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353 lines
11 KiB
353 lines
11 KiB
*> \brief \b ZGEHRD
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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 ZGEHRD + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgehrd.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/zgehrd.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/zgehrd.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 ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER IHI, ILO, INFO, LDA, LWORK, N
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* ..
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* .. Array Arguments ..
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* COMPLEX*16 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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*> ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by
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*> an 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 ZGEBAL; otherwise they should be
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*> set to 1 and N respectively. See Further Details.
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*> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
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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*16 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*16 array, dimension (N-1)
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*> The scalar factors of the elementary reflectors (see Further
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*> Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
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*> zero.
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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*16 array, dimension (LWORK)
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*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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*> \endverbatim
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The length of the array WORK. LWORK >= max(1,N).
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*> For good performance, LWORK should generally be larger.
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*>
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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*> this value as the first entry of the WORK array, and no error
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*> message related to LWORK is issued by XERBLA.
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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 complex16GEcomputational
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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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*>
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*> This file is a slight modification of LAPACK-3.0's ZGEHRD
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*> subroutine incorporating improvements proposed by Quintana-Orti and
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*> Van de Geijn (2006). (See ZLAHR2.)
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, 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, LWORK, N
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* ..
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* .. Array Arguments ..
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COMPLEX*16 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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INTEGER NBMAX, LDT, TSIZE
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PARAMETER ( NBMAX = 64, LDT = NBMAX+1,
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$ TSIZE = LDT*NBMAX )
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COMPLEX*16 ZERO, ONE
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PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
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$ ONE = ( 1.0D+0, 0.0D+0 ) )
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* ..
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* .. Local Scalars ..
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LOGICAL LQUERY
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INTEGER I, IB, IINFO, IWT, J, LDWORK, LWKOPT, NB,
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$ NBMIN, NH, NX
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COMPLEX*16 EI
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* ..
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* .. External Subroutines ..
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EXTERNAL ZAXPY, ZGEHD2, ZGEMM, ZLAHR2, ZLARFB, ZTRMM,
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$ XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN
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* ..
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* .. External Functions ..
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INTEGER ILAENV
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EXTERNAL ILAENV
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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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LQUERY = ( LWORK.EQ.-1 )
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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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ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
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INFO = -8
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END IF
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*
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IF( INFO.EQ.0 ) THEN
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*
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* Compute the workspace requirements
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*
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NB = MIN( NBMAX, ILAENV( 1, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) )
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LWKOPT = N*NB + TSIZE
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WORK( 1 ) = LWKOPT
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ENDIF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZGEHRD', -INFO )
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RETURN
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ELSE IF( LQUERY ) THEN
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RETURN
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END IF
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*
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* Set elements 1:ILO-1 and IHI:N-1 of TAU to zero
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*
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DO 10 I = 1, ILO - 1
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TAU( I ) = ZERO
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10 CONTINUE
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DO 20 I = MAX( 1, IHI ), N - 1
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TAU( I ) = ZERO
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20 CONTINUE
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*
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* Quick return if possible
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*
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NH = IHI - ILO + 1
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IF( NH.LE.1 ) THEN
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WORK( 1 ) = 1
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RETURN
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END IF
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*
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* Determine the block size
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*
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NB = MIN( NBMAX, ILAENV( 1, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) )
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NBMIN = 2
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IF( NB.GT.1 .AND. NB.LT.NH ) THEN
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*
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* Determine when to cross over from blocked to unblocked code
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* (last block is always handled by unblocked code)
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*
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NX = MAX( NB, ILAENV( 3, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) )
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IF( NX.LT.NH ) THEN
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*
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* Determine if workspace is large enough for blocked code
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*
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IF( LWORK.LT.N*NB+TSIZE ) THEN
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*
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* Not enough workspace to use optimal NB: determine the
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* minimum value of NB, and reduce NB or force use of
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* unblocked code
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*
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NBMIN = MAX( 2, ILAENV( 2, 'ZGEHRD', ' ', N, ILO, IHI,
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$ -1 ) )
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IF( LWORK.GE.(N*NBMIN + TSIZE) ) THEN
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NB = (LWORK-TSIZE) / N
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ELSE
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NB = 1
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END IF
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END IF
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END IF
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END IF
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LDWORK = N
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*
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IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN
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*
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* Use unblocked code below
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*
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I = ILO
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*
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ELSE
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*
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* Use blocked code
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*
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IWT = 1 + N*NB
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DO 40 I = ILO, IHI - 1 - NX, NB
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IB = MIN( NB, IHI-I )
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*
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* Reduce columns i:i+ib-1 to Hessenberg form, returning the
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* matrices V and T of the block reflector H = I - V*T*V**H
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* which performs the reduction, and also the matrix Y = A*V*T
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*
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CALL ZLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ),
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$ WORK( IWT ), LDT, WORK, LDWORK )
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*
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* Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
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* right, computing A := A - Y * V**H. V(i+ib,ib-1) must be set
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* to 1
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*
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EI = A( I+IB, I+IB-1 )
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A( I+IB, I+IB-1 ) = ONE
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CALL ZGEMM( 'No transpose', 'Conjugate transpose',
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$ IHI, IHI-I-IB+1,
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$ IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE,
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$ A( 1, I+IB ), LDA )
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A( I+IB, I+IB-1 ) = EI
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*
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* Apply the block reflector H to A(1:i,i+1:i+ib-1) from the
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* right
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*
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CALL ZTRMM( 'Right', 'Lower', 'Conjugate transpose',
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$ 'Unit', I, IB-1,
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$ ONE, A( I+1, I ), LDA, WORK, LDWORK )
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DO 30 J = 0, IB-2
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CALL ZAXPY( I, -ONE, WORK( LDWORK*J+1 ), 1,
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$ A( 1, I+J+1 ), 1 )
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30 CONTINUE
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*
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* Apply the block reflector H to A(i+1:ihi,i+ib:n) from the
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* left
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*
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CALL ZLARFB( 'Left', 'Conjugate transpose', 'Forward',
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$ 'Columnwise',
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$ IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA,
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$ WORK( IWT ), LDT, A( I+1, I+IB ), LDA,
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$ WORK, LDWORK )
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40 CONTINUE
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END IF
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*
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* Use unblocked code to reduce the rest of the matrix
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*
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CALL ZGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO )
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WORK( 1 ) = LWKOPT
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*
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RETURN
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*
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* End of ZGEHRD
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*
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END
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