ZGEHD2(3)      LAPACK routine of NEC Numeric Library Collection      ZGEHD2(3)



NAME
       ZGEHD2

SYNOPSIS
       SUBROUTINE ZGEHD2 (N, ILO, IHI, A, LDA, TAU, WORK, INFO)



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




ARGUMENTS
           N         (input)
                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           ILO       (input)
                     ILO is INTEGER

           IHI       (input)
                     IHI is 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.
                     1 <= ILO <= IHI <= max(1,N).

           A         (input/output)
                     A is 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 overwritten with the upper Hessenberg matrix H, and the
                     elements below the first subdiagonal, with the array TAU,
                     represent the unitary matrix Q as a product of elementary
                     reflectors. See Further Details.

           LDA       (input)
                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,N).

           TAU       (output)
                     TAU is COMPLEX*16 array, dimension (N-1)
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK      (output)
                     WORK is COMPLEX*16 array, dimension (N)

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value.






FURTHER DETAILS
             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**H

             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
             modified element of the upper Hessenberg matrix H, and vi denotes an
             element of the vector defining H(i).



LAPACK routine                  31 October 2017                      ZGEHD2(3)