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



NAME
       ZGEHRD

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



PURPOSE
            ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by
            an 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 <= N, if N > 0; ILO=1 and IHI=0, if N=0.

           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). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
                     zero.

           WORK      (output)
                     WORK is COMPLEX*16 array, dimension (LWORK)
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK     (input)
                     LWORK is INTEGER
                     The length of the array WORK.  LWORK >= max(1,N).
                     For optimum performance LWORK >= N*NB, where NB is the
                     optimal blocksize.

                     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)
                     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).

             This file is a slight modification of LAPACK-3.0's DGEHRD
             subroutine incorporating improvements proposed by Quintana-Orti and
             Van de Geijn (2006).



LAPACK routine                  31 October 2017                      ZGEHRD(3)