SYNOPSIS

SUBROUTINE DHSEIN(

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

    

CHARACTER EIGSRC, INITV, SIDE

    

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

    

LOGICAL SELECT( * )

    

INTEGER IFAILL( * ), IFAILR( * )

    

DOUBLE PRECISION H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ), WORK( * ), WR( * )

PURPOSE

DHSEIN 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 * 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 (WR,WI):
= 'Q': the eigenvalues were found using DHSEQR; 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 DHSEIN 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, DHSEIN 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 eigenvalue 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 conjugate (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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N)

WI (input) DOUBLE PRECISION 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 eigenvalues are perturbed slightly in searching for independent eigenvectors.

VL (input/output) DOUBLE PRECISION 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(s) 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. 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 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) DOUBLE PRECISION 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(s) 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. 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 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; each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns.

WORK (workspace) DOUBLE PRECISION array, dimension ((N+2)*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 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 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 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 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|.