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NAME

        ZGEHD2  -  reduce a complex general matrix A to upper Hessenberg form H
        by a unitary similarity transformation
 

SYNOPSIS

        SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
 
            INTEGER        IHI, ILO, INFO, LDA, N
 
            COMPLEX*16     A( LDA, * ), TAU( * ), WORK( * )
 

PURPOSE

        ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H by
        a unitary similarity transformation: Q’ * A * Q = H .
 

ARGUMENTS

        N       (input) INTEGER
                The order of the matrix A.  N >= 0.
 
        ILO     (input) INTEGER
                IHI      (input)  INTEGER It is assumed that A 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 ZGEBAL; otherwise they
                should be set to 1 and N respectively. See Further Details.
 
        A       (input/output) COMPLEX*16 array, dimension (LDA,N)
                On entry, the n by n general matrix to be  reduced.   On  exit,
                the upper triangle and the first subdiagonal of A are overwrit‐
                ten with the upper Hessenberg matrix H, and the elements  below
                the  first  subdiagonal, with the array TAU, represent the uni‐
                tary matrix Q as a product of elementary reflectors.  See  Fur‐
                ther Details.  LDA     (input) INTEGER The leading dimension of
                the array A.  LDA >= max(1,N).
 
        TAU     (output) COMPLEX*16 array, dimension (N-1)
                The scalar factors of the elementary  reflectors  (see  Further
                Details).
 
        WORK    (workspace) COMPLEX*16 array, dimension (N)
 
        INFO    (output) INTEGER
                = 0:  successful exit
                < 0:  if INFO = -i, the i-th argument had an illegal value.
        The  matrix  Q  is  represented  as  a  product of (ihi-ilo) elementary
        reflectors
 
           Q = H(ilo) H(ilo+1) . . . H(ihi-1).
 
        Each H(i) has the form
 
           H(i) = I - tau * v * v’
 
        where tau is a complex scalar, and v is a complex vector with v(1:i)  =
        0,  v(i+1)  =  1  and  v(ihi+1:n)  = 0; v(i+2:ihi) is stored on exit in
        A(i+2:ihi,i), and tau in TAU(i).
 
        The contents of A are illustrated by the following example, with n = 7,
        ilo = 2 and ihi = 6:
 
        on entry,                        on exit,
 
        ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a ) (     a
        a   a   a   a   a )    (      a   h   h   h   h   a ) (     a    a    a
        a    a    a )    (      h   h   h   h   h   h ) (     a   a   a   a   a
        a )    (      v2  h   h   h   h   h ) (     a   a   a    a    a    a  )
        (       v2   v3   h    h    h    h ) (     a   a   a   a   a   a )    (
        v2   v3   v4   h    h    h  )  (                           a   )      (
        a )
 
        where  a denotes an element of the original matrix A, h denotes a modi‐
        fied element of the upper Hessenberg matrix H, and vi denotes  an  ele‐
        ment of the vector defining H(i).
 

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