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

NAME

dhsein - use inverse iteration to find specified right and/or left eigenvectors of a real upper Hessenberg matrix H

SYNOPSIS

  SUBROUTINE DHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI, 
 *      VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL, IFAILR, INFO)
  CHARACTER * 1 SIDE, EIGSRC, INITV
  INTEGER N, LDH, LDVL, LDVR, MM, M, INFO
  INTEGER IFAILL(*), IFAILR(*)
  LOGICAL SELECT(*)
  DOUBLE PRECISION H(LDH,*), WR(*), WI(*), VL(LDVL,*), VR(LDVR,*), WORK(*)

  SUBROUTINE DHSEIN_64( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, 
 *      WI, VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL, IFAILR, INFO)
  CHARACTER * 1 SIDE, EIGSRC, INITV
  INTEGER*8 N, LDH, LDVL, LDVR, MM, M, INFO
  INTEGER*8 IFAILL(*), IFAILR(*)
  LOGICAL*8 SELECT(*)
  DOUBLE PRECISION H(LDH,*), WR(*), WI(*), VL(LDVL,*), VR(LDVR,*), WORK(*)

F95 INTERFACE

  SUBROUTINE HSEIN( SIDE, EIGSRC, INITV, SELECT, [N], H, [LDH], WR, 
 *       WI, VL, [LDVL], VR, [LDVR], MM, M, [WORK], IFAILL, IFAILR, [INFO])
  CHARACTER(LEN=1) :: SIDE, EIGSRC, INITV
  INTEGER :: N, LDH, LDVL, LDVR, MM, M, INFO
  INTEGER, DIMENSION(:) :: IFAILL, IFAILR
  LOGICAL, DIMENSION(:) :: SELECT
  REAL(8), DIMENSION(:) :: WR, WI, WORK
  REAL(8), DIMENSION(:,:) :: H, VL, VR

  SUBROUTINE HSEIN_64( SIDE, EIGSRC, INITV, SELECT, [N], H, [LDH], WR, 
 *       WI, VL, [LDVL], VR, [LDVR], MM, M, [WORK], IFAILL, IFAILR, [INFO])
  CHARACTER(LEN=1) :: SIDE, EIGSRC, INITV
  INTEGER(8) :: N, LDH, LDVL, LDVR, MM, M, INFO
  INTEGER(8), DIMENSION(:) :: IFAILL, IFAILR
  LOGICAL(8), DIMENSION(:) :: SELECT
  REAL(8), DIMENSION(:) :: WR, WI, WORK
  REAL(8), DIMENSION(:,:) :: H, VL, VR

C INTERFACE

#include <sunperf.h>

void dhsein(char side, char eigsrc, char initv, logical *select, int n, double *h, int ldh, double *wr, double *wi, double *vl, int ldvl, double *vr, int ldvr, int mm, int *m, int *ifaill, int *ifailr, int *info);

void dhsein_64(char side, char eigsrc, char initv, logical *select, long n, double *h, long ldh, double *wr, double *wi, double *vl, long ldvl, double *vr, long ldvr, long mm, long *m, long *ifaill, long *ifailr, long *info);

PURPOSE

dhsein uses inverse iteration to find specified right and/or left eigenvectors of a real upper Hessenberg matrix H.

The right eigenvector x and the left eigenvector y of the matrix H corresponding to an eigenvalue w are defined by:

             H * x = w * x,     y**h * H = w * y**h

where y**h denotes the conjugate transpose of the vector y.

ARGUMENTS

• SIDE (input)
  

   = 'R': compute right eigenvectors only;

   = 'L': compute left eigenvectors only;

   = 'B': compute both right and left eigenvectors.

• EIGSRC (input)
  Specifies the source of eigenvalues supplied in (WR,WI):

   = 'Q': the eigenvalues were found using SHSEQR; thus, if
  H has zero subdiagonal elements, and so is
  block-triangular, then the j-th eigenvalue can be
  assumed to be an eigenvalue of the block containing
  the j-th row/column.  This property allows SHSEIN to
  perform inverse iteration on just one diagonal block.
   = 'N': no assumptions are made on the correspondence
  between eigenvalues and diagonal blocks.  In this
  case, SHSEIN must always perform inverse iteration
  using the whole matrix H.

• INITV (input)
  

   = 'N': no initial vectors are supplied;

   = 'U': user-supplied initial vectors are stored in the arrays
  VL and/or VR.

• SELECT (input/output)
  Specifies the eigenvectors to be computed. To select the real eigenvector corresponding to a real eigenvalue WR(j), SELECT(j) must be set to .TRUE.. To select the complex eigenvector corresponding to a complex eigenvalue (WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)), either SELECT(j) or SELECT(j+1) or both must be set to .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is .FALSE..

• N (input)
  The order of the matrix H. N > = 0.

• H (input)
  The upper Hessenberg matrix H.

• LDH (input)
  The leading dimension of the array H. LDH > = max(1,N).

• WR (input/output)
  On entry, the real and imaginary parts of the eigenvalues of H; a complex conjugate pair of eigenvalues must be stored in consecutive elements of WR and WI. On exit, WR may have been altered since close eigenvalues are perturbed slightly in searching for independent eigenvectors.

• WI (input)
  See the description of WR.

• VL (input/output)
  On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must contain starting vectors for the inverse iteration for the left eigenvectors; the starting vector for each eigenvector must be in the same column(s) in which the eigenvector will be stored. On exit, if SIDE = 'L' or 'B', the left eigenvectors specified by SELECT will be stored consecutively in the columns of VL, in the same order as their eigenvalues. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part and the second the imaginary part. If SIDE = 'R', VL is not referenced.

• LDVL (input)
  The leading dimension of the array VL. LDVL > = max(1,N) if SIDE = 'L' or 'B'; LDVL > = 1 otherwise.

• VR (input/output)
  On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must contain starting vectors for the inverse iteration for the right eigenvectors; the starting vector for each eigenvector must be in the same column(s) in which the eigenvector will be stored. On exit, if SIDE = 'R' or 'B', the right eigenvectors specified by SELECT will be stored consecutively in the columns of VR, in the same order as their eigenvalues. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part and the second the imaginary part. If SIDE = 'L', VR is not referenced.

• LDVR (input)
  The leading dimension of the array VR. LDVR > = max(1,N) if SIDE = 'R' or 'B'; LDVR > = 1 otherwise.

• MM (input)
  The number of columns in the arrays VL and/or VR. MM > = M.

• M (output)
  The number of columns in the arrays VL and/or VR required to store the eigenvectors; each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns.

• WORK (workspace)
  dimension((N+2)*N)

• IFAILL (output)
  If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left eigenvector in the i-th column of VL (corresponding to the eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the eigenvector converged satisfactorily. If the i-th and (i+1)th columns of VL hold a complex eigenvector, then IFAILL(i) and IFAILL(i+1) are set to the same value. If SIDE = 'R', IFAILL is not referenced.

• IFAILR (output)
  If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right eigenvector in the i-th column of VR (corresponding to the eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the eigenvector converged satisfactorily. If the i-th and (i+1)th columns of VR hold a complex eigenvector, then IFAILR(i) and IFAILR(i+1) are set to the same value. If SIDE = 'L', IFAILR is not referenced.

• INFO (output)
  

   = 0:  successful exit

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

   > 0:  if INFO  = i, i is the number of eigenvectors which
  failed to converge; see IFAILL and IFAILR for further
  details.

FURTHER DETAILS

Each eigenvector is normalized so that the element of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x|+|y|.