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|
SUBROUTINE PZGEQPF( M, N, A, IA, JA, DESCA, IPIV, TAU, WORK,
$ LWORK, RWORK, LRWORK, INFO )
*
* -- ScaLAPACK routine (version 2.1) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* November 20, 2019
*
* .. Scalar Arguments ..
INTEGER IA, JA, INFO, LRWORK, LWORK, M, N
* ..
* .. Array Arguments ..
INTEGER DESCA( * ), IPIV( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* PZGEQPF computes a QR factorization with column pivoting of a
* M-by-N distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1):
*
* sub( A ) * P = Q * R.
*
* 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
* =========
*
* M (global input) INTEGER
* The number of rows to be operated on, i.e. the number of rows
* of the distributed submatrix sub( A ). M >= 0.
*
* N (global input) INTEGER
* The number of columns to be operated on, i.e. the number of
* columns of the distributed submatrix sub( A ). N >= 0.
*
* A (local input/local output) COMPLEX*16 pointer into the
* local memory to an array of dimension (LLD_A, LOCc(JA+N-1)).
* On entry, the local pieces of the M-by-N distributed matrix
* sub( A ) which is to be factored. On exit, the elements on
* and above the diagonal of sub( A ) contain the min(M,N) by N
* upper trapezoidal matrix R (R is upper triangular if M >= N);
* the elements below the diagonal, with the array TAU, repre-
* sent the unitary matrix Q as a product of elementary
* reflectors (see Further Details).
*
* IA (global input) INTEGER
* The row index in the global array A indicating the first
* row of sub( A ).
*
* JA (global input) INTEGER
* The column index in the global array A indicating the
* first column of sub( A ).
*
* DESCA (global and local input) INTEGER array of dimension DLEN_.
* The array descriptor for the distributed matrix A.
*
* IPIV (local output) INTEGER array, dimension LOCc(JA+N-1).
* On exit, if IPIV(I) = K, the local i-th column of sub( A )*P
* was the global K-th column of sub( A ). IPIV is tied to the
* distributed matrix A.
*
* TAU (local output) COMPLEX*16, array, dimension
* LOCc(JA+MIN(M,N)-1). This array contains the scalar factors
* TAU of the elementary reflectors. TAU is tied to the
* distributed matrix A.
*
* WORK (local workspace/local output) COMPLEX*16 array,
* dimension (LWORK)
* On exit, WORK(1) returns the minimal and optimal LWORK.
*
* LWORK (local or global input) INTEGER
* The dimension of the array WORK.
* LWORK is local input and must be at least
* LWORK >= MAX(3,Mp0 + Nq0).
*
* If LWORK = -1, then LWORK is global input and a workspace
* query is assumed; the routine only calculates the minimum
* and optimal size for all work arrays. Each of these
* values is returned in the first entry of the corresponding
* work array, and no error message is issued by PXERBLA.
*
* RWORK (local workspace/local output) DOUBLE PRECISION array,
* dimension (LRWORK)
* On exit, RWORK(1) returns the minimal and optimal LRWORK.
*
* LRWORK (local or global input) INTEGER
* The dimension of the array RWORK.
* LRWORK is local input and must be at least
* LRWORK >= LOCc(JA+N-1)+Nq0.
*
* IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ),
* IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ),
* IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ),
* Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ),
* Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ),
* LOCc(JA+N-1) = NUMROC( JA+N-1, NB_A, MYCOL, CSRC_A, NPCOL )
*
* and NUMROC, INDXG2P are ScaLAPACK tool functions;
* MYROW, MYCOL, NPROW and NPCOL can be determined by calling
* the subroutine BLACS_GRIDINFO.
*
* If LRWORK = -1, then LRWORK is global input and a workspace
* query is assumed; the routine only calculates the minimum
* and optimal size for all work arrays. Each of these
* values is returned in the first entry of the corresponding
* work array, and no error message is issued by PXERBLA.
*
*
* INFO (global output) INTEGER
* = 0: successful exit
* < 0: If the i-th argument is an array and the j-entry had
* an illegal value, then INFO = -(i*100+j), if the i-th
* argument is a scalar and had an illegal value, then
* INFO = -i.
*
* Further Details
* ===============
*
* The matrix Q is represented as a product of elementary reflectors
*
* Q = H(1) H(2) . . . H(n)
*
* Each H(i) has the form
*
* H = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
* A(ia+i-1:ia+m-1,ja+i-1).
*
* The matrix P is represented in jpvt as follows: If
* jpvt(j) = i
* then the jth column of P is the ith canonical unit vector.
*
* References
* ==========
*
* For modifications introduced in Scalapack 2.1
* LAWN 295
* New robust ScaLAPACK routine for computing the QR factorization with column pivoting
* Zvonimir Bujanovic, Zlatko Drmac
* http://www.netlib.org/lapack/lawnspdf/lawn295.pdf
*
* =====================================================================
*
* .. Parameters ..
INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
$ LLD_, MB_, M_, NB_, N_, RSRC_
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 )
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IACOL, IAROW, ICOFF, ICTXT, ICURROW,
$ ICURCOL, II, IIA, IOFFA, IPCOL, IROFF, ITEMP,
$ J, JB, JJ, JJA, JJPVT, JN, KB, K, KK, KSTART,
$ KSTEP, LDA, LL, LRWMIN, LWMIN, MN, MP, MYCOL,
$ MYROW, NPCOL, NPROW, NQ, NQ0, PVT
DOUBLE PRECISION TEMP, TEMP2, TOL3Z
COMPLEX*16 AJJ, ALPHA
* ..
* .. Local Arrays ..
INTEGER DESCN( DLEN_ ), IDUM1( 2 ), IDUM2( 2 )
* ..
* .. External Subroutines ..
EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, IGERV2D,
$ IGESD2D, INFOG1L, INFOG2L, PCHK1MAT, PDAMAX,
$ PDZNRM2, PXERBLA, PZELSET,
$ PZLARFC, PZLARFG, ZCOPY, ZGEBR2D,
$ ZGEBS2D, ZGERV2D, ZGESD2D, ZLARFG,
$ ZSWAP
* ..
* .. External Functions ..
INTEGER ICEIL, INDXG2P, NUMROC
EXTERNAL ICEIL, INDXG2P, NUMROC
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DCMPLX, DCONJG, IDINT, MAX, MIN, MOD, SQRT
* ..
* .. Executable Statements ..
*
* Get grid parameters
*
ICTXT = DESCA( CTXT_ )
CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
*
* Test the input parameters
*
INFO = 0
IF( NPROW.EQ.-1 ) THEN
INFO = -(600+CTXT_)
ELSE
CALL CHK1MAT( M, 1, N, 2, IA, JA, DESCA, 6, INFO )
IF( INFO.EQ.0 ) THEN
IROFF = MOD( IA-1, DESCA( MB_ ) )
ICOFF = MOD( JA-1, DESCA( NB_ ) )
IAROW = INDXG2P( IA, DESCA( MB_ ), MYROW, DESCA( RSRC_ ),
$ NPROW )
IACOL = INDXG2P( JA, DESCA( NB_ ), MYCOL, DESCA( CSRC_ ),
$ NPCOL )
MP = NUMROC( M+IROFF, DESCA( MB_ ), MYROW, IAROW, NPROW )
NQ = NUMROC( N+ICOFF, DESCA( NB_ ), MYCOL, IACOL, NPCOL )
NQ0 = NUMROC( JA+N-1, DESCA( NB_ ), MYCOL, DESCA( CSRC_ ),
$ NPCOL )
LWMIN = MAX( 3, MP + NQ )
LRWMIN = NQ0 + NQ
*
WORK( 1 ) = DCMPLX( DBLE( LWMIN ) )
RWORK( 1 ) = DBLE( LRWMIN )
LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 )
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -10
ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
IF( LWORK.EQ.-1 ) THEN
IDUM1( 1 ) = -1
ELSE
IDUM1( 1 ) = 1
END IF
IDUM2( 1 ) = 10
IF( LRWORK.EQ.-1 ) THEN
IDUM1( 2 ) = -1
ELSE
IDUM1( 2 ) = 1
END IF
IDUM2( 2 ) = 12
CALL PCHK1MAT( M, 1, N, 2, IA, JA, DESCA, 6, 2, IDUM1, IDUM2,
$ INFO )
END IF
*
IF( INFO.NE.0 ) THEN
CALL PXERBLA( ICTXT, 'PZGEQPF', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
CALL INFOG2L( IA, JA, DESCA, NPROW, NPCOL, MYROW, MYCOL, IIA, JJA,
$ IAROW, IACOL )
IF( MYROW.EQ.IAROW )
$ MP = MP - IROFF
IF( MYCOL.EQ.IACOL )
$ NQ = NQ - ICOFF
MN = MIN( M, N )
TOL3Z = SQRT( DLAMCH('Epsilon') )
*
* Initialize the array of pivots
*
LDA = DESCA( LLD_ )
JN = MIN( ICEIL( JA, DESCA( NB_ ) ) * DESCA( NB_ ), JA+N-1 )
KSTEP = NPCOL * DESCA( NB_ )
*
IF( MYCOL.EQ.IACOL ) THEN
*
* Handle first block separately
*
JB = JN - JA + 1
DO 10 LL = JJA, JJA+JB-1
IPIV( LL ) = JA + LL - JJA
10 CONTINUE
KSTART = JN + KSTEP - DESCA( NB_ )
*
* Loop over remaining block of columns
*
DO 30 KK = JJA+JB, JJA+NQ-1, DESCA( NB_ )
KB = MIN( JJA+NQ-KK, DESCA( NB_ ) )
DO 20 LL = KK, KK+KB-1
IPIV( LL ) = KSTART+LL-KK+1
20 CONTINUE
KSTART = KSTART + KSTEP
30 CONTINUE
ELSE
KSTART = JN + ( MOD( MYCOL-IACOL+NPCOL, NPCOL )-1 )*
$ DESCA( NB_ )
DO 50 KK = JJA, JJA+NQ-1, DESCA( NB_ )
KB = MIN( JJA+NQ-KK, DESCA( NB_ ) )
DO 40 LL = KK, KK+KB-1
IPIV( LL ) = KSTART+LL-KK+1
40 CONTINUE
KSTART = KSTART + KSTEP
50 CONTINUE
END IF
*
* Initialize partial column norms, handle first block separately
*
CALL DESCSET( DESCN, 1, DESCA( N_ ), 1, DESCA( NB_ ), MYROW,
$ DESCA( CSRC_ ), ICTXT, 1 )
*
JJ = JJA
IF( MYCOL.EQ.IACOL ) THEN
DO 60 KK = 0, JB-1
CALL PDZNRM2( M, RWORK( JJ+KK ), A, IA, JA+KK, DESCA, 1 )
RWORK( NQ+JJ+KK ) = RWORK( JJ+KK )
60 CONTINUE
JJ = JJ + JB
END IF
ICURCOL = MOD( IACOL+1, NPCOL )
*
* Loop over the remaining blocks of columns
*
DO 80 J = JN+1, JA+N-1, DESCA( NB_ )
JB = MIN( JA+N-J, DESCA( NB_ ) )
*
IF( MYCOL.EQ.ICURCOL ) THEN
DO 70 KK = 0, JB-1
CALL PDZNRM2( M, RWORK( JJ+KK ), A, IA, J+KK, DESCA, 1 )
RWORK( NQ+JJ+KK ) = RWORK( JJ+KK )
70 CONTINUE
JJ = JJ + JB
END IF
ICURCOL = MOD( ICURCOL+1, NPCOL )
80 CONTINUE
*
* Compute factorization
*
DO 120 J = JA, JA+MN-1
I = IA + J - JA
*
CALL INFOG1L( J, DESCA( NB_ ), NPCOL, MYCOL, DESCA( CSRC_ ),
$ JJ, ICURCOL )
K = JA + N - J
IF( K.GT.1 ) THEN
CALL PDAMAX( K, TEMP, PVT, RWORK, 1, J, DESCN,
$ DESCN( M_ ) )
ELSE
PVT = J
END IF
IF( J.NE.PVT ) THEN
CALL INFOG1L( PVT, DESCA( NB_ ), NPCOL, MYCOL,
$ DESCA( CSRC_ ), JJPVT, IPCOL )
IF( ICURCOL.EQ.IPCOL ) THEN
IF( MYCOL.EQ.ICURCOL ) THEN
CALL ZSWAP( MP, A( IIA+(JJ-1)*LDA ), 1,
$ A( IIA+(JJPVT-1)*LDA ), 1 )
ITEMP = IPIV( JJPVT )
IPIV( JJPVT ) = IPIV( JJ )
IPIV( JJ ) = ITEMP
RWORK( JJPVT ) = RWORK( JJ )
RWORK( NQ+JJPVT ) = RWORK( NQ+JJ )
END IF
ELSE
IF( MYCOL.EQ.ICURCOL ) THEN
*
CALL ZGESD2D( ICTXT, MP, 1, A( IIA+(JJ-1)*LDA ), LDA,
$ MYROW, IPCOL )
WORK( 1 ) = DCMPLX( DBLE( IPIV( JJ ) ) )
WORK( 2 ) = DCMPLX( RWORK( JJ ) )
WORK( 3 ) = DCMPLX( RWORK( JJ + NQ ) )
CALL ZGESD2D( ICTXT, 3, 1, WORK, 3, MYROW, IPCOL )
*
CALL ZGERV2D( ICTXT, MP, 1, A( IIA+(JJ-1)*LDA ), LDA,
$ MYROW, IPCOL )
CALL IGERV2D( ICTXT, 1, 1, IPIV( JJ ), 1, MYROW,
$ IPCOL )
*
ELSE IF( MYCOL.EQ.IPCOL ) THEN
*
CALL ZGESD2D( ICTXT, MP, 1, A( IIA+(JJPVT-1)*LDA ),
$ LDA, MYROW, ICURCOL )
CALL IGESD2D( ICTXT, 1, 1, IPIV( JJPVT ), 1, MYROW,
$ ICURCOL )
*
CALL ZGERV2D( ICTXT, MP, 1, A( IIA+(JJPVT-1)*LDA ),
$ LDA, MYROW, ICURCOL )
CALL ZGERV2D( ICTXT, 3, 1, WORK, 3, MYROW, ICURCOL )
IPIV( JJPVT ) = IDINT( DBLE( WORK( 1 ) ) )
RWORK( JJPVT ) = DBLE( WORK( 2 ) )
RWORK( JJPVT+NQ ) = DBLE( WORK( 3 ) )
*
END IF
*
END IF
*
END IF
*
* Generate elementary reflector H(i)
*
CALL INFOG1L( I, DESCA( MB_ ), NPROW, MYROW, DESCA( RSRC_ ),
$ II, ICURROW )
IF( DESCA( M_ ).EQ.1 ) THEN
IF( MYROW.EQ.ICURROW ) THEN
IF( MYCOL.EQ.ICURCOL ) THEN
IOFFA = II+(JJ-1)*DESCA( LLD_ )
AJJ = A( IOFFA )
CALL ZLARFG( 1, AJJ, A( IOFFA ), 1, TAU( JJ ) )
IF( N.GT.1 ) THEN
ALPHA = CMPLX( ONE ) - DCONJG( TAU( JJ ) )
CALL ZGEBS2D( ICTXT, 'Rowwise', ' ', 1, 1, ALPHA,
$ 1 )
CALL ZSCAL( NQ-JJ, ALPHA, A( IOFFA+DESCA( LLD_ ) ),
$ DESCA( LLD_ ) )
END IF
CALL ZGEBS2D( ICTXT, 'Columnwise', ' ', 1, 1,
$ TAU( JJ ), 1 )
A( IOFFA ) = AJJ
ELSE
IF( N.GT.1 ) THEN
CALL ZGEBR2D( ICTXT, 'Rowwise', ' ', 1, 1, ALPHA,
$ 1, ICURROW, ICURCOL )
CALL ZSCAL( NQ-JJ+1, ALPHA, A( I ), DESCA( LLD_ ) )
END IF
END IF
ELSE IF( MYCOL.EQ.ICURCOL ) THEN
CALL ZGEBR2D( ICTXT, 'Columnwise', ' ', 1, 1, TAU( JJ ),
$ 1, ICURROW, ICURCOL )
END IF
*
ELSE
*
CALL PZLARFG( M-J+JA, AJJ, I, J, A, MIN( I+1, IA+M-1 ), J,
$ DESCA, 1, TAU )
IF( J.LT.JA+N-1 ) THEN
*
* Apply H(i) to A(ia+j-ja:ia+m-1,j+1:ja+n-1) from the left
*
CALL PZELSET( A, I, J, DESCA, DCMPLX( ONE ) )
CALL PZLARFC( 'Left', M-J+JA, JA+N-1-J, A, I, J, DESCA,
$ 1, TAU, A, I, J+1, DESCA, WORK )
END IF
CALL PZELSET( A, I, J, DESCA, AJJ )
*
END IF
*
* Update partial columns norms
*
IF( MYCOL.EQ.ICURCOL )
$ JJ = JJ + 1
IF( MOD( J, DESCA( NB_ ) ).EQ.0 )
$ ICURCOL = MOD( ICURCOL+1, NPCOL )
IF( (JJA+NQ-JJ).GT.0 ) THEN
IF( MYROW.EQ.ICURROW ) THEN
CALL ZGEBS2D( ICTXT, 'Columnwise', ' ', 1, JJA+NQ-JJ,
$ A( II+( MIN( JJA+NQ-1, JJ )-1 )*LDA ),
$ LDA )
CALL ZCOPY( JJA+NQ-JJ, A( II+( MIN( JJA+NQ-1, JJ )
$ -1)*LDA ), LDA, WORK( MIN( JJA+NQ-1, JJ ) ),
$ 1 )
ELSE
CALL ZGEBR2D( ICTXT, 'Columnwise', ' ', JJA+NQ-JJ, 1,
$ WORK( MIN( JJA+NQ-1, JJ ) ), MAX( 1, NQ ),
$ ICURROW, MYCOL )
END IF
END IF
*
JN = MIN( ICEIL( J+1, DESCA( NB_ ) ) * DESCA( NB_ ),
$ JA + N - 1 )
IF( MYCOL.EQ.ICURCOL ) THEN
DO 90 LL = JJ, JJ + JN - J - 1
IF( RWORK( LL ).NE.ZERO ) THEN
TEMP = ABS( WORK( LL ) ) / RWORK( LL )
TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
TEMP2 = TEMP * ( RWORK( LL ) / RWORK( NQ+LL ) )**2
IF( TEMP2.LE.TOL3Z ) THEN
IF( IA+M-1.GT.I ) THEN
CALL PDZNRM2( IA+M-I-1, RWORK( LL ), A,
$ I+1, J+LL-JJ+1, DESCA, 1 )
RWORK( NQ+LL ) = RWORK( LL )
ELSE
RWORK( LL ) = ZERO
RWORK( NQ+LL ) = ZERO
END IF
ELSE
RWORK( LL ) = RWORK( LL ) * SQRT( TEMP )
END IF
END IF
90 CONTINUE
JJ = JJ + JN - J
END IF
ICURCOL = MOD( ICURCOL+1, NPCOL )
*
DO 110 K = JN+1, JA+N-1, DESCA( NB_ )
KB = MIN( JA+N-K, DESCA( NB_ ) )
*
IF( MYCOL.EQ.ICURCOL ) THEN
DO 100 LL = JJ, JJ+KB-1
IF( RWORK(LL).NE.ZERO ) THEN
TEMP = ABS( WORK( LL ) ) / RWORK( LL )
TEMP = MAX( ZERO, ( ONE+TEMP )*( ONE-TEMP ) )
TEMP2 = TEMP * ( RWORK( LL ) / RWORK( NQ+LL ) )**2
IF( TEMP2.LE.TOL3Z ) THEN
IF( IA+M-1.GT.I ) THEN
CALL PDZNRM2( IA+M-I-1, RWORK( LL ), A,
$ I+1, K+LL-JJ, DESCA, 1 )
RWORK( NQ+LL ) = RWORK( LL )
ELSE
RWORK( LL ) = ZERO
RWORK( NQ+LL ) = ZERO
END IF
ELSE
RWORK( LL ) = RWORK( LL ) * SQRT( TEMP )
END IF
END IF
100 CONTINUE
JJ = JJ + KB
END IF
ICURCOL = MOD( ICURCOL+1, NPCOL )
*
110 CONTINUE
*
120 CONTINUE
*
WORK( 1 ) = DCMPLX( DBLE( LWMIN ) )
RWORK( 1 ) = DBLE( LRWMIN )
*
RETURN
*
* End of PZGEQPF
*
END
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