RECURSIVE SUBROUTINE PDLAQR0( WANTT, WANTZ, N, ILO, IHI, H, $ DESCH, WR, WI, ILOZ, IHIZ, Z, DESCZ, WORK, LWORK, $ IWORK, LIWORK, INFO, RECLEVEL ) * * Contribution from the Department of Computing Science and HPC2N, * Umea University, Sweden * * -- ScaLAPACK auxiliary routine (version 2.0.1) -- * University of Tennessee, Knoxville, Oak Ridge National Laboratory, * Univ. of Colorado Denver and University of California, Berkeley. * January, 2012 * IMPLICIT NONE * * .. Scalar Arguments .. INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LIWORK, LWORK, N, $ RECLEVEL LOGICAL WANTT, WANTZ * .. * .. Array Arguments .. INTEGER DESCH( * ), DESCZ( * ), IWORK( * ) DOUBLE PRECISION H( * ), WI( N ), WORK( * ), WR( N ), $ Z( * ) * .. * * Purpose * ======= * * PDLAQR0 computes the eigenvalues of a Hessenberg matrix H * and, optionally, the matrices T and Z from the Schur decomposition * H = Z*T*Z**T, where T is an upper quasi-triangular matrix (the * Schur form), and Z is the orthogonal matrix of Schur vectors. * * Optionally Z may be postmultiplied into an input orthogonal * matrix Q so that this routine can give the Schur factorization * of a matrix A which has been reduced to the Hessenberg form H * by the orthogonal matrix Q: * A = Q * H * Q**T = (QZ) * T * (QZ)**T. * * Notes * ===== * * Each global data object is described by an associated description * vector. This vector stores the information required to establish * the mapping between an object element and its corresponding process * and memory location. * * Let A be a generic term for any 2D block cyclicly distributed array. * Such a global array has an associated description vector DESCA. * In the following comments, the character _ should be read as * "of the global array". * * NOTATION STORED IN EXPLANATION * --------------- -------------- -------------------------------------- * DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, * DTYPE_A = 1. * CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating * the BLACS process grid A is distribu- * ted over. The context itself is glo- * bal, but the handle (the integer * value) may vary. * M_A (global) DESCA( M_ ) The number of rows in the global * array A. * N_A (global) DESCA( N_ ) The number of columns in the global * array A. * MB_A (global) DESCA( MB_ ) The blocking factor used to distribute * the rows of the array. * NB_A (global) DESCA( NB_ ) The blocking factor used to distribute * the columns of the array. * RSRC_A (global) DESCA( RSRC_ ) The process row over which the first * row of the array A is distributed. * CSRC_A (global) DESCA( CSRC_ ) The process column over which the * first column of the array A is * distributed. * LLD_A (local) DESCA( LLD_ ) The leading dimension of the local * array. LLD_A >= MAX(1,LOCr(M_A)). * * Let K be the number of rows or columns of a distributed matrix, * and assume that its process grid has dimension p x q. * LOCr( K ) denotes the number of elements of K that a process * would receive if K were distributed over the p processes of its * process column. * Similarly, LOCc( K ) denotes the number of elements of K that a * process would receive if K were distributed over the q processes of * its process row. * The values of LOCr() and LOCc() may be determined via a call to the * ScaLAPACK tool function, NUMROC: * LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ), * LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). * An upper bound for these quantities may be computed by: * LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A * LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A * * Arguments * ========= * * WANTT (global input) LOGICAL * = .TRUE. : the full Schur form T is required; * = .FALSE.: only eigenvalues are required. * * WANTZ (global input) LOGICAL * = .TRUE. : the matrix of Schur vectors Z is required; * = .FALSE.: Schur vectors are not required. * * N (global input) INTEGER * The order of the Hessenberg matrix H (and Z if WANTZ). * N >= 0. * * ILO (global input) INTEGER * IHI (global input) INTEGER * It is assumed that H is already upper triangular in rows * and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally * set by a previous call to PDGEBAL, and then passed to PDGEHRD * when the matrix output by PDGEBAL is reduced to Hessenberg * form. Otherwise ILO and IHI should be set to 1 and N * respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. * If N = 0, then ILO = 1 and IHI = 0. * * H (global input/output) DOUBLE PRECISION array, dimension * (DESCH(LLD_),*) * On entry, the upper Hessenberg matrix H. * On exit, if JOB = 'S', H is upper quasi-triangular in * rows and columns ILO:IHI, with 1-by-1 and 2-by-2 blocks on * the main diagonal. The 2-by-2 diagonal blocks (corresponding * to complex conjugate pairs of eigenvalues) are returned in * standard form, with H(i,i) = H(i+1,i+1) and * H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the * contents of H are unspecified on exit. * * DESCH (global and local input) INTEGER array of dimension DLEN_. * The array descriptor for the distributed matrix H. * * WR (global output) DOUBLE PRECISION array, dimension (N) * WI (global output) DOUBLE PRECISION array, dimension (N) * The real and imaginary parts, respectively, of the computed * eigenvalues ILO to IHI are stored in the corresponding * elements of WR and WI. If two eigenvalues are computed as a * complex conjugate pair, they are stored in consecutive * elements of WR and WI, say the i-th and (i+1)th, with * WI(i) > 0 and WI(i+1) < 0. If JOB = 'S', the * eigenvalues are stored in the same order as on the diagonal * of the Schur form returned in H. * * Z (global input/output) DOUBLE PRECISION array. * If COMPZ = 'V', on entry Z must contain the current * matrix Z of accumulated transformations from, e.g., PDGEHRD, * and on exit Z has been updated; transformations are applied * only to the submatrix Z(ILO:IHI,ILO:IHI). * If COMPZ = 'N', Z is not referenced. * If COMPZ = 'I', on entry Z need not be set and on exit, * if INFO = 0, Z contains the orthogonal matrix Z of the Schur * vectors of H. * * DESCZ (global and local input) INTEGER array of dimension DLEN_. * The array descriptor for the distributed matrix Z. * * WORK (local workspace) DOUBLE PRECISION array, dimension(DWORK) * * LWORK (local input) INTEGER * The length of the workspace array WORK. * * IWORK (local workspace) INTEGER array, dimension (LIWORK) * * LIWORK (local input) INTEGER * The length of the workspace array IWORK. * * INFO (output) INTEGER * = 0: successful exit * .LT. 0: if INFO = -i, the i-th argument had an illegal * value * .GT. 0: if INFO = i, PDLAQR0 failed to compute all of * the eigenvalues. Elements 1:ilo-1 and i+1:n of WR * and WI contain those eigenvalues which have been * successfully computed. (Failures are rare.) * * If INFO .GT. 0 and JOB = 'E', then on exit, the * remaining unconverged eigenvalues are the eigen- * values of the upper Hessenberg matrix rows and * columns ILO through INFO of the final, output * value of H. * * If INFO .GT. 0 and JOB = 'S', then on exit * * (*) (initial value of H)*U = U*(final value of H) * * where U is an orthogonal matrix. The final * value of H is upper Hessenberg and quasi-triangular * in rows and columns INFO+1 through IHI. * * If INFO .GT. 0 and COMPZ = 'V', then on exit * * (final value of Z) = (initial value of Z)*U * * where U is the orthogonal matrix in (*) (regard- * less of the value of JOB.) * * If INFO .GT. 0 and COMPZ = 'I', then on exit * (final value of Z) = U * where U is the orthogonal matrix in (*) (regard- * less of the value of JOB.) * * If INFO .GT. 0 and COMPZ = 'N', then Z is not * accessed. * * ================================================================ * Based on contributions by * Robert Granat, Department of Computing Science and HPC2N, * Umea University, Sweden. * ================================================================ * * Restrictions: The block size in H and Z must be square and larger * than or equal to six (6) due to restrictions in PDLAQR1, PDLAQR5 * and DLAQR6. Moreover, H and Z need to be distributed identically * with the same context. * * ================================================================ * References: * K. Braman, R. Byers, and R. Mathias, * The Multi-Shift QR Algorithm Part I: Maintaining Well Focused * Shifts, and Level 3 Performance. * SIAM J. Matrix Anal. Appl., 23(4):929--947, 2002. * * K. Braman, R. Byers, and R. Mathias, * The Multi-Shift QR Algorithm Part II: Aggressive Early * Deflation. * SIAM J. Matrix Anal. Appl., 23(4):948--973, 2002. * * R. Granat, B. Kagstrom, and D. Kressner, * A Novel Parallel QR Algorithm for Hybrid Distributed Momory HPC * Systems. * SIAM J. Sci. Comput., 32(4):2345--2378, 2010. * * ================================================================ * * .. Parameters .. * * ==== Exceptional deflation windows: try to cure rare * . slow convergence by increasing the size of the * . deflation window after KEXNW iterations. ===== * * ==== Exceptional shifts: try to cure rare slow convergence * . with ad-hoc exceptional shifts every KEXSH iterations. * . The constants WILK1 and WILK2 are used to form the * . exceptional shifts. ==== * INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_, $ LLD_, MB_, M_, NB_, N_, RSRC_ INTEGER RECMAX PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, $ CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, $ RSRC_ = 7, CSRC_ = 8, LLD_ = 9, RECMAX = 3 ) INTEGER NTINY PARAMETER ( NTINY = 11 ) INTEGER KEXNW, KEXSH PARAMETER ( KEXNW = 5, KEXSH = 6 ) DOUBLE PRECISION WILK1, WILK2 PARAMETER ( WILK1 = 0.75D0, WILK2 = -0.4375D0 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) * .. * .. Local Scalars .. DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP, ELEM, T0, $ ELEM1, ELEM2, ELEM3, ALPHA, SDSUM, STAMP INTEGER I, J, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS, $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS, $ LWKOPT, NDFL, NH, NHO, NIBBLE, NMIN, NS, NSMAX, $ NSR, NVE, NW, NWMAX, NWR, LLDH, LLDZ, II, JJ, $ ICTXT, NPROW, NPCOL, MYROW, MYCOL, IPV, IPT, $ IPW, IPWRK, VROWS, VCOLS, TROWS, TCOLS, WROWS, $ WCOLS, HRSRC, HCSRC, NB, IS, IE, NPROCS, KK, $ IROFFH, ICOFFH, HRSRC3, HCSRC3, NWIN, TOTIT, $ SWEEP, JW, TOTNS, LIWKOPT, NPMIN, ICTXT_NEW, $ MYROW_NEW, MYCOL_NEW LOGICAL NWINC, SORTED, LQUERY, RECURSION CHARACTER JBCMPZ*2 * .. * .. External Functions .. INTEGER PILAENVX, NUMROC, INDXG2P, ICEIL, BLACS_PNUM EXTERNAL PILAENVX, NUMROC, INDXG2P, ICEIL, BLACS_PNUM * .. * .. Local Arrays .. INTEGER DESCV( DLEN_ ), DESCT( DLEN_ ), DESCW( DLEN_ ), $ PMAP( 64*64 ) DOUBLE PRECISION ZDUM( 1, 1 ) * .. * .. External Subroutines .. EXTERNAL PDLACPY, PDLAQR1, DLANV2, PDLAQR3, PDLAQR5, $ PDELGET, DLAQR0, DLASET, PDGEMR2D * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD * .. * .. Executable Statements .. INFO = 0 ICTXT = DESCH( CTXT_ ) CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) NPROCS = NPROW*NPCOL RECURSION = RECLEVEL .LT. RECMAX * * Quick return for N = 0: nothing to do. * IF( N.EQ.0 ) THEN WORK( 1 ) = ONE IWORK( 1 ) = 1 RETURN END IF * * Set up job flags for PILAENV. * IF( WANTT ) THEN JBCMPZ( 1: 1 ) = 'S' ELSE JBCMPZ( 1: 1 ) = 'E' END IF IF( WANTZ ) THEN JBCMPZ( 2: 2 ) = 'V' ELSE JBCMPZ( 2: 2 ) = 'N' END IF * * Check if workspace query * LQUERY = LWORK.EQ.-1 .OR. LIWORK.EQ.-1 * * Extract local leading dimensions and block factors of matrices * H and Z * LLDH = DESCH( LLD_ ) LLDZ = DESCZ( LLD_ ) NB = DESCH( MB_ ) * * Tiny (sub-) matrices must use PDLAQR1. (Stops recursion) * IF( N.LE.NTINY ) THEN * * Estimate optimal workspace. * CALL PDLAQR1( WANTT, WANTZ, N, ILO, IHI, H, DESCH, WR, WI, $ ILOZ, IHIZ, Z, DESCZ, WORK, LWORK, IWORK, LIWORK, INFO ) LWKOPT = INT( WORK(1) ) LIWKOPT = IWORK(1) * * Completely local matrices uses LAPACK. (Stops recursion) * ELSEIF( N.LE.NB ) THEN IF( MYROW.EQ.DESCH(RSRC_) .AND. MYCOL.EQ.DESCH(CSRC_) ) THEN CALL DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, DESCH(LLD_), $ WR, WI, ILOZ, IHIZ, Z, DESCZ(LLD_), WORK, LWORK, INFO ) IF( N.GT.2 ) $ CALL DLASET( 'L', N-2, N-2, ZERO, ZERO, H(3), $ DESCH(LLD_) ) LWKOPT = INT( WORK(1) ) LIWKOPT = 1 ELSE LWKOPT = 1 LIWKOPT = 1 END IF * * Do one more step of recursion * ELSE * * Zero out iteration and sweep counters for debugging purposes * TOTIT = 0 SWEEP = 0 TOTNS = 0 * * Use small bulge multi-shift QR with aggressive early * deflation on larger-than-tiny matrices. * * Hope for the best. * INFO = 0 * * NWR = recommended deflation window size. At this * point, N .GT. NTINY = 11, so there is enough * subdiagonal workspace for NWR.GE.2 as required. * (In fact, there is enough subdiagonal space for * NWR.GE.3.) * NWR = PILAENVX( ICTXT, 13, 'PDLAQR0', JBCMPZ, N, ILO, IHI, $ LWORK ) NWR = MAX( 2, NWR ) NWR = MIN( IHI-ILO+1, NWR ) NW = NWR * * NSR = recommended number of simultaneous shifts. * At this point N .GT. NTINY = 11, so there is at * enough subdiagonal workspace for NSR to be even * and greater than or equal to two as required. * NWIN = PILAENVX( ICTXT, 19, 'PDLAQR0', JBCMPZ, N, NB, NB, NB ) NSR = PILAENVX( ICTXT, 15, 'PDLAQR0', JBCMPZ, N, ILO, IHI, $ MAX(NWIN,NB) ) NSR = MIN( NSR, IHI-ILO ) NSR = MAX( 2, NSR-MOD( NSR, 2 ) ) * * Estimate optimal workspace * LWKOPT = 3*ICEIL(NWR,NPROW)*ICEIL(NWR,NPCOL) * * Workspace query call to PDLAQR3 * CALL PDLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, $ DESCH, ILOZ, IHIZ, Z, DESCZ, LS, LD, WR, WI, H, $ DESCH, N, H, DESCH, N, H, DESCH, WORK, -1, IWORK, $ LIWORK, RECLEVEL ) LWKOPT = LWKOPT + INT( WORK( 1 ) ) LIWKOPT = IWORK( 1 ) * * Workspace query call to PDLAQR5 * CALL PDLAQR5( WANTT, WANTZ, 2, N, 1, N, N, WR, WI, H, $ DESCH, ILOZ, IHIZ, Z, DESCZ, WORK, -1, IWORK, $ LIWORK ) * * Optimal workspace = MAX(PDLAQR3, PDLAQR5) * LWKOPT = MAX( LWKOPT, INT( WORK( 1 ) ) ) LIWKOPT = MAX( LIWKOPT, IWORK( 1 ) ) * * Quick return in case of workspace query. * IF( LQUERY ) THEN WORK( 1 ) = DBLE( LWKOPT ) IWORK( 1 ) = LIWKOPT RETURN END IF * * PDLAQR1/PDLAQR0 crossover point. * NMIN = PILAENVX( ICTXT, 12, 'PDLAQR0', JBCMPZ, N, ILO, IHI, $ LWORK ) NMIN = MAX( NTINY, NMIN ) * * Nibble crossover point. * NIBBLE = PILAENVX( ICTXT, 14, 'PDLAQR0', JBCMPZ, N, ILO, IHI, $ LWORK ) NIBBLE = MAX( 0, NIBBLE ) * * Accumulate reflections during ttswp? Use block * 2-by-2 structure during matrix-matrix multiply? * KACC22 = PILAENVX( ICTXT, 16, 'PDLAQR0', JBCMPZ, N, ILO, IHI, $ LWORK ) KACC22 = MAX( 1, KACC22 ) KACC22 = MIN( 2, KACC22 ) * * NWMAX = the largest possible deflation window for * which there is sufficient workspace. * NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 ) * * NSMAX = the Largest number of simultaneous shifts * for which there is sufficient workspace. * NSMAX = MIN( ( N+6 ) / 9, LWORK - LWORK/3 ) NSMAX = NSMAX - MOD( NSMAX, 2 ) * * NDFL: an iteration count restarted at deflation. * NDFL = 1 * * ITMAX = iteration limit * ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) ) * * Last row and column in the active block. * KBOT = IHI * * Main Loop. * DO 110 IT = 1, ITMAX TOTIT = TOTIT + 1 * * Done when KBOT falls below ILO. * IF( KBOT.LT.ILO ) $ GO TO 120 * * Locate active block. * DO 10 K = KBOT, ILO + 1, -1 CALL INFOG2L( K, K-1, DESCH, NPROW, NPCOL, MYROW, MYCOL, $ II, JJ, HRSRC, HCSRC ) IF( MYROW.EQ.HRSRC .AND. MYCOL.EQ.HCSRC ) THEN IF( H( II + (JJ-1)*LLDH ).EQ.ZERO ) $ GO TO 20 END IF 10 CONTINUE K = ILO 20 CONTINUE KTOP = K IF( NPROCS.GT.1 ) $ CALL IGAMX2D( ICTXT, 'All', '1-Tree', 1, 1, KTOP, 1, $ -1, -1, -1, -1, -1 ) * * Select deflation window size. * NH = KBOT - KTOP + 1 IF( NH.LE.NTINY ) THEN NW = NH ELSEIF( NDFL.LT.KEXNW .OR. NH.LT.NW ) THEN * * Typical deflation window. If possible and * advisable, nibble the entire active block. * If not, use size NWR or NWR+1 depending upon * which has the smaller corresponding subdiagonal * entry (a heuristic). * NWINC = .TRUE. IF( NH.LE.MIN( NMIN, NWMAX ) ) THEN NW = NH ELSE NW = MIN( NWR, NH, NWMAX ) IF( NW.LT.NWMAX ) THEN IF( NW.GE.NH-1 ) THEN NW = NH ELSE KWTOP = KBOT - NW + 1 CALL PDELGET( 'All', '1-Tree', ELEM1, H, KWTOP, $ KWTOP-1, DESCH ) CALL PDELGET( 'All', '1-Tree', ELEM2, H, $ KWTOP-1, KWTOP-2, DESCH ) IF( ABS( ELEM1 ).GT.ABS( ELEM2 ) ) NW = NW + 1 END IF END IF END IF ELSE * * Exceptional deflation window. If there have * been no deflations in KEXNW or more iterations, * then vary the deflation window size. At first, * because, larger windows are, in general, more * powerful than smaller ones, rapidly increase the * window up to the maximum reasonable and possible. * Then maybe try a slightly smaller window. * IF( NWINC .AND. NW.LT.MIN( NWMAX, NH ) ) THEN NW = MIN( NWMAX, NH, 2*NW ) ELSE NWINC = .FALSE. IF( NW.EQ.NH .AND. NH.GT.2 ) $ NW = NH - 1 END IF END IF * * Aggressive early deflation: * split workspace into * - an NW-by-NW work array V for orthogonal matrix * - an NW-by-at-least-NW-but-more-is-better * (NW-by-NHO) horizontal work array for Schur factor * - an at-least-NW-but-more-is-better (NVE-by-NW) * vertical work array for matrix multiplications * - align T, V and W with the deflation window * KV = N - NW + 1 KT = NW + 1 NHO = ( N-NW-1 ) - KT + 1 KWV = NW + 2 NVE = ( N-NW ) - KWV + 1 * JW = MIN( NW, KBOT-KTOP+1 ) KWTOP = KBOT - JW + 1 IROFFH = MOD( KWTOP - 1, NB ) ICOFFH = IROFFH HRSRC = INDXG2P( KWTOP, NB, MYROW, DESCH(RSRC_), NPROW ) HCSRC = INDXG2P( KWTOP, NB, MYCOL, DESCH(CSRC_), NPCOL ) VROWS = NUMROC( JW+IROFFH, NB, MYROW, HRSRC, NPROW ) VCOLS = NUMROC( JW+ICOFFH, NB, MYCOL, HCSRC, NPCOL ) CALL DESCINIT( DESCV, JW+IROFFH, JW+ICOFFH, NB, NB, $ HRSRC, HCSRC, ICTXT, MAX(1, VROWS), INFO ) * TROWS = NUMROC( JW+IROFFH, NB, MYROW, HRSRC, NPROW ) TCOLS = NUMROC( JW+ICOFFH, NB, MYCOL, HCSRC, NPCOL ) CALL DESCINIT( DESCT, JW+IROFFH, JW+ICOFFH, NB, NB, $ HRSRC, HCSRC, ICTXT, MAX(1, TROWS), INFO ) WROWS = NUMROC( JW+IROFFH, NB, MYROW, HRSRC, NPROW ) WCOLS = NUMROC( JW+ICOFFH, NB, MYCOL, HCSRC, NPCOL ) CALL DESCINIT( DESCW, JW+IROFFH, JW+ICOFFH, NB, NB, $ HRSRC, HCSRC, ICTXT, MAX(1, WROWS), INFO ) * IPV = 1 IPT = IPV + DESCV( LLD_ ) * VCOLS IPW = IPT + DESCT( LLD_ ) * TCOLS IPWRK = IPW + DESCW( LLD_ ) * WCOLS * * Aggressive early deflation * IWORK(1) = IT CALL PDLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, $ DESCH, ILOZ, IHIZ, Z, DESCZ, LS, LD, WR, WI, $ WORK(IPV), DESCV, NHO, WORK(IPT), DESCT, NVE, $ WORK(IPW), DESCW, WORK(IPWRK), LWORK-IPWRK+1, $ IWORK, LIWORK, RECLEVEL ) * * Adjust KBOT accounting for new deflations. * KBOT = KBOT - LD * * KS points to the shifts. * KS = KBOT - LS + 1 * * Skip an expensive QR sweep if there is a (partly * heuristic) reason to expect that many eigenvalues * will deflate without it. Here, the QR sweep is * skipped if many eigenvalues have just been deflated * or if the remaining active block is small. * IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT- $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN * * NS = nominal number of simultaneous shifts. * This may be lowered (slightly) if PDLAQR3 * did not provide that many shifts. * NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) ) NS = NS - MOD( NS, 2 ) * * If there have been no deflations * in a multiple of KEXSH iterations, * then try exceptional shifts. * Otherwise use shifts provided by * PDLAQR3 above or from the eigenvalues * of a trailing principal submatrix. * IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN KS = KBOT - NS + 1 DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2 CALL PDELGET( 'All', '1-Tree', ELEM1, H, I, I-1, $ DESCH ) CALL PDELGET( 'All', '1-Tree', ELEM2, H, I-1, I-2, $ DESCH ) CALL PDELGET( 'All', '1-Tree', ELEM3, H, I, I, $ DESCH ) SS = ABS( ELEM1 ) + ABS( ELEM2 ) AA = WILK1*SS + ELEM3 BB = SS CC = WILK2*SS DD = AA CALL DLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ), $ WR( I ), WI( I ), CS, SN ) 30 CONTINUE IF( KS.EQ.KTOP ) THEN CALL PDELGET( 'All', '1-Tree', ELEM1, H, KS+1, $ KS+1, DESCH ) WR( KS+1 ) = ELEM1 WI( KS+1 ) = ZERO WR( KS ) = WR( KS+1 ) WI( KS ) = WI( KS+1 ) END IF ELSE * * Got NS/2 or fewer shifts? Use PDLAQR0 or * PDLAQR1 on a trailing principal submatrix to * get more. * IF( KBOT-KS+1.LE.NS / 2 ) THEN KS = KBOT - NS + 1 KT = N - NS + 1 NPMIN = PILAENVX( ICTXT, 23, 'PDLAQR0', 'EN', NS, $ NB, NPROW, NPCOL ) c c Temporarily force NPMIN <= 8 since only PDLAQR1 is used. c NPMIN = MIN(NPMIN, 8) IF( MIN(NPROW, NPCOL).LE.NPMIN+1 .OR. $ RECLEVEL.GE.1 ) THEN * * The window is large enough. Compute the Schur * decomposition with all processors. * IROFFH = MOD( KS - 1, NB ) ICOFFH = IROFFH IF( NS.GT.NMIN ) THEN HRSRC = INDXG2P( KS, NB, MYROW, DESCH(RSRC_), $ NPROW ) HCSRC = INDXG2P( KS, NB, MYROW, DESCH(CSRC_), $ NPCOL ) ELSE HRSRC = 0 HCSRC = 0 END IF TROWS = NUMROC( NS+IROFFH, NB, MYROW, HRSRC, $ NPROW ) TCOLS = NUMROC( NS+ICOFFH, NB, MYCOL, HCSRC, $ NPCOL ) CALL DESCINIT( DESCT, NS+IROFFH, NS+ICOFFH, NB, $ NB, HRSRC, HCSRC, ICTXT, MAX(1, TROWS), $ INFO ) IPT = 1 IPWRK = IPT + DESCT(LLD_) * TCOLS * IF( NS.GT.NMIN .AND. RECURSION ) THEN CALL PDLACPY( 'All', NS, NS, H, KS, KS, $ DESCH, WORK(IPT), 1+IROFFH, 1+ICOFFH, $ DESCT ) CALL PDLAQR0( .FALSE., .FALSE., IROFFH+NS, $ 1+IROFFH, IROFFH+NS, WORK(IPT), $ DESCT, WR( KS-IROFFH ), $ WI( KS-IROFFH ), 1, 1, ZDUM, $ DESCZ, WORK( IPWRK ), $ LWORK-IPWRK+1, IWORK, LIWORK, $ INF, RECLEVEL+1 ) ELSE CALL PDLAMVE( 'All', NS, NS, H, KS, KS, $ DESCH, WORK(IPT), 1+IROFFH, 1+ICOFFH, $ DESCT, WORK(IPWRK) ) CALL PDLAQR1( .FALSE., .FALSE., IROFFH+NS, $ 1+IROFFH, IROFFH+NS, WORK(IPT), $ DESCT, WR( KS-IROFFH ), $ WI( KS-IROFFH ), 1+IROFFH, IROFFH+NS, $ ZDUM, DESCZ, WORK( IPWRK ), $ LWORK-IPWRK+1, IWORK, LIWORK, INF ) END IF ELSE * * The window is too small. Redistribute the AED * window to a subgrid and do the computation on * the subgrid. * ICTXT_NEW = ICTXT DO 50 I = 0, NPMIN-1 DO 40 J = 0, NPMIN-1 PMAP( J+1+I*NPMIN ) = $ BLACS_PNUM( ICTXT, I, J ) 40 CONTINUE 50 CONTINUE CALL BLACS_GRIDMAP( ICTXT_NEW, PMAP, NPMIN, $ NPMIN, NPMIN ) CALL BLACS_GRIDINFO( ICTXT_NEW, NPMIN, NPMIN, $ MYROW_NEW, MYCOL_NEW ) IF( MYROW.GE.NPMIN .OR. MYCOL.GE.NPMIN ) $ ICTXT_NEW = -1 * IF( ICTXT_NEW.GE.0 ) THEN TROWS = NUMROC( NS, NB, MYROW_NEW, 0, NPMIN ) TCOLS = NUMROC( NS, NB, MYCOL_NEW, 0, NPMIN ) CALL DESCINIT( DESCT, NS, NS, NB, NB, 0, 0, $ ICTXT_NEW, MAX(1,TROWS), INFO ) IPT = 1 IPWRK = IPT + DESCT(LLD_) * TCOLS ELSE IPT = 1 IPWRK = 2 DESCT( CTXT_ ) = -1 INF = 0 END IF CALL PDGEMR2D( NS, NS, H, KS, KS, DESCH, $ WORK(IPT), 1, 1, DESCT, ICTXT ) * c c This part is still not perfect. c Either PDLAQR0 or PDLAQR1 can work, but not both. c c NMIN = PILAENVX( ICTXT_NEW, 12, 'PDLAQR0', c $ 'EN', NS, 1, NS, LWORK ) IF( ICTXT_NEW.GE.0 ) THEN c IF( NS.GT.NMIN .AND. RECLEVEL.LT.1 ) THEN c CALL PDLAQR0( .FALSE., .FALSE., NS, 1, c $ NS, WORK(IPT), DESCT, WR( KS ), c $ WI( KS ), 1, 1, ZDUM, DESCT, c $ WORK( IPWRK ), LWORK-IPWRK+1, IWORK, c $ LIWORK, INF, RECLEVEL+1 ) c ELSE CALL PDLAQR1( .FALSE., .FALSE., NS, 1, $ NS, WORK(IPT), DESCT, WR( KS ), $ WI( KS ), 1, NS, ZDUM, DESCT, $ WORK( IPWRK ), LWORK-IPWRK+1, IWORK, $ LIWORK, INF ) c END IF CALL BLACS_GRIDEXIT( ICTXT_NEW ) END IF IF( MYROW+MYCOL.GT.0 ) THEN DO 60 J = 0, NS-1 WR( KS+J ) = ZERO WI( KS+J ) = ZERO 60 CONTINUE END IF CALL IGAMN2D( ICTXT, 'All', '1-Tree', 1, 1, INF, $ 1, -1, -1, -1, -1, -1 ) CALL DGSUM2D( ICTXT, 'All', ' ', NS, 1, WR(KS), $ NS, -1, -1 ) CALL DGSUM2D( ICTXT, 'All', ' ', NS, 1, WI(KS), $ NS, -1, -1 ) END IF KS = KS + INF * * In case of a rare QR failure use * eigenvalues of the trailing 2-by-2 * principal submatrix. * IF( KS.GE.KBOT ) THEN CALL PDELGET( 'All', '1-Tree', AA, H, KBOT-1, $ KBOT-1, DESCH ) CALL PDELGET( 'All', '1-Tree', CC, H, KBOT, $ KBOT-1, DESCH ) CALL PDELGET( 'All', '1-Tree', BB, H, KBOT-1, $ KBOT, DESCH ) CALL PDELGET( 'All', '1-Tree', DD, H, KBOT, $ KBOT, DESCH ) CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ), $ WI( KBOT-1 ), WR( KBOT ), $ WI( KBOT ), CS, SN ) KS = KBOT - 1 END IF END IF * IF( KBOT-KS+1.GT.NS ) THEN * * Sort the shifts (helps a little) * Bubble sort keeps complex conjugate * pairs together. * SORTED = .FALSE. DO 80 K = KBOT, KS + 1, -1 IF( SORTED ) $ GO TO 90 SORTED = .TRUE. DO 70 I = KS, K - 1 IF( ABS( WR( I ) )+ABS( WI( I ) ).LT. $ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN SORTED = .FALSE. * SWAP = WR( I ) WR( I ) = WR( I+1 ) WR( I+1 ) = SWAP * SWAP = WI( I ) WI( I ) = WI( I+1 ) WI( I+1 ) = SWAP END IF 70 CONTINUE 80 CONTINUE 90 CONTINUE END IF * * Shuffle shifts into pairs of real shifts * and pairs of complex conjugate shifts * assuming complex conjugate shifts are * already adjacent to one another. (Yes, * they are.) * DO 100 I = KBOT, KS + 2, -2 IF( WI( I ).NE.-WI( I-1 ) ) THEN * SWAP = WR( I ) WR( I ) = WR( I-1 ) WR( I-1 ) = WR( I-2 ) WR( I-2 ) = SWAP * SWAP = WI( I ) WI( I ) = WI( I-1 ) WI( I-1 ) = WI( I-2 ) WI( I-2 ) = SWAP END IF 100 CONTINUE END IF * * If there are only two shifts and both are * real, then use only one. * IF( KBOT-KS+1.EQ.2 ) THEN IF( WI( KBOT ).EQ.ZERO ) THEN CALL PDELGET( 'All', '1-Tree', ELEM, H, KBOT, $ KBOT, DESCH ) IF( ABS( WR( KBOT )-ELEM ).LT. $ ABS( WR( KBOT-1 )-ELEM ) ) THEN WR( KBOT-1 ) = WR( KBOT ) ELSE WR( KBOT ) = WR( KBOT-1 ) END IF END IF END IF * * Use up to NS of the the smallest magnatiude * shifts. If there aren't NS shifts available, * then use them all, possibly dropping one to * make the number of shifts even. * NS = MIN( NS, KBOT-KS+1 ) NS = NS - MOD( NS, 2 ) KS = KBOT - NS + 1 * * Small-bulge multi-shift QR sweep. * TOTNS = TOTNS + NS SWEEP = SWEEP + 1 CALL PDLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, $ NS, WR( KS ), WI( KS ), H, DESCH, ILOZ, IHIZ, Z, $ DESCZ, WORK, LWORK, IWORK, LIWORK ) END IF * * Note progress (or the lack of it). * IF( LD.GT.0 ) THEN NDFL = 1 ELSE NDFL = NDFL + 1 END IF * * End of main loop. 110 CONTINUE * * Iteration limit exceeded. Set INFO to show where * the problem occurred and exit. * INFO = KBOT 120 CONTINUE END IF * * Return the optimal value of LWORK. * WORK( 1 ) = DBLE( LWKOPT ) IWORK( 1 ) = LIWKOPT IF( .NOT. LQUERY ) THEN IWORK( 1 ) = TOTIT IWORK( 2 ) = SWEEP IWORK( 3 ) = TOTNS END IF RETURN * * End of PDLAQR0 * END