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dlaqr5.f
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*> \brief \b DLAQR5 performs a single small-bulge multi-shift QR sweep.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQR5 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqr5.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqr5.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqr5.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS,
* SR, SI, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U,
* LDU, NV, WV, LDWV, NH, WH, LDWH )
*
* .. Scalar Arguments ..
* INTEGER IHIZ, ILOZ, KACC22, KBOT, KTOP, LDH, LDU, LDV,
* $ LDWH, LDWV, LDZ, N, NH, NSHFTS, NV
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), U( LDU, * ),
* $ V( LDV, * ), WH( LDWH, * ), WV( LDWV, * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQR5, called by DLAQR0, performs a
*> single small-bulge multi-shift QR sweep.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> WANTT = .true. if the quasi-triangular Schur factor
*> is being computed. WANTT is set to .false. otherwise.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> WANTZ = .true. if the orthogonal Schur factor is being
*> computed. WANTZ is set to .false. otherwise.
*> \endverbatim
*>
*> \param[in] KACC22
*> \verbatim
*> KACC22 is INTEGER with value 0, 1, or 2.
*> Specifies the computation mode of far-from-diagonal
*> orthogonal updates.
*> = 0: DLAQR5 does not accumulate reflections and does not
*> use matrix-matrix multiply to update far-from-diagonal
*> matrix entries.
*> = 1: DLAQR5 accumulates reflections and uses matrix-matrix
*> multiply to update the far-from-diagonal matrix entries.
*> = 2: Same as KACC22 = 1. This option used to enable exploiting
*> the 2-by-2 structure during matrix multiplications, but
*> this is no longer supported.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> N is the order of the Hessenberg matrix H upon which this
*> subroutine operates.
*> \endverbatim
*>
*> \param[in] KTOP
*> \verbatim
*> KTOP is INTEGER
*> \endverbatim
*>
*> \param[in] KBOT
*> \verbatim
*> KBOT is INTEGER
*> These are the first and last rows and columns of an
*> isolated diagonal block upon which the QR sweep is to be
*> applied. It is assumed without a check that
*> either KTOP = 1 or H(KTOP,KTOP-1) = 0
*> and
*> either KBOT = N or H(KBOT+1,KBOT) = 0.
*> \endverbatim
*>
*> \param[in] NSHFTS
*> \verbatim
*> NSHFTS is INTEGER
*> NSHFTS gives the number of simultaneous shifts. NSHFTS
*> must be positive and even.
*> \endverbatim
*>
*> \param[in,out] SR
*> \verbatim
*> SR is DOUBLE PRECISION array, dimension (NSHFTS)
*> \endverbatim
*>
*> \param[in,out] SI
*> \verbatim
*> SI is DOUBLE PRECISION array, dimension (NSHFTS)
*> SR contains the real parts and SI contains the imaginary
*> parts of the NSHFTS shifts of origin that define the
*> multi-shift QR sweep. On output SR and SI may be
*> reordered.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is DOUBLE PRECISION array, dimension (LDH,N)
*> On input H contains a Hessenberg matrix. On output a
*> multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
*> to the isolated diagonal block in rows and columns KTOP
*> through KBOT.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> LDH is the leading dimension of H just as declared in the
*> calling procedure. LDH >= MAX(1,N).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (LDZ,IHIZ)
*> If WANTZ = .TRUE., then the QR Sweep orthogonal
*> similarity transformation is accumulated into
*> Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
*> If WANTZ = .FALSE., then Z is unreferenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> LDA is the leading dimension of Z just as declared in
*> the calling procedure. LDZ >= N.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension (LDV,NSHFTS/2)
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> LDV is the leading dimension of V as declared in the
*> calling procedure. LDV >= 3.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU,2*NSHFTS)
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> LDU is the leading dimension of U just as declared in the
*> in the calling subroutine. LDU >= 2*NSHFTS.
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*> NV is INTEGER
*> NV is the number of rows in WV agailable for workspace.
*> NV >= 1.
*> \endverbatim
*>
*> \param[out] WV
*> \verbatim
*> WV is DOUBLE PRECISION array, dimension (LDWV,2*NSHFTS)
*> \endverbatim
*>
*> \param[in] LDWV
*> \verbatim
*> LDWV is INTEGER
*> LDWV is the leading dimension of WV as declared in the
*> in the calling subroutine. LDWV >= NV.
*> \endverbatim
*
*> \param[in] NH
*> \verbatim
*> NH is INTEGER
*> NH is the number of columns in array WH available for
*> workspace. NH >= 1.
*> \endverbatim
*>
*> \param[out] WH
*> \verbatim
*> WH is DOUBLE PRECISION array, dimension (LDWH,NH)
*> \endverbatim
*>
*> \param[in] LDWH
*> \verbatim
*> LDWH is INTEGER
*> Leading dimension of WH just as declared in the
*> calling procedure. LDWH >= 2*NSHFTS.
*> \endverbatim
*>
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*>
*> Lars Karlsson, Daniel Kressner, and Bruno Lang
*>
*> Thijs Steel, Department of Computer science,
*> KU Leuven, Belgium
*
*> \par References:
* ================
*>
*> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*> Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*> 929--947, 2002.
*>
*> Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed
*> chains of bulges in multishift QR algorithms.
*> ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).
*>
* =====================================================================
SUBROUTINE DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS,
$ SR, SI, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U,
$ LDU, NV, WV, LDWV, NH, WH, LDWH )
IMPLICIT NONE
*
* -- LAPACK auxiliary 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 IHIZ, ILOZ, KACC22, KBOT, KTOP, LDH, LDU, LDV,
$ LDWH, LDWV, LDZ, N, NH, NSHFTS, NV
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
DOUBLE PRECISION H( LDH, * ), SI( * ), SR( * ), U( LDU, * ),
$ V( LDV, * ), WH( LDWH, * ), WV( LDWV, * ),
$ Z( LDZ, * )
* ..
*
* ================================================================
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION ALPHA, BETA, H11, H12, H21, H22, REFSUM,
$ SAFMAX, SAFMIN, SCL, SMLNUM, SWAP, T1, T2,
$ T3, TST1, TST2, ULP
INTEGER I, I2, I4, INCOL, J, JBOT, JCOL, JLEN,
$ JROW, JTOP, K, K1, KDU, KMS, KRCOL,
$ M, M22, MBOT, MTOP, NBMPS, NDCOL,
$ NS, NU
LOGICAL ACCUM, BMP22
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
*
INTRINSIC ABS, DBLE, MAX, MIN, MOD
* ..
* .. Local Arrays ..
DOUBLE PRECISION VT( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLAQR1, DLARFG, DLASET, DTRMM
* ..
* .. Executable Statements ..
*
* ==== If there are no shifts, then there is nothing to do. ====
*
IF( NSHFTS.LT.2 )
$ RETURN
*
* ==== If the active block is empty or 1-by-1, then there
* . is nothing to do. ====
*
IF( KTOP.GE.KBOT )
$ RETURN
*
* ==== Shuffle shifts into pairs of real shifts and pairs
* . of complex conjugate shifts assuming complex
* . conjugate shifts are already adjacent to one
* . another. ====
*
DO 10 I = 1, NSHFTS - 2, 2
IF( SI( I ).NE.-SI( I+1 ) ) THEN
*
SWAP = SR( I )
SR( I ) = SR( I+1 )
SR( I+1 ) = SR( I+2 )
SR( I+2 ) = SWAP
*
SWAP = SI( I )
SI( I ) = SI( I+1 )
SI( I+1 ) = SI( I+2 )
SI( I+2 ) = SWAP
END IF
10 CONTINUE
*
* ==== NSHFTS is supposed to be even, but if it is odd,
* . then simply reduce it by one. The shuffle above
* . ensures that the dropped shift is real and that
* . the remaining shifts are paired. ====
*
NS = NSHFTS - MOD( NSHFTS, 2 )
*
* ==== Machine constants for deflation ====
*
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
SAFMAX = ONE / SAFMIN
ULP = DLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( DBLE( N ) / ULP )
*
* ==== Use accumulated reflections to update far-from-diagonal
* . entries ? ====
*
ACCUM = ( KACC22.EQ.1 ) .OR. ( KACC22.EQ.2 )
*
* ==== clear trash ====
*
IF( KTOP+2.LE.KBOT )
$ H( KTOP+2, KTOP ) = ZERO
*
* ==== NBMPS = number of 2-shift bulges in the chain ====
*
NBMPS = NS / 2
*
* ==== KDU = width of slab ====
*
KDU = 4*NBMPS
*
* ==== Create and chase chains of NBMPS bulges ====
*
DO 180 INCOL = KTOP - 2*NBMPS + 1, KBOT - 2, 2*NBMPS
*
* JTOP = Index from which updates from the right start.
*
IF( ACCUM ) THEN
JTOP = MAX( KTOP, INCOL )
ELSE IF( WANTT ) THEN
JTOP = 1
ELSE
JTOP = KTOP
END IF
*
NDCOL = INCOL + KDU
IF( ACCUM )
$ CALL DLASET( 'ALL', KDU, KDU, ZERO, ONE, U, LDU )
*
* ==== Near-the-diagonal bulge chase. The following loop
* . performs the near-the-diagonal part of a small bulge
* . multi-shift QR sweep. Each 4*NBMPS column diagonal
* . chunk extends from column INCOL to column NDCOL
* . (including both column INCOL and column NDCOL). The
* . following loop chases a 2*NBMPS+1 column long chain of
* . NBMPS bulges 2*NBMPS columns to the right. (INCOL
* . may be less than KTOP and and NDCOL may be greater than
* . KBOT indicating phantom columns from which to chase
* . bulges before they are actually introduced or to which
* . to chase bulges beyond column KBOT.) ====
*
DO 145 KRCOL = INCOL, MIN( INCOL+2*NBMPS-1, KBOT-2 )
*
* ==== Bulges number MTOP to MBOT are active double implicit
* . shift bulges. There may or may not also be small
* . 2-by-2 bulge, if there is room. The inactive bulges
* . (if any) must wait until the active bulges have moved
* . down the diagonal to make room. The phantom matrix
* . paradigm described above helps keep track. ====
*
MTOP = MAX( 1, ( KTOP-KRCOL ) / 2+1 )
MBOT = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 2 )
M22 = MBOT + 1
BMP22 = ( MBOT.LT.NBMPS ) .AND. ( KRCOL+2*( M22-1 ) ).EQ.
$ ( KBOT-2 )
*
* ==== Generate reflections to chase the chain right
* . one column. (The minimum value of K is KTOP-1.) ====
*
IF ( BMP22 ) THEN
*
* ==== Special case: 2-by-2 reflection at bottom treated
* . separately ====
*
K = KRCOL + 2*( M22-1 )
IF( K.EQ.KTOP-1 ) THEN
CALL DLAQR1( 2, H( K+1, K+1 ), LDH, SR( 2*M22-1 ),
$ SI( 2*M22-1 ), SR( 2*M22 ), SI( 2*M22 ),
$ V( 1, M22 ) )
BETA = V( 1, M22 )
CALL DLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
ELSE
BETA = H( K+1, K )
V( 2, M22 ) = H( K+2, K )
CALL DLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
H( K+1, K ) = BETA
H( K+2, K ) = ZERO
END IF
*
* ==== Perform update from right within
* . computational window. ====
*
T1 = V( 1, M22 )
T2 = T1*V( 2, M22 )
DO 30 J = JTOP, MIN( KBOT, K+3 )
REFSUM = H( J, K+1 ) + V( 2, M22 )*H( J, K+2 )
H( J, K+1 ) = H( J, K+1 ) - REFSUM*T1
H( J, K+2 ) = H( J, K+2 ) - REFSUM*T2
30 CONTINUE
*
* ==== Perform update from left within
* . computational window. ====
*
IF( ACCUM ) THEN
JBOT = MIN( NDCOL, KBOT )
ELSE IF( WANTT ) THEN
JBOT = N
ELSE
JBOT = KBOT
END IF
T1 = V( 1, M22 )
T2 = T1*V( 2, M22 )
DO 40 J = K+1, JBOT
REFSUM = H( K+1, J ) + V( 2, M22 )*H( K+2, J )
H( K+1, J ) = H( K+1, J ) - REFSUM*T1
H( K+2, J ) = H( K+2, J ) - REFSUM*T2
40 CONTINUE
*
* ==== The following convergence test requires that
* . the tradition small-compared-to-nearby-diagonals
* . criterion and the Ahues & Tisseur (LAWN 122, 1997)
* . criteria both be satisfied. The latter improves
* . accuracy in some examples. Falling back on an
* . alternate convergence criterion when TST1 or TST2
* . is zero (as done here) is traditional but probably
* . unnecessary. ====
*
IF( K.GE.KTOP ) THEN
IF( H( K+1, K ).NE.ZERO ) THEN
TST1 = ABS( H( K, K ) ) + ABS( H( K+1, K+1 ) )
IF( TST1.EQ.ZERO ) THEN
IF( K.GE.KTOP+1 )
$ TST1 = TST1 + ABS( H( K, K-1 ) )
IF( K.GE.KTOP+2 )
$ TST1 = TST1 + ABS( H( K, K-2 ) )
IF( K.GE.KTOP+3 )
$ TST1 = TST1 + ABS( H( K, K-3 ) )
IF( K.LE.KBOT-2 )
$ TST1 = TST1 + ABS( H( K+2, K+1 ) )
IF( K.LE.KBOT-3 )
$ TST1 = TST1 + ABS( H( K+3, K+1 ) )
IF( K.LE.KBOT-4 )
$ TST1 = TST1 + ABS( H( K+4, K+1 ) )
END IF
IF( ABS( H( K+1, K ) )
$ .LE.MAX( SMLNUM, ULP*TST1 ) ) THEN
H12 = MAX( ABS( H( K+1, K ) ),
$ ABS( H( K, K+1 ) ) )
H21 = MIN( ABS( H( K+1, K ) ),
$ ABS( H( K, K+1 ) ) )
H11 = MAX( ABS( H( K+1, K+1 ) ),
$ ABS( H( K, K )-H( K+1, K+1 ) ) )
H22 = MIN( ABS( H( K+1, K+1 ) ),
$ ABS( H( K, K )-H( K+1, K+1 ) ) )
SCL = H11 + H12
TST2 = H22*( H11 / SCL )
*
IF( TST2.EQ.ZERO .OR. H21*( H12 / SCL ).LE.
$ MAX( SMLNUM, ULP*TST2 ) ) THEN
H( K+1, K ) = ZERO
END IF
END IF
END IF
END IF
*
* ==== Accumulate orthogonal transformations. ====
*
IF( ACCUM ) THEN
KMS = K - INCOL
T1 = V( 1, M22 )
T2 = T1*V( 2, M22 )
DO 50 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = U( J, KMS+1 ) + V( 2, M22 )*U( J, KMS+2 )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM*T1
U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*T2
50 CONTINUE
ELSE IF( WANTZ ) THEN
T1 = V( 1, M22 )
T2 = T1*V( 2, M22 )
DO 60 J = ILOZ, IHIZ
REFSUM = Z( J, K+1 )+V( 2, M22 )*Z( J, K+2 )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM*T1
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*T2
60 CONTINUE
END IF
END IF
*
* ==== Normal case: Chain of 3-by-3 reflections ====
*
DO 80 M = MBOT, MTOP, -1
K = KRCOL + 2*( M-1 )
IF( K.EQ.KTOP-1 ) THEN
CALL DLAQR1( 3, H( KTOP, KTOP ), LDH, SR( 2*M-1 ),
$ SI( 2*M-1 ), SR( 2*M ), SI( 2*M ),
$ V( 1, M ) )
ALPHA = V( 1, M )
CALL DLARFG( 3, ALPHA, V( 2, M ), 1, V( 1, M ) )
ELSE
*
* ==== Perform delayed transformation of row below
* . Mth bulge. Exploit fact that first two elements
* . of row are actually zero. ====
*
T1 = V( 1, M )
T2 = T1*V( 2, M )
T3 = T1*V( 3, M )
REFSUM = V( 3, M )*H( K+3, K+2 )
H( K+3, K ) = -REFSUM*T1
H( K+3, K+1 ) = -REFSUM*T2
H( K+3, K+2 ) = H( K+3, K+2 ) - REFSUM*T3
*
* ==== Calculate reflection to move
* . Mth bulge one step. ====
*
BETA = H( K+1, K )
V( 2, M ) = H( K+2, K )
V( 3, M ) = H( K+3, K )
CALL DLARFG( 3, BETA, V( 2, M ), 1, V( 1, M ) )
*
* ==== A Bulge may collapse because of vigilant
* . deflation or destructive underflow. In the
* . underflow case, try the two-small-subdiagonals
* . trick to try to reinflate the bulge. ====
*
IF( H( K+3, K ).NE.ZERO .OR. H( K+3, K+1 ).NE.
$ ZERO .OR. H( K+3, K+2 ).EQ.ZERO ) THEN
*
* ==== Typical case: not collapsed (yet). ====
*
H( K+1, K ) = BETA
H( K+2, K ) = ZERO
H( K+3, K ) = ZERO
ELSE
*
* ==== Atypical case: collapsed. Attempt to
* . reintroduce ignoring H(K+1,K) and H(K+2,K).
* . If the fill resulting from the new
* . reflector is too large, then abandon it.
* . Otherwise, use the new one. ====
*
CALL DLAQR1( 3, H( K+1, K+1 ), LDH, SR( 2*M-1 ),
$ SI( 2*M-1 ), SR( 2*M ), SI( 2*M ),
$ VT )
ALPHA = VT( 1 )
CALL DLARFG( 3, ALPHA, VT( 2 ), 1, VT( 1 ) )
T1 = VT( 1 )
T2 = T1*VT( 2 )
T3 = T1*VT( 3 )
REFSUM = H( K+1, K ) + VT( 2 )*H( K+2, K )
*
IF( ABS( H( K+2, K )-REFSUM*T2 )+
$ ABS( REFSUM*T3 ).GT.ULP*
$ ( ABS( H( K, K ) )+ABS( H( K+1,
$ K+1 ) )+ABS( H( K+2, K+2 ) ) ) ) THEN
*
* ==== Starting a new bulge here would
* . create non-negligible fill. Use
* . the old one with trepidation. ====
*
H( K+1, K ) = BETA
H( K+2, K ) = ZERO
H( K+3, K ) = ZERO
ELSE
*
* ==== Starting a new bulge here would
* . create only negligible fill.
* . Replace the old reflector with
* . the new one. ====
*
H( K+1, K ) = H( K+1, K ) - REFSUM*T1
H( K+2, K ) = ZERO
H( K+3, K ) = ZERO
V( 1, M ) = VT( 1 )
V( 2, M ) = VT( 2 )
V( 3, M ) = VT( 3 )
END IF
END IF
END IF
*
* ==== Apply reflection from the right and
* . the first column of update from the left.
* . These updates are required for the vigilant
* . deflation check. We still delay most of the
* . updates from the left for efficiency. ====
*
T1 = V( 1, M )
T2 = T1*V( 2, M )
T3 = T1*V( 3, M )
DO 70 J = JTOP, MIN( KBOT, K+3 )
REFSUM = H( J, K+1 ) + V( 2, M )*H( J, K+2 )
$ + V( 3, M )*H( J, K+3 )
H( J, K+1 ) = H( J, K+1 ) - REFSUM*T1
H( J, K+2 ) = H( J, K+2 ) - REFSUM*T2
H( J, K+3 ) = H( J, K+3 ) - REFSUM*T3
70 CONTINUE
*
* ==== Perform update from left for subsequent
* . column. ====
*
REFSUM = H( K+1, K+1 ) + V( 2, M )*H( K+2, K+1 )
$ + V( 3, M )*H( K+3, K+1 )
H( K+1, K+1 ) = H( K+1, K+1 ) - REFSUM*T1
H( K+2, K+1 ) = H( K+2, K+1 ) - REFSUM*T2
H( K+3, K+1 ) = H( K+3, K+1 ) - REFSUM*T3
*
* ==== The following convergence test requires that
* . the tradition small-compared-to-nearby-diagonals
* . criterion and the Ahues & Tisseur (LAWN 122, 1997)
* . criteria both be satisfied. The latter improves
* . accuracy in some examples. Falling back on an
* . alternate convergence criterion when TST1 or TST2
* . is zero (as done here) is traditional but probably
* . unnecessary. ====
*
IF( K.LT.KTOP)
$ CYCLE
IF( H( K+1, K ).NE.ZERO ) THEN
TST1 = ABS( H( K, K ) ) + ABS( H( K+1, K+1 ) )
IF( TST1.EQ.ZERO ) THEN
IF( K.GE.KTOP+1 )
$ TST1 = TST1 + ABS( H( K, K-1 ) )
IF( K.GE.KTOP+2 )
$ TST1 = TST1 + ABS( H( K, K-2 ) )
IF( K.GE.KTOP+3 )
$ TST1 = TST1 + ABS( H( K, K-3 ) )
IF( K.LE.KBOT-2 )
$ TST1 = TST1 + ABS( H( K+2, K+1 ) )
IF( K.LE.KBOT-3 )
$ TST1 = TST1 + ABS( H( K+3, K+1 ) )
IF( K.LE.KBOT-4 )
$ TST1 = TST1 + ABS( H( K+4, K+1 ) )
END IF
IF( ABS( H( K+1, K ) ).LE.MAX( SMLNUM, ULP*TST1 ) )
$ THEN
H12 = MAX( ABS( H( K+1, K ) ), ABS( H( K, K+1 ) ) )
H21 = MIN( ABS( H( K+1, K ) ), ABS( H( K, K+1 ) ) )
H11 = MAX( ABS( H( K+1, K+1 ) ),
$ ABS( H( K, K )-H( K+1, K+1 ) ) )
H22 = MIN( ABS( H( K+1, K+1 ) ),
$ ABS( H( K, K )-H( K+1, K+1 ) ) )
SCL = H11 + H12
TST2 = H22*( H11 / SCL )
*
IF( TST2.EQ.ZERO .OR. H21*( H12 / SCL ).LE.
$ MAX( SMLNUM, ULP*TST2 ) ) THEN
H( K+1, K ) = ZERO
END IF
END IF
END IF
80 CONTINUE
*
* ==== Multiply H by reflections from the left ====
*
IF( ACCUM ) THEN
JBOT = MIN( NDCOL, KBOT )
ELSE IF( WANTT ) THEN
JBOT = N
ELSE
JBOT = KBOT
END IF
*
DO 100 M = MBOT, MTOP, -1
K = KRCOL + 2*( M-1 )
T1 = V( 1, M )
T2 = T1*V( 2, M )
T3 = T1*V( 3, M )
DO 90 J = MAX( KTOP, KRCOL + 2*M ), JBOT
REFSUM = H( K+1, J ) + V( 2, M )*H( K+2, J )
$ + V( 3, M )*H( K+3, J )
H( K+1, J ) = H( K+1, J ) - REFSUM*T1
H( K+2, J ) = H( K+2, J ) - REFSUM*T2
H( K+3, J ) = H( K+3, J ) - REFSUM*T3
90 CONTINUE
100 CONTINUE
*
* ==== Accumulate orthogonal transformations. ====
*
IF( ACCUM ) THEN
*
* ==== Accumulate U. (If needed, update Z later
* . with an efficient matrix-matrix
* . multiply.) ====
*
DO 120 M = MBOT, MTOP, -1
K = KRCOL + 2*( M-1 )
KMS = K - INCOL
I2 = MAX( 1, KTOP-INCOL )
I2 = MAX( I2, KMS-(KRCOL-INCOL)+1 )
I4 = MIN( KDU, KRCOL + 2*( MBOT-1 ) - INCOL + 5 )
T1 = V( 1, M )
T2 = T1*V( 2, M )
T3 = T1*V( 3, M )
DO 110 J = I2, I4
REFSUM = U( J, KMS+1 ) + V( 2, M )*U( J, KMS+2 )
$ + V( 3, M )*U( J, KMS+3 )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM*T1
U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*T2
U( J, KMS+3 ) = U( J, KMS+3 ) - REFSUM*T3
110 CONTINUE
120 CONTINUE
ELSE IF( WANTZ ) THEN
*
* ==== U is not accumulated, so update Z
* . now by multiplying by reflections
* . from the right. ====
*
DO 140 M = MBOT, MTOP, -1
K = KRCOL + 2*( M-1 )
T1 = V( 1, M )
T2 = T1*V( 2, M )
T3 = T1*V( 3, M )
DO 130 J = ILOZ, IHIZ
REFSUM = Z( J, K+1 ) + V( 2, M )*Z( J, K+2 )
$ + V( 3, M )*Z( J, K+3 )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM*T1
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*T2
Z( J, K+3 ) = Z( J, K+3 ) - REFSUM*T3
130 CONTINUE
140 CONTINUE
END IF
*
* ==== End of near-the-diagonal bulge chase. ====
*
145 CONTINUE
*
* ==== Use U (if accumulated) to update far-from-diagonal
* . entries in H. If required, use U to update Z as
* . well. ====
*
IF( ACCUM ) THEN
IF( WANTT ) THEN
JTOP = 1
JBOT = N
ELSE
JTOP = KTOP
JBOT = KBOT
END IF
K1 = MAX( 1, KTOP-INCOL )
NU = ( KDU-MAX( 0, NDCOL-KBOT ) ) - K1 + 1
*
* ==== Horizontal Multiply ====
*
DO 150 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
JLEN = MIN( NH, JBOT-JCOL+1 )
CALL DGEMM( 'C', 'N', NU, JLEN, NU, ONE, U( K1, K1 ),
$ LDU, H( INCOL+K1, JCOL ), LDH, ZERO, WH,
$ LDWH )
CALL DLACPY( 'ALL', NU, JLEN, WH, LDWH,
$ H( INCOL+K1, JCOL ), LDH )
150 CONTINUE
*
* ==== Vertical multiply ====
*
DO 160 JROW = JTOP, MAX( KTOP, INCOL ) - 1, NV
JLEN = MIN( NV, MAX( KTOP, INCOL )-JROW )
CALL DGEMM( 'N', 'N', JLEN, NU, NU, ONE,
$ H( JROW, INCOL+K1 ), LDH, U( K1, K1 ),
$ LDU, ZERO, WV, LDWV )
CALL DLACPY( 'ALL', JLEN, NU, WV, LDWV,
$ H( JROW, INCOL+K1 ), LDH )
160 CONTINUE
*
* ==== Z multiply (also vertical) ====
*
IF( WANTZ ) THEN
DO 170 JROW = ILOZ, IHIZ, NV
JLEN = MIN( NV, IHIZ-JROW+1 )
CALL DGEMM( 'N', 'N', JLEN, NU, NU, ONE,
$ Z( JROW, INCOL+K1 ), LDZ, U( K1, K1 ),
$ LDU, ZERO, WV, LDWV )
CALL DLACPY( 'ALL', JLEN, NU, WV, LDWV,
$ Z( JROW, INCOL+K1 ), LDZ )
170 CONTINUE
END IF
END IF
180 CONTINUE
*
* ==== End of DLAQR5 ====
*
END