sgehd2()

subroutine sgehd2	(	integer	n,
		integer	ilo,
		integer	ihi,
		real, dimension( lda, * )	a,
		integer	lda,
		real, dimension( * )	tau,
		real, dimension( * )	work,
		integer	info
	)

SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

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Purpose:

 SGEHD2 reduces a real general matrix A to upper Hessenberg form H by
 an orthogonal similarity transformation:  Q**T * A * Q = H .

Parameters

[in]	N	N is INTEGER The order of the matrix A. N >= 0.
[in]	ILO	ILO is INTEGER
[in]	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 <= max(1,N).
[in,out]	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.
[in]	LDA	LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	TAU	TAU is REAL array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]	WORK	WORK is REAL array, dimension (N)
[out]	INFO	INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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**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).

Definition at line 148 of file sgehd2.f.

*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IHI, ILO, INFO, LDA, N
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), TAU( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE
      parameter( one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      REAL               AII
*     ..
*     .. External Subroutines ..
      EXTERNAL           slarf, slarfg, 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( 'SGEHD2', -info )
         RETURN
      END IF
*
      DO 10 i = ilo, ihi - 1
*
*        Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
*
         CALL slarfg( 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 slarf( '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 slarf( '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 SGEHD2
*

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