DHSEQR - compute the eigenvalues of a real upper Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors

SUBROUTINE DHSEQR(

JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ, WORK, LWORK, INFO )

CHARACTER COMPZ, JOB

INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N

DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), Z( LDZ, * )

DHSEQR computes the eigenvalues of a real upper Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors. Optionally Z may be postmultiplied into an input orthogonal matrix Q, so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.

JOB (input) CHARACTER*1

= 'E': compute eigenvalues only;

= 'S': compute eigenvalues and the Schur form T.

COMPZ (input) CHARACTER*1

= 'N': no Schur vectors are computed;

= 'I': Z is initialized to the unit matrix and the matrix Z of Schur vectors of H is returned; = 'V': Z must contain an orthogonal matrix Q on entry, and the product Q*Z is returned.

N (input) INTEGER

The order of the matrix H. N >= 0.

ILO (input) INTEGER

IHI (input) INTEGER It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to DGEBAL, and then passed to SGEHRD when the matrix output by DGEBAL is reduced to Hessenberg form. Otherwise ILO and IHI should be set to 1 and N respectively. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

H (input/output) DOUBLE PRECISION array, dimension (LDH,N)

On entry, the upper Hessenberg matrix H. On exit, if JOB = 'S', H contains the upper quasi-triangular matrix T from the Schur decomposition (the Schur form); 2-by-2 diagonal blocks (corresponding to complex conjugate pairs of eigenvalues) are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0. If JOB = 'E', the contents of H are unspecified on exit.

LDH (input) INTEGER

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

WR (output) DOUBLE PRECISION array, dimension (N)

WI (output) DOUBLE PRECISION array, dimension (N) The real and imaginary parts, respectively, of the computed eigenvalues. If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and WI(i+1) < 0. If JOB = 'S', the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).

Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)

If COMPZ = 'N': Z is not referenced.

If COMPZ = 'I': on entry, Z need not be set, and on exit, Z contains the orthogonal matrix Z of the Schur vectors of H. If COMPZ = 'V': on entry Z must contain an N-by-N matrix Q, which is assumed to be equal to the unit matrix except for the submatrix Z(ILO:IHI,ILO:IHI); on exit Z contains Q*Z. Normally Q is the orthogonal matrix generated by DORGHR after the call to DGEHRD which formed the Hessenberg matrix H.

LDZ (input) INTEGER

The leading dimension of the array Z. LDZ >= max(1,N) if COMPZ = 'I' or 'V'; LDZ >= 1 otherwise.

WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >= max(1,N).

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

INFO (output) INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, DHSEQR failed to compute all of the eigenvalues in a total of 30*(IHI-ILO+1) iterations; elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed.