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Merge pull request #3206 from martin-frbg/lapack480535 

Import packing improvements to LAPACK xLAQR from Reference-LAPACK (PR 480+535)
This commit is contained in:
Martin Kroeker 2021-04-30 21:42:44 +02:00 committed by GitHub
parent c9a82f54d1 87d2e314db
commit dc6b04c375
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16 changed files with 1238 additions and 1641 deletions
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4
lapack-netlib/SRC/chseqr.f
View File

@ -320,10 +320,10 @@
* . CLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
PARAMETER ( NTINY = 15 )
*
* ==== NL allocates some local workspace to help small matrices
* . through a rare CLAHQR failure. NL > NTINY = 11 is
* . through a rare CLAHQR failure. NL > NTINY = 15 is
* . required and NL <= NMIN = ILAENV(ISPEC=12,...) is recom-
* . mended. (The default value of NMIN is 75.) Using NL = 49
* . allows up to six simultaneous shifts and a 16-by-16

16
lapack-netlib/SRC/claqr0.f
View File

@ -260,7 +260,7 @@
* . CLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
PARAMETER ( NTINY = 15 )
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by varying the size of the
@ -355,22 +355,22 @@
END IF
*
* ==== NWR = recommended deflation window size. At this
* . point, N .GT. NTINY = 11, so there is enough
* . point, N .GT. NTINY = 15, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
* . NWR.GE.3.) ====
* . NWR.GE.4.) ====
*
NWR = ILAENV( 13, 'CLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
* ==== NSR = recommended number of simultaneous shifts.
* . At this point N .GT. NTINY = 11, so there is at
* . At this point N .GT. NTINY = 15, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'CLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
@ -418,7 +418,7 @@
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
@ -558,7 +558,7 @@
*
* ==== Got NS/2 or fewer shifts? Use CLAQR4 or
* . CLAHQR on a trailing principal submatrix to
* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
@ -659,7 +659,7 @@
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
KDU = 3*NS - 3
KDU = 2*NS
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1

16
lapack-netlib/SRC/claqr4.f
View File

@ -270,7 +270,7 @@
* . CLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
PARAMETER ( NTINY = 15 )
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by varying the size of the
@ -365,22 +365,22 @@
END IF
*
* ==== NWR = recommended deflation window size. At this
* . point, N .GT. NTINY = 11, so there is enough
* . point, N .GT. NTINY = 15, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
* . NWR.GE.3.) ====
* . NWR.GE.4.) ====
*
NWR = ILAENV( 13, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
* ==== NSR = recommended number of simultaneous shifts.
* . At this point N .GT. NTINY = 11, so there is at
* . At this point N .GT. NTINY = 15, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
@ -428,7 +428,7 @@
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
@ -568,7 +568,7 @@
*
* ==== Got NS/2 or fewer shifts? Use CLAHQR
* . on a trailing principal submatrix to
* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
@ -663,7 +663,7 @@
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
KDU = 3*NS - 3
KDU = 2*NS
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1

617
lapack-netlib/SRC/claqr5.f
View File

@ -69,10 +69,9 @@
*> matrix entries.
*> = 1: CLAQR5 accumulates reflections and uses matrix-matrix
*> multiply to update the far-from-diagonal matrix entries.
*> = 2: CLAQR5 accumulates reflections, uses matrix-matrix
*> multiply to update the far-from-diagonal matrix entries,
*> and takes advantage of 2-by-2 block structure during
*> matrix multiplies.
*> = 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
@ -170,14 +169,14 @@
*>
*> \param[out] U
*> \verbatim
*> U is COMPLEX array, dimension (LDU,3*NSHFTS-3)
*> U is COMPLEX 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 >= 3*NSHFTS-3.
*> in the calling subroutine. LDU >= 2*NSHFTS.
*> \endverbatim
*>
*> \param[in] NV
@ -189,7 +188,7 @@
*>
*> \param[out] WV
*> \verbatim
*> WV is COMPLEX array, dimension (LDWV,3*NSHFTS-3)
*> WV is COMPLEX array, dimension (LDWV,2*NSHFTS)
*> \endverbatim
*>
*> \param[in] LDWV
@ -215,7 +214,7 @@
*> \verbatim
*> LDWH is INTEGER
*> Leading dimension of WH just as declared in the
*> calling procedure. LDWH >= 3*NSHFTS-3.
*> calling procedure. LDWH >= 2*NSHFTS.
*> \endverbatim
*>
* Authors:
@ -226,7 +225,7 @@
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*> \date January 2021
*
*> \ingroup complexOTHERauxiliary
*
@ -235,6 +234,11 @@
*>
*> 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:
* ================
@ -244,10 +248,15 @@
*> 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 CLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, S,
$ H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U, LDU, NV,
$ WV, LDWV, NH, WH, LDWH )
IMPLICIT NONE
*
* -- LAPACK auxiliary routine (version 3.7.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
@ -276,11 +285,11 @@
COMPLEX ALPHA, BETA, CDUM, REFSUM
REAL H11, H12, H21, H22, SAFMAX, SAFMIN, SCL,
$ SMLNUM, TST1, TST2, ULP
INTEGER I2, I4, INCOL, J, J2, J4, JBOT, JCOL, JLEN,
$ JROW, JTOP, K, K1, KDU, KMS, KNZ, KRCOL, KZS,
$ M, M22, MBOT, MEND, MSTART, MTOP, NBMPS, NDCOL,
INTEGER I2, I4, INCOL, J, JBOT, JCOL, JLEN,
$ JROW, JTOP, K, K1, KDU, KMS, KRCOL,
$ M, M22, MBOT, MTOP, NBMPS, NDCOL,
$ NS, NU
LOGICAL ACCUM, BLK22, BMP22
LOGICAL ACCUM, BMP22
* ..
* .. External Functions ..
REAL SLAMCH
@ -334,10 +343,6 @@
*
ACCUM = ( KACC22.EQ.1 ) .OR. ( KACC22.EQ.2 )
*
* ==== If so, exploit the 2-by-2 block structure? ====
*
BLK22 = ( NS.GT.2 ) .AND. ( KACC22.EQ.2 )
*
* ==== clear trash ====
*
IF( KTOP+2.LE.KBOT )
@ -349,28 +354,39 @@
*
* ==== KDU = width of slab ====
*
KDU = 6*NBMPS - 3
KDU = 4*NBMPS
*
* ==== Create and chase chains of NBMPS bulges ====
*
DO 210 INCOL = 3*( 1-NBMPS ) + KTOP - 1, KBOT - 2, 3*NBMPS - 2
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 CLASET( '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 6*NBMPS-2 column diagonal
* . 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 3*NBMPS column long chain of
* . NBMPS bulges 3*NBMPS-2 columns to the right. (INCOL
* . 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 140 KRCOL = INCOL, MIN( INCOL+3*NBMPS-3, KBOT-2 )
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
@ -379,23 +395,155 @@
* . down the diagonal to make room. The phantom matrix
* . paradigm described above helps keep track. ====
*
MTOP = MAX( 1, ( ( KTOP-1 )-KRCOL+2 ) / 3+1 )
MBOT = MIN( NBMPS, ( KBOT-KRCOL ) / 3 )
MTOP = MAX( 1, ( KTOP-KRCOL ) / 2+1 )
MBOT = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 2 )
M22 = MBOT + 1
BMP22 = ( MBOT.LT.NBMPS ) .AND. ( KRCOL+3*( M22-1 ) ).EQ.
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.) ====
*
DO 10 M = MTOP, MBOT
K = KRCOL + 3*( M-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 CLAQR1( 2, H( K+1, K+1 ), LDH, S( 2*M22-1 ),
$ S( 2*M22 ), V( 1, M22 ) )
BETA = V( 1, M22 )
CALL CLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
ELSE
BETA = H( K+1, K )
V( 2, M22 ) = H( K+2, K )
CALL CLARFG( 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. ====
*
DO 30 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M22 )*( H( J, K+1 )+V( 2, M22 )*
$ H( J, K+2 ) )
H( J, K+1 ) = H( J, K+1 ) - REFSUM
H( J, K+2 ) = H( J, K+2 ) -
$ REFSUM*CONJG( V( 2, M22 ) )
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
DO 40 J = K+1, JBOT
REFSUM = CONJG( V( 1, M22 ) )*
$ ( H( K+1, J )+CONJG( V( 2, M22 ) )*
$ H( K+2, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M22 )
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 = CABS1( H( K, K ) ) + CABS1( H( K+1, K+1 ) )
IF( TST1.EQ.RZERO ) THEN
IF( K.GE.KTOP+1 )
$ TST1 = TST1 + CABS1( H( K, K-1 ) )
IF( K.GE.KTOP+2 )
$ TST1 = TST1 + CABS1( H( K, K-2 ) )
IF( K.GE.KTOP+3 )
$ TST1 = TST1 + CABS1( H( K, K-3 ) )
IF( K.LE.KBOT-2 )
$ TST1 = TST1 + CABS1( H( K+2, K+1 ) )
IF( K.LE.KBOT-3 )
$ TST1 = TST1 + CABS1( H( K+3, K+1 ) )
IF( K.LE.KBOT-4 )
$ TST1 = TST1 + CABS1( H( K+4, K+1 ) )
END IF
IF( CABS1( H( K+1, K ) )
$ .LE.MAX( SMLNUM, ULP*TST1 ) ) THEN
H12 = MAX( CABS1( H( K+1, K ) ),
$ CABS1( H( K, K+1 ) ) )
H21 = MIN( CABS1( H( K+1, K ) ),
$ CABS1( H( K, K+1 ) ) )
H11 = MAX( CABS1( H( K+1, K+1 ) ),
$ CABS1( H( K, K )-H( K+1, K+1 ) ) )
H22 = MIN( CABS1( H( K+1, K+1 ) ),
$ CABS1( H( K, K )-H( K+1, K+1 ) ) )
SCL = H11 + H12
TST2 = H22*( H11 / SCL )
*
IF( TST2.EQ.RZERO .OR. H21*( H12 / SCL ).LE.
$ MAX( SMLNUM, ULP*TST2 ) )H( K+1, K ) = ZERO
END IF
END IF
END IF
*
* ==== Accumulate orthogonal transformations. ====
*
IF( ACCUM ) THEN
KMS = K - INCOL
DO 50 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M22 )*( U( J, KMS+1 )+
$ V( 2, M22 )*U( J, KMS+2 ) )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM
U( J, KMS+2 ) = U( J, KMS+2 ) -
$ REFSUM*CONJG( V( 2, M22 ) )
50 CONTINUE
ELSE IF( WANTZ ) THEN
DO 60 J = ILOZ, IHIZ
REFSUM = V( 1, M22 )*( Z( J, K+1 )+V( 2, M22 )*
$ Z( J, K+2 ) )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM
Z( J, K+2 ) = Z( J, K+2 ) -
$ REFSUM*CONJG( V( 2, M22 ) )
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 CLAQR1( 3, H( KTOP, KTOP ), LDH, S( 2*M-1 ),
$ S( 2*M ), V( 1, M ) )
ALPHA = V( 1, M )
CALL CLARFG( 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. ====
*
REFSUM = V( 1, M )*V( 3, M )*H( K+3, K+2 )
H( K+3, K ) = -REFSUM
H( K+3, K+1 ) = -REFSUM*CONJG( V( 2, M ) )
H( K+3, K+2 ) = H( K+3, K+2 ) -
$ REFSUM*CONJG( V( 3, M ) )
*
* ==== 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 )
@ -444,7 +592,7 @@
H( K+3, K ) = ZERO
ELSE
*
* ==== Stating a new bulge here would
* ==== Starting a new bulge here would
* . create only negligible fill.
* . Replace the old reflector with
* . the new one. ====
@ -458,73 +606,14 @@
END IF
END IF
END IF
10 CONTINUE
*
* ==== Generate a 2-by-2 reflection, if needed. ====
* ==== 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. ====
*
K = KRCOL + 3*( M22-1 )
IF( BMP22 ) THEN
IF( K.EQ.KTOP-1 ) THEN
CALL CLAQR1( 2, H( K+1, K+1 ), LDH, S( 2*M22-1 ),
$ S( 2*M22 ), V( 1, M22 ) )
BETA = V( 1, M22 )
CALL CLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
ELSE
BETA = H( K+1, K )
V( 2, M22 ) = H( K+2, K )
CALL CLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
H( K+1, K ) = BETA
H( K+2, K ) = ZERO
END IF
END IF
*
* ==== 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 30 J = MAX( KTOP, KRCOL ), JBOT
MEND = MIN( MBOT, ( J-KRCOL+2 ) / 3 )
DO 20 M = MTOP, MEND
K = KRCOL + 3*( M-1 )
REFSUM = CONJG( V( 1, M ) )*
$ ( H( K+1, J )+CONJG( V( 2, M ) )*H( K+2, J )+
$ CONJG( V( 3, M ) )*H( K+3, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M )
H( K+3, J ) = H( K+3, J ) - REFSUM*V( 3, M )
20 CONTINUE
30 CONTINUE
IF( BMP22 ) THEN
K = KRCOL + 3*( M22-1 )
DO 40 J = MAX( K+1, KTOP ), JBOT
REFSUM = CONJG( V( 1, M22 ) )*
$ ( H( K+1, J )+CONJG( V( 2, M22 ) )*
$ H( K+2, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M22 )
40 CONTINUE
END IF
*
* ==== Multiply H by reflections from the right.
* . Delay filling in the last row until the
* . vigilant deflation check is complete. ====
*
IF( ACCUM ) THEN
JTOP = MAX( KTOP, INCOL )
ELSE IF( WANTT ) THEN
JTOP = 1
ELSE
JTOP = KTOP
END IF
DO 80 M = MTOP, MBOT
IF( V( 1, M ).NE.ZERO ) THEN
K = KRCOL + 3*( M-1 )
DO 50 J = JTOP, MIN( KBOT, K+3 )
DO 70 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M )*( 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
@ -532,89 +621,17 @@
$ REFSUM*CONJG( V( 2, M ) )
H( J, K+3 ) = H( J, K+3 ) -
$ REFSUM*CONJG( V( 3, M ) )
50 CONTINUE
*
IF( ACCUM ) THEN
*
* ==== Accumulate U. (If necessary, update Z later
* . with with an efficient matrix-matrix
* . multiply.) ====
*
KMS = K - INCOL
DO 60 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M )*( 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
U( J, KMS+2 ) = U( J, KMS+2 ) -
$ REFSUM*CONJG( V( 2, M ) )
U( J, KMS+3 ) = U( J, KMS+3 ) -
$ REFSUM*CONJG( V( 3, M ) )
60 CONTINUE
ELSE IF( WANTZ ) THEN
*
* ==== U is not accumulated, so update Z
* . now by multiplying by reflections
* . from the right. ====
*
DO 70 J = ILOZ, IHIZ
REFSUM = V( 1, M )*( 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
Z( J, K+2 ) = Z( J, K+2 ) -
$ REFSUM*CONJG( V( 2, M ) )
Z( J, K+3 ) = Z( J, K+3 ) -
$ REFSUM*CONJG( V( 3, M ) )
70 CONTINUE
END IF
END IF
80 CONTINUE
*
* ==== Special case: 2-by-2 reflection (if needed) ====
* ==== Perform update from left for subsequent
* . column. ====
*
K = KRCOL + 3*( M22-1 )
IF( BMP22 ) THEN
IF ( V( 1, M22 ).NE.ZERO ) THEN
DO 90 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M22 )*( H( J, K+1 )+V( 2, M22 )*
$ H( J, K+2 ) )
H( J, K+1 ) = H( J, K+1 ) - REFSUM
H( J, K+2 ) = H( J, K+2 ) -
$ REFSUM*CONJG( V( 2, M22 ) )
90 CONTINUE
*
IF( ACCUM ) THEN
KMS = K - INCOL
DO 100 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M22 )*( U( J, KMS+1 )+
$ V( 2, M22 )*U( J, KMS+2 ) )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM
U( J, KMS+2 ) = U( J, KMS+2 ) -
$ REFSUM*CONJG( V( 2, M22 ) )
100 CONTINUE
ELSE IF( WANTZ ) THEN
DO 110 J = ILOZ, IHIZ
REFSUM = V( 1, M22 )*( Z( J, K+1 )+V( 2, M22 )*
$ Z( J, K+2 ) )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM
Z( J, K+2 ) = Z( J, K+2 ) -
$ REFSUM*CONJG( V( 2, M22 ) )
110 CONTINUE
END IF
END IF
END IF
*
* ==== Vigilant deflation check ====
*
MSTART = MTOP
IF( KRCOL+3*( MSTART-1 ).LT.KTOP )
$ MSTART = MSTART + 1
MEND = MBOT
IF( BMP22 )
$ MEND = MEND + 1
IF( KRCOL.EQ.KBOT-2 )
$ MEND = MEND + 1
DO 120 M = MSTART, MEND
K = MIN( KBOT-1, KRCOL+3*( M-1 ) )
REFSUM = CONJG( V( 1, M ) )*( H( K+1, K+1 )
$ +CONJG( V( 2, M ) )*H( K+2, K+1 )
$ +CONJG( V( 3, M ) )*H( K+3, K+1 ) )
H( K+1, K+1 ) = H( K+1, K+1 ) - REFSUM
H( K+2, K+1 ) = H( K+2, K+1 ) - REFSUM*V( 2, M )
H( K+3, K+1 ) = H( K+3, K+1 ) - REFSUM*V( 3, M )
*
* ==== The following convergence test requires that
* . the tradition small-compared-to-nearby-diagonals
@ -625,6 +642,8 @@
* . is zero (as done here) is traditional but probably
* . unnecessary. ====
*
IF( K.LT.KTOP)
$ CYCLE
IF( H( K+1, K ).NE.ZERO ) THEN
TST1 = CABS1( H( K, K ) ) + CABS1( H( K+1, K+1 ) )
IF( TST1.EQ.RZERO ) THEN
@ -658,22 +677,77 @@
$ MAX( SMLNUM, ULP*TST2 ) )H( K+1, K ) = ZERO
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 )
DO 90 J = MAX( KTOP, KRCOL + 2*M ), JBOT
REFSUM = CONJG( V( 1, M ) )*
$ ( H( K+1, J )+CONJG( V( 2, M ) )*
$ H( K+2, J )+CONJG( V( 3, M ) )*H( K+3, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M )
H( K+3, J ) = H( K+3, J ) - REFSUM*V( 3, M )
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 )
DO 110 J = I2, I4
REFSUM = V( 1, M )*( 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
U( J, KMS+2 ) = U( J, KMS+2 ) -
$ REFSUM*CONJG( V( 2, M ) )
U( J, KMS+3 ) = U( J, KMS+3 ) -
$ REFSUM*CONJG( V( 3, M ) )
110 CONTINUE
120 CONTINUE
ELSE IF( WANTZ ) THEN
*
* ==== Fill in the last row of each bulge. ====
* ==== U is not accumulated, so update Z
* . now by multiplying by reflections
* . from the right. ====
*
MEND = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 3 )
DO 130 M = MTOP, MEND
K = KRCOL + 3*( M-1 )
REFSUM = V( 1, M )*V( 3, M )*H( K+4, K+3 )
H( K+4, K+1 ) = -REFSUM
H( K+4, K+2 ) = -REFSUM*CONJG( V( 2, M ) )
H( K+4, K+3 ) = H( K+4, K+3 ) - REFSUM*CONJG( V( 3, M ) )
DO 140 M = MBOT, MTOP, -1
K = KRCOL + 2*( M-1 )
DO 130 J = ILOZ, IHIZ
REFSUM = V( 1, M )*( 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
Z( J, K+2 ) = Z( J, K+2 ) -
$ REFSUM*CONJG( V( 2, M ) )
Z( J, K+3 ) = Z( J, K+3 ) -
$ REFSUM*CONJG( V( 3, M ) )
130 CONTINUE
140 CONTINUE
END IF
*
* ==== End of near-the-diagonal bulge chase. ====
*
140 CONTINUE
145 CONTINUE
*
* ==== Use U (if accumulated) to update far-from-diagonal
* . entries in H. If required, use U to update Z as
@ -687,18 +761,6 @@
JTOP = KTOP
JBOT = KBOT
END IF
IF( ( .NOT.BLK22 ) .OR. ( INCOL.LT.KTOP ) .OR.
$ ( NDCOL.GT.KBOT ) .OR. ( NS.LE.2 ) ) THEN
*
* ==== Updates not exploiting the 2-by-2 block
* . structure of U. K1 and NU keep track of
* . the location and size of U in the special
* . cases of introducing bulges and chasing
* . bulges off the bottom. In these special
* . cases and in case the number of shifts
* . is NS = 2, there is no 2-by-2 block
* . structure to exploit. ====
*
K1 = MAX( 1, KTOP-INCOL )
NU = ( KDU-MAX( 0, NDCOL-KBOT ) ) - K1 + 1
*
@ -736,172 +798,9 @@
$ Z( JROW, INCOL+K1 ), LDZ )
170 CONTINUE
END IF
ELSE
*
* ==== Updates exploiting U's 2-by-2 block structure.
* . (I2, I4, J2, J4 are the last rows and columns
* . of the blocks.) ====
*
I2 = ( KDU+1 ) / 2
I4 = KDU
J2 = I4 - I2
J4 = KDU
*
* ==== KZS and KNZ deal with the band of zeros
* . along the diagonal of one of the triangular
* . blocks. ====
*
KZS = ( J4-J2 ) - ( NS+1 )
KNZ = NS + 1
*
* ==== Horizontal multiply ====
*
DO 180 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
JLEN = MIN( NH, JBOT-JCOL+1 )
*
* ==== Copy bottom of H to top+KZS of scratch ====
* (The first KZS rows get multiplied by zero.) ====
*
CALL CLACPY( 'ALL', KNZ, JLEN, H( INCOL+1+J2, JCOL ),
$ LDH, WH( KZS+1, 1 ), LDWH )
*
* ==== Multiply by U21**H ====
*
CALL CLASET( 'ALL', KZS, JLEN, ZERO, ZERO, WH, LDWH )
CALL CTRMM( 'L', 'U', 'C', 'N', KNZ, JLEN, ONE,
$ U( J2+1, 1+KZS ), LDU, WH( KZS+1, 1 ),
$ LDWH )
*
* ==== Multiply top of H by U11**H ====
*
CALL CGEMM( 'C', 'N', I2, JLEN, J2, ONE, U, LDU,
$ H( INCOL+1, JCOL ), LDH, ONE, WH, LDWH )
*
* ==== Copy top of H to bottom of WH ====
*
CALL CLACPY( 'ALL', J2, JLEN, H( INCOL+1, JCOL ), LDH,
$ WH( I2+1, 1 ), LDWH )
*
* ==== Multiply by U21**H ====
*
CALL CTRMM( 'L', 'L', 'C', 'N', J2, JLEN, ONE,
$ U( 1, I2+1 ), LDU, WH( I2+1, 1 ), LDWH )
*
* ==== Multiply by U22 ====
*
CALL CGEMM( 'C', 'N', I4-I2, JLEN, J4-J2, ONE,
$ U( J2+1, I2+1 ), LDU,
$ H( INCOL+1+J2, JCOL ), LDH, ONE,
$ WH( I2+1, 1 ), LDWH )
*
* ==== Copy it back ====
*
CALL CLACPY( 'ALL', KDU, JLEN, WH, LDWH,
$ H( INCOL+1, JCOL ), LDH )
END IF
180 CONTINUE
*
* ==== Vertical multiply ====
*
DO 190 JROW = JTOP, MAX( INCOL, KTOP ) - 1, NV
JLEN = MIN( NV, MAX( INCOL, KTOP )-JROW )
*
* ==== Copy right of H to scratch (the first KZS
* . columns get multiplied by zero) ====
*
CALL CLACPY( 'ALL', JLEN, KNZ, H( JROW, INCOL+1+J2 ),
$ LDH, WV( 1, 1+KZS ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL CLASET( 'ALL', JLEN, KZS, ZERO, ZERO, WV, LDWV )
CALL CTRMM( 'R', 'U', 'N', 'N', JLEN, KNZ, ONE,
$ U( J2+1, 1+KZS ), LDU, WV( 1, 1+KZS ),
$ LDWV )
*
* ==== Multiply by U11 ====
*
CALL CGEMM( 'N', 'N', JLEN, I2, J2, ONE,
$ H( JROW, INCOL+1 ), LDH, U, LDU, ONE, WV,
$ LDWV )
*
* ==== Copy left of H to right of scratch ====
*
CALL CLACPY( 'ALL', JLEN, J2, H( JROW, INCOL+1 ), LDH,
$ WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL CTRMM( 'R', 'L', 'N', 'N', JLEN, I4-I2, ONE,
$ U( 1, I2+1 ), LDU, WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U22 ====
*
CALL CGEMM( 'N', 'N', JLEN, I4-I2, J4-J2, ONE,
$ H( JROW, INCOL+1+J2 ), LDH,
$ U( J2+1, I2+1 ), LDU, ONE, WV( 1, 1+I2 ),
$ LDWV )
*
* ==== Copy it back ====
*
CALL CLACPY( 'ALL', JLEN, KDU, WV, LDWV,
$ H( JROW, INCOL+1 ), LDH )
190 CONTINUE
*
* ==== Multiply Z (also vertical) ====
*
IF( WANTZ ) THEN
DO 200 JROW = ILOZ, IHIZ, NV
JLEN = MIN( NV, IHIZ-JROW+1 )
*
* ==== Copy right of Z to left of scratch (first
* . KZS columns get multiplied by zero) ====
*
CALL CLACPY( 'ALL', JLEN, KNZ,
$ Z( JROW, INCOL+1+J2 ), LDZ,
$ WV( 1, 1+KZS ), LDWV )
*
* ==== Multiply by U12 ====
*
CALL CLASET( 'ALL', JLEN, KZS, ZERO, ZERO, WV,
$ LDWV )
CALL CTRMM( 'R', 'U', 'N', 'N', JLEN, KNZ, ONE,
$ U( J2+1, 1+KZS ), LDU, WV( 1, 1+KZS ),
$ LDWV )
*
* ==== Multiply by U11 ====
*
CALL CGEMM( 'N', 'N', JLEN, I2, J2, ONE,
$ Z( JROW, INCOL+1 ), LDZ, U, LDU, ONE,
$ WV, LDWV )
*
* ==== Copy left of Z to right of scratch ====
*
CALL CLACPY( 'ALL', JLEN, J2, Z( JROW, INCOL+1 ),
$ LDZ, WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL CTRMM( 'R', 'L', 'N', 'N', JLEN, I4-I2, ONE,
$ U( 1, I2+1 ), LDU, WV( 1, 1+I2 ),
$ LDWV )
*
* ==== Multiply by U22 ====
*
CALL CGEMM( 'N', 'N', JLEN, I4-I2, J4-J2, ONE,
$ Z( JROW, INCOL+1+J2 ), LDZ,
$ U( J2+1, I2+1 ), LDU, ONE,
$ WV( 1, 1+I2 ), LDWV )
*
* ==== Copy the result back to Z ====
*
CALL CLACPY( 'ALL', JLEN, KDU, WV, LDWV,
$ Z( JROW, INCOL+1 ), LDZ )
200 CONTINUE
END IF
END IF
END IF
210 CONTINUE
*
* ==== End of CLAQR5 ====
*
END

4
lapack-netlib/SRC/dhseqr.f
View File

@ -338,10 +338,10 @@
* . DLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
PARAMETER ( NTINY = 15 )
*
* ==== NL allocates some local workspace to help small matrices
* . through a rare DLAHQR failure. NL > NTINY = 11 is
* . through a rare DLAHQR failure. NL > NTINY = 15 is
* . required and NL <= NMIN = ILAENV(ISPEC=12,...) is recom-
* . mended. (The default value of NMIN is 75.) Using NL = 49
* . allows up to six simultaneous shifts and a 16-by-16

16
lapack-netlib/SRC/dlaqr0.f
View File

@ -278,7 +278,7 @@
* . DLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
PARAMETER ( NTINY = 15 )
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by varying the size of the
@ -362,22 +362,22 @@
END IF
*
* ==== NWR = recommended deflation window size. At this
* . point, N .GT. NTINY = 11, so there is enough
* . point, N .GT. NTINY = 15, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
* . NWR.GE.3.) ====
* . NWR.GE.4.) ====
*
NWR = ILAENV( 13, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
* ==== NSR = recommended number of simultaneous shifts.
* . At this point N .GT. NTINY = 11, so there is at
* . At this point N .GT. NTINY = 15, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
@ -425,7 +425,7 @@
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
@ -576,7 +576,7 @@
*
* ==== Got NS/2 or fewer shifts? Use DLAQR4 or
* . DLAHQR on a trailing principal submatrix to
* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
@ -698,7 +698,7 @@
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
KDU = 3*NS - 3
KDU = 2*NS
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1

16
lapack-netlib/SRC/dlaqr4.f
View File

@ -284,7 +284,7 @@
* . DLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
PARAMETER ( NTINY = 15 )
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by varying the size of the
@ -368,22 +368,22 @@
END IF
*
* ==== NWR = recommended deflation window size. At this
* . point, N .GT. NTINY = 11, so there is enough
* . point, N .GT. NTINY = 15, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
* . NWR.GE.3.) ====
* . NWR.GE.4.) ====
*
NWR = ILAENV( 13, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
* ==== NSR = recommended number of simultaneous shifts.
* . At this point N .GT. NTINY = 11, so there is at
* . At this point N .GT. NTINY = 15, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'DLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
@ -431,7 +431,7 @@
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
@ -582,7 +582,7 @@
*
* ==== Got NS/2 or fewer shifts? Use DLAHQR
* . on a trailing principal submatrix to
* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
@ -697,7 +697,7 @@
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
KDU = 3*NS - 3
KDU = 2*NS
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1

624
lapack-netlib/SRC/dlaqr5.f
View File

@ -70,10 +70,9 @@
*> matrix entries.
*> = 1: DLAQR5 accumulates reflections and uses matrix-matrix
*> multiply to update the far-from-diagonal matrix entries.
*> = 2: DLAQR5 accumulates reflections, uses matrix-matrix
*> multiply to update the far-from-diagonal matrix entries,
*> and takes advantage of 2-by-2 block structure during
*> matrix multiplies.
*> = 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
@ -178,14 +177,14 @@
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU,3*NSHFTS-3)
*> 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 >= 3*NSHFTS-3.
*> in the calling subroutine. LDU >= 2*NSHFTS.
*> \endverbatim
*>
*> \param[in] NV
@ -197,7 +196,7 @@
*>
*> \param[out] WV
*> \verbatim
*> WV is DOUBLE PRECISION array, dimension (LDWV,3*NSHFTS-3)
*> WV is DOUBLE PRECISION array, dimension (LDWV,2*NSHFTS)
*> \endverbatim
*>
*> \param[in] LDWV
@ -223,7 +222,7 @@
*> \verbatim
*> LDWH is INTEGER
*> Leading dimension of WH just as declared in the
*> calling procedure. LDWH >= 3*NSHFTS-3.
*> calling procedure. LDWH >= 2*NSHFTS.
*> \endverbatim
*>
* Authors:
@ -234,7 +233,7 @@
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*> \date January 2021
*
*> \ingroup doubleOTHERauxiliary
*
@ -243,6 +242,11 @@
*>
*> 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:
* ================
@ -252,10 +256,15 @@
*> 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 (version 3.7.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
@ -282,11 +291,11 @@
DOUBLE PRECISION ALPHA, BETA, H11, H12, H21, H22, REFSUM,
$ SAFMAX, SAFMIN, SCL, SMLNUM, SWAP, TST1, TST2,
$ ULP
INTEGER I, I2, I4, INCOL, J, J2, J4, JBOT, JCOL, JLEN,
$ JROW, JTOP, K, K1, KDU, KMS, KNZ, KRCOL, KZS,
$ M, M22, MBOT, MEND, MSTART, MTOP, NBMPS, NDCOL,
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, BLK22, BMP22
LOGICAL ACCUM, BMP22
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
@ -356,10 +365,6 @@
*
ACCUM = ( KACC22.EQ.1 ) .OR. ( KACC22.EQ.2 )
*
* ==== If so, exploit the 2-by-2 block structure? ====
*
BLK22 = ( NS.GT.2 ) .AND. ( KACC22.EQ.2 )
*
* ==== clear trash ====
*
IF( KTOP+2.LE.KBOT )
@ -371,28 +376,39 @@
*
* ==== KDU = width of slab ====
*
KDU = 6*NBMPS - 3
KDU = 4*NBMPS
*
* ==== Create and chase chains of NBMPS bulges ====
*
DO 220 INCOL = 3*( 1-NBMPS ) + KTOP - 1, KBOT - 2, 3*NBMPS - 2
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 6*NBMPS-2 column diagonal
* . 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 3*NBMPS column long chain of
* . NBMPS bulges 3*NBMPS-2 columns to the right. (INCOL
* . 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 150 KRCOL = INCOL, MIN( INCOL+3*NBMPS-3, KBOT-2 )
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
@ -401,17 +417,134 @@
* . down the diagonal to make room. The phantom matrix
* . paradigm described above helps keep track. ====
*
MTOP = MAX( 1, ( ( KTOP-1 )-KRCOL+2 ) / 3+1 )
MBOT = MIN( NBMPS, ( KBOT-KRCOL ) / 3 )
MTOP = MAX( 1, ( KTOP-KRCOL ) / 2+1 )
MBOT = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 2 )
M22 = MBOT + 1
BMP22 = ( MBOT.LT.NBMPS ) .AND. ( KRCOL+3*( M22-1 ) ).EQ.
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.) ====
*
DO 20 M = MTOP, MBOT
K = KRCOL + 3*( M-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. ====
*
DO 30 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M22 )*( H( J, K+1 )+V( 2, M22 )*
$ H( J, K+2 ) )
H( J, K+1 ) = H( J, K+1 ) - REFSUM
H( J, K+2 ) = H( J, K+2 ) - REFSUM*V( 2, M22 )
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
DO 40 J = K+1, JBOT
REFSUM = V( 1, M22 )*( H( K+1, J )+V( 2, M22 )*
$ H( K+2, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M22 )
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
DO 50 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M22 )*( U( J, KMS+1 )+
$ V( 2, M22 )*U( J, KMS+2 ) )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM
U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*V( 2, M22 )
50 CONTINUE
ELSE IF( WANTZ ) THEN
DO 60 J = ILOZ, IHIZ
REFSUM = V( 1, M22 )*( Z( J, K+1 )+V( 2, M22 )*
$ Z( J, K+2 ) )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*V( 2, M22 )
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 ),
@ -419,6 +552,19 @@
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. ====
*
REFSUM = V( 1, M )*V( 3, M )*H( K+3, K+2 )
H( K+3, K ) = -REFSUM
H( K+3, K+1 ) = -REFSUM*V( 2, M )
H( K+3, K+2 ) = H( K+3, K+2 ) - REFSUM*V( 3, M )
*
* ==== 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 )
@ -467,7 +613,7 @@
H( K+3, K ) = ZERO
ELSE
*
* ==== Stating a new bulge here would
* ==== Starting a new bulge here would
* . create only negligible fill.
* . Replace the old reflector with
* . the new one. ====
@ -481,154 +627,29 @@
END IF
END IF
END IF
20 CONTINUE
*
* ==== Generate a 2-by-2 reflection, if needed. ====
* ==== 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. ====
*
K = KRCOL + 3*( M22-1 )
IF( BMP22 ) THEN
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
END IF
*
* ==== 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 40 J = MAX( KTOP, KRCOL ), JBOT
MEND = MIN( MBOT, ( J-KRCOL+2 ) / 3 )
DO 30 M = MTOP, MEND
K = KRCOL + 3*( M-1 )
REFSUM = V( 1, M )*( 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
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M )
H( K+3, J ) = H( K+3, J ) - REFSUM*V( 3, M )
30 CONTINUE
40 CONTINUE
IF( BMP22 ) THEN
K = KRCOL + 3*( M22-1 )
DO 50 J = MAX( K+1, KTOP ), JBOT
REFSUM = V( 1, M22 )*( H( K+1, J )+V( 2, M22 )*
$ H( K+2, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M22 )
50 CONTINUE
END IF
*
* ==== Multiply H by reflections from the right.
* . Delay filling in the last row until the
* . vigilant deflation check is complete. ====
*
IF( ACCUM ) THEN
JTOP = MAX( KTOP, INCOL )
ELSE IF( WANTT ) THEN
JTOP = 1
ELSE
JTOP = KTOP
END IF
DO 90 M = MTOP, MBOT
IF( V( 1, M ).NE.ZERO ) THEN
K = KRCOL + 3*( M-1 )
DO 60 J = JTOP, MIN( KBOT, K+3 )
DO 70 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M )*( 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
H( J, K+2 ) = H( J, K+2 ) - REFSUM*V( 2, M )
H( J, K+3 ) = H( J, K+3 ) - REFSUM*V( 3, M )
60 CONTINUE
*
IF( ACCUM ) THEN
*
* ==== Accumulate U. (If necessary, update Z later
* . with with an efficient matrix-matrix
* . multiply.) ====
*
KMS = K - INCOL
DO 70 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M )*( 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
U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*V( 2, M )
U( J, KMS+3 ) = U( J, KMS+3 ) - REFSUM*V( 3, M )
70 CONTINUE
ELSE IF( WANTZ ) THEN
*
* ==== U is not accumulated, so update Z
* . now by multiplying by reflections
* . from the right. ====
* ==== Perform update from left for subsequent
* . column. ====
*
DO 80 J = ILOZ, IHIZ
REFSUM = V( 1, M )*( 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
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*V( 2, M )
Z( J, K+3 ) = Z( J, K+3 ) - REFSUM*V( 3, M )
80 CONTINUE
END IF
END IF
90 CONTINUE
*
* ==== Special case: 2-by-2 reflection (if needed) ====
*
K = KRCOL + 3*( M22-1 )
IF( BMP22 ) THEN
IF ( V( 1, M22 ).NE.ZERO ) THEN
DO 100 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M22 )*( H( J, K+1 )+V( 2, M22 )*
$ H( J, K+2 ) )
H( J, K+1 ) = H( J, K+1 ) - REFSUM
H( J, K+2 ) = H( J, K+2 ) - REFSUM*V( 2, M22 )
100 CONTINUE
*
IF( ACCUM ) THEN
KMS = K - INCOL
DO 110 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M22 )*( U( J, KMS+1 )+
$ V( 2, M22 )*U( J, KMS+2 ) )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM
U( J, KMS+2 ) = U( J, KMS+2 ) -
$ REFSUM*V( 2, M22 )
110 CONTINUE
ELSE IF( WANTZ ) THEN
DO 120 J = ILOZ, IHIZ
REFSUM = V( 1, M22 )*( Z( J, K+1 )+V( 2, M22 )*
$ Z( J, K+2 ) )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*V( 2, M22 )
120 CONTINUE
END IF
END IF
END IF
*
* ==== Vigilant deflation check ====
*
MSTART = MTOP
IF( KRCOL+3*( MSTART-1 ).LT.KTOP )
$ MSTART = MSTART + 1
MEND = MBOT
IF( BMP22 )
$ MEND = MEND + 1
IF( KRCOL.EQ.KBOT-2 )
$ MEND = MEND + 1
DO 130 M = MSTART, MEND
K = MIN( KBOT-1, KRCOL+3*( M-1 ) )
REFSUM = V( 1, M )*( 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
H( K+2, K+1 ) = H( K+2, K+1 ) - REFSUM*V( 2, M )
H( K+3, K+1 ) = H( K+3, K+1 ) - REFSUM*V( 3, M )
*
* ==== The following convergence test requires that
* . the tradition small-compared-to-nearby-diagonals
@ -639,6 +660,8 @@
* . 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
@ -667,25 +690,77 @@
TST2 = H22*( H11 / SCL )
*
IF( TST2.EQ.ZERO .OR. H21*( H12 / SCL ).LE.
$ MAX( SMLNUM, ULP*TST2 ) )H( K+1, K ) = ZERO
$ 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 )
DO 90 J = MAX( KTOP, KRCOL + 2*M ), JBOT
REFSUM = V( 1, M )*( 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
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M )
H( K+3, J ) = H( K+3, J ) - REFSUM*V( 3, M )
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 )
DO 110 J = I2, I4
REFSUM = V( 1, M )*( 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
U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*V( 2, M )
U( J, KMS+3 ) = U( J, KMS+3 ) - REFSUM*V( 3, M )
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 )
DO 130 J = ILOZ, IHIZ
REFSUM = V( 1, M )*( 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
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*V( 2, M )
Z( J, K+3 ) = Z( J, K+3 ) - REFSUM*V( 3, M )
130 CONTINUE
*
* ==== Fill in the last row of each bulge. ====
*
MEND = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 3 )
DO 140 M = MTOP, MEND
K = KRCOL + 3*( M-1 )
REFSUM = V( 1, M )*V( 3, M )*H( K+4, K+3 )
H( K+4, K+1 ) = -REFSUM
H( K+4, K+2 ) = -REFSUM*V( 2, M )
H( K+4, K+3 ) = H( K+4, K+3 ) - REFSUM*V( 3, M )
140 CONTINUE
END IF
*
* ==== End of near-the-diagonal bulge chase. ====
*
150 CONTINUE
145 CONTINUE
*
* ==== Use U (if accumulated) to update far-from-diagonal
* . entries in H. If required, use U to update Z as
@ -699,220 +774,45 @@
JTOP = KTOP
JBOT = KBOT
END IF
IF( ( .NOT.BLK22 ) .OR. ( INCOL.LT.KTOP ) .OR.
$ ( NDCOL.GT.KBOT ) .OR. ( NS.LE.2 ) ) THEN
*
* ==== Updates not exploiting the 2-by-2 block
* . structure of U. K1 and NU keep track of
* . the location and size of U in the special
* . cases of introducing bulges and chasing
* . bulges off the bottom. In these special
* . cases and in case the number of shifts
* . is NS = 2, there is no 2-by-2 block
* . structure to exploit. ====
*
K1 = MAX( 1, KTOP-INCOL )
NU = ( KDU-MAX( 0, NDCOL-KBOT ) ) - K1 + 1
*
* ==== Horizontal Multiply ====
*
DO 160 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
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 )
160 CONTINUE
150 CONTINUE
*
* ==== Vertical multiply ====
*
DO 170 JROW = JTOP, MAX( KTOP, INCOL ) - 1, NV
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 )
170 CONTINUE
160 CONTINUE
*
* ==== Z multiply (also vertical) ====
*
IF( WANTZ ) THEN
DO 180 JROW = ILOZ, IHIZ, NV
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 IF
ELSE
*
* ==== Updates exploiting U's 2-by-2 block structure.
* . (I2, I4, J2, J4 are the last rows and columns
* . of the blocks.) ====
*
I2 = ( KDU+1 ) / 2
I4 = KDU
J2 = I4 - I2
J4 = KDU
*
* ==== KZS and KNZ deal with the band of zeros
* . along the diagonal of one of the triangular
* . blocks. ====
*
KZS = ( J4-J2 ) - ( NS+1 )
KNZ = NS + 1
*
* ==== Horizontal multiply ====
*
DO 190 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
JLEN = MIN( NH, JBOT-JCOL+1 )
*
* ==== Copy bottom of H to top+KZS of scratch ====
* (The first KZS rows get multiplied by zero.) ====
*
CALL DLACPY( 'ALL', KNZ, JLEN, H( INCOL+1+J2, JCOL ),
$ LDH, WH( KZS+1, 1 ), LDWH )
*
* ==== Multiply by U21**T ====
*
CALL DLASET( 'ALL', KZS, JLEN, ZERO, ZERO, WH, LDWH )
CALL DTRMM( 'L', 'U', 'C', 'N', KNZ, JLEN, ONE,
$ U( J2+1, 1+KZS ), LDU, WH( KZS+1, 1 ),
$ LDWH )
*
* ==== Multiply top of H by U11**T ====
*
CALL DGEMM( 'C', 'N', I2, JLEN, J2, ONE, U, LDU,
$ H( INCOL+1, JCOL ), LDH, ONE, WH, LDWH )
*
* ==== Copy top of H to bottom of WH ====
*
CALL DLACPY( 'ALL', J2, JLEN, H( INCOL+1, JCOL ), LDH,
$ WH( I2+1, 1 ), LDWH )
*
* ==== Multiply by U21**T ====
*
CALL DTRMM( 'L', 'L', 'C', 'N', J2, JLEN, ONE,
$ U( 1, I2+1 ), LDU, WH( I2+1, 1 ), LDWH )
*
* ==== Multiply by U22 ====
*
CALL DGEMM( 'C', 'N', I4-I2, JLEN, J4-J2, ONE,
$ U( J2+1, I2+1 ), LDU,
$ H( INCOL+1+J2, JCOL ), LDH, ONE,
$ WH( I2+1, 1 ), LDWH )
*
* ==== Copy it back ====
*
CALL DLACPY( 'ALL', KDU, JLEN, WH, LDWH,
$ H( INCOL+1, JCOL ), LDH )
190 CONTINUE
*
* ==== Vertical multiply ====
*
DO 200 JROW = JTOP, MAX( INCOL, KTOP ) - 1, NV
JLEN = MIN( NV, MAX( INCOL, KTOP )-JROW )
*
* ==== Copy right of H to scratch (the first KZS
* . columns get multiplied by zero) ====
*
CALL DLACPY( 'ALL', JLEN, KNZ, H( JROW, INCOL+1+J2 ),
$ LDH, WV( 1, 1+KZS ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL DLASET( 'ALL', JLEN, KZS, ZERO, ZERO, WV, LDWV )
CALL DTRMM( 'R', 'U', 'N', 'N', JLEN, KNZ, ONE,
$ U( J2+1, 1+KZS ), LDU, WV( 1, 1+KZS ),
$ LDWV )
*
* ==== Multiply by U11 ====
*
CALL DGEMM( 'N', 'N', JLEN, I2, J2, ONE,
$ H( JROW, INCOL+1 ), LDH, U, LDU, ONE, WV,
$ LDWV )
*
* ==== Copy left of H to right of scratch ====
*
CALL DLACPY( 'ALL', JLEN, J2, H( JROW, INCOL+1 ), LDH,
$ WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL DTRMM( 'R', 'L', 'N', 'N', JLEN, I4-I2, ONE,
$ U( 1, I2+1 ), LDU, WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U22 ====
*
CALL DGEMM( 'N', 'N', JLEN, I4-I2, J4-J2, ONE,
$ H( JROW, INCOL+1+J2 ), LDH,
$ U( J2+1, I2+1 ), LDU, ONE, WV( 1, 1+I2 ),
$ LDWV )
*
* ==== Copy it back ====
*
CALL DLACPY( 'ALL', JLEN, KDU, WV, LDWV,
$ H( JROW, INCOL+1 ), LDH )
200 CONTINUE
*
* ==== Multiply Z (also vertical) ====
*
IF( WANTZ ) THEN
DO 210 JROW = ILOZ, IHIZ, NV
JLEN = MIN( NV, IHIZ-JROW+1 )
*
* ==== Copy right of Z to left of scratch (first
* . KZS columns get multiplied by zero) ====
*
CALL DLACPY( 'ALL', JLEN, KNZ,
$ Z( JROW, INCOL+1+J2 ), LDZ,
$ WV( 1, 1+KZS ), LDWV )
*
* ==== Multiply by U12 ====
*
CALL DLASET( 'ALL', JLEN, KZS, ZERO, ZERO, WV,
$ LDWV )
CALL DTRMM( 'R', 'U', 'N', 'N', JLEN, KNZ, ONE,
$ U( J2+1, 1+KZS ), LDU, WV( 1, 1+KZS ),
$ LDWV )
*
* ==== Multiply by U11 ====
*
CALL DGEMM( 'N', 'N', JLEN, I2, J2, ONE,
$ Z( JROW, INCOL+1 ), LDZ, U, LDU, ONE,
$ WV, LDWV )
*
* ==== Copy left of Z to right of scratch ====
*
CALL DLACPY( 'ALL', JLEN, J2, Z( JROW, INCOL+1 ),
$ LDZ, WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL DTRMM( 'R', 'L', 'N', 'N', JLEN, I4-I2, ONE,
$ U( 1, I2+1 ), LDU, WV( 1, 1+I2 ),
$ LDWV )
*
* ==== Multiply by U22 ====
*
CALL DGEMM( 'N', 'N', JLEN, I4-I2, J4-J2, ONE,
$ Z( JROW, INCOL+1+J2 ), LDZ,
$ U( J2+1, I2+1 ), LDU, ONE,
$ WV( 1, 1+I2 ), LDWV )
*
* ==== Copy the result back to Z ====
*
CALL DLACPY( 'ALL', JLEN, KDU, WV, LDWV,
$ Z( JROW, INCOL+1 ), LDZ )
210 CONTINUE
END IF
END IF
END IF
220 CONTINUE
*
* ==== End of DLAQR5 ====
*

4
lapack-netlib/SRC/shseqr.f
View File

@ -338,10 +338,10 @@
* . SLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
PARAMETER ( NTINY = 15 )
*
* ==== NL allocates some local workspace to help small matrices
* . through a rare SLAHQR failure. NL > NTINY = 11 is
* . through a rare SLAHQR failure. NL > NTINY = 15 is
* . required and NL <= NMIN = ILAENV(ISPEC=12,...) is recom-
* . mended. (The default value of NMIN is 75.) Using NL = 49
* . allows up to six simultaneous shifts and a 16-by-16

16
lapack-netlib/SRC/slaqr0.f
View File

@ -277,7 +277,7 @@
* . SLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
PARAMETER ( NTINY = 15 )
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by varying the size of the
@ -361,22 +361,22 @@
END IF
*
* ==== NWR = recommended deflation window size. At this
* . point, N .GT. NTINY = 11, so there is enough
* . point, N .GT. NTINY = 15, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
* . NWR.GE.3.) ====
* . NWR.GE.4.) ====
*
NWR = ILAENV( 13, 'SLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
* ==== NSR = recommended number of simultaneous shifts.
* . At this point N .GT. NTINY = 11, so there is at
* . At this point N .GT. NTINY = 15, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'SLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
@ -424,7 +424,7 @@
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
@ -575,7 +575,7 @@
*
* ==== Got NS/2 or fewer shifts? Use SLAQR4 or
* . SLAHQR on a trailing principal submatrix to
* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
@ -697,7 +697,7 @@
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
KDU = 3*NS - 3
KDU = 2*NS
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1

16
lapack-netlib/SRC/slaqr4.f
View File

@ -287,7 +287,7 @@
* . SLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
PARAMETER ( NTINY = 15 )
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by varying the size of the
@ -371,22 +371,22 @@
END IF
*
* ==== NWR = recommended deflation window size. At this
* . point, N .GT. NTINY = 11, so there is enough
* . point, N .GT. NTINY = 15, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
* . NWR.GE.3.) ====
* . NWR.GE.4.) ====
*
NWR = ILAENV( 13, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
* ==== NSR = recommended number of simultaneous shifts.
* . At this point N .GT. NTINY = 11, so there is at
* . At this point N .GT. NTINY = 15, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
@ -434,7 +434,7 @@
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
@ -585,7 +585,7 @@
*
* ==== Got NS/2 or fewer shifts? Use SLAHQR
* . on a trailing principal submatrix to
* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
@ -700,7 +700,7 @@
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
KDU = 3*NS - 3
KDU = 2*NS
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1

624
lapack-netlib/SRC/slaqr5.f
View File

@ -70,10 +70,9 @@
*> matrix entries.
*> = 1: SLAQR5 accumulates reflections and uses matrix-matrix
*> multiply to update the far-from-diagonal matrix entries.
*> = 2: SLAQR5 accumulates reflections, uses matrix-matrix
*> multiply to update the far-from-diagonal matrix entries,
*> and takes advantage of 2-by-2 block structure during
*> matrix multiplies.
*> = 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
@ -178,14 +177,14 @@
*>
*> \param[out] U
*> \verbatim
*> U is REAL array, dimension (LDU,3*NSHFTS-3)
*> U is REAL 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 >= 3*NSHFTS-3.
*> in the calling subroutine. LDU >= 2*NSHFTS.
*> \endverbatim
*>
*> \param[in] NV
@ -197,7 +196,7 @@
*>
*> \param[out] WV
*> \verbatim
*> WV is REAL array, dimension (LDWV,3*NSHFTS-3)
*> WV is REAL array, dimension (LDWV,2*NSHFTS)
*> \endverbatim
*>
*> \param[in] LDWV
@ -223,7 +222,7 @@
*> \verbatim
*> LDWH is INTEGER
*> Leading dimension of WH just as declared in the
*> calling procedure. LDWH >= 3*NSHFTS-3.
*> calling procedure. LDWH >= 2*NSHFTS.
*> \endverbatim
*>
* Authors:
@ -234,7 +233,7 @@
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*> \date January 2021
*
*> \ingroup realOTHERauxiliary
*
@ -243,6 +242,11 @@
*>
*> 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:
* ================
@ -252,10 +256,15 @@
*> 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 SLAQR5( 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 (version 3.7.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
@ -282,11 +291,11 @@
REAL ALPHA, BETA, H11, H12, H21, H22, REFSUM,
$ SAFMAX, SAFMIN, SCL, SMLNUM, SWAP, TST1, TST2,
$ ULP
INTEGER I, I2, I4, INCOL, J, J2, J4, JBOT, JCOL, JLEN,
$ JROW, JTOP, K, K1, KDU, KMS, KNZ, KRCOL, KZS,
$ M, M22, MBOT, MEND, MSTART, MTOP, NBMPS, NDCOL,
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, BLK22, BMP22
LOGICAL ACCUM, BMP22
* ..
* .. External Functions ..
REAL SLAMCH
@ -356,10 +365,6 @@
*
ACCUM = ( KACC22.EQ.1 ) .OR. ( KACC22.EQ.2 )
*
* ==== If so, exploit the 2-by-2 block structure? ====
*
BLK22 = ( NS.GT.2 ) .AND. ( KACC22.EQ.2 )
*
* ==== clear trash ====
*
IF( KTOP+2.LE.KBOT )
@ -371,28 +376,39 @@
*
* ==== KDU = width of slab ====
*
KDU = 6*NBMPS - 3
KDU = 4*NBMPS
*
* ==== Create and chase chains of NBMPS bulges ====
*
DO 220 INCOL = 3*( 1-NBMPS ) + KTOP - 1, KBOT - 2, 3*NBMPS - 2
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 SLASET( '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 6*NBMPS-2 column diagonal
* . 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 3*NBMPS column long chain of
* . NBMPS bulges 3*NBMPS-2 columns to the right. (INCOL
* . following loop chases a 2*NBMPS+1 column long chain of
* . NBMPS bulges 2*NBMPS-1 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 150 KRCOL = INCOL, MIN( INCOL+3*NBMPS-3, KBOT-2 )
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
@ -401,17 +417,134 @@
* . down the diagonal to make room. The phantom matrix
* . paradigm described above helps keep track. ====
*
MTOP = MAX( 1, ( ( KTOP-1 )-KRCOL+2 ) / 3+1 )
MBOT = MIN( NBMPS, ( KBOT-KRCOL ) / 3 )
MTOP = MAX( 1, ( KTOP-KRCOL ) / 2+1 )
MBOT = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 2 )
M22 = MBOT + 1
BMP22 = ( MBOT.LT.NBMPS ) .AND. ( KRCOL+3*( M22-1 ) ).EQ.
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.) ====
*
DO 20 M = MTOP, MBOT
K = KRCOL + 3*( M-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 SLAQR1( 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 SLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
ELSE
BETA = H( K+1, K )
V( 2, M22 ) = H( K+2, K )
CALL SLARFG( 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. ====
*
DO 30 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M22 )*( H( J, K+1 )+V( 2, M22 )*
$ H( J, K+2 ) )
H( J, K+1 ) = H( J, K+1 ) - REFSUM
H( J, K+2 ) = H( J, K+2 ) - REFSUM*V( 2, M22 )
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
DO 40 J = K+1, JBOT
REFSUM = V( 1, M22 )*( H( K+1, J )+V( 2, M22 )*
$ H( K+2, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M22 )
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
DO 50 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M22 )*( U( J, KMS+1 )+
$ V( 2, M22 )*U( J, KMS+2 ) )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM
U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*V( 2, M22 )
50 CONTINUE
ELSE IF( WANTZ ) THEN
DO 60 J = ILOZ, IHIZ
REFSUM = V( 1, M22 )*( Z( J, K+1 )+V( 2, M22 )*
$ Z( J, K+2 ) )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*V( 2, M22 )
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 SLAQR1( 3, H( KTOP, KTOP ), LDH, SR( 2*M-1 ),
$ SI( 2*M-1 ), SR( 2*M ), SI( 2*M ),
@ -419,6 +552,19 @@
ALPHA = V( 1, M )
CALL SLARFG( 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. ====
*
REFSUM = V( 1, M )*V( 3, M )*H( K+3, K+2 )
H( K+3, K ) = -REFSUM
H( K+3, K+1 ) = -REFSUM*V( 2, M )
H( K+3, K+2 ) = H( K+3, K+2 ) - REFSUM*V( 3, M )
*
* ==== 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 )
@ -467,7 +613,7 @@
H( K+3, K ) = ZERO
ELSE
*
* ==== Stating a new bulge here would
* ==== Starting a new bulge here would
* . create only negligible fill.
* . Replace the old reflector with
* . the new one. ====
@ -481,154 +627,29 @@
END IF
END IF
END IF
20 CONTINUE
*
* ==== Generate a 2-by-2 reflection, if needed. ====
* ==== 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. ====
*
K = KRCOL + 3*( M22-1 )
IF( BMP22 ) THEN
IF( K.EQ.KTOP-1 ) THEN
CALL SLAQR1( 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 SLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
ELSE
BETA = H( K+1, K )
V( 2, M22 ) = H( K+2, K )
CALL SLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
H( K+1, K ) = BETA
H( K+2, K ) = ZERO
END IF
END IF
*
* ==== 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 40 J = MAX( KTOP, KRCOL ), JBOT
MEND = MIN( MBOT, ( J-KRCOL+2 ) / 3 )
DO 30 M = MTOP, MEND
K = KRCOL + 3*( M-1 )
REFSUM = V( 1, M )*( 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
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M )
H( K+3, J ) = H( K+3, J ) - REFSUM*V( 3, M )
30 CONTINUE
40 CONTINUE
IF( BMP22 ) THEN
K = KRCOL + 3*( M22-1 )
DO 50 J = MAX( K+1, KTOP ), JBOT
REFSUM = V( 1, M22 )*( H( K+1, J )+V( 2, M22 )*
$ H( K+2, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M22 )
50 CONTINUE
END IF
*
* ==== Multiply H by reflections from the right.
* . Delay filling in the last row until the
* . vigilant deflation check is complete. ====
*
IF( ACCUM ) THEN
JTOP = MAX( KTOP, INCOL )
ELSE IF( WANTT ) THEN
JTOP = 1
ELSE
JTOP = KTOP
END IF
DO 90 M = MTOP, MBOT
IF( V( 1, M ).NE.ZERO ) THEN
K = KRCOL + 3*( M-1 )
DO 60 J = JTOP, MIN( KBOT, K+3 )
DO 70 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M )*( 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
H( J, K+2 ) = H( J, K+2 ) - REFSUM*V( 2, M )
H( J, K+3 ) = H( J, K+3 ) - REFSUM*V( 3, M )
60 CONTINUE
*
IF( ACCUM ) THEN
*
* ==== Accumulate U. (If necessary, update Z later
* . with with an efficient matrix-matrix
* . multiply.) ====
*
KMS = K - INCOL
DO 70 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M )*( 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
U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*V( 2, M )
U( J, KMS+3 ) = U( J, KMS+3 ) - REFSUM*V( 3, M )
70 CONTINUE
ELSE IF( WANTZ ) THEN
*
* ==== U is not accumulated, so update Z
* . now by multiplying by reflections
* . from the right. ====
* ==== Perform update from left for subsequent
* . column. ====
*
DO 80 J = ILOZ, IHIZ
REFSUM = V( 1, M )*( 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
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*V( 2, M )
Z( J, K+3 ) = Z( J, K+3 ) - REFSUM*V( 3, M )
80 CONTINUE
END IF
END IF
90 CONTINUE
*
* ==== Special case: 2-by-2 reflection (if needed) ====
*
K = KRCOL + 3*( M22-1 )
IF( BMP22 ) THEN
IF ( V( 1, M22 ).NE.ZERO ) THEN
DO 100 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M22 )*( H( J, K+1 )+V( 2, M22 )*
$ H( J, K+2 ) )
H( J, K+1 ) = H( J, K+1 ) - REFSUM
H( J, K+2 ) = H( J, K+2 ) - REFSUM*V( 2, M22 )
100 CONTINUE
*
IF( ACCUM ) THEN
KMS = K - INCOL
DO 110 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M22 )*( U( J, KMS+1 )+
$ V( 2, M22 )*U( J, KMS+2 ) )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM
U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*
$ V( 2, M22 )
110 CONTINUE
ELSE IF( WANTZ ) THEN
DO 120 J = ILOZ, IHIZ
REFSUM = V( 1, M22 )*( Z( J, K+1 )+V( 2, M22 )*
$ Z( J, K+2 ) )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*V( 2, M22 )
120 CONTINUE
END IF
END IF
END IF
*
* ==== Vigilant deflation check ====
*
MSTART = MTOP
IF( KRCOL+3*( MSTART-1 ).LT.KTOP )
$ MSTART = MSTART + 1
MEND = MBOT
IF( BMP22 )
$ MEND = MEND + 1
IF( KRCOL.EQ.KBOT-2 )
$ MEND = MEND + 1
DO 130 M = MSTART, MEND
K = MIN( KBOT-1, KRCOL+3*( M-1 ) )
REFSUM = V( 1, M )*( 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
H( K+2, K+1 ) = H( K+2, K+1 ) - REFSUM*V( 2, M )
H( K+3, K+1 ) = H( K+3, K+1 ) - REFSUM*V( 3, M )
*
* ==== The following convergence test requires that
* . the tradition small-compared-to-nearby-diagonals
@ -639,6 +660,8 @@
* . 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
@ -667,25 +690,77 @@
TST2 = H22*( H11 / SCL )
*
IF( TST2.EQ.ZERO .OR. H21*( H12 / SCL ).LE.
$ MAX( SMLNUM, ULP*TST2 ) )H( K+1, K ) = ZERO
$ 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 )
DO 90 J = MAX( KTOP, KRCOL + 2*M ), JBOT
REFSUM = V( 1, M )*( 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
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M )
H( K+3, J ) = H( K+3, J ) - REFSUM*V( 3, M )
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 )
DO 110 J = I2, I4
REFSUM = V( 1, M )*( 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
U( J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*V( 2, M )
U( J, KMS+3 ) = U( J, KMS+3 ) - REFSUM*V( 3, M )
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 )
DO 130 J = ILOZ, IHIZ
REFSUM = V( 1, M )*( 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
Z( J, K+2 ) = Z( J, K+2 ) - REFSUM*V( 2, M )
Z( J, K+3 ) = Z( J, K+3 ) - REFSUM*V( 3, M )
130 CONTINUE
*
* ==== Fill in the last row of each bulge. ====
*
MEND = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 3 )
DO 140 M = MTOP, MEND
K = KRCOL + 3*( M-1 )
REFSUM = V( 1, M )*V( 3, M )*H( K+4, K+3 )
H( K+4, K+1 ) = -REFSUM
H( K+4, K+2 ) = -REFSUM*V( 2, M )
H( K+4, K+3 ) = H( K+4, K+3 ) - REFSUM*V( 3, M )
140 CONTINUE
END IF
*
* ==== End of near-the-diagonal bulge chase. ====
*
150 CONTINUE
145 CONTINUE
*
* ==== Use U (if accumulated) to update far-from-diagonal
* . entries in H. If required, use U to update Z as
@ -699,220 +774,45 @@
JTOP = KTOP
JBOT = KBOT
END IF
IF( ( .NOT.BLK22 ) .OR. ( INCOL.LT.KTOP ) .OR.
$ ( NDCOL.GT.KBOT ) .OR. ( NS.LE.2 ) ) THEN
*
* ==== Updates not exploiting the 2-by-2 block
* . structure of U. K1 and NU keep track of
* . the location and size of U in the special
* . cases of introducing bulges and chasing
* . bulges off the bottom. In these special
* . cases and in case the number of shifts
* . is NS = 2, there is no 2-by-2 block
* . structure to exploit. ====
*
K1 = MAX( 1, KTOP-INCOL )
NU = ( KDU-MAX( 0, NDCOL-KBOT ) ) - K1 + 1
*
* ==== Horizontal Multiply ====
*
DO 160 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
DO 150 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
JLEN = MIN( NH, JBOT-JCOL+1 )
CALL SGEMM( 'C', 'N', NU, JLEN, NU, ONE, U( K1, K1 ),
$ LDU, H( INCOL+K1, JCOL ), LDH, ZERO, WH,
$ LDWH )
CALL SLACPY( 'ALL', NU, JLEN, WH, LDWH,
$ H( INCOL+K1, JCOL ), LDH )
160 CONTINUE
150 CONTINUE
*
* ==== Vertical multiply ====
*
DO 170 JROW = JTOP, MAX( KTOP, INCOL ) - 1, NV
DO 160 JROW = JTOP, MAX( KTOP, INCOL ) - 1, NV
JLEN = MIN( NV, MAX( KTOP, INCOL )-JROW )
CALL SGEMM( 'N', 'N', JLEN, NU, NU, ONE,
$ H( JROW, INCOL+K1 ), LDH, U( K1, K1 ),
$ LDU, ZERO, WV, LDWV )
CALL SLACPY( 'ALL', JLEN, NU, WV, LDWV,
$ H( JROW, INCOL+K1 ), LDH )
170 CONTINUE
160 CONTINUE
*
* ==== Z multiply (also vertical) ====
*
IF( WANTZ ) THEN
DO 180 JROW = ILOZ, IHIZ, NV
DO 170 JROW = ILOZ, IHIZ, NV
JLEN = MIN( NV, IHIZ-JROW+1 )
CALL SGEMM( 'N', 'N', JLEN, NU, NU, ONE,
$ Z( JROW, INCOL+K1 ), LDZ, U( K1, K1 ),
$ LDU, ZERO, WV, LDWV )
CALL SLACPY( 'ALL', JLEN, NU, WV, LDWV,
$ Z( JROW, INCOL+K1 ), LDZ )
170 CONTINUE
END IF
END IF
180 CONTINUE
END IF
ELSE
*
* ==== Updates exploiting U's 2-by-2 block structure.
* . (I2, I4, J2, J4 are the last rows and columns
* . of the blocks.) ====
*
I2 = ( KDU+1 ) / 2
I4 = KDU
J2 = I4 - I2
J4 = KDU
*
* ==== KZS and KNZ deal with the band of zeros
* . along the diagonal of one of the triangular
* . blocks. ====
*
KZS = ( J4-J2 ) - ( NS+1 )
KNZ = NS + 1
*
* ==== Horizontal multiply ====
*
DO 190 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
JLEN = MIN( NH, JBOT-JCOL+1 )
*
* ==== Copy bottom of H to top+KZS of scratch ====
* (The first KZS rows get multiplied by zero.) ====
*
CALL SLACPY( 'ALL', KNZ, JLEN, H( INCOL+1+J2, JCOL ),
$ LDH, WH( KZS+1, 1 ), LDWH )
*
* ==== Multiply by U21**T ====
*
CALL SLASET( 'ALL', KZS, JLEN, ZERO, ZERO, WH, LDWH )
CALL STRMM( 'L', 'U', 'C', 'N', KNZ, JLEN, ONE,
$ U( J2+1, 1+KZS ), LDU, WH( KZS+1, 1 ),
$ LDWH )
*
* ==== Multiply top of H by U11**T ====
*
CALL SGEMM( 'C', 'N', I2, JLEN, J2, ONE, U, LDU,
$ H( INCOL+1, JCOL ), LDH, ONE, WH, LDWH )
*
* ==== Copy top of H to bottom of WH ====
*
CALL SLACPY( 'ALL', J2, JLEN, H( INCOL+1, JCOL ), LDH,
$ WH( I2+1, 1 ), LDWH )
*
* ==== Multiply by U21**T ====
*
CALL STRMM( 'L', 'L', 'C', 'N', J2, JLEN, ONE,
$ U( 1, I2+1 ), LDU, WH( I2+1, 1 ), LDWH )
*
* ==== Multiply by U22 ====
*
CALL SGEMM( 'C', 'N', I4-I2, JLEN, J4-J2, ONE,
$ U( J2+1, I2+1 ), LDU,
$ H( INCOL+1+J2, JCOL ), LDH, ONE,
$ WH( I2+1, 1 ), LDWH )
*
* ==== Copy it back ====
*
CALL SLACPY( 'ALL', KDU, JLEN, WH, LDWH,
$ H( INCOL+1, JCOL ), LDH )
190 CONTINUE
*
* ==== Vertical multiply ====
*
DO 200 JROW = JTOP, MAX( INCOL, KTOP ) - 1, NV
JLEN = MIN( NV, MAX( INCOL, KTOP )-JROW )
*
* ==== Copy right of H to scratch (the first KZS
* . columns get multiplied by zero) ====
*
CALL SLACPY( 'ALL', JLEN, KNZ, H( JROW, INCOL+1+J2 ),
$ LDH, WV( 1, 1+KZS ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL SLASET( 'ALL', JLEN, KZS, ZERO, ZERO, WV, LDWV )
CALL STRMM( 'R', 'U', 'N', 'N', JLEN, KNZ, ONE,
$ U( J2+1, 1+KZS ), LDU, WV( 1, 1+KZS ),
$ LDWV )
*
* ==== Multiply by U11 ====
*
CALL SGEMM( 'N', 'N', JLEN, I2, J2, ONE,
$ H( JROW, INCOL+1 ), LDH, U, LDU, ONE, WV,
$ LDWV )
*
* ==== Copy left of H to right of scratch ====
*
CALL SLACPY( 'ALL', JLEN, J2, H( JROW, INCOL+1 ), LDH,
$ WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL STRMM( 'R', 'L', 'N', 'N', JLEN, I4-I2, ONE,
$ U( 1, I2+1 ), LDU, WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U22 ====
*
CALL SGEMM( 'N', 'N', JLEN, I4-I2, J4-J2, ONE,
$ H( JROW, INCOL+1+J2 ), LDH,
$ U( J2+1, I2+1 ), LDU, ONE, WV( 1, 1+I2 ),
$ LDWV )
*
* ==== Copy it back ====
*
CALL SLACPY( 'ALL', JLEN, KDU, WV, LDWV,
$ H( JROW, INCOL+1 ), LDH )
200 CONTINUE
*
* ==== Multiply Z (also vertical) ====
*
IF( WANTZ ) THEN
DO 210 JROW = ILOZ, IHIZ, NV
JLEN = MIN( NV, IHIZ-JROW+1 )
*
* ==== Copy right of Z to left of scratch (first
* . KZS columns get multiplied by zero) ====
*
CALL SLACPY( 'ALL', JLEN, KNZ,
$ Z( JROW, INCOL+1+J2 ), LDZ,
$ WV( 1, 1+KZS ), LDWV )
*
* ==== Multiply by U12 ====
*
CALL SLASET( 'ALL', JLEN, KZS, ZERO, ZERO, WV,
$ LDWV )
CALL STRMM( 'R', 'U', 'N', 'N', JLEN, KNZ, ONE,
$ U( J2+1, 1+KZS ), LDU, WV( 1, 1+KZS ),
$ LDWV )
*
* ==== Multiply by U11 ====
*
CALL SGEMM( 'N', 'N', JLEN, I2, J2, ONE,
$ Z( JROW, INCOL+1 ), LDZ, U, LDU, ONE,
$ WV, LDWV )
*
* ==== Copy left of Z to right of scratch ====
*
CALL SLACPY( 'ALL', JLEN, J2, Z( JROW, INCOL+1 ),
$ LDZ, WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL STRMM( 'R', 'L', 'N', 'N', JLEN, I4-I2, ONE,
$ U( 1, I2+1 ), LDU, WV( 1, 1+I2 ),
$ LDWV )
*
* ==== Multiply by U22 ====
*
CALL SGEMM( 'N', 'N', JLEN, I4-I2, J4-J2, ONE,
$ Z( JROW, INCOL+1+J2 ), LDZ,
$ U( J2+1, I2+1 ), LDU, ONE,
$ WV( 1, 1+I2 ), LDWV )
*
* ==== Copy the result back to Z ====
*
CALL SLACPY( 'ALL', JLEN, KDU, WV, LDWV,
$ Z( JROW, INCOL+1 ), LDZ )
210 CONTINUE
END IF
END IF
END IF
220 CONTINUE
*
* ==== End of SLAQR5 ====
*

4
lapack-netlib/SRC/zhseqr.f
View File

@ -320,10 +320,10 @@
* . ZLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
PARAMETER ( NTINY = 15 )
*
* ==== NL allocates some local workspace to help small matrices
* . through a rare ZLAHQR failure. NL > NTINY = 11 is
* . through a rare ZLAHQR failure. NL > NTINY = 15 is
* . required and NL <= NMIN = ILAENV(ISPEC=12,...) is recom-
* . mended. (The default value of NMIN is 75.) Using NL = 49
* . allows up to six simultaneous shifts and a 16-by-16

16
lapack-netlib/SRC/zlaqr0.f
View File

@ -262,7 +262,7 @@
* . ZLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
PARAMETER ( NTINY = 15 )
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by varying the size of the
@ -357,22 +357,22 @@
END IF
*
* ==== NWR = recommended deflation window size. At this
* . point, N .GT. NTINY = 11, so there is enough
* . point, N .GT. NTINY = 15, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
* . NWR.GE.3.) ====
* . NWR.GE.4.) ====
*
NWR = ILAENV( 13, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
* ==== NSR = recommended number of simultaneous shifts.
* . At this point N .GT. NTINY = 11, so there is at
* . At this point N .GT. NTINY = 15, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'ZLAQR0', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
@ -420,7 +420,7 @@
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
@ -560,7 +560,7 @@
*
* ==== Got NS/2 or fewer shifts? Use ZLAQR4 or
* . ZLAHQR on a trailing principal submatrix to
* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
@ -661,7 +661,7 @@
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
KDU = 3*NS - 3
KDU = 2*NS
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1

16
lapack-netlib/SRC/zlaqr4.f
View File

@ -268,7 +268,7 @@
* . ZLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
PARAMETER ( NTINY = 15 )
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by varying the size of the
@ -363,22 +363,22 @@
END IF
*
* ==== NWR = recommended deflation window size. At this
* . point, N .GT. NTINY = 11, so there is enough
* . point, N .GT. NTINY = 15, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
* . NWR.GE.3.) ====
* . NWR.GE.4.) ====
*
NWR = ILAENV( 13, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
*
* ==== NSR = recommended number of simultaneous shifts.
* . At this point N .GT. NTINY = 11, so there is at
* . At this point N .GT. NTINY = 15, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'ZLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
NSR = MIN( NSR, ( N-3 ) / 6, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
@ -426,7 +426,7 @@
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
NSMAX = MIN( ( N-3 ) / 6, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
@ -566,7 +566,7 @@
*
* ==== Got NS/2 or fewer shifts? Use ZLAHQR
* . on a trailing principal submatrix to
* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
@ -661,7 +661,7 @@
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
KDU = 3*NS - 3
KDU = 2*NS
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1

616
lapack-netlib/SRC/zlaqr5.f
View File

@ -69,10 +69,9 @@
*> matrix entries.
*> = 1: ZLAQR5 accumulates reflections and uses matrix-matrix
*> multiply to update the far-from-diagonal matrix entries.
*> = 2: ZLAQR5 accumulates reflections, uses matrix-matrix
*> multiply to update the far-from-diagonal matrix entries,
*> and takes advantage of 2-by-2 block structure during
*> matrix multiplies.
*> = 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
@ -170,14 +169,14 @@
*>
*> \param[out] U
*> \verbatim
*> U is COMPLEX*16 array, dimension (LDU,3*NSHFTS-3)
*> U is COMPLEX*16 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 >= 3*NSHFTS-3.
*> in the calling subroutine. LDU >= 2*NSHFTS.
*> \endverbatim
*>
*> \param[in] NV
@ -189,7 +188,7 @@
*>
*> \param[out] WV
*> \verbatim
*> WV is COMPLEX*16 array, dimension (LDWV,3*NSHFTS-3)
*> WV is COMPLEX*16 array, dimension (LDWV,2*NSHFTS)
*> \endverbatim
*>
*> \param[in] LDWV
@ -215,7 +214,7 @@
*> \verbatim
*> LDWH is INTEGER
*> Leading dimension of WH just as declared in the
*> calling procedure. LDWH >= 3*NSHFTS-3.
*> calling procedure. LDWH >= 2*NSHFTS.
*> \endverbatim
*>
* Authors:
@ -226,7 +225,7 @@
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*> \date January 2021
*
*> \ingroup complex16OTHERauxiliary
*
@ -235,6 +234,11 @@
*>
*> 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:
* ================
@ -244,10 +248,15 @@
*> 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 ZLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, S,
$ H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U, LDU, NV,
$ WV, LDWV, NH, WH, LDWH )
IMPLICIT NONE
*
* -- LAPACK auxiliary routine (version 3.7.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
@ -276,11 +285,11 @@
COMPLEX*16 ALPHA, BETA, CDUM, REFSUM
DOUBLE PRECISION H11, H12, H21, H22, SAFMAX, SAFMIN, SCL,
$ SMLNUM, TST1, TST2, ULP
INTEGER I2, I4, INCOL, J, J2, J4, JBOT, JCOL, JLEN,
$ JROW, JTOP, K, K1, KDU, KMS, KNZ, KRCOL, KZS,
$ M, M22, MBOT, MEND, MSTART, MTOP, NBMPS, NDCOL,
INTEGER I2, I4, INCOL, J, JBOT, JCOL, JLEN,
$ JROW, JTOP, K, K1, KDU, KMS, KRCOL,
$ M, M22, MBOT, MTOP, NBMPS, NDCOL,
$ NS, NU
LOGICAL ACCUM, BLK22, BMP22
LOGICAL ACCUM, BMP22
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
@ -334,10 +343,6 @@
*
ACCUM = ( KACC22.EQ.1 ) .OR. ( KACC22.EQ.2 )
*
* ==== If so, exploit the 2-by-2 block structure? ====
*
BLK22 = ( NS.GT.2 ) .AND. ( KACC22.EQ.2 )
*
* ==== clear trash ====
*
IF( KTOP+2.LE.KBOT )
@ -349,28 +354,39 @@
*
* ==== KDU = width of slab ====
*
KDU = 6*NBMPS - 3
KDU = 4*NBMPS
*
* ==== Create and chase chains of NBMPS bulges ====
*
DO 210 INCOL = 3*( 1-NBMPS ) + KTOP - 1, KBOT - 2, 3*NBMPS - 2
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 ZLASET( '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 6*NBMPS-2 column diagonal
* . 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 3*NBMPS column long chain of
* . NBMPS bulges 3*NBMPS-2 columns to the right. (INCOL
* . 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 140 KRCOL = INCOL, MIN( INCOL+3*NBMPS-3, KBOT-2 )
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
@ -379,23 +395,155 @@
* . down the diagonal to make room. The phantom matrix
* . paradigm described above helps keep track. ====
*
MTOP = MAX( 1, ( ( KTOP-1 )-KRCOL+2 ) / 3+1 )
MBOT = MIN( NBMPS, ( KBOT-KRCOL ) / 3 )
MTOP = MAX( 1, ( KTOP-KRCOL ) / 2+1 )
MBOT = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 2 )
M22 = MBOT + 1
BMP22 = ( MBOT.LT.NBMPS ) .AND. ( KRCOL+3*( M22-1 ) ).EQ.
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.) ====
*
DO 10 M = MTOP, MBOT
K = KRCOL + 3*( M-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 ZLAQR1( 2, H( K+1, K+1 ), LDH, S( 2*M22-1 ),
$ S( 2*M22 ), V( 1, M22 ) )
BETA = V( 1, M22 )
CALL ZLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
ELSE
BETA = H( K+1, K )
V( 2, M22 ) = H( K+2, K )
CALL ZLARFG( 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. ====
*
DO 30 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M22 )*( H( J, K+1 )+V( 2, M22 )*
$ H( J, K+2 ) )
H( J, K+1 ) = H( J, K+1 ) - REFSUM
H( J, K+2 ) = H( J, K+2 ) -
$ REFSUM*DCONJG( V( 2, M22 ) )
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
DO 40 J = K+1, JBOT
REFSUM = DCONJG( V( 1, M22 ) )*
$ ( H( K+1, J )+DCONJG( V( 2, M22 ) )*
$ H( K+2, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M22 )
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 = CABS1( H( K, K ) ) + CABS1( H( K+1, K+1 ) )
IF( TST1.EQ.RZERO ) THEN
IF( K.GE.KTOP+1 )
$ TST1 = TST1 + CABS1( H( K, K-1 ) )
IF( K.GE.KTOP+2 )
$ TST1 = TST1 + CABS1( H( K, K-2 ) )
IF( K.GE.KTOP+3 )
$ TST1 = TST1 + CABS1( H( K, K-3 ) )
IF( K.LE.KBOT-2 )
$ TST1 = TST1 + CABS1( H( K+2, K+1 ) )
IF( K.LE.KBOT-3 )
$ TST1 = TST1 + CABS1( H( K+3, K+1 ) )
IF( K.LE.KBOT-4 )
$ TST1 = TST1 + CABS1( H( K+4, K+1 ) )
END IF
IF( CABS1( H( K+1, K ) )
$ .LE.MAX( SMLNUM, ULP*TST1 ) ) THEN
H12 = MAX( CABS1( H( K+1, K ) ),
$ CABS1( H( K, K+1 ) ) )
H21 = MIN( CABS1( H( K+1, K ) ),
$ CABS1( H( K, K+1 ) ) )
H11 = MAX( CABS1( H( K+1, K+1 ) ),
$ CABS1( H( K, K )-H( K+1, K+1 ) ) )
H22 = MIN( CABS1( H( K+1, K+1 ) ),
$ CABS1( H( K, K )-H( K+1, K+1 ) ) )
SCL = H11 + H12
TST2 = H22*( H11 / SCL )
*
IF( TST2.EQ.RZERO .OR. H21*( H12 / SCL ).LE.
$ MAX( SMLNUM, ULP*TST2 ) )H( K+1, K ) = ZERO
END IF
END IF
END IF
*
* ==== Accumulate orthogonal transformations. ====
*
IF( ACCUM ) THEN
KMS = K - INCOL
DO 50 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M22 )*( U( J, KMS+1 )+
$ V( 2, M22 )*U( J, KMS+2 ) )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM
U( J, KMS+2 ) = U( J, KMS+2 ) -
$ REFSUM*DCONJG( V( 2, M22 ) )
50 CONTINUE
ELSE IF( WANTZ ) THEN
DO 60 J = ILOZ, IHIZ
REFSUM = V( 1, M22 )*( Z( J, K+1 )+V( 2, M22 )*
$ Z( J, K+2 ) )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM
Z( J, K+2 ) = Z( J, K+2 ) -
$ REFSUM*DCONJG( V( 2, M22 ) )
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 ZLAQR1( 3, H( KTOP, KTOP ), LDH, S( 2*M-1 ),
$ S( 2*M ), V( 1, M ) )
ALPHA = V( 1, M )
CALL ZLARFG( 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. ====
*
REFSUM = V( 1, M )*V( 3, M )*H( K+3, K+2 )
H( K+3, K ) = -REFSUM
H( K+3, K+1 ) = -REFSUM*DCONJG( V( 2, M ) )
H( K+3, K+2 ) = H( K+3, K+2 ) -
$ REFSUM*DCONJG( V( 3, M ) )
*
* ==== 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 )
@ -444,7 +592,7 @@
H( K+3, K ) = ZERO
ELSE
*
* ==== Stating a new bulge here would
* ==== Starting a new bulge here would
* . create only negligible fill.
* . Replace the old reflector with
* . the new one. ====
@ -458,73 +606,14 @@
END IF
END IF
END IF
10 CONTINUE
*
* ==== Generate a 2-by-2 reflection, if needed. ====
* ==== 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. ====
*
K = KRCOL + 3*( M22-1 )
IF( BMP22 ) THEN
IF( K.EQ.KTOP-1 ) THEN
CALL ZLAQR1( 2, H( K+1, K+1 ), LDH, S( 2*M22-1 ),
$ S( 2*M22 ), V( 1, M22 ) )
BETA = V( 1, M22 )
CALL ZLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
ELSE
BETA = H( K+1, K )
V( 2, M22 ) = H( K+2, K )
CALL ZLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
H( K+1, K ) = BETA
H( K+2, K ) = ZERO
END IF
END IF
*
* ==== 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 30 J = MAX( KTOP, KRCOL ), JBOT
MEND = MIN( MBOT, ( J-KRCOL+2 ) / 3 )
DO 20 M = MTOP, MEND
K = KRCOL + 3*( M-1 )
REFSUM = DCONJG( V( 1, M ) )*
$ ( H( K+1, J )+DCONJG( V( 2, M ) )*
$ H( K+2, J )+DCONJG( V( 3, M ) )*H( K+3, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M )
H( K+3, J ) = H( K+3, J ) - REFSUM*V( 3, M )
20 CONTINUE
30 CONTINUE
IF( BMP22 ) THEN
K = KRCOL + 3*( M22-1 )
DO 40 J = MAX( K+1, KTOP ), JBOT
REFSUM = DCONJG( V( 1, M22 ) )*
$ ( H( K+1, J )+DCONJG( V( 2, M22 ) )*
$ H( K+2, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M22 )
40 CONTINUE
END IF
*
* ==== Multiply H by reflections from the right.
* . Delay filling in the last row until the
* . vigilant deflation check is complete. ====
*
IF( ACCUM ) THEN
JTOP = MAX( KTOP, INCOL )
ELSE IF( WANTT ) THEN
JTOP = 1
ELSE
JTOP = KTOP
END IF
DO 80 M = MTOP, MBOT
IF( V( 1, M ).NE.ZERO ) THEN
K = KRCOL + 3*( M-1 )
DO 50 J = JTOP, MIN( KBOT, K+3 )
DO 70 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M )*( 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
@ -532,89 +621,17 @@
$ REFSUM*DCONJG( V( 2, M ) )
H( J, K+3 ) = H( J, K+3 ) -
$ REFSUM*DCONJG( V( 3, M ) )
50 CONTINUE
*
IF( ACCUM ) THEN
*
* ==== Accumulate U. (If necessary, update Z later
* . with with an efficient matrix-matrix
* . multiply.) ====
*
KMS = K - INCOL
DO 60 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M )*( 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
U( J, KMS+2 ) = U( J, KMS+2 ) -
$ REFSUM*DCONJG( V( 2, M ) )
U( J, KMS+3 ) = U( J, KMS+3 ) -
$ REFSUM*DCONJG( V( 3, M ) )
60 CONTINUE
ELSE IF( WANTZ ) THEN
*
* ==== U is not accumulated, so update Z
* . now by multiplying by reflections
* . from the right. ====
*
DO 70 J = ILOZ, IHIZ
REFSUM = V( 1, M )*( 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
Z( J, K+2 ) = Z( J, K+2 ) -
$ REFSUM*DCONJG( V( 2, M ) )
Z( J, K+3 ) = Z( J, K+3 ) -
$ REFSUM*DCONJG( V( 3, M ) )
70 CONTINUE
END IF
END IF
80 CONTINUE
*
* ==== Special case: 2-by-2 reflection (if needed) ====
* ==== Perform update from left for subsequent
* . column. ====
*
K = KRCOL + 3*( M22-1 )
IF( BMP22 ) THEN
IF ( V( 1, M22 ).NE.ZERO ) THEN
DO 90 J = JTOP, MIN( KBOT, K+3 )
REFSUM = V( 1, M22 )*( H( J, K+1 )+V( 2, M22 )*
$ H( J, K+2 ) )
H( J, K+1 ) = H( J, K+1 ) - REFSUM
H( J, K+2 ) = H( J, K+2 ) -
$ REFSUM*DCONJG( V( 2, M22 ) )
90 CONTINUE
*
IF( ACCUM ) THEN
KMS = K - INCOL
DO 100 J = MAX( 1, KTOP-INCOL ), KDU
REFSUM = V( 1, M22 )*( U( J, KMS+1 )+
$ V( 2, M22 )*U( J, KMS+2 ) )
U( J, KMS+1 ) = U( J, KMS+1 ) - REFSUM
U( J, KMS+2 ) = U( J, KMS+2 ) -
$ REFSUM*DCONJG( V( 2, M22 ) )
100 CONTINUE
ELSE IF( WANTZ ) THEN
DO 110 J = ILOZ, IHIZ
REFSUM = V( 1, M22 )*( Z( J, K+1 )+V( 2, M22 )*
$ Z( J, K+2 ) )
Z( J, K+1 ) = Z( J, K+1 ) - REFSUM
Z( J, K+2 ) = Z( J, K+2 ) -
$ REFSUM*DCONJG( V( 2, M22 ) )
110 CONTINUE
END IF
END IF
END IF
*
* ==== Vigilant deflation check ====
*
MSTART = MTOP
IF( KRCOL+3*( MSTART-1 ).LT.KTOP )
$ MSTART = MSTART + 1
MEND = MBOT
IF( BMP22 )
$ MEND = MEND + 1
IF( KRCOL.EQ.KBOT-2 )
$ MEND = MEND + 1
DO 120 M = MSTART, MEND
K = MIN( KBOT-1, KRCOL+3*( M-1 ) )
REFSUM = DCONJG( V( 1, M ) )*( H( K+1, K+1 )
$ +DCONJG( V( 2, M ) )*H( K+2, K+1 )
$ +DCONJG( V( 3, M ) )*H( K+3, K+1 ) )
H( K+1, K+1 ) = H( K+1, K+1 ) - REFSUM
H( K+2, K+1 ) = H( K+2, K+1 ) - REFSUM*V( 2, M )
H( K+3, K+1 ) = H( K+3, K+1 ) - REFSUM*V( 3, M )
*
* ==== The following convergence test requires that
* . the tradition small-compared-to-nearby-diagonals
@ -625,6 +642,8 @@
* . is zero (as done here) is traditional but probably
* . unnecessary. ====
*
IF( K.LT.KTOP)
$ CYCLE
IF( H( K+1, K ).NE.ZERO ) THEN
TST1 = CABS1( H( K, K ) ) + CABS1( H( K+1, K+1 ) )
IF( TST1.EQ.RZERO ) THEN
@ -658,23 +677,77 @@
$ MAX( SMLNUM, ULP*TST2 ) )H( K+1, K ) = ZERO
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 )
DO 90 J = MAX( KTOP, KRCOL + 2*M ), JBOT
REFSUM = DCONJG( V( 1, M ) )*
$ ( H( K+1, J )+DCONJG( V( 2, M ) )*
$ H( K+2, J )+DCONJG( V( 3, M ) )*H( K+3, J ) )
H( K+1, J ) = H( K+1, J ) - REFSUM
H( K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M )
H( K+3, J ) = H( K+3, J ) - REFSUM*V( 3, M )
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 )
DO 110 J = I2, I4
REFSUM = V( 1, M )*( 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
U( J, KMS+2 ) = U( J, KMS+2 ) -
$ REFSUM*DCONJG( V( 2, M ) )
U( J, KMS+3 ) = U( J, KMS+3 ) -
$ REFSUM*DCONJG( V( 3, M ) )
110 CONTINUE
120 CONTINUE
ELSE IF( WANTZ ) THEN
*
* ==== Fill in the last row of each bulge. ====
* ==== U is not accumulated, so update Z
* . now by multiplying by reflections
* . from the right. ====
*
MEND = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 3 )
DO 130 M = MTOP, MEND
K = KRCOL + 3*( M-1 )
REFSUM = V( 1, M )*V( 3, M )*H( K+4, K+3 )
H( K+4, K+1 ) = -REFSUM
H( K+4, K+2 ) = -REFSUM*DCONJG( V( 2, M ) )
H( K+4, K+3 ) = H( K+4, K+3 ) -
DO 140 M = MBOT, MTOP, -1
K = KRCOL + 2*( M-1 )
DO 130 J = ILOZ, IHIZ
REFSUM = V( 1, M )*( 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
Z( J, K+2 ) = Z( J, K+2 ) -
$ REFSUM*DCONJG( V( 2, M ) )
Z( J, K+3 ) = Z( J, K+3 ) -
$ REFSUM*DCONJG( V( 3, M ) )
130 CONTINUE
140 CONTINUE
END IF
*
* ==== End of near-the-diagonal bulge chase. ====
*
140 CONTINUE
145 CONTINUE
*
* ==== Use U (if accumulated) to update far-from-diagonal
* . entries in H. If required, use U to update Z as
@ -688,18 +761,6 @@
JTOP = KTOP
JBOT = KBOT
END IF
IF( ( .NOT.BLK22 ) .OR. ( INCOL.LT.KTOP ) .OR.
$ ( NDCOL.GT.KBOT ) .OR. ( NS.LE.2 ) ) THEN
*
* ==== Updates not exploiting the 2-by-2 block
* . structure of U. K1 and NU keep track of
* . the location and size of U in the special
* . cases of introducing bulges and chasing
* . bulges off the bottom. In these special
* . cases and in case the number of shifts
* . is NS = 2, there is no 2-by-2 block
* . structure to exploit. ====
*
K1 = MAX( 1, KTOP-INCOL )
NU = ( KDU-MAX( 0, NDCOL-KBOT ) ) - K1 + 1
*
@ -737,172 +798,9 @@
$ Z( JROW, INCOL+K1 ), LDZ )
170 CONTINUE
END IF
ELSE
*
* ==== Updates exploiting U's 2-by-2 block structure.
* . (I2, I4, J2, J4 are the last rows and columns
* . of the blocks.) ====
*
I2 = ( KDU+1 ) / 2
I4 = KDU
J2 = I4 - I2
J4 = KDU
*
* ==== KZS and KNZ deal with the band of zeros
* . along the diagonal of one of the triangular
* . blocks. ====
*
KZS = ( J4-J2 ) - ( NS+1 )
KNZ = NS + 1
*
* ==== Horizontal multiply ====
*
DO 180 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
JLEN = MIN( NH, JBOT-JCOL+1 )
*
* ==== Copy bottom of H to top+KZS of scratch ====
* (The first KZS rows get multiplied by zero.) ====
*
CALL ZLACPY( 'ALL', KNZ, JLEN, H( INCOL+1+J2, JCOL ),
$ LDH, WH( KZS+1, 1 ), LDWH )
*
* ==== Multiply by U21**H ====
*
CALL ZLASET( 'ALL', KZS, JLEN, ZERO, ZERO, WH, LDWH )
CALL ZTRMM( 'L', 'U', 'C', 'N', KNZ, JLEN, ONE,
$ U( J2+1, 1+KZS ), LDU, WH( KZS+1, 1 ),
$ LDWH )
*
* ==== Multiply top of H by U11**H ====
*
CALL ZGEMM( 'C', 'N', I2, JLEN, J2, ONE, U, LDU,
$ H( INCOL+1, JCOL ), LDH, ONE, WH, LDWH )
*
* ==== Copy top of H to bottom of WH ====
*
CALL ZLACPY( 'ALL', J2, JLEN, H( INCOL+1, JCOL ), LDH,
$ WH( I2+1, 1 ), LDWH )
*
* ==== Multiply by U21**H ====
*
CALL ZTRMM( 'L', 'L', 'C', 'N', J2, JLEN, ONE,
$ U( 1, I2+1 ), LDU, WH( I2+1, 1 ), LDWH )
*
* ==== Multiply by U22 ====
*
CALL ZGEMM( 'C', 'N', I4-I2, JLEN, J4-J2, ONE,
$ U( J2+1, I2+1 ), LDU,
$ H( INCOL+1+J2, JCOL ), LDH, ONE,
$ WH( I2+1, 1 ), LDWH )
*
* ==== Copy it back ====
*
CALL ZLACPY( 'ALL', KDU, JLEN, WH, LDWH,
$ H( INCOL+1, JCOL ), LDH )
END IF
180 CONTINUE
*
* ==== Vertical multiply ====
*
DO 190 JROW = JTOP, MAX( INCOL, KTOP ) - 1, NV
JLEN = MIN( NV, MAX( INCOL, KTOP )-JROW )
*
* ==== Copy right of H to scratch (the first KZS
* . columns get multiplied by zero) ====
*
CALL ZLACPY( 'ALL', JLEN, KNZ, H( JROW, INCOL+1+J2 ),
$ LDH, WV( 1, 1+KZS ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL ZLASET( 'ALL', JLEN, KZS, ZERO, ZERO, WV, LDWV )
CALL ZTRMM( 'R', 'U', 'N', 'N', JLEN, KNZ, ONE,
$ U( J2+1, 1+KZS ), LDU, WV( 1, 1+KZS ),
$ LDWV )
*
* ==== Multiply by U11 ====
*
CALL ZGEMM( 'N', 'N', JLEN, I2, J2, ONE,
$ H( JROW, INCOL+1 ), LDH, U, LDU, ONE, WV,
$ LDWV )
*
* ==== Copy left of H to right of scratch ====
*
CALL ZLACPY( 'ALL', JLEN, J2, H( JROW, INCOL+1 ), LDH,
$ WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL ZTRMM( 'R', 'L', 'N', 'N', JLEN, I4-I2, ONE,
$ U( 1, I2+1 ), LDU, WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U22 ====
*
CALL ZGEMM( 'N', 'N', JLEN, I4-I2, J4-J2, ONE,
$ H( JROW, INCOL+1+J2 ), LDH,
$ U( J2+1, I2+1 ), LDU, ONE, WV( 1, 1+I2 ),
$ LDWV )
*
* ==== Copy it back ====
*
CALL ZLACPY( 'ALL', JLEN, KDU, WV, LDWV,
$ H( JROW, INCOL+1 ), LDH )
190 CONTINUE
*
* ==== Multiply Z (also vertical) ====
*
IF( WANTZ ) THEN
DO 200 JROW = ILOZ, IHIZ, NV
JLEN = MIN( NV, IHIZ-JROW+1 )
*
* ==== Copy right of Z to left of scratch (first
* . KZS columns get multiplied by zero) ====
*
CALL ZLACPY( 'ALL', JLEN, KNZ,
$ Z( JROW, INCOL+1+J2 ), LDZ,
$ WV( 1, 1+KZS ), LDWV )
*
* ==== Multiply by U12 ====
*
CALL ZLASET( 'ALL', JLEN, KZS, ZERO, ZERO, WV,
$ LDWV )
CALL ZTRMM( 'R', 'U', 'N', 'N', JLEN, KNZ, ONE,
$ U( J2+1, 1+KZS ), LDU, WV( 1, 1+KZS ),
$ LDWV )
*
* ==== Multiply by U11 ====
*
CALL ZGEMM( 'N', 'N', JLEN, I2, J2, ONE,
$ Z( JROW, INCOL+1 ), LDZ, U, LDU, ONE,
$ WV, LDWV )
*
* ==== Copy left of Z to right of scratch ====
*
CALL ZLACPY( 'ALL', JLEN, J2, Z( JROW, INCOL+1 ),
$ LDZ, WV( 1, 1+I2 ), LDWV )
*
* ==== Multiply by U21 ====
*
CALL ZTRMM( 'R', 'L', 'N', 'N', JLEN, I4-I2, ONE,
$ U( 1, I2+1 ), LDU, WV( 1, 1+I2 ),
$ LDWV )
*
* ==== Multiply by U22 ====
*
CALL ZGEMM( 'N', 'N', JLEN, I4-I2, J4-J2, ONE,
$ Z( JROW, INCOL+1+J2 ), LDZ,
$ U( J2+1, I2+1 ), LDU, ONE,
$ WV( 1, 1+I2 ), LDWV )
*
* ==== Copy the result back to Z ====
*
CALL ZLACPY( 'ALL', JLEN, KDU, WV, LDWV,
$ Z( JROW, INCOL+1 ), LDZ )
200 CONTINUE
END IF
END IF
END IF
210 CONTINUE
*
* ==== End of ZLAQR5 ====
*
END