cgehd2.f

          SUBROUTINE CGEHD2( 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 ..
          COMPLEX            A( LDA, * ), TAU( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  CGEHD2 reduces a complex general matrix A to upper Hessenberg form H
    *  by a unitary 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 CGEBAL; otherwise they should be
    *          set to 1 and N respectively. See Further Details.
    *          1 <= ILO <= IHI <= max(1,N).
    *
    *  A       (input/output) COMPLEX 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 unitary 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) COMPLEX array, dimension (N-1)
    *          The scalar factors of the elementary reflectors (see Further
    *          Details).
    *
    *  WORK    (workspace) COMPLEX 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 complex scalar, and v is a complex 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 ..
          COMPLEX            ONE
          PARAMETER          ( ONE = ( 1.0E+0, 0.0E+0 ) )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I
          COMPLEX            ALPHA
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           CLARF, CLARFG, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          CONJG, 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( 'CGEHD2', -INFO )
             RETURN
          END IF
    *
          DO 10 I = ILO, IHI - 1
    *
    *        Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
    *
             ALPHA = A( I+1, I )
             CALL CLARFG( IHI-I, ALPHA, A( MIN( I+2, N ), I ), 1, TAU( I ) )
             A( I+1, I ) = ONE
    *
    *        Apply H(i) to A(1:ihi,i+1:ihi) from the right
    *
             CALL CLARF( '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 CLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1,
         $               CONJG( TAU( I ) ), A( I+1, I+1 ), LDA, WORK )
    *
             A( I+1, I ) = ALPHA
       10 CONTINUE
    *
          RETURN
    *
    *     End of CGEHD2
    *
          END