• NAME
• SYNOPSIS
• F95 INTERFACE
• C INTERFACE
• PURPOSE
• ARGUMENTS

NAME

zheevx - compute selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A

SYNOPSIS

  SUBROUTINE ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, 
 *      ABTOL, NFOUND, W, Z, LDZ, WORK, LDWORK, WORK2, IWORK3, IFAIL, 
 *      INFO)
  CHARACTER * 1 JOBZ, RANGE, UPLO
  DOUBLE COMPLEX A(LDA,*), Z(LDZ,*), WORK(*)
  INTEGER N, LDA, IL, IU, NFOUND, LDZ, LDWORK, INFO
  INTEGER IWORK3(*), IFAIL(*)
  DOUBLE PRECISION VL, VU, ABTOL
  DOUBLE PRECISION W(*), WORK2(*)

  SUBROUTINE ZHEEVX_64( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, 
 *      ABTOL, NFOUND, W, Z, LDZ, WORK, LDWORK, WORK2, IWORK3, IFAIL, 
 *      INFO)
  CHARACTER * 1 JOBZ, RANGE, UPLO
  DOUBLE COMPLEX A(LDA,*), Z(LDZ,*), WORK(*)
  INTEGER*8 N, LDA, IL, IU, NFOUND, LDZ, LDWORK, INFO
  INTEGER*8 IWORK3(*), IFAIL(*)
  DOUBLE PRECISION VL, VU, ABTOL
  DOUBLE PRECISION W(*), WORK2(*)

F95 INTERFACE

  SUBROUTINE HEEVX( JOBZ, RANGE, UPLO, [N], A, [LDA], VL, VU, IL, IU, 
 *       ABTOL, [NFOUND], W, Z, [LDZ], [WORK], [LDWORK], [WORK2], [IWORK3], 
 *       IFAIL, [INFO])
  CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
  COMPLEX(8), DIMENSION(:) :: WORK
  COMPLEX(8), DIMENSION(:,:) :: A, Z
  INTEGER :: N, LDA, IL, IU, NFOUND, LDZ, LDWORK, INFO
  INTEGER, DIMENSION(:) :: IWORK3, IFAIL
  REAL(8) :: VL, VU, ABTOL
  REAL(8), DIMENSION(:) :: W, WORK2

  SUBROUTINE HEEVX_64( JOBZ, RANGE, UPLO, [N], A, [LDA], VL, VU, IL, 
 *       IU, ABTOL, [NFOUND], W, Z, [LDZ], [WORK], [LDWORK], [WORK2], 
 *       [IWORK3], IFAIL, [INFO])
  CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
  COMPLEX(8), DIMENSION(:) :: WORK
  COMPLEX(8), DIMENSION(:,:) :: A, Z
  INTEGER(8) :: N, LDA, IL, IU, NFOUND, LDZ, LDWORK, INFO
  INTEGER(8), DIMENSION(:) :: IWORK3, IFAIL
  REAL(8) :: VL, VU, ABTOL
  REAL(8), DIMENSION(:) :: W, WORK2

C INTERFACE

#include <sunperf.h>

void zheevx(char jobz, char range, char uplo, int n, doublecomplex *a, int lda, double vl, double vu, int il, int iu, double abtol, int *nfound, double *w, doublecomplex *z, int ldz, int *ifail, int *info);

void zheevx_64(char jobz, char range, char uplo, long n, doublecomplex *a, long lda, double vl, double vu, long il, long iu, double abtol, long *nfound, double *w, doublecomplex *z, long ldz, long *ifail, long *info);

PURPOSE

zheevx computes selected eigenvalues and, optionally, eigenvectors of a complex 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.

ARGUMENTS

• 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 triangle of A is stored;

   = 'L':  Lower triangle of A is stored.

• N (input)
  The order of the matrix A. 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).

• 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'.

• ABTOL (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

  ABTOL + EPS * max( |a|,|b| ) ,

  where EPS is the machine precision. If ABTOL 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 ABTOL 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 ABTOL to 2*SLAMCH('S').

  See ``Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,'' by Demmel and Kahan, LAPACK Working Note #3.

• NFOUND (input)
  The total number of eigenvalues found. 0 < = NFOUND < = N. If RANGE = 'A', NFOUND = N, and if RANGE = 'I', NFOUND = IU-IL+1.

• W (output)
  On normal exit, the first NFOUND elements contain the selected eigenvalues in ascending order.

• Z (output)
  If JOBZ = 'V', then if INFO = 0, the first NFOUND 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). 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. If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,NFOUND) columns are supplied in the array Z; if RANGE = 'V', the exact value of NFOUND 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 LDWORK.

• LDWORK (input)
  The length of the array WORK. LDWORK > = max(1,2*N). For optimal efficiency, LDWORK > = (NB+1)*N, where NB is the max of the blocksize for CHETRD and for CUNMTR as returned by ILAENV.

  If LDWORK = -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 LDWORK is issued by XERBLA.

• WORK2 (workspace)
  dimension(7*N)

• IWORK3 (workspace)
  dimension(5*N)

• IFAIL (output)
  If JOBZ = 'V', then if INFO = 0, the first NFOUND 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:  if INFO  = i, then i eigenvectors failed to converge.
  Their indices are stored in array IFAIL.