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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 ) * * -- LAPACK auxiliary routine (version 3.3.0) -- * Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.. * November 2010 * * .. Scalar Arguments .. INTEGER IHIZ, ILOZ, KACC22, KBOT, KTOP, LDH, LDU, LDV, $ LDWH, LDWV, LDZ, N, NH, NSHFTS, NV LOGICAL WANTT, WANTZ * .. * .. Array Arguments .. COMPLEX*16 H( LDH, * ), S( * ), U( LDU, * ), V( LDV, * ), $ WH( LDWH, * ), WV( LDWV, * ), Z( LDZ, * ) * .. * * This auxiliary subroutine called by ZLAQR0 performs a * single small-bulge multi-shift QR sweep. * * WANTT (input) logical scalar * WANTT = .true. if the triangular Schur factor * is being computed. WANTT is set to .false. otherwise. * * WANTZ (input) logical scalar * WANTZ = .true. if the unitary 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: ZLAQR5 does not accumulate reflections and does not * use matrix-matrix multiply to update far-from-diagonal * 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. * * 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. * * S (input/output) COMPLEX*16 array of size (NSHFTS) * S contains the shifts of origin that define the multi- * shift QR sweep. On output S may be reordered. * * H (input/output) COMPLEX*16 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) COMPLEX*16 array of size (LDZ,IHI) * If WANTZ = .TRUE., then the QR Sweep unitary * 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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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 .. COMPLEX*16 ZERO, ONE PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ), $ ONE = ( 1.0d0, 0.0d0 ) ) DOUBLE PRECISION RZERO, RONE PARAMETER ( RZERO = 0.0d0, RONE = 1.0d0 ) * .. * .. Local Scalars .. 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, $ NS, NU LOGICAL ACCUM, BLK22, BMP22 * .. * .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH * .. * .. Intrinsic Functions .. * INTRINSIC ABS, DBLE, DCONJG, DIMAG, MAX, MIN, MOD * .. * .. Local Arrays .. COMPLEX*16 VT( 3 ) * .. * .. External Subroutines .. EXTERNAL DLABAD, ZGEMM, ZLACPY, ZLAQR1, ZLARFG, ZLASET, $ ZTRMM * .. * .. Statement Functions .. DOUBLE PRECISION CABS1 * .. * .. Statement Function definitions .. CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) ) * .. * .. 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 * * ==== NSHFTS is supposed to be even, but if it is odd, * . then simply reduce it by one. ==== * NS = NSHFTS - MOD( NSHFTS, 2 ) * * ==== Machine constants for deflation ==== * SAFMIN = DLAMCH( 'SAFE MINIMUM' ) SAFMAX = RONE / SAFMIN CALL DLABAD( SAFMIN, SAFMAX ) ULP = DLAMCH( 'PRECISION' ) SMLNUM = SAFMIN*( DBLE( N ) / ULP ) * * ==== Use accumulated reflections to update far-from-diagonal * . entries ? ==== * ACCUM = ( KACC22.EQ.1 ) .OR. ( KACC22.EQ.2 ) * * ==== 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 210 INCOL = 3*( 1-NBMPS ) + KTOP - 1, KBOT - 2, 3*NBMPS - 2 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 * . 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 140 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 10 M = MTOP, MBOT K = KRCOL + 3*( 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 BETA = H( K+1, K ) V( 2, M ) = H( K+2, K ) V( 3, M ) = H( K+3, K ) CALL ZLARFG( 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 ZLAQR1( 3, H( K+1, K+1 ), LDH, S( 2*M-1 ), $ S( 2*M ), VT ) ALPHA = VT( 1 ) CALL ZLARFG( 3, ALPHA, VT( 2 ), 1, VT( 1 ) ) REFSUM = DCONJG( VT( 1 ) )* $ ( H( K+1, K )+DCONJG( VT( 2 ) )* $ H( K+2, K ) ) * IF( CABS1( H( K+2, K )-REFSUM*VT( 2 ) )+ $ CABS1( REFSUM*VT( 3 ) ).GT.ULP* $ ( CABS1( H( K, K ) )+CABS1( H( K+1, $ K+1 ) )+CABS1( 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 * * ==== Stating 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 10 CONTINUE * * ==== Generate a 2-by-2 reflection, if needed. ==== * 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 ) 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*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) ==== * 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 ) ) * * ==== 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 = 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 120 CONTINUE * * ==== Fill in the last row of each bulge. ==== * 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 ) - $ REFSUM*DCONJG( V( 3, M ) ) 130 CONTINUE * * ==== End of near-the-diagonal bulge chase. ==== * 140 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 150 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH JLEN = MIN( NH, JBOT-JCOL+1 ) CALL ZGEMM( 'C', 'N', NU, JLEN, NU, ONE, U( K1, K1 ), $ LDU, H( INCOL+K1, JCOL ), LDH, ZERO, WH, $ LDWH ) CALL ZLACPY( '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 ZGEMM( 'N', 'N', JLEN, NU, NU, ONE, $ H( JROW, INCOL+K1 ), LDH, U( K1, K1 ), $ LDU, ZERO, WV, LDWV ) CALL ZLACPY( '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 ZGEMM( 'N', 'N', JLEN, NU, NU, ONE, $ Z( JROW, INCOL+K1 ), LDZ, U( K1, K1 ), $ LDU, ZERO, WV, LDWV ) CALL ZLACPY( 'ALL', JLEN, NU, WV, LDWV, $ 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 ) 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 |