# zhsein.f - Man Page

## Synopsis

### Functions/Subroutines

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

## Function/Subroutine Documentation

### subroutine zhsein (character SIDE, character EIGSRC, character INITV, logical, dimension( * ) SELECT, integer N, complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( * ) W, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, integer MM, integer M, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IFAILL, integer, dimension( * ) IFAILR, integer INFO)

**ZHSEIN**

**Purpose:**

ZHSEIN 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.

**Parameters:***SIDE*SIDE is CHARACTER*1 = 'R': compute right eigenvectors only; = 'L': compute left eigenvectors only; = 'B': compute both right and left eigenvectors.

*EIGSRC*EIGSRC is CHARACTER*1 Specifies the source of eigenvalues supplied in W: = 'Q': the eigenvalues were found using ZHSEQR; 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 ZHSEIN 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, ZHSEIN must always perform inverse iteration using the whole matrix H.

*INITV*INITV is CHARACTER*1 = 'N': no initial vectors are supplied; = 'U': user-supplied initial vectors are stored in the arrays VL and/or VR.

*SELECT*SELECT is 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*N is INTEGER The order of the matrix H. N >= 0.

*H*H is COMPLEX*16 array, dimension (LDH,N) The upper Hessenberg matrix H. If a NaN is detected in H, the routine will return with INFO=-6.

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

*W*W is COMPLEX*16 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*VL is COMPLEX*16 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*LDVL is INTEGER The leading dimension of the array VL. LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.

*VR*VR is COMPLEX*16 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*LDVR is INTEGER The leading dimension of the array VR. LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.

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

*M*M is 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*WORK is COMPLEX*16 array, dimension (N*N)

*RWORK*RWORK is DOUBLE PRECISION array, dimension (N)

*IFAILL*IFAILL is 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*IFAILR is 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*INFO is 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.

**Author:**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**December 2016

**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|.

Definition at line 247 of file zhsein.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zhsein(3) is an alias of zhsein.f(3).

Tue Nov 14 2017 Version 3.8.0 LAPACK