.TH ZHSEIN 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " .SH NAME ZHSEIN - inverse iteration to find specified right and/or left eigenvectors of a complex upper Hessenberg matrix H .SH SYNOPSIS .TP 19 SUBROUTINE ZHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL, LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL, IFAILR, INFO ) .TP 19 .ti +4 CHARACTER EIGSRC, INITV, SIDE .TP 19 .ti +4 INTEGER INFO, LDH, LDVL, LDVR, M, MM, N .TP 19 .ti +4 LOGICAL SELECT( * ) .TP 19 .ti +4 INTEGER IFAILL( * ), IFAILR( * ) .TP 19 .ti +4 DOUBLE PRECISION RWORK( * ) .TP 19 .ti +4 COMPLEX*16 H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ), W( * ), WORK( * ) .SH 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: .br H * x = w * x, y**h * H = w * y**h .br where y**h denotes the conjugate transpose of the vector y. .SH ARGUMENTS .TP 8 SIDE (input) CHARACTER*1 = \(aqR\(aq: compute right eigenvectors only; .br = \(aqL\(aq: compute left eigenvectors only; .br = \(aqB\(aq: compute both right and left eigenvectors. .TP 8 EIGSRC (input) CHARACTER*1 .br Specifies the source of eigenvalues supplied in W: .br = \(aqQ\(aq: 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. = \(aqN\(aq: 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. .TP 8 INITV (input) CHARACTER*1 = \(aqN\(aq: no initial vectors are supplied; .br = \(aqU\(aq: user-supplied initial vectors are stored in the arrays VL and/or VR. .TP 8 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.. .TP 8 N (input) INTEGER The order of the matrix H. N >= 0. .TP 8 H (input) COMPLEX*16 array, dimension (LDH,N) The upper Hessenberg matrix H. .TP 8 LDH (input) INTEGER The leading dimension of the array H. LDH >= max(1,N). .TP 8 W (input/output) 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. .TP 8 VL (input/output) COMPLEX*16 array, dimension (LDVL,MM) On entry, if INITV = \(aqU\(aq and SIDE = \(aqL\(aq or \(aqB\(aq, 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 = \(aqL\(aq or \(aqB\(aq, the left eigenvectors specified by SELECT will be stored consecutively in the columns of VL, in the same order as their eigenvalues. If SIDE = \(aqR\(aq, VL is not referenced. .TP 8 LDVL (input) INTEGER The leading dimension of the array VL. LDVL >= max(1,N) if SIDE = \(aqL\(aq or \(aqB\(aq; LDVL >= 1 otherwise. .TP 8 VR (input/output) COMPLEX*16 array, dimension (LDVR,MM) On entry, if INITV = \(aqU\(aq and SIDE = \(aqR\(aq or \(aqB\(aq, 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 = \(aqR\(aq or \(aqB\(aq, the right eigenvectors specified by SELECT will be stored consecutively in the columns of VR, in the same order as their eigenvalues. If SIDE = \(aqL\(aq, VR is not referenced. .TP 8 LDVR (input) INTEGER The leading dimension of the array VR. LDVR >= max(1,N) if SIDE = \(aqR\(aq or \(aqB\(aq; LDVR >= 1 otherwise. .TP 8 MM (input) INTEGER The number of columns in the arrays VL and/or VR. MM >= M. .TP 8 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). .TP 8 WORK (workspace) COMPLEX*16 array, dimension (N*N) .TP 8 RWORK (workspace) DOUBLE PRECISION array, dimension (N) .TP 8 IFAILL (output) INTEGER array, dimension (MM) If SIDE = \(aqL\(aq or \(aqB\(aq, 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 = \(aqR\(aq, IFAILL is not referenced. .TP 8 IFAILR (output) INTEGER array, dimension (MM) If SIDE = \(aqR\(aq or \(aqB\(aq, 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 = \(aqL\(aq, IFAILR is not referenced. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .br > 0: if INFO = i, i is the number of eigenvectors which failed to converge; see IFAILL and IFAILR for further details. .SH 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|. .br