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NAME SHSEIN - use inverse iteration to find specified right and/or left eigenvectors of a real upper Hessenberg matrix H SYNOPSIS SUBROUTINE SHSEIN( JOB, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI, VL, LDVL, VR, LDVR, MM, M, WORK, IFAILL, IFAILR, INFO ) CHARACTER EIGSRC, INITV, JOB INTEGER INFO, LDH, LDVL, LDVR, M, MM, N LOGICAL SELECT( * ) INTEGER IFAILL( * ), IFAILR( * ) REAL H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ), WORK( * ), WR( * ) PURPOSE SHSEIN 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 = 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 (WR,WI): = 'Q': the eigenvalues were found using SHSEQR; thus, if H has zero subdiagonal entries, 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) CHARACTER*1 = 'N': no initial vectors are supplied; = 'U': user-supplied initial vectors are stored in the arrays VL and/or VR. SELECT (input/output) LOGICAL array, dimension(N) Specifies the eigenvectors to be computed. To select the real eigenvector corresponding to a real eigen- value 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 con- jugate (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) INTEGER The order of the matrix H. N >= 0. H (input) REAL array, dimension (LDH,N) The upper Hessenberg matrix H. LDH (input) INTEGER The leading dimension of the array H. LDH >= max(1,N). WR (input/output) REAL array, dimension (N) WI (input) REAL array, dimension (N) 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 eigen- values are perturbed slightly in searching for independent eigenvectors. VL (input/output) REAL 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(s) 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 eigen- values. 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 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) REAL 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(s) in which the eigenvector will be stored. On exit, if JOB = 'R' or 'B', the right eigenvectors speci- fied by SELECT will be stored consecutively in the columns of VR, in the same order as their eigen- values. 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 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; each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns. WORK (workspace) REAL array, dimension ((N+2)*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 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 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 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 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|.