFAIL is not referenced.

26: 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, then i eigenvectors failed to converge. Their indices are stored in array JFAIL. Please see ABSTOL.

INFO>N: F08UFF (DPBSTF) 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.