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LAPACK version 3.0
dgehd2(3)

PURPOSE
  DGEHD2 - reduce a real general matrix A to upper Hessenberg form H by an
  orthogonal similarity transformation

SYNTAX
  SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )

      INTEGER	     IHI, ILO, INFO, LDA, N

      DOUBLE	     PRECISION A( LDA, * ), TAU( * ), WORK( * )

DESCRIPTION
  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.

  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).