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LAPACK version 3.0
dgehrd(3)

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

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

      INTEGER	     IHI, ILO, INFO, LDA, LWORK, N

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

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

  WORK	  (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
	  On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK	  (input) 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) 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).