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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December 2016

  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 DGEHRD


  subroutine incorporating improvements proposed by Quintana-Orti and


  Van de Geijn (2006). (See DLAHR2.)