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NAME ZHSEIN - use inverse iteration to find specified right and/or left eigenvectors of a complex upper Hessenberg matrix H SYNOPSIS SUBROUTINE ZHSEIN( JOB, EIGSRC, INITV, SELECT, N, H, LDH, W, VL, LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL, IFAILR, INFO ) CHARACTER EIGSRC, INITV, JOB INTEGER INFO, LDH, LDVL, LDVR, M, MM, N LOGICAL SELECT( * ) INTEGER IFAILL( * ), IFAILR( * ) DOUBLE PRECISION RWORK( * ) COMPLEX*16 H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ), W( * ), WORK( * ) 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 = w y' where y' denotes the conjugate transpose of the vector y. ARGUMENTS JOB (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 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 (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*16 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*16 array, dimension (N) On entry, the eigenvalues of H. On exit, the real parts of W may have been altered since close eigen- values are perturbed slightly in searching for independent eigenvectors. VL (input/output) COMPLEX*16 array, dimension (LDVL,MM) On entry, if INITV = 'U' and JOB = 'L' or 'B', VL must contain starting vectors for the inverse itera- tion 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 JOB = '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 JOB = 'R', VL is not referenced. LDVL (input) INTEGER The leading dimension of the array VL. LDVL >= max(1,N) if JOB = 'L' or 'B'; LDVL >= 1 otherwise. VR (input/output) COMPLEX*16 array, dimension (LDVR,MM) On entry, if INITV = 'U' and JOB = 'R' or 'B', VR must contain starting vectors for the inverse itera- tion 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 JOB = '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 JOB = 'L', VR is not referenced. LDVR (input) INTEGER The leading dimension of the array VR. LDVR >= max(1,N) if JOB = '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*16 array, dimension (N*N) RWORK (workspace) DOUBLE PRECISION array, dimension (N) IFAILL (output) INTEGER array, dimension (MM) If JOB = '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 satisfac- torily. If JOB = 'R', IFAILL is not referenced. IFAILR (output) INTEGER array, dimension (MM) If JOB = '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 satisfac- torily. If JOB = '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 larg- est magnitude has magnitude 1; here the magnitude of a com- plex number (x,y) is taken to be |x|+|y|.