F08UPF (ZHBGVX) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08UPF (ZHBGVX)

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.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08UPF (ZHBGVX) computes selected the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form
Az=λBz ,  
where A and B are Hermitian and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.

2  Specification

SUBROUTINE F08UPF ( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, JFAIL, INFO)
INTEGER  N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, IWORK(5*N), 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  Arguments

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:     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: N0.
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: KA0.
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: KAKB0.
7:     ABLDAB* – 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-kaij;
  • 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 ​jiminn,j+ka.
On exit: the contents of AB are overwritten.
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: LDABKA+1.
9:     BBLDBB* – 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 positive definite 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-kbij;
  • 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 ​jiminn,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: LDBBKB+1.
11:   QLDQ* – COMPLEX (KIND=nag_wp) arrayOutput
Note: the second dimension of the array Q must be at least max1,N if JOBZ='V', and at least 1 otherwise.
On exit: if JOBZ='V', the n by n matrix, Q used in the reduction of the standard form, i.e., Cx=λx, from symmetric banded 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='V', LDQ max1,N ;
  • otherwise LDQ1.
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=1toN, 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. 0MN.
If RANGE='A', M=N.
If RANGE='I', M=IU-IL+1.
19:   WN – REAL (KIND=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
20:   ZLDZ* – 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', 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 LDZ1.
22:   WORKN – COMPLEX (KIND=nag_wp) arrayWorkspace
23:   RWORK7×N – REAL (KIND=nag_wp) arrayWorkspace
24:   IWORK5×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=1toN, 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

INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
INFO=1toN
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 1iN, 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.

7  Accuracy

If B 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 B differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of B would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.

8  Parallelism and Performance

F08UPF (ZHBGVX) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F08UPF (ZHBGVX) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The total number of floating-point operations is proportional to n3  if JOBZ='V' and RANGE='A', and assuming that nka , is approximately proportional to n2 ka  if JOBZ='N'. Otherwise the number of floating-point operations depends upon the number of eigenvectors computed.
The real analogue of this routine is F08UBF (DSBGVX).

10  Example

This example finds the eigenvalues in the half-open interval 0.0,2.0 , and corresponding eigenvectors, of the generalized band Hermitian eigenproblem Az = λ Bz , where
A = -1.13i+0.00 1.94-2.10i -1.40+0.25i 0.00i+0.00 1.94+2.10i -1.91i+0.00 -0.82-0.89i -0.67+0.34i -1.40-0.25i -0.82+0.89i -1.87i+0.00 -1.10-0.16i 0.00i+0.00 -0.67-0.34i -1.10+0.16i 0.50i+0.00  
and
B = 9.89i+0.00 1.08-1.73i 0.00i+0.00 0.00i+0.00 1.08+1.73i 1.69i+0.00 -0.04+0.29i 0.00i+0.00 0.00i+0.00 -0.04-0.29i 2.65i+0.00 -0.33+2.24i 0.00i+0.00 0.00i+0.00 -0.33-2.24i 2.17i+0.00 .  

10.1  Program Text

Program Text (f08upfe.f90)

10.2  Program Data

Program Data (f08upfe.d)

10.3  Program Results

Program Results (f08upfe.r)


F08UPF (ZHBGVX) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016