---
name: "chsein"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/chsein/3lapack"
---

# chsein(3lapack)

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

## NAME

CHSEIN - uses inverse iteration to find specified right and/or left eigenvectors of a complex upper Hessenberg matrix H

## SYNOPSIS

- SUBROUTINE CHSEIN(

  SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL, LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL, IFAILR, INFO )

- CHARACTER

  EIGSRC, INITV, SIDE

- INTEGER

  INFO, LDH, LDVL, LDVR, M, MM, N

- LOGICAL

  SELECT( \* )

- INTEGER

  IFAILL( \* ), IFAILR( \* )

- REAL

  RWORK( \* )

- COMPLEX

  H( LDH, \* ), VL( LDVL, \* ), VR( LDVR, \* ), W( \* ), WORK( \* )

## PURPOSE

CHSEIN uses inverse iteration to find specified right and/or left eigenvectors of a complex 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) CHARACTER\*1

  \= 'R': compute right eigenvectors only;\
   \= 'L': compute left eigenvectors only;\
   \= 'B': compute both right and left eigenvectors.

- EIGSRC (input) CHARACTER\*1

  Specifies the source of eigenvalues supplied in W:\
   \= 'Q': the eigenvalues were found using CHSEQR; 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 CHSEIN 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, CHSEIN must always perform inverse iteration using the whole matrix H.

- INITV (input) CHARACTER\*1

  \= 'N': no initial vectors are supplied;\
   \= 'U': user-supplied initial vectors are stored in the arrays VL and/or VR.

- SELECT (input) LOGICAL array, dimension (N)

  Specifies the eigenvectors to be computed. To select the eigenvector corresponding to the eigenvalue W(j), SELECT(j) must be set to .TRUE..

- N (input) INTEGER

  The order of the matrix H.  N \>= 0.

- H (input) COMPLEX array, dimension (LDH,N)

  The upper Hessenberg matrix H.

- LDH (input) INTEGER

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

- W (input/output) COMPLEX array, dimension (N)

  On entry, the eigenvalues of H. On exit, the real parts of W may have been altered since close eigenvalues are perturbed slightly in searching for independent eigenvectors.

- VL (input/output) COMPLEX array, dimension (LDVL,MM)

  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 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. If SIDE = 'R', VL is not referenced.

- LDVL (input) INTEGER

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

- VR (input/output) COMPLEX array, dimension (LDVR,MM)

  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 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. If SIDE = 'L', VR is not referenced.

- LDVR (input) INTEGER

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

- MM (input) INTEGER

  The number of columns in the arrays VL and/or VR. MM \>= M.

- M (output) INTEGER

  The number of columns in the arrays VL and/or VR required to store the eigenvectors (= the number of .TRUE. elements in SELECT).

- WORK (workspace) COMPLEX array, dimension (N\*N)

  - RWORK (workspace) REAL array, dimension (N)

  - IFAILL (output) INTEGER array, dimension (MM)

  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 SIDE = 'R', IFAILL is not referenced.

- IFAILR (output) INTEGER array, dimension (MM)

  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 SIDE = 'L', IFAILR is not referenced.

- INFO (output) INTEGER

  \= 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\|.\