*> \brief \b SLAQR5 performs a single small-bulge multi-shift QR sweep. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLAQR5 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * 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 ) * * .. Scalar Arguments .. * INTEGER IHIZ, ILOZ, KACC22, KBOT, KTOP, LDH, LDU, LDV, * $ LDWH, LDWV, LDZ, N, NH, NSHFTS, NV * LOGICAL WANTT, WANTZ * .. * .. Array Arguments .. * REAL H( LDH, * ), SI( * ), SR( * ), U( LDU, * ), * $ V( LDV, * ), WH( LDWH, * ), WV( LDWV, * ), * $ Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLAQR5, called by SLAQR0, performs a *> single small-bulge multi-shift QR sweep. *> \endverbatim * * Arguments: * ========== * *> \param[in] WANTT *> \verbatim *> WANTT is LOGICAL *> WANTT = .true. if the quasi-triangular Schur factor *> is being computed. WANTT is set to .false. otherwise. *> \endverbatim *> *> \param[in] WANTZ *> \verbatim *> WANTZ is LOGICAL *> WANTZ = .true. if the orthogonal Schur factor is being *> computed. WANTZ is set to .false. otherwise. *> \endverbatim *> *> \param[in] KACC22 *> \verbatim *> KACC22 is INTEGER with value 0, 1, or 2. *> Specifies the computation mode of far-from-diagonal *> orthogonal updates. *> = 0: SLAQR5 does not accumulate reflections and does not *> use matrix-matrix multiply to update far-from-diagonal *> matrix entries. *> = 1: SLAQR5 accumulates reflections and uses matrix-matrix *> multiply to update the far-from-diagonal matrix entries. *> = 2: Same as KACC22 = 1. This option used to enable exploiting *> the 2-by-2 structure during matrix multiplications, but *> this is no longer supported. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> N is the order of the Hessenberg matrix H upon which this *> subroutine operates. *> \endverbatim *> *> \param[in] KTOP *> \verbatim *> KTOP is INTEGER *> \endverbatim *> *> \param[in] KBOT *> \verbatim *> KBOT is INTEGER *> These are the first and last rows and columns of an *> isolated diagonal block upon which the QR sweep is to be *> applied. It is assumed without a check that *> either KTOP = 1 or H(KTOP,KTOP-1) = 0 *> and *> either KBOT = N or H(KBOT+1,KBOT) = 0. *> \endverbatim *> *> \param[in] NSHFTS *> \verbatim *> NSHFTS is INTEGER *> NSHFTS gives the number of simultaneous shifts. NSHFTS *> must be positive and even. *> \endverbatim *> *> \param[in,out] SR *> \verbatim *> SR is REAL array, dimension (NSHFTS) *> \endverbatim *> *> \param[in,out] SI *> \verbatim *> SI is REAL array, dimension (NSHFTS) *> SR contains the real parts and SI contains the imaginary *> parts of the NSHFTS shifts of origin that define the *> multi-shift QR sweep. On output SR and SI may be *> reordered. *> \endverbatim *> *> \param[in,out] H *> \verbatim *> H is REAL array, dimension (LDH,N) *> On input H contains a Hessenberg matrix. On output a *> multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied *> to the isolated diagonal block in rows and columns KTOP *> through KBOT. *> \endverbatim *> *> \param[in] LDH *> \verbatim *> LDH is INTEGER *> LDH is the leading dimension of H just as declared in the *> calling procedure. LDH >= MAX(1,N). *> \endverbatim *> *> \param[in] ILOZ *> \verbatim *> ILOZ is INTEGER *> \endverbatim *> *> \param[in] IHIZ *> \verbatim *> IHIZ is INTEGER *> Specify the rows of Z to which transformations must be *> applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N *> \endverbatim *> *> \param[in,out] Z *> \verbatim *> Z is REAL array, dimension (LDZ,IHIZ) *> If WANTZ = .TRUE., then the QR Sweep orthogonal *> similarity transformation is accumulated into *> Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. *> If WANTZ = .FALSE., then Z is unreferenced. *> \endverbatim *> *> \param[in] LDZ *> \verbatim *> LDZ is INTEGER *> LDA is the leading dimension of Z just as declared in *> the calling procedure. LDZ >= N. *> \endverbatim *> *> \param[out] V *> \verbatim *> V is REAL array, dimension (LDV,NSHFTS/2) *> \endverbatim *> *> \param[in] LDV *> \verbatim *> LDV is INTEGER *> LDV is the leading dimension of V as declared in the *> calling procedure. LDV >= 3. *> \endverbatim *> *> \param[out] U *> \verbatim *> U is 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 >= 2*NSHFTS. *> \endverbatim *> *> \param[in] NV *> \verbatim *> NV is INTEGER *> NV is the number of rows in WV agailable for workspace. *> NV >= 1. *> \endverbatim *> *> \param[out] WV *> \verbatim *> WV is REAL array, dimension (LDWV,2*NSHFTS) *> \endverbatim *> *> \param[in] LDWV *> \verbatim *> LDWV is INTEGER *> LDWV is the leading dimension of WV as declared in the *> in the calling subroutine. LDWV >= NV. *> \endverbatim * *> \param[in] NH *> \verbatim *> NH is INTEGER *> NH is the number of columns in array WH available for *> workspace. NH >= 1. *> \endverbatim *> *> \param[out] WH *> \verbatim *> WH is REAL array, dimension (LDWH,NH) *> \endverbatim *> *> \param[in] LDWH *> \verbatim *> LDWH is INTEGER *> Leading dimension of WH just as declared in the *> calling procedure. LDWH >= 2*NSHFTS. *> \endverbatim *> * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup realOTHERauxiliary * *> \par Contributors: * ================== *> *> Karen Braman and Ralph Byers, Department of Mathematics, *> University of Kansas, USA *> *> Lars Karlsson, Daniel Kressner, and Bruno Lang *> *> Thijs Steel, Department of Computer science, *> KU Leuven, Belgium * *> \par References: * ================ *> *> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR *> Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 *> Performance, SIAM Journal of Matrix Analysis, volume 23, pages *> 929--947, 2002. *> *> Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed *> chains of bulges in multishift QR algorithms. *> ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014). *> * ===================================================================== SUBROUTINE 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 -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. INTEGER IHIZ, ILOZ, KACC22, KBOT, KTOP, LDH, LDU, LDV, $ LDWH, LDWV, LDZ, N, NH, NSHFTS, NV LOGICAL WANTT, WANTZ * .. * .. Array Arguments .. REAL H( LDH, * ), SI( * ), SR( * ), U( LDU, * ), $ V( LDV, * ), WH( LDWH, * ), WV( LDWV, * ), $ Z( LDZ, * ) * .. * * ================================================================ * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0e0, ONE = 1.0e0 ) * .. * .. Local Scalars .. REAL ALPHA, BETA, H11, H12, H21, H22, REFSUM, $ SAFMAX, SAFMIN, SCL, SMLNUM, SWAP, TST1, TST2, $ ULP INTEGER I, I2, I4, INCOL, J, JBOT, JCOL, JLEN, $ JROW, JTOP, K, K1, KDU, KMS, KRCOL, $ M, M22, MBOT, MTOP, NBMPS, NDCOL, $ NS, NU LOGICAL ACCUM, BMP22 * .. * .. External Functions .. REAL SLAMCH EXTERNAL SLAMCH * .. * .. Intrinsic Functions .. * INTRINSIC ABS, MAX, MIN, MOD, REAL * .. * .. Local Arrays .. REAL VT( 3 ) * .. * .. External Subroutines .. EXTERNAL SGEMM, SLABAD, SLACPY, SLAQR1, SLARFG, SLASET, $ STRMM * .. * .. Executable Statements .. * * ==== If there are no shifts, then there is nothing to do. ==== * IF( NSHFTS.LT.2 ) $ RETURN * * ==== If the active block is empty or 1-by-1, then there * . is nothing to do. ==== * IF( KTOP.GE.KBOT ) $ RETURN * * ==== Shuffle shifts into pairs of real shifts and pairs * . of complex conjugate shifts assuming complex * . conjugate shifts are already adjacent to one * . another. ==== * DO 10 I = 1, NSHFTS - 2, 2 IF( SI( I ).NE.-SI( I+1 ) ) THEN * SWAP = SR( I ) SR( I ) = SR( I+1 ) SR( I+1 ) = SR( I+2 ) SR( I+2 ) = SWAP * SWAP = SI( I ) SI( I ) = SI( I+1 ) SI( I+1 ) = SI( I+2 ) SI( I+2 ) = SWAP END IF 10 CONTINUE * * ==== NSHFTS is supposed to be even, but if it is odd, * . then simply reduce it by one. The shuffle above * . ensures that the dropped shift is real and that * . the remaining shifts are paired. ==== * NS = NSHFTS - MOD( NSHFTS, 2 ) * * ==== Machine constants for deflation ==== * SAFMIN = SLAMCH( 'SAFE MINIMUM' ) SAFMAX = ONE / SAFMIN CALL SLABAD( SAFMIN, SAFMAX ) ULP = SLAMCH( 'PRECISION' ) SMLNUM = SAFMIN*( REAL( N ) / ULP ) * * ==== Use accumulated reflections to update far-from-diagonal * . entries ? ==== * ACCUM = ( KACC22.EQ.1 ) .OR. ( KACC22.EQ.2 ) * * ==== clear trash ==== * IF( KTOP+2.LE.KBOT ) $ H( KTOP+2, KTOP ) = ZERO * * ==== NBMPS = number of 2-shift bulges in the chain ==== * NBMPS = NS / 2 * * ==== KDU = width of slab ==== * KDU = 4*NBMPS * * ==== Create and chase chains of NBMPS bulges ==== * DO 180 INCOL = KTOP - 2*NBMPS + 1, KBOT - 2, 2*NBMPS * * JTOP = Index from which updates from the right start. * IF( ACCUM ) THEN JTOP = MAX( KTOP, INCOL ) ELSE IF( WANTT ) THEN JTOP = 1 ELSE JTOP = KTOP END IF * NDCOL = INCOL + KDU IF( ACCUM ) $ CALL 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 4*NBMPS column diagonal * . chunk extends from column INCOL to column NDCOL * . (including both column INCOL and column NDCOL). The * . following loop chases a 2*NBMPS+1 column long chain of * . NBMPS bulges 2*NBMPS-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 145 KRCOL = INCOL, MIN( INCOL+2*NBMPS-1, KBOT-2 ) * * ==== Bulges number MTOP to MBOT are active double implicit * . shift bulges. There may or may not also be small * . 2-by-2 bulge, if there is room. The inactive bulges * . (if any) must wait until the active bulges have moved * . down the diagonal to make room. The phantom matrix * . paradigm described above helps keep track. ==== * MTOP = MAX( 1, ( KTOP-KRCOL ) / 2+1 ) MBOT = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 2 ) M22 = MBOT + 1 BMP22 = ( MBOT.LT.NBMPS ) .AND. ( KRCOL+2*( M22-1 ) ).EQ. $ ( KBOT-2 ) * * ==== Generate reflections to chase the chain right * . one column. (The minimum value of K is KTOP-1.) ==== * IF ( BMP22 ) THEN * * ==== Special case: 2-by-2 reflection at bottom treated * . separately ==== * K = KRCOL + 2*( M22-1 ) IF( K.EQ.KTOP-1 ) THEN CALL 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 ), $ V( 1, M ) ) 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 ) CALL SLARFG( 3, BETA, V( 2, M ), 1, V( 1, M ) ) * * ==== A Bulge may collapse because of vigilant * . deflation or destructive underflow. In the * . underflow case, try the two-small-subdiagonals * . trick to try to reinflate the bulge. ==== * IF( H( K+3, K ).NE.ZERO .OR. H( K+3, K+1 ).NE. $ ZERO .OR. H( K+3, K+2 ).EQ.ZERO ) THEN * * ==== Typical case: not collapsed (yet). ==== * H( K+1, K ) = BETA H( K+2, K ) = ZERO H( K+3, K ) = ZERO ELSE * * ==== Atypical case: collapsed. Attempt to * . reintroduce ignoring H(K+1,K) and H(K+2,K). * . If the fill resulting from the new * . reflector is too large, then abandon it. * . Otherwise, use the new one. ==== * CALL SLAQR1( 3, H( K+1, K+1 ), LDH, SR( 2*M-1 ), $ SI( 2*M-1 ), SR( 2*M ), SI( 2*M ), $ VT ) ALPHA = VT( 1 ) CALL SLARFG( 3, ALPHA, VT( 2 ), 1, VT( 1 ) ) REFSUM = VT( 1 )*( H( K+1, K )+VT( 2 )* $ H( K+2, K ) ) * IF( ABS( H( K+2, K )-REFSUM*VT( 2 ) )+ $ ABS( REFSUM*VT( 3 ) ).GT.ULP* $ ( ABS( H( K, K ) )+ABS( H( K+1, $ K+1 ) )+ABS( H( K+2, K+2 ) ) ) ) THEN * * ==== Starting a new bulge here would * . create non-negligible fill. Use * . the old one with trepidation. ==== * H( K+1, K ) = BETA H( K+2, K ) = ZERO H( K+3, K ) = ZERO ELSE * * ==== Starting a new bulge here would * . create only negligible fill. * . Replace the old reflector with * . the new one. ==== * H( K+1, K ) = H( K+1, K ) - REFSUM H( K+2, K ) = ZERO H( K+3, K ) = ZERO V( 1, M ) = VT( 1 ) V( 2, M ) = VT( 2 ) V( 3, M ) = VT( 3 ) END IF END IF END IF * * ==== Apply reflection from the right and * . the first column of update from the left. * . These updates are required for the vigilant * . deflation check. We still delay most of the * . updates from the left for efficiency. ==== * 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 ) 70 CONTINUE * * ==== Perform update from left for subsequent * . column. ==== * 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 * . criterion and the Ahues & Tisseur (LAWN 122, 1997) * . criteria both be satisfied. The latter improves * . accuracy in some examples. Falling back on an * . alternate convergence criterion when TST1 or TST2 * . is zero (as done here) is traditional but probably * . unnecessary. ==== * IF( K.LT.KTOP) $ CYCLE IF( H( K+1, K ).NE.ZERO ) THEN TST1 = ABS( H( K, K ) ) + ABS( H( K+1, K+1 ) ) IF( TST1.EQ.ZERO ) THEN IF( K.GE.KTOP+1 ) $ TST1 = TST1 + ABS( H( K, K-1 ) ) IF( K.GE.KTOP+2 ) $ TST1 = TST1 + ABS( H( K, K-2 ) ) IF( K.GE.KTOP+3 ) $ TST1 = TST1 + ABS( H( K, K-3 ) ) IF( K.LE.KBOT-2 ) $ TST1 = TST1 + ABS( H( K+2, K+1 ) ) IF( K.LE.KBOT-3 ) $ TST1 = TST1 + ABS( H( K+3, K+1 ) ) IF( K.LE.KBOT-4 ) $ TST1 = TST1 + ABS( H( K+4, K+1 ) ) END IF IF( ABS( H( K+1, K ) ).LE.MAX( SMLNUM, ULP*TST1 ) ) $ THEN H12 = MAX( ABS( H( K+1, K ) ), ABS( H( K, K+1 ) ) ) H21 = MIN( ABS( H( K+1, K ) ), ABS( H( K, K+1 ) ) ) H11 = MAX( ABS( H( K+1, K+1 ) ), $ ABS( H( K, K )-H( K+1, K+1 ) ) ) H22 = MIN( ABS( H( K+1, K+1 ) ), $ ABS( H( K, K )-H( K+1, K+1 ) ) ) SCL = H11 + H12 TST2 = H22*( H11 / SCL ) * IF( TST2.EQ.ZERO .OR. H21*( H12 / SCL ).LE. $ MAX( SMLNUM, ULP*TST2 ) ) THEN H( K+1, K ) = ZERO END IF END IF END IF 80 CONTINUE * * ==== Multiply H by reflections from the left ==== * IF( ACCUM ) THEN JBOT = MIN( NDCOL, KBOT ) ELSE IF( WANTT ) THEN JBOT = N ELSE JBOT = KBOT END IF * DO 100 M = MBOT, MTOP, -1 K = KRCOL + 2*( M-1 ) 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 140 CONTINUE END IF * * ==== End of near-the-diagonal bulge chase. ==== * 145 CONTINUE * * ==== Use U (if accumulated) to update far-from-diagonal * . entries in H. If required, use U to update Z as * . well. ==== * IF( ACCUM ) THEN IF( WANTT ) THEN JTOP = 1 JBOT = N ELSE JTOP = KTOP JBOT = KBOT END IF K1 = MAX( 1, KTOP-INCOL ) NU = ( KDU-MAX( 0, NDCOL-KBOT ) ) - K1 + 1 * * ==== Horizontal Multiply ==== * DO 150 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH JLEN = MIN( NH, JBOT-JCOL+1 ) CALL 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 ) 150 CONTINUE * * ==== Vertical multiply ==== * 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 ) 160 CONTINUE * * ==== Z multiply (also vertical) ==== * IF( WANTZ ) THEN 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 of SLAQR5 ==== * END