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|
SUBROUTINE PCTREVC( SIDE, HOWMNY, SELECT, N, T, DESCT, VL, DESCVL,
$ VR, DESCVR, MM, M, WORK, RWORK, INFO )
*
* -- ScaLAPACK routine (version 1.7) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* July 31, 2001
*
* .. Scalar Arguments ..
CHARACTER HOWMNY, SIDE
INTEGER INFO, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
INTEGER DESCT( * ), DESCVL( * ), DESCVR( * )
REAL RWORK( * )
COMPLEX T( * ), VL( * ), VR( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* PCTREVC computes some or all of the right and/or left eigenvectors of
* a complex upper triangular matrix T in parallel.
*
* The right eigenvector x and the left eigenvector y of T corresponding
* to an eigenvalue w are defined by:
*
* T*x = w*x, y'*T = w*y'
*
* where y' denotes the conjugate transpose of the vector y.
*
* If all eigenvectors are requested, the routine may either return the
* matrices X and/or Y of right or left eigenvectors of T, or the
* products Q*X and/or Q*Y, where Q is an input unitary
* matrix. If T was obtained from the Schur factorization of an
* original matrix A = Q*T*Q', then Q*X and Q*Y are the matrices of
* right or left eigenvectors of A.
*
* 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 r x c.
* LOCr( K ) denotes the number of elements of K that a process
* would receive if K were distributed over the r 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 c 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
* =========
*
* SIDE (global input) CHARACTER*1
* = 'R': compute right eigenvectors only;
* = 'L': compute left eigenvectors only;
* = 'B': compute both right and left eigenvectors.
*
* HOWMNY (global input) CHARACTER*1
* = 'A': compute all right and/or left eigenvectors;
* = 'B': compute all right and/or left eigenvectors,
* and backtransform them using the input matrices
* supplied in VR and/or VL;
* = 'S': compute selected right and/or left eigenvectors,
* specified by the logical array SELECT.
*
* SELECT (global input) LOGICAL array, dimension (N)
* If HOWMNY = 'S', SELECT specifies the eigenvectors to be
* computed.
* If HOWMNY = 'A' or 'B', SELECT is not referenced.
* To select the eigenvector corresponding to the j-th
* eigenvalue, SELECT(j) must be set to .TRUE..
*
* N (global input) INTEGER
* The order of the matrix T. N >= 0.
*
* T (global input/output) COMPLEX array, dimension
* (DESCT(LLD_),*)
* The upper triangular matrix T. T is modified, but restored
* on exit.
*
* DESCT (global and local input) INTEGER array of dimension DLEN_.
* The array descriptor for the distributed matrix T.
*
* VL (global input/output) COMPLEX array, dimension
* (DESCVL(LLD_),MM)
* On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
* contain an N-by-N matrix Q (usually the unitary matrix Q of
* Schur vectors returned by CHSEQR).
* On exit, if SIDE = 'L' or 'B', VL contains:
* if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
* if HOWMNY = 'B', the matrix Q*Y;
* if HOWMNY = 'S', the left eigenvectors of T specified by
* SELECT, stored consecutively in the columns
* of VL, in the same order as their
* eigenvalues.
* If SIDE = 'R', VL is not referenced.
*
* DESCVL (global and local input) INTEGER array of dimension DLEN_.
* The array descriptor for the distributed matrix VL.
*
* VR (global input/output) COMPLEX array, dimension
* (DESCVR(LLD_),MM)
* On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
* contain an N-by-N matrix Q (usually the unitary matrix Q of
* Schur vectors returned by CHSEQR).
* On exit, if SIDE = 'R' or 'B', VR contains:
* if HOWMNY = 'A', the matrix X of right eigenvectors of T;
* if HOWMNY = 'B', the matrix Q*X;
* if HOWMNY = 'S', the right eigenvectors of T specified by
* SELECT, stored consecutively in the columns
* of VR, in the same order as their
* eigenvalues.
* If SIDE = 'L', VR is not referenced.
*
* DESCVR (global and local input) INTEGER array of dimension DLEN_.
* The array descriptor for the distributed matrix VR.
*
* MM (global input) INTEGER
* The number of columns in the arrays VL and/or VR. MM >= M.
*
* M (global output) INTEGER
* The number of columns in the arrays VL and/or VR actually
* used to store the eigenvectors. If HOWMNY = 'A' or 'B', M
* is set to N. Each selected eigenvector occupies one
* column.
*
* WORK (local workspace) COMPLEX array,
* dimension ( 2*DESCT(LLD_) )
* Additional workspace may be required if PCLATTRS is updated
* to use WORK.
*
* RWORK (local workspace) REAL array,
* dimension ( DESCT(LLD_) )
*
* INFO (global output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* The algorithm used in this program is basically backward (forward)
* substitution. It is the hope that scaling would be used to make the
* the code robust against possible overflow. But scaling has not yet
* been implemented in PCLATTRS which is called by this routine to solve
* the triangular systems. PCLATTRS just calls PCTRSV.
*
* Each eigenvector is normalized so that the element of largest
* magnitude has magnitude 1; here the magnitude of a complex number
* (x,y) is taken to be |x| + |y|.
*
* Further Details
* ===============
*
* Implemented by Mark R. Fahey, June, 2000
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
$ CONE = ( 1.0E+0, 0.0E+0 ) )
INTEGER BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_,
$ MB_, NB_, RSRC_, CSRC_, LLD_
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 )
* ..
* .. Local Scalars ..
LOGICAL ALLV, BOTHV, LEFTV, OVER, RIGHTV, SOMEV
INTEGER CONTXT, CSRC, I, ICOL, II, IROW, IS, ITMP1,
$ ITMP2, J, K, KI, LDT, LDVL, LDVR, LDW, MB,
$ MYCOL, MYROW, NB, NPCOL, NPROW, RSRC
REAL SELF
REAL OVFL, REMAXD, SCALE, SMIN, SMLNUM, ULP, UNFL
COMPLEX CDUM, REMAXC, SHIFT
* ..
* .. Local Arrays ..
INTEGER DESCW( DLEN_ )
* ..
* .. External Functions ..
LOGICAL LSAME
REAL PSLAMCH
EXTERNAL LSAME, PSLAMCH
* ..
* .. External Subroutines ..
EXTERNAL BLACS_GRIDINFO, DESCINIT, SGSUM2D, IGAMN2D,
$ INFOG2L, PSLABAD, PSCASUM, PXERBLA, PCAMAX,
$ PCCOPY, PCSSCAL, PCGEMV, PCLASET, PCLATTRS,
$ CGSUM2D
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, REAL, CMPLX, CONJG, AIMAG, MAX
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Statement Function definitions ..
CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( CDUM ) )
* ..
* .. Executable Statements ..
*
* This is just to keep ftnchek happy
IF( BLOCK_CYCLIC_2D*CSRC_*CTXT_*DLEN_*DTYPE_*LLD_*MB_*M_*NB_*N_*
$ RSRC_.LT.0 )RETURN
*
CONTXT = DESCT( CTXT_ )
RSRC = DESCT( RSRC_ )
CSRC = DESCT( CSRC_ )
MB = DESCT( MB_ )
NB = DESCT( NB_ )
LDT = DESCT( LLD_ )
LDW = LDT
LDVR = DESCVR( LLD_ )
LDVL = DESCVL( LLD_ )
*
CALL BLACS_GRIDINFO( CONTXT, NPROW, NPCOL, MYROW, MYCOL )
SELF = MYROW*NPCOL + MYCOL
*
* Decode and test the input parameters
*
BOTHV = LSAME( SIDE, 'B' )
RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV
LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV
*
ALLV = LSAME( HOWMNY, 'A' )
OVER = LSAME( HOWMNY, 'B' ) .OR. LSAME( HOWMNY, 'O' )
SOMEV = LSAME( HOWMNY, 'S' )
*
* Set M to the number of columns required to store the selected
* eigenvectors.
*
IF( SOMEV ) THEN
M = 0
DO 10 J = 1, N
IF( SELECT( J ) )
$ M = M + 1
10 CONTINUE
ELSE
M = N
END IF
*
INFO = 0
IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN
INFO = -1
ELSE IF( .NOT.ALLV .AND. .NOT.OVER .AND. .NOT.SOMEV ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( MM.LT.M ) THEN
INFO = -11
END IF
CALL IGAMN2D( CONTXT, 'ALL', ' ', 1, 1, INFO, 1, ITMP1, ITMP2, -1,
$ -1, -1 )
IF( INFO.LT.0 ) THEN
CALL PXERBLA( CONTXT, 'PCTREVC', -INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 )
$ RETURN
*
* Set the constants to control overflow.
*
UNFL = PSLAMCH( CONTXT, 'Safe minimum' )
OVFL = ONE / UNFL
CALL PSLABAD( CONTXT, UNFL, OVFL )
ULP = PSLAMCH( CONTXT, 'Precision' )
SMLNUM = UNFL*( N / ULP )
*
* Store the diagonal elements of T in working array WORK( LDW+1 ).
*
DO 20 I = 1, N
CALL INFOG2L( I, I, DESCT, NPROW, NPCOL, MYROW, MYCOL, IROW,
$ ICOL, ITMP1, ITMP2 )
IF( ( MYROW.EQ.ITMP1 ) .AND. ( MYCOL.EQ.ITMP2 ) ) THEN
WORK( LDW+IROW ) = T( ( ICOL-1 )*LDT+IROW )
END IF
20 CONTINUE
*
* Compute 1-norm of each column of strictly upper triangular
* part of T to control overflow in triangular solver. Computed,
* but not used. For use in PCLATTRS.
*
RWORK( 1 ) = ZERO
DO 30 J = 2, N
CALL PSCASUM( J-1, RWORK( J ), T, 1, J, DESCT, 1 )
30 CONTINUE
* I replicate the norms in RWORK. Should they be distributed
* over the process rows?
CALL SGSUM2D( CONTXT, 'Row', ' ', N, 1, RWORK, 1, -1, -1 )
*
IF( RIGHTV ) THEN
*
* Compute right eigenvectors.
*
* Need to set the distribution pattern of WORK
*
CALL DESCINIT( DESCW, N, 1, NB, 1, RSRC, CSRC, CONTXT, LDW,
$ INFO )
*
IS = M
DO 70 KI = N, 1, -1
*
IF( SOMEV ) THEN
IF( .NOT.SELECT( KI ) )
$ GO TO 70
END IF
*
SMIN = ZERO
SHIFT = CZERO
CALL INFOG2L( KI, KI, DESCT, NPROW, NPCOL, MYROW, MYCOL,
$ IROW, ICOL, ITMP1, ITMP2 )
IF( ( MYROW.EQ.ITMP1 ) .AND. ( MYCOL.EQ.ITMP2 ) ) THEN
SHIFT = T( ( ICOL-1 )*LDT+IROW )
SMIN = MAX( ULP*( CABS1( SHIFT ) ), SMLNUM )
END IF
CALL SGSUM2D( CONTXT, 'ALL', ' ', 1, 1, SMIN, 1, -1, -1 )
CALL CGSUM2D( CONTXT, 'ALL', ' ', 1, 1, SHIFT, 1, -1, -1 )
*
CALL INFOG2L( 1, 1, DESCW, NPROW, NPCOL, MYROW, MYCOL, IROW,
$ ICOL, ITMP1, ITMP2 )
IF( ( MYROW.EQ.ITMP1 ) .AND. ( MYCOL.EQ.ITMP2 ) ) THEN
WORK( 1 ) = CONE
END IF
*
* Form right-hand side. Distribute rhs onto first column
* of processor grid.
*
IF( KI.GT.1 ) THEN
CALL PCCOPY( KI-1, T, 1, KI, DESCT, 1, WORK, 1, 1, DESCW,
$ 1 )
END IF
DO 40 K = 1, KI - 1
CALL INFOG2L( K, 1, DESCW, NPROW, NPCOL, MYROW, MYCOL,
$ IROW, ICOL, ITMP1, ITMP2 )
IF( MYROW.EQ.ITMP1 .AND. MYCOL.EQ.ITMP2 ) THEN
WORK( IROW ) = -WORK( IROW )
END IF
40 CONTINUE
*
* Solve the triangular system:
* (T(1:KI-1,1:KI-1) - T(KI,KI))*X = SCALE*WORK.
*
DO 50 K = 1, KI - 1
CALL INFOG2L( K, K, DESCT, NPROW, NPCOL, MYROW, MYCOL,
$ IROW, ICOL, ITMP1, ITMP2 )
IF( ( MYROW.EQ.ITMP1 ) .AND. ( MYCOL.EQ.ITMP2 ) ) THEN
T( ( ICOL-1 )*LDT+IROW ) = T( ( ICOL-1 )*LDT+IROW ) -
$ SHIFT
IF( CABS1( T( ( ICOL-1 )*LDT+IROW ) ).LT.SMIN ) THEN
T( ( ICOL-1 )*LDT+IROW ) = CMPLX( SMIN )
END IF
END IF
50 CONTINUE
*
IF( KI.GT.1 ) THEN
CALL PCLATTRS( 'Upper', 'No transpose', 'Non-unit', 'Y',
$ KI-1, T, 1, 1, DESCT, WORK, 1, 1, DESCW,
$ SCALE, RWORK, INFO )
CALL INFOG2L( KI, 1, DESCW, NPROW, NPCOL, MYROW, MYCOL,
$ IROW, ICOL, ITMP1, ITMP2 )
IF( MYROW.EQ.ITMP1 .AND. MYCOL.EQ.ITMP2 ) THEN
WORK( IROW ) = CMPLX( SCALE )
END IF
END IF
*
* Copy the vector x or Q*x to VR and normalize.
*
IF( .NOT.OVER ) THEN
CALL PCCOPY( KI, WORK, 1, 1, DESCW, 1, VR, 1, IS, DESCVR,
$ 1 )
*
CALL PCAMAX( KI, REMAXC, II, VR, 1, IS, DESCVR, 1 )
REMAXD = ONE / MAX( CABS1( REMAXC ), UNFL )
CALL PCSSCAL( KI, REMAXD, VR, 1, IS, DESCVR, 1 )
*
CALL PCLASET( ' ', N-KI, 1, CZERO, CZERO, VR, KI+1, IS,
$ DESCVR )
ELSE
IF( KI.GT.1 )
$ CALL PCGEMV( 'N', N, KI-1, CONE, VR, 1, 1, DESCVR,
$ WORK, 1, 1, DESCW, 1, CMPLX( SCALE ),
$ VR, 1, KI, DESCVR, 1 )
*
CALL PCAMAX( N, REMAXC, II, VR, 1, KI, DESCVR, 1 )
REMAXD = ONE / MAX( CABS1( REMAXC ), UNFL )
CALL PCSSCAL( N, REMAXD, VR, 1, KI, DESCVR, 1 )
END IF
*
* Set back the original diagonal elements of T.
*
DO 60 K = 1, KI - 1
CALL INFOG2L( K, K, DESCT, NPROW, NPCOL, MYROW, MYCOL,
$ IROW, ICOL, ITMP1, ITMP2 )
IF( ( MYROW.EQ.ITMP1 ) .AND. ( MYCOL.EQ.ITMP2 ) ) THEN
T( ( ICOL-1 )*LDT+IROW ) = WORK( LDW+IROW )
END IF
60 CONTINUE
*
IS = IS - 1
70 CONTINUE
END IF
*
IF( LEFTV ) THEN
*
* Compute left eigenvectors.
*
* Need to set the distribution pattern of WORK
*
CALL DESCINIT( DESCW, N, 1, MB, 1, RSRC, CSRC, CONTXT, LDW,
$ INFO )
*
IS = 1
DO 110 KI = 1, N
*
IF( SOMEV ) THEN
IF( .NOT.SELECT( KI ) )
$ GO TO 110
END IF
*
SMIN = ZERO
SHIFT = CZERO
CALL INFOG2L( KI, KI, DESCT, NPROW, NPCOL, MYROW, MYCOL,
$ IROW, ICOL, ITMP1, ITMP2 )
IF( ( MYROW.EQ.ITMP1 ) .AND. ( MYCOL.EQ.ITMP2 ) ) THEN
SHIFT = T( ( ICOL-1 )*LDT+IROW )
SMIN = MAX( ULP*( CABS1( SHIFT ) ), SMLNUM )
END IF
CALL SGSUM2D( CONTXT, 'ALL', ' ', 1, 1, SMIN, 1, -1, -1 )
CALL CGSUM2D( CONTXT, 'ALL', ' ', 1, 1, SHIFT, 1, -1, -1 )
*
CALL INFOG2L( N, 1, DESCW, NPROW, NPCOL, MYROW, MYCOL, IROW,
$ ICOL, ITMP1, ITMP2 )
IF( ( MYROW.EQ.ITMP1 ) .AND. ( MYCOL.EQ.ITMP2 ) ) THEN
WORK( IROW ) = CONE
END IF
*
* Form right-hand side.
*
IF( KI.LT.N ) THEN
CALL PCCOPY( N-KI, T, KI, KI+1, DESCT, N, WORK, KI+1, 1,
$ DESCW, 1 )
END IF
DO 80 K = KI + 1, N
CALL INFOG2L( K, 1, DESCW, NPROW, NPCOL, MYROW, MYCOL,
$ IROW, ICOL, ITMP1, ITMP2 )
IF( MYROW.EQ.ITMP1 .AND. MYCOL.EQ.ITMP2 ) THEN
WORK( IROW ) = -CONJG( WORK( IROW ) )
END IF
80 CONTINUE
*
* Solve the triangular system:
* (T(KI+1:N,KI+1:N) - T(KI,KI))'*X = SCALE*WORK.
*
DO 90 K = KI + 1, N
CALL INFOG2L( K, K, DESCT, NPROW, NPCOL, MYROW, MYCOL,
$ IROW, ICOL, ITMP1, ITMP2 )
IF( ( MYROW.EQ.ITMP1 ) .AND. ( MYCOL.EQ.ITMP2 ) ) THEN
T( ( ICOL-1 )*LDT+IROW ) = T( ( ICOL-1 )*LDT+IROW ) -
$ SHIFT
IF( CABS1( T( ( ICOL-1 )*LDT+IROW ) ).LT.SMIN )
$ T( ( ICOL-1 )*LDT+IROW ) = CMPLX( SMIN )
END IF
90 CONTINUE
*
IF( KI.LT.N ) THEN
CALL PCLATTRS( 'Upper', 'Conjugate transpose', 'Nonunit',
$ 'Y', N-KI, T, KI+1, KI+1, DESCT, WORK,
$ KI+1, 1, DESCW, SCALE, RWORK, INFO )
CALL INFOG2L( KI, 1, DESCW, NPROW, NPCOL, MYROW, MYCOL,
$ IROW, ICOL, ITMP1, ITMP2 )
IF( MYROW.EQ.ITMP1 .AND. MYCOL.EQ.ITMP2 ) THEN
WORK( IROW ) = CMPLX( SCALE )
END IF
END IF
*
* Copy the vector x or Q*x to VL and normalize.
*
IF( .NOT.OVER ) THEN
CALL PCCOPY( N-KI+1, WORK, KI, 1, DESCW, 1, VL, KI, IS,
$ DESCVL, 1 )
*
CALL PCAMAX( N-KI+1, REMAXC, II, VL, KI, IS, DESCVL, 1 )
REMAXD = ONE / MAX( CABS1( REMAXC ), UNFL )
CALL PCSSCAL( N-KI+1, REMAXD, VL, KI, IS, DESCVL, 1 )
*
CALL PCLASET( ' ', KI-1, 1, CZERO, CZERO, VL, 1, IS,
$ DESCVL )
ELSE
IF( KI.LT.N )
$ CALL PCGEMV( 'N', N, N-KI, CONE, VL, 1, KI+1, DESCVL,
$ WORK, KI+1, 1, DESCW, 1, CMPLX( SCALE ),
$ VL, 1, KI, DESCVL, 1 )
*
CALL PCAMAX( N, REMAXC, II, VL, 1, KI, DESCVL, 1 )
REMAXD = ONE / MAX( CABS1( REMAXC ), UNFL )
CALL PCSSCAL( N, REMAXD, VL, 1, KI, DESCVL, 1 )
END IF
*
* Set back the original diagonal elements of T.
*
DO 100 K = KI + 1, N
CALL INFOG2L( K, K, DESCT, NPROW, NPCOL, MYROW, MYCOL,
$ IROW, ICOL, ITMP1, ITMP2 )
IF( ( MYROW.EQ.ITMP1 ) .AND. ( MYCOL.EQ.ITMP2 ) ) THEN
T( ( ICOL-1 )*LDT+IROW ) = WORK( LDW+IROW )
END IF
100 CONTINUE
*
IS = IS + 1
110 CONTINUE
END IF
*
RETURN
*
* End of PCTREVC
*
END
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