RECURSIVE SUBROUTINE PSLAQR0( 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( * )
REAL H( * ), WI( N ), WORK( * ), WR( N ),
$ Z( * )
* ..
*
* Purpose
* =======
*
* PSLAQR0 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 PSGEBAL, and then passed to PSGEHRD
* when the matrix output by PSGEBAL 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) REAL 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) REAL array, dimension (N)
* WI (global output) REAL 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) REAL array.
* If COMPZ = 'V', on entry Z must contain the current
* matrix Z of accumulated transformations from, e.g., PSGEHRD,
* 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) REAL 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, PSLAQR0 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 PSLAQR1, PSLAQR5
* and SLAQR6. 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 )
REAL WILK1, WILK2
PARAMETER ( WILK1 = 0.75E0, WILK2 = -0.4375E0 )
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0e0, ONE = 1.0e0 )
* ..
* .. Local Scalars ..
REAL 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 )
REAL ZDUM( 1, 1 )
* ..
* .. External Subroutines ..
EXTERNAL PSLACPY, PSLAQR1, SLANV2, PSLAQR3, PSLAQR5,
$ PSELGET, SLAQR0, SLASET, PSGEMR2D
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, FLOAT, 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 PSLAQR1. (Stops recursion)
*
IF( N.LE.NTINY ) THEN
*
* Estimate optimal workspace.
*
CALL PSLAQR1( 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 SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, DESCH(LLD_),
$ WR, WI, ILOZ, IHIZ, Z, DESCZ(LLD_), WORK, LWORK, INFO )
IF( N.GT.2 )
$ CALL SLASET( '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, 'PSLAQR0', 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, 'PSLAQR0', JBCMPZ, N, NB, NB, NB )
NSR = PILAENVX( ICTXT, 15, 'PSLAQR0', 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 PSLAQR3
*
CALL PSLAQR3( 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 PSLAQR5
*
CALL PSLAQR5( WANTT, WANTZ, 2, N, 1, N, N, WR, WI, H,
$ DESCH, ILOZ, IHIZ, Z, DESCZ, WORK, -1, IWORK,
$ LIWORK )
*
* Optimal workspace = MAX(PSLAQR3, PSLAQR5)
*
LWKOPT = MAX( LWKOPT, INT( WORK( 1 ) ) )
LIWKOPT = MAX( LIWKOPT, IWORK( 1 ) )
*
* Quick return in case of workspace query.
*
IF( LQUERY ) THEN
WORK( 1 ) = FLOAT( LWKOPT )
IWORK( 1 ) = LIWKOPT
RETURN
END IF
*
* PSLAQR1/PSLAQR0 crossover point.
*
NMIN = PILAENVX( ICTXT, 12, 'PSLAQR0', JBCMPZ, N, ILO, IHI,
$ LWORK )
NMIN = MAX( NTINY, NMIN )
*
* Nibble crossover point.
*
NIBBLE = PILAENVX( ICTXT, 14, 'PSLAQR0', 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, 'PSLAQR0', 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 PSELGET( 'All', '1-Tree', ELEM1, H, KWTOP,
$ KWTOP-1, DESCH )
CALL PSELGET( '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 PSLAQR3( 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 PSLAQR3
* 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
* PSLAQR3 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 PSELGET( 'All', '1-Tree', ELEM1, H, I, I-1,
$ DESCH )
CALL PSELGET( 'All', '1-Tree', ELEM2, H, I-1, I-2,
$ DESCH )
CALL PSELGET( '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 SLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
$ WR( I ), WI( I ), CS, SN )
30 CONTINUE
IF( KS.EQ.KTOP ) THEN
CALL PSELGET( '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 PSLAQR0 or
* PSLAQR1 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, 'PSLAQR0', 'EN', NS,
$ NB, NPROW, NPCOL )
c
c Temporarily force NPMIN <= 8 since only PSLAQR1 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 PSLACPY( 'All', NS, NS, H, KS, KS,
$ DESCH, WORK(IPT), 1+IROFFH, 1+ICOFFH,
$ DESCT )
CALL PSLAQR0( .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 PSLAMVE( 'All', NS, NS, H, KS, KS,
$ DESCH, WORK(IPT), 1+IROFFH, 1+ICOFFH,
$ DESCT, WORK(IPWRK) )
CALL PSLAQR1( .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 PSGEMR2D( NS, NS, H, KS, KS, DESCH,
$ WORK(IPT), 1, 1, DESCT, ICTXT )
*
c
c This part is still not perfect.
c Either PSLAQR0 or PSLAQR1 can work, but not both.
c
c NMIN = PILAENVX( ICTXT_NEW, 12, 'PSLAQR0',
c $ 'EN', NS, 1, NS, LWORK )
IF( ICTXT_NEW.GE.0 ) THEN
c IF( NS.GT.NMIN .AND. RECLEVEL.LT.1 ) THEN
c CALL PSLAQR0( .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 PSLAQR1( .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 SGSUM2D( ICTXT, 'All', ' ', NS, 1, WR(KS),
$ NS, -1, -1 )
CALL SGSUM2D( 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 PSELGET( 'All', '1-Tree', AA, H, KBOT-1,
$ KBOT-1, DESCH )
CALL PSELGET( 'All', '1-Tree', CC, H, KBOT,
$ KBOT-1, DESCH )
CALL PSELGET( 'All', '1-Tree', BB, H, KBOT-1,
$ KBOT, DESCH )
CALL PSELGET( 'All', '1-Tree', DD, H, KBOT,
$ KBOT, DESCH )
CALL SLANV2( 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 PSELGET( '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 PSLAQR5( 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 ) = FLOAT( LWKOPT )
IWORK( 1 ) = LIWKOPT
IF( .NOT. LQUERY ) THEN
IWORK( 1 ) = TOTIT
IWORK( 2 ) = SWEEP
IWORK( 3 ) = TOTNS
END IF
RETURN
*
* End of PSLAQR0
*
END