NAG Library Routine Document

F08FPF (ZHEEVX)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F08FPF (ZHEEVX) computes selected eigenvalues and, optionally, eigenvectors of a complex n by n Hermitian matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

2 Specification

SUBROUTINE F08FPF (

JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, JFAIL, INFO)

INTEGER	N, LDA, IL, IU, M, LDZ, LWORK, IWORK(), JFAIL(), INFO
REAL (KIND=nag_wp)	VL, VU, ABSTOL, W(), RWORK()
COMPLEX (KIND=nag_wp)	A(LDA,), Z(LDZ,), WORK(max(1,LWORK))
CHARACTER(1)	JOBZ, RANGE, UPLO

The routine may be called by its LAPACK name zheevx.

3 Description

The Hermitian matrix A is first reduced to real tridiagonal form, using unitary similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.

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

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

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 triangular part of A is stored.

If UPLO='L', the lower triangular part of A is stored.

Constraint: UPLO='U' or 'L'.

4: N – INTEGERInput

On entry: n, the order of the matrix A.

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: the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') 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 F08FPF (ZHEEVX) is called.

Constraint: LDA≥max1,N.

7: VL – REAL (KIND=nag_wp)Input

8: 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.

9: IL – INTEGERInput

10: 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.

11: 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 A 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).

12: 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.

13: W(*) – REAL (KIND=nag_wp) arrayOutput

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

On exit: the first M elements contain the selected eigenvalues in ascending order.

14: Z(LDZ,*) – COMPLEX (KIND=nag_wp) arrayOutput

Note: the second dimension of the array Z must be at least max1,M if JOBZ='V', and at least 1 otherwise.

On exit: if JOBZ='V', then if INFO=0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the ith column of Z holding the eigenvector associated with Wi.

If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in JFAIL.

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

Note: you must ensure that at least max1,M columns are supplied in the array Z; if RANGE='V', the exact value of M is not known in advance and an upper bound of at least N must be used.

15: LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08FPF (ZHEEVX) is called.

Constraints:

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

16: 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.

17: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08FPF (ZHEEVX) 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.