SHSEQR(3)

NAME
SHSEQR - compute the eigenvalues of a 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

SYNOPSIS

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

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

CHARACTER
    COMPZ, JOB

REAL
    H( LDH, * ), WI( * ), WORK( * ), WR( * ), Z( LDZ, * )

PURPOSE
   SHSEQR computes the eigenvalues of a 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.

ARGUMENTS

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 .GE. 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 SGEBAL, and then passed to SGEHRD when the matrix output by SGEBAL is reduced to Hessenberg form. Otherwise ILO and IHI should be set to 1 and N respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0.

H     (input/output) REAL array, dimension (LDH,N)
    On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and JOB = 'S', then 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).LT.0. If INFO = 0 and JOB = 'E', the contents of H are unspecified on exit. (The output value of H when INFO.GT.0 is given under the description of INFO below.) Unlike earlier versions of SHSEQR, this subroutine may explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

LDH   (input) INTEGER
    The leading dimension of the array H. LDH .GE. max(1,N).

WR    (output) REAL array, dimension (N)
    WI (output) REAL 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) .GT. 0 and WI(i+1) .LT. 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) REAL array, dimension (LDZ,N)
    If COMPZ = 'N', Z is not referenced. If COMPZ = 'I', on entry Z need not be set and on exit, if INFO = 0, 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, if INFO = 0, Z contains Q*Z. Normally Q is the orthogonal matrix generated by SORGHR after the call to SGEHRD which formed the Hessenberg matrix H. (The output value of Z when INFO.GT.0 is given under the description of INFO below.)

LDZ   (input) INTEGER
    The leading dimension of the array Z. if COMPZ = 'I' or COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1.

WORK  (workspace/output) REAL array, dimension (LWORK)
    On exit, if INFO = 0, WORK(1) returns an estimate of the optimal value for LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK .GE. max(1,N) is sufficient, but LWORK typically as large as 6*N may be required for optimal performance. A workspace query to determine the optimal workspace size is recommended. If LWORK = -1, then SHSEQR does a workspace query. In this case, SHSEQR checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed.

INFO  (output) INTEGER
    = 0: successful exit
    value
    the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.) If INFO .GT. 0 and JOB = 'E', then on exit, the remaining unconverged eigenvalues are the eigen- values of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H. If INFO .GT. 0 and JOB = 'S', then on exit

(*)  (initial value of H)*U  = U*(final value of H)
    where U is an orthogonal matrix. The final value of H is upper Hessenberg and quasi-triangular in rows and columns INFO+1 through IHI. If INFO .GT. 0 and COMPZ = 'V', then on exit (final value of Z) = (initial value of Z)*U where U is the orthogonal matrix in (*) (regard- less of the value of JOB.) If INFO .GT. 0 and COMPZ = 'I', then on exit (final value of Z) = U where U is the orthogonal matrix in (*) (regard- less of the value of JOB.) If INFO .GT. 0 and COMPZ = 'N', then Z is not accessed.