!>
!> SGEHRD reduces a real general matrix A to upper Hessenberg form H by
!> an orthogonal similarity transformation:  Q**T * A * Q = H .
!> 

N

!>          N is INTEGER
!>          The order of the matrix A.  N >= 0.
!> 

ILO

!>          ILO is INTEGER
!> 

IHI

!>          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 SGEBAL; 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

!>          A is REAL 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 orthogonal matrix Q as a product of elementary
!>          reflectors. See Further Details.
!> 

LDA

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

TAU

!>          TAU is REAL 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

!>          WORK is REAL array, dimension (LWORK)
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!> 

LWORK

!>          LWORK is INTEGER
!>          The length of the array WORK.  LWORK >= max(1,N).
!>          For good performance, LWORK should generally be larger.
!>
!>          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

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

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!>
!>  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**T
!>
!>  where tau is a real scalar, and v is a real 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 SGEHRD
!>  subroutine incorporating improvements proposed by Quintana-Orti and
!>  Van de Geijn (2006). (See SLAHR2.)
!>