```      SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
INTEGER            IHI, ILO, INFO, LDA, N
*     ..
*     .. Array Arguments ..
DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
*  an orthogonal 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 DGEBAL; otherwise they should be
*          set to 1 and N respectively. See Further Details.
*          1 <= ILO <= IHI <= max(1,N).
*
*  A       (input/output) DOUBLE PRECISION 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     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
*          The scalar factors of the elementary reflectors (see Further
*          Details).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (N)
*
*  INFO    (output) 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'
*
*  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).
*
*  =====================================================================
*
*     .. Parameters ..
DOUBLE PRECISION   ONE
PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
INTEGER            I
DOUBLE PRECISION   AII
*     ..
*     .. External Subroutines ..
EXTERNAL           DLARF, DLARFG, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -2
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEHD2', -INFO )
RETURN
END IF
*
DO 10 I = ILO, IHI - 1
*
*        Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
*
CALL DLARFG( IHI-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
\$                TAU( I ) )
AII = A( I+1, I )
A( I+1, I ) = ONE
*
*        Apply H(i) to A(1:ihi,i+1:ihi) from the right
*
CALL DLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),
\$               A( 1, I+1 ), LDA, WORK )
*
*        Apply H(i) to A(i+1:ihi,i+1:n) from the left
*
CALL DLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1, TAU( I ),
\$               A( I+1, I+1 ), LDA, WORK )
*
A( I+1, I ) = AII
10 CONTINUE
*
RETURN
*
*     End of DGEHD2
*
END

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