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 ) * * -- LAPACK auxiliary routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. 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, * ) * .. * * This auxiliary subroutine called by SLAQR0 performs a * single small-bulge multi-shift QR sweep. * * WANTT (input) logical scalar * WANTT = .true. if the quasi-triangular Schur factor * is being computed. WANTT is set to .false. otherwise. * * WANTZ (input) logical scalar * WANTZ = .true. if the orthogonal Schur factor is being * computed. WANTZ is set to .false. otherwise. * * KACC22 (input) 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: 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. * * N (input) integer scalar * N is the order of the Hessenberg matrix H upon which this * subroutine operates. * * KTOP (input) integer scalar * KBOT (input) integer scalar * 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. * * NSHFTS (input) integer scalar * NSHFTS gives the number of simultaneous shifts. NSHFTS * must be positive and even. * * SR (input) REAL array of size (NSHFTS) * SI (input) REAL array of size (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. * * H (input/output) REAL array of size (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. * * LDH (input) integer scalar * LDH is the leading dimension of H just as declared in the * calling procedure. LDH.GE.MAX(1,N). * * ILOZ (input) INTEGER * IHIZ (input) INTEGER * Specify the rows of Z to which transformations must be * applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N * * Z (input/output) REAL array of size (LDZ,IHI) * If WANTZ = .TRUE., then the QR Sweep orthogonal * similarity transformation is accumulated into * Z(ILOZ:IHIZ,ILO:IHI) from the right. * If WANTZ = .FALSE., then Z is unreferenced. * * LDZ (input) integer scalar * LDA is the leading dimension of Z just as declared in * the calling procedure. LDZ.GE.N. * * V (workspace) REAL array of size (LDV,NSHFTS/2) * * LDV (input) integer scalar * LDV is the leading dimension of V as declared in the * calling procedure. LDV.GE.3. * * U (workspace) REAL array of size * (LDU,3*NSHFTS-3) * * LDU (input) integer scalar * LDU is the leading dimension of U just as declared in the * in the calling subroutine. LDU.GE.3*NSHFTS-3. * * NH (input) integer scalar * NH is the number of columns in array WH available for * workspace. NH.GE.1. * * WH (workspace) REAL array of size (LDWH,NH) * * LDWH (input) integer scalar * Leading dimension of WH just as declared in the * calling procedure. LDWH.GE.3*NSHFTS-3. * * NV (input) integer scalar * NV is the number of rows in WV agailable for workspace. * NV.GE.1. * * WV (workspace) REAL array of size * (LDWV,3*NSHFTS-3) * * LDWV (input) integer scalar * LDWV is the leading dimension of WV as declared in the * in the calling subroutine. LDWV.GE.NV. * * * ================================================================ * Based on contributions by * Karen Braman and Ralph Byers, Department of Mathematics, * University of Kansas, USA * * ============================================================ * Reference: * * 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. * * ============================================================ * .. 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, J2, J4, JBOT, JCOL, JLEN, $ JROW, JTOP, K, K1, KDU, KMS, KNZ, KRCOL, KZS, $ M, M22, MBOT, MEND, MSTART, MTOP, NBMPS, NDCOL, $ NS, NU LOGICAL ACCUM, BLK22, 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 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 ) * * ==== If so, exploit the 2-by-2 block structure? ==== * BLK22 = ( NS.GT.2 ) .AND. ( 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 = 6*NBMPS - 3 * * ==== Create and chase chains of NBMPS bulges ==== * DO 220 INCOL = 3*( 1-NBMPS ) + KTOP - 1, KBOT - 2, 3*NBMPS - 2 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 * . 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 * . 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 ) * * ==== 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-1 )-KRCOL+2 ) / 3+1 ) MBOT = MIN( NBMPS, ( KBOT-KRCOL ) / 3 ) M22 = MBOT + 1 BMP22 = ( MBOT.LT.NBMPS ) .AND. ( KRCOL+3*( 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( 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 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. (The * . initial bulge is always collapsed.) Use * . the two-small-subdiagonals trick to try * . to get it started again. If V(2,M).NE.0 and * . V(3,M) = H(K+3,K+1) = H(K+3,K+2) = 0, then * . this bulge is collapsing into a zero * . subdiagonal. It will be restarted next * . trip through the loop.) * IF( V( 1, M ).NE.ZERO .AND. $ ( V( 3, M ).NE.ZERO .OR. ( H( K+3, $ K+1 ).EQ.ZERO .AND. 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). 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 ) SCL = ABS( VT( 1 ) ) + ABS( VT( 2 ) ) + $ ABS( VT( 3 ) ) IF( SCL.NE.ZERO ) THEN VT( 1 ) = VT( 1 ) / SCL VT( 2 ) = VT( 2 ) / SCL VT( 3 ) = VT( 3 ) / SCL END IF * * ==== The following is the traditional and * . conservative two-small-subdiagonals * . test. ==== * . IF( ABS( H( K+1, K ) )*( ABS( VT( 2 ) )+ $ ABS( VT( 3 ) ) ).GT.ULP*ABS( VT( 1 ) )* $ ( 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. If * . the old reflector is diagonal (only * . possible with underflows), then * . change it to I. Otherwise, use * . it with trepidation. ==== * IF( V( 2, M ).EQ.ZERO .AND. V( 3, M ).EQ.ZERO ) $ THEN V( 1, M ) = ZERO ELSE H( K+1, K ) = BETA H( K+2, K ) = ZERO H( K+3, K ) = ZERO END IF ELSE * * ==== Stating a new bulge here would * . create only negligible fill. * . Replace the old reflector with * . the new one. ==== * ALPHA = VT( 1 ) CALL SLARFG( 3, ALPHA, VT( 2 ), 1, VT( 1 ) ) REFSUM = H( K+1, K ) + H( K+2, K )*VT( 2 ) + $ H( K+3, K )*VT( 3 ) H( K+1, K ) = H( K+1, K ) - VT( 1 )*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 20 CONTINUE * * ==== Generate a 2-by-2 reflection, if needed. ==== * 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 ELSE * * ==== Initialize V(1,M22) here to avoid possible undefined * . variable problems later. ==== * V( 1, M22 ) = ZERO 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 ) 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. ==== * 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 .AND. ( 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 * * ==== 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 ) ) * * ==== 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( 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 ) )H( K+1, K ) = ZERO END IF END IF 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 of near-the-diagonal bulge chase. ==== * 150 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 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 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 * * ==== Vertical multiply ==== * DO 170 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 * * ==== Z multiply (also vertical) ==== * IF( WANTZ ) THEN DO 180 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 ) 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' ==== * 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' ==== * CALL SGEMM( 'C', 'N', I2, JLEN, J2, ONE, U, LDU, $ H( INCOL+1, JCOL ), LDH, ONE, WH, LDWH ) * * ==== Copy top of H bottom of WH ==== * CALL SLACPY( 'ALL', J2, JLEN, H( INCOL+1, JCOL ), LDH, $ WH( I2+1, 1 ), LDWH ) * * ==== Multiply by U21' ==== * 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 ==== * END