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221 lines
6.4 KiB
221 lines
6.4 KiB
*> \brief \b CGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download CGEHD2 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgehd2.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgehd2.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgehd2.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE CGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER IHI, ILO, INFO, LDA, N
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* ..
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* .. Array Arguments ..
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* COMPLEX A( LDA, * ), TAU( * ), WORK( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> CGEHD2 reduces a complex general matrix A to upper Hessenberg form H
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*> by a unitary similarity transformation: Q**H * A * Q = H .
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] ILO
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*> \verbatim
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*> ILO is INTEGER
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*> \endverbatim
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*>
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*> \param[in] IHI
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*> \verbatim
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*> IHI is INTEGER
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*>
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*> It is assumed that A is already upper triangular in rows
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*> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
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*> set by a previous call to CGEBAL; otherwise they should be
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*> set to 1 and N respectively. See Further Details.
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*> 1 <= ILO <= IHI <= max(1,N).
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is COMPLEX array, dimension (LDA,N)
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*> On entry, the n by n general matrix to be reduced.
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*> On exit, the upper triangle and the first subdiagonal of A
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*> are overwritten with the upper Hessenberg matrix H, and the
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*> elements below the first subdiagonal, with the array TAU,
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*> represent the unitary matrix Q as a product of elementary
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*> reflectors. See Further Details.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max(1,N).
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*> \endverbatim
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*>
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*> \param[out] TAU
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*> \verbatim
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*> TAU is COMPLEX array, dimension (N-1)
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*> The scalar factors of the elementary reflectors (see Further
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*> Details).
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is COMPLEX array, dimension (N)
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup complexGEcomputational
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*
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> The matrix Q is represented as a product of (ihi-ilo) elementary
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*> reflectors
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*>
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*> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
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*>
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*> Each H(i) has the form
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*>
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*> H(i) = I - tau * v * v**H
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*>
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*> where tau is a complex scalar, and v is a complex vector with
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*> v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
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*> exit in A(i+2:ihi,i), and tau in TAU(i).
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*>
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*> The contents of A are illustrated by the following example, with
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*> n = 7, ilo = 2 and ihi = 6:
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*>
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*> on entry, on exit,
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*>
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*> ( a a a a a a a ) ( a a h h h h a )
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*> ( a a a a a a ) ( a h h h h a )
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*> ( a a a a a a ) ( h h h h h h )
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*> ( a a a a a a ) ( v2 h h h h h )
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*> ( a a a a a a ) ( v2 v3 h h h h )
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*> ( a a a a a a ) ( v2 v3 v4 h h h )
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*> ( a ) ( a )
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*>
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*> where a denotes an element of the original matrix A, h denotes a
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*> modified element of the upper Hessenberg matrix H, and vi denotes an
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*> element of the vector defining H(i).
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE CGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
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*
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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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INTEGER IHI, ILO, INFO, LDA, N
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* ..
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* .. Array Arguments ..
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COMPLEX A( LDA, * ), TAU( * ), WORK( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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COMPLEX ONE
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PARAMETER ( ONE = ( 1.0E+0, 0.0E+0 ) )
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* ..
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* .. Local Scalars ..
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INTEGER I
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COMPLEX ALPHA
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* ..
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* .. External Subroutines ..
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EXTERNAL CLARF, CLARFG, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC CONJG, MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters
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*
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INFO = 0
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IF( N.LT.0 ) THEN
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INFO = -1
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ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
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INFO = -2
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ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
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INFO = -3
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -5
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'CGEHD2', -INFO )
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RETURN
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END IF
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*
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DO 10 I = ILO, IHI - 1
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*
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* Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
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*
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ALPHA = A( I+1, I )
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CALL CLARFG( IHI-I, ALPHA, A( MIN( I+2, N ), I ), 1, TAU( I ) )
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A( I+1, I ) = ONE
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*
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* Apply H(i) to A(1:ihi,i+1:ihi) from the right
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*
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CALL CLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),
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$ A( 1, I+1 ), LDA, WORK )
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*
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* Apply H(i)**H to A(i+1:ihi,i+1:n) from the left
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*
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CALL CLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1,
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$ CONJG( TAU( I ) ), A( I+1, I+1 ), LDA, WORK )
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*
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A( I+1, I ) = ALPHA
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10 CONTINUE
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*
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RETURN
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*
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* End of CGEHD2
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*
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END
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