*> \brief \b ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
* COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H
*> by a unitary similarity transformation: Q**H * A * Q = H .
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> 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 ZGEBAL; otherwise they should be
*> set to 1 and N respectively. See Further Details.
*> 1 <= ILO <= IHI <= max(1,N).
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is COMPLEX*16 array, dimension (N-1)
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX*16 array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16GEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> 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**H
*>
*> 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).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
*
* -- 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 ..
COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX*16 ONE
PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
INTEGER I
COMPLEX*16 ALPHA
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZLARF, ZLARFG
* ..
* .. Intrinsic Functions ..
INTRINSIC DCONJG, 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( 'ZGEHD2', -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 ZLARFG( 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 ZLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),
$ A( 1, I+1 ), LDA, WORK )
*
* Apply H(i)**H to A(i+1:ihi,i+1:n) from the left
*
CALL ZLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1,
$ DCONJG( TAU( I ) ), A( I+1, I+1 ), LDA, WORK )
*
A( I+1, I ) = ALPHA
10 CONTINUE
*
RETURN
*
* End of ZGEHD2
*
END