NAME

zhegvx - compute selected eigenvalues, and optionally, eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x

SYNOPSIS

  SUBROUTINE ZHEGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, 
 *      VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, 
 *      IFAIL, INFO)
  CHARACTER * 1 JOBZ, RANGE, UPLO
  DOUBLE COMPLEX A(LDA,*), B(LDB,*), Z(LDZ,*), WORK(*)
  INTEGER ITYPE, N, LDA, LDB, IL, IU, M, LDZ, LWORK, INFO
  INTEGER IWORK(*), IFAIL(*)
  DOUBLE PRECISION VL, VU, ABSTOL
  DOUBLE PRECISION W(*), RWORK(*)

  SUBROUTINE ZHEGVX_64( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, 
 *      VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, 
 *      IFAIL, INFO)
  CHARACTER * 1 JOBZ, RANGE, UPLO
  DOUBLE COMPLEX A(LDA,*), B(LDB,*), Z(LDZ,*), WORK(*)
  INTEGER*8 ITYPE, N, LDA, LDB, IL, IU, M, LDZ, LWORK, INFO
  INTEGER*8 IWORK(*), IFAIL(*)
  DOUBLE PRECISION VL, VU, ABSTOL
  DOUBLE PRECISION W(*), RWORK(*)

F95 INTERFACE

  SUBROUTINE HEGVX( ITYPE, JOBZ, RANGE, UPLO, [N], A, [LDA], B, [LDB], 
 *       VL, VU, IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK], [LWORK], [RWORK], 
 *       [IWORK], IFAIL, [INFO])
  CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
  COMPLEX(8), DIMENSION(:) :: WORK
  COMPLEX(8), DIMENSION(:,:) :: A, B, Z
  INTEGER :: ITYPE, N, LDA, LDB, IL, IU, M, LDZ, LWORK, INFO
  INTEGER, DIMENSION(:) :: IWORK, IFAIL
  REAL(8) :: VL, VU, ABSTOL
  REAL(8), DIMENSION(:) :: W, RWORK

  SUBROUTINE HEGVX_64( ITYPE, JOBZ, RANGE, UPLO, [N], A, [LDA], B, 
 *       [LDB], VL, VU, IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK], [LWORK], 
 *       [RWORK], [IWORK], IFAIL, [INFO])
  CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
  COMPLEX(8), DIMENSION(:) :: WORK
  COMPLEX(8), DIMENSION(:,:) :: A, B, Z
  INTEGER(8) :: ITYPE, N, LDA, LDB, IL, IU, M, LDZ, LWORK, INFO
  INTEGER(8), DIMENSION(:) :: IWORK, IFAIL
  REAL(8) :: VL, VU, ABSTOL
  REAL(8), DIMENSION(:) :: W, RWORK

void zhegvx(int itype, char jobz, char range, char uplo, int n, doublecomplex *a, int lda, doublecomplex *b, int ldb, double vl, double vu, int il, int iu, double abstol, int *m, double *w, doublecomplex *z, int ldz, int *ifail, int *info);

void zhegvx_64(long itype, char jobz, char range, char uplo, long n, doublecomplex *a, long lda, doublecomplex *b, long ldb, double vl, double vu, long il, long iu, double abstol, long *m, double *w, doublecomplex *z, long ldz, long *ifail, long *info);

PURPOSE

zhegvx computes selected eigenvalues, and optionally, eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

ARGUMENTS

ITYPE (input)
Specifies the problem type to be solved:

 = 1:  A*x  = (lambda)*B*x

 = 2:  A*B*x  = (lambda)*x

 = 3:  B*A*x  = (lambda)*x

JOBZ (input)

 = 'N':  Compute eigenvalues only;

 = 'V':  Compute eigenvalues and eigenvectors.

RANGE (input)

 = 'A': all eigenvalues will be found.

 = 'V': all eigenvalues in the half-open interval (VL,VU]
will be found.
 = 'I': the IL-th through IU-th eigenvalues will be found.

UPLO (input)

 = 'U':  Upper triangles of A and B are stored;

 = 'L':  Lower triangles of A and B are stored.

N (input)
The order of the matrices A and B. N > = 0.
A (input/output)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
On exit, the lower triangle (if UPLO ='L') or the upper triangle (if UPLO ='U') of A, including the diagonal, is destroyed.
LDA (input)
The leading dimension of the array A. LDA > = max(1,N).
B (input/output)
On entry, the Hermitian 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 = U**H*U or B = L*L**H.
LDB (input)
The leading dimension of the array B. LDB > = max(1,N).
VL (input)
If RANGE ='V', the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
VU (input)
If RANGE ='V', the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
IL (input)
If RANGE ='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 < = IL < = IU < = N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
IU (input)
If RANGE ='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned. 1 < = IL < = IU < = N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.
ABSTOL (input)
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 + EPS * max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of 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*SLAMCH('S'), not zero. If this routine returns with INFO >0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S').
M (output)
The total number of eigenvalues found. 0 < = M < = N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
W (output)
The first M elements contain the selected eigenvalues in ascending order.
Z (output)
If JOBZ = 'N', then Z is not referenced. 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 i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3, Z**T*inv(B)*Z = I.
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 IFAIL. Note: the user must ensure that at least max(1,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 must be used.
LDZ (input)
The leading dimension of the array Z. LDZ > = 1, and if JOBZ = 'V', LDZ > = max(1,N).
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The length of the array WORK. LWORK > = max(1,2*N-1). For optimal efficiency, LWORK > = (NB+1)*N, where NB is the blocksize for CHETRD returned by ILAENV.
If LWORK = -1, then 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 by XERBLA.
RWORK (workspace)
dimension(7*N)
IWORK (workspace)
dimension(5*N)
IFAIL (output)
If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.

INFO (output)

 = 0:  successful exit

 < 0:  if INFO  = -i, the i-th argument had an illegal value

 > 0:  CPOTRF or CHEEVX returned an error code:

 < = N:  if INFO  = i, CHEEVX failed to converge;
i eigenvectors failed to converge.  Their indices
are stored in array IFAIL.
 > 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.

FURTHER DETAILS

Based on contributions by