F08SQF (ZHEGVD) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08SQF (ZHEGVD)

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

F08SQF (ZHEGVD) computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form
Az=λBz ,   ABz=λz   or   BAz=λz ,  
where A and B are Hermitian and B is also positive definite. If eigenvectors are desired, it uses a divide-and-conquer algorithm.

2  Specification

SUBROUTINE F08SQF ( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
INTEGER  ITYPE, N, LDA, LDB, LWORK, LRWORK, IWORK(max(1,LIWORK)), LIWORK, INFO
REAL (KIND=nag_wp)  W(N), RWORK(max(1,LRWORK))
COMPLEX (KIND=nag_wp)  A(LDA,*), B(LDB,*), WORK(max(1,LWORK))
CHARACTER(1)  JOBZ, UPLO
The routine may be called by its LAPACK name zhegvd.

3  Description

F08SQF (ZHEGVD) first performs a Cholesky factorization of the matrix B as B=UHU , when UPLO='U' or B=LLH , 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
ZH A Z = Λ   and   ZH 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-H = Λ   and   ZH B Z = I ,  
and for B A z = λ z  we have
ZH A Z = Λ   and   ZH 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  Arguments

1:     ITYPE – INTEGERInput
On entry: specifies the problem type to be solved.
ITYPE=1
Az=λBz.
ITYPE=2
ABz=λz.
ITYPE=3
BAz=λz.
Constraint: ITYPE=1, 2 or 3.
2:     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'.
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:     ALDA* – 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: if JOBZ='V', A contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:
  • if ITYPE=1 or 2, ZHBZ=I;
  • if ITYPE=3, ZHB-1Z=I.
If JOBZ='N', the upper triangle (if UPLO='U') or the lower triangle (if UPLO='L') of A, including the diagonal, is overwritten.
6:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08SQF (ZHEGVD) is called.
Constraint: LDAmax1,N.
7:     BLDB* – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least max1,N.
On entry: the n by n Hermitian matrix B.
  • If UPLO='U', the upper triangular part of B must be stored and the elements of the array below the diagonal are not referenced.
  • If UPLO='L', the lower triangular part of B must be stored and the elements of the array above the diagonal are not referenced.
On exit: the triangular factor U or L from the Cholesky factorization B=UHU or B=LLH.
8:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08SQF (ZHEGVD) is called.
Constraint: LDBmax1,N.
9:     WN – REAL (KIND=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
10:   WORKmax1,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.
11:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08SQF (ZHEGVD) is called.
If LWORK=-1, a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK, LRWORK 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 N1, LWORK1;
  • if JOBZ='N' and N>1, LWORKN+1;
  • if JOBZ='V' and N>1, LWORK2×N+N2.
12:   RWORKmax1,LRWORK – REAL (KIND=nag_wp) arrayWorkspace
On exit: if INFO=0, RWORK1 returns the optimal LRWORK.
13:   LRWORK – INTEGERInput
On entry: the dimension of the array RWORK as declared in the (sub)program from which F08SQF (ZHEGVD) is called.
If LRWORK=-1, a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK, LRWORK or LIWORK is issued.
Constraints:
  • if N1, LRWORK1;
  • if JOBZ='N' and N>1, LRWORKN;
  • if JOBZ='V' and N>1, LRWORK1+5×N+2×N2.
14:   IWORKmax1,LIWORK – INTEGER arrayWorkspace
On exit: if INFO=0, IWORK1 returns the optimal LIWORK.
15:   LIWORK – INTEGERInput
On entry: the dimension of the array IWORK as declared in the (sub)program from which F08SQF (ZHEGVD) is called.
If LIWORK=-1, a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK, LRWORK or LIWORK is issued.
Constraints:
  • if N1, LIWORK1;
  • if JOBZ='N' and N>1, LIWORK1;
  • if JOBZ='V' and N>1, LIWORK3+5×N.
16:   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, F08FQF (ZHEEVD) failed to converge; i i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
INFO>N
F07FRF (ZPOTRF) 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.
The example program below illustrates the computation of approximate error bounds.

8  Parallelism and Performance

F08SQF (ZHEGVD) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F08SQF (ZHEGVD) 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 .
The real analogue of this routine is F08SCF (DSYGVD).

10  Example

This example finds all the eigenvalues and eigenvectors of the generalized Hermitian eigenproblem ABz=λz , where
A = -7.36i+0.00 0.77-0.43i -0.64-0.92i 3.01-6.97i 0.77+0.43i 3.49i+0.00 2.19+4.45i 1.90+3.73i -0.64+0.92i 2.19-4.45i 0.12i+0.00 2.88-3.17i 3.01+6.97i 1.90-3.73i 2.88+3.17i -2.54i+0.00  
and
B = 3.23i+0.00 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58i+0.00 -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09i+0.00 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29i+0.00 ,  
together with an estimate of the condition number of B, and approximate error bounds for the computed eigenvalues and eigenvectors.
The example program for F08SNF (ZHEGV) illustrates solving a generalized Hermitian eigenproblem of the form Az=λBz .

10.1  Program Text

Program Text (f08sqfe.f90)

10.2  Program Data

Program Data (f08sqfe.d)

10.3  Program Results

Program Results (f08sqfe.r)


F08SQF (ZHEGVD) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

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