SUBROUTINE PZGEBRD( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP,
$ WORK, LWORK, INFO )
*
* -- ScaLAPACK routine (version 1.5) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* May 1, 1997
*
* .. Scalar Arguments ..
INTEGER IA, INFO, JA, LWORK, M, N
* ..
* .. Array Arguments ..
INTEGER DESCA( * )
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 A( * ), TAUP( * ), TAUQ( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* PZGEBRD reduces a complex general M-by-N distributed matrix
* sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal
* form B by an unitary transformation: Q' * sub( A ) * P = B.
*
* If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal.
*
* 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, this array contains the local pieces of the
* general distributed matrix sub( A ). On exit, if M >= N,
* the diagonal and the first superdiagonal of sub( A ) are
* overwritten with the upper bidiagonal matrix B; the elements
* below the diagonal, with the array TAUQ, represent the
* unitary matrix Q as a product of elementary reflectors, and
* the elements above the first superdiagonal, with the array
* TAUP, represent the orthogonal matrix P as a product of
* elementary reflectors. If M < N, the diagonal and the first
* subdiagonal are overwritten with the lower bidiagonal
* matrix B; the elements below the first subdiagonal, with the
* array TAUQ, represent the unitary matrix Q as a product of
* elementary reflectors, and the elements above the diagonal,
* with the array TAUP, represent the orthogonal matrix P 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.
*
* D (local output) DOUBLE PRECISION array, dimension
* LOCc(JA+MIN(M,N)-1) if M >= N; LOCr(IA+MIN(M,N)-1) otherwise.
* The distributed diagonal elements of the bidiagonal matrix
* B: D(i) = A(i,i). D is tied to the distributed matrix A.
*
* E (local output) DOUBLE PRECISION array, dimension
* LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise.
* The distributed off-diagonal elements of the bidiagonal
* distributed matrix B:
* if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
* if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
* E is tied to the distributed matrix A.
*
* TAUQ (local output) COMPLEX*16 array dimension
* LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary
* reflectors which represent the unitary matrix Q. TAUQ is
* tied to the distributed matrix A. See Further Details.
*
* TAUP (local output) COMPLEX*16 array, dimension
* LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary
* reflectors which represent the unitary matrix P. TAUP is
* tied to the distributed matrix A. See Further Details.
*
* 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 >= NB*( MpA0 + NqA0 + 1 ) + NqA0
*
* where NB = MB_A = NB_A,
* IROFFA = MOD( IA-1, NB ), ICOFFA = MOD( JA-1, NB ),
* IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ),
* IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A, NPCOL ),
* MpA0 = NUMROC( M+IROFFA, NB, MYROW, IAROW, NPROW ),
* NqA0 = NUMROC( N+ICOFFA, NB, MYCOL, IACOL, NPCOL ).
*
* INDXG2P and NUMROC are ScaLAPACK tool functions;
* MYROW, MYCOL, NPROW and NPCOL can be determined by calling
* the subroutine BLACS_GRIDINFO.
*
* 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.
*
* 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 matrices Q and P are represented as products of elementary
* reflectors:
*
* If m >= n,
*
* Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
*
* Each H(i) and G(i) has the form:
*
* H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
*
* where tauq and taup are complex scalars, and v and u are complex
* vectors;
* v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
* A(ia+i:ia+m-1,ja+i-1);
* u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
* A(ia+i-1,ja+i+1:ja+n-1);
* tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1).
*
* If m < n,
*
* Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
*
* Each H(i) and G(i) has the form:
*
* H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
*
* where tauq and taup are complex scalars, and v and u are complex
* vectors;
* v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in
* A(ia+i+1:ia+m-1,ja+i-1);
* u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in
* A(ia+i-1,ja+i:ja+n-1);
* tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1).
*
* The contents of sub( A ) on exit are illustrated by the following
* examples:
*
* m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
*
* ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
* ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
* ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
* ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
* ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
* ( v1 v2 v3 v4 v5 )
*
* where d and e denote diagonal and off-diagonal elements of B, vi
* denotes an element of the vector defining H(i), and ui an element of
* the vector defining G(i).
*
* Alignment requirements
* ======================
*
* The distributed submatrix sub( A ) must verify some alignment proper-
* ties, namely the following expressions should be true:
* ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA )
*
* =====================================================================
*
* .. 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 )
COMPLEX*16 ONE
PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
CHARACTER COLCTOP, ROWCTOP
INTEGER I, IACOL, IAROW, ICTXT, IINFO, IOFF, IPW, IPY,
$ IW, J, JB, JS, JW, K, L, LWMIN, MN, MP, MYCOL,
$ MYROW, NB, NPCOL, NPROW, NQ
* ..
* .. Local Arrays ..
INTEGER DESCWX( DLEN_ ), DESCWY( DLEN_ ), IDUM1( 1 ),
$ IDUM2( 1 )
* ..
* .. External Subroutines ..
EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, PCHK1MAT,
$ PTOPGET, PTOPSET, PXERBLA, PZELSET, PZGEBD2,
$ PZGEMM, PZLABRD
* ..
* .. External Functions ..
INTEGER INDXG2L, INDXG2P, NUMROC
EXTERNAL INDXG2L, INDXG2P, NUMROC
* ..
* .. Intrinsic Functions ..
INTRINSIC DCMPLX, DBLE, MAX, MIN, MOD
* ..
* .. 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
NB = DESCA( MB_ )
IOFF = MOD( IA-1, DESCA( MB_ ) )
IAROW = INDXG2P( IA, NB, MYROW, DESCA( RSRC_ ), NPROW )
IACOL = INDXG2P( JA, NB, MYCOL, DESCA( CSRC_ ), NPCOL )
MP = NUMROC( M+IOFF, NB, MYROW, IAROW, NPROW )
NQ = NUMROC( N+IOFF, NB, MYCOL, IACOL, NPCOL )
LWMIN = NB*( MP+NQ+1 ) + NQ
*
WORK( 1 ) = DCMPLX( DBLE( LWMIN ) )
LQUERY = ( LWORK.EQ.-1 )
IF( IOFF.NE.MOD( JA-1, DESCA( NB_ ) ) ) THEN
INFO = -5
ELSE IF( NB.NE.DESCA( NB_ ) ) THEN
INFO = -(600+NB_)
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
IF( LQUERY ) THEN
IDUM1( 1 ) = -1
ELSE
IDUM1( 1 ) = 1
END IF
IDUM2( 1 ) = 12
CALL PCHK1MAT( M, 1, N, 2, IA, JA, DESCA, 6, 1, IDUM1, IDUM2,
$ INFO )
END IF
*
IF( INFO.LT.0 ) THEN
CALL PXERBLA( ICTXT, 'PZGEBRD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
MN = MIN( M, N )
IF( MN.EQ.0 )
$ RETURN
*
* Initialize parameters.
*
CALL PTOPGET( ICTXT, 'Combine', 'Columnwise', COLCTOP )
CALL PTOPGET( ICTXT, 'Combine', 'Rowwise', ROWCTOP )
CALL PTOPSET( ICTXT, 'Combine', 'Columnwise', '1-tree' )
CALL PTOPSET( ICTXT, 'Combine', 'Rowwise', '1-tree' )
*
IPY = MP * NB + 1
IPW = NQ * NB + IPY
*
CALL DESCSET( DESCWX, M+IOFF, NB, NB, NB, IAROW, IACOL, ICTXT,
$ MAX( 1, MP ) )
CALL DESCSET( DESCWY, NB, N+IOFF, NB, NB, IAROW, IACOL, ICTXT,
$ NB )
*
MP = NUMROC( M+IA-1, NB, MYROW, DESCA( RSRC_ ), NPROW )
NQ = NUMROC( N+JA-1, NB, MYCOL, DESCA( CSRC_ ), NPCOL )
K = 1
JB = NB - IOFF
IW = IOFF + 1
JW = IOFF + 1
*
DO 10 L = 1, MN+IOFF-NB, NB
I = IA + K - 1
J = JA + K - 1
*
* Reduce rows and columns i:i+nb-1 to bidiagonal form and return
* the matrices X and Y which are needed to update the unreduced
* part of the matrix.
*
CALL PZLABRD( M-K+1, N-K+1, JB, A, I, J, DESCA, D, E, TAUQ,
$ TAUP, WORK, IW, JW, DESCWX, WORK( IPY ), IW,
$ JW, DESCWY, WORK( IPW ) )
*
* Update the trailing submatrix A(i+nb:ia+m-1,j+nb:ja+n-1), using
* an update of the form A := A - V*Y' - X*U'.
*
CALL PZGEMM( 'No transpose', 'No transpose', M-K-JB+1,
$ N-K-JB+1, JB, -ONE, A, I+JB, J, DESCA,
$ WORK( IPY ), IW, JW+JB, DESCWY, ONE, A, I+JB,
$ J+JB, DESCA )
CALL PZGEMM( 'No transpose', 'No transpose', M-K-JB+1,
$ N-K-JB+1, JB, -ONE, WORK, IW+JB, JW, DESCWX, A, I,
$ J+JB, DESCA, ONE, A, I+JB, J+JB, DESCA )
*
* Copy last off-diagonal elements of B back into sub( A ).
*
IF( M.GE.N ) THEN
JS = MIN( INDXG2L( I+JB-1, NB, 0, DESCA( RSRC_ ), NPROW ),
$ MP )
CALL PZELSET( A, I+JB-1, J+JB, DESCA, DCMPLX( E( JS ) ) )
ELSE
JS = MIN( INDXG2L( J+JB-1, NB, 0, DESCA( CSRC_ ), NPCOL ),
$ NQ )
CALL PZELSET( A, I+JB, J+JB-1, DESCA, DCMPLX( E( JS ) ) )
END IF
*
K = K + JB
JB = NB
IW = 1
JW = 1
DESCWX( M_ ) = DESCWX( M_ ) - JB
DESCWX( RSRC_ ) = MOD( DESCWX( RSRC_ ) + 1, NPROW )
DESCWX( CSRC_ ) = MOD( DESCWX( CSRC_ ) + 1, NPCOL )
DESCWY( N_ ) = DESCWY( N_ ) - JB
DESCWY( RSRC_ ) = MOD( DESCWY( RSRC_ ) + 1, NPROW )
DESCWY( CSRC_ ) = MOD( DESCWY( CSRC_ ) + 1, NPCOL )
*
10 CONTINUE
*
* Use unblocked code to reduce the remainder of the matrix.
*
CALL PZGEBD2( M-K+1, N-K+1, A, IA+K-1, JA+K-1, DESCA, D, E, TAUQ,
$ TAUP, WORK, LWORK, IINFO )
*
CALL PTOPSET( ICTXT, 'Combine', 'Columnwise', COLCTOP )
CALL PTOPSET( ICTXT, 'Combine', 'Rowwise', ROWCTOP )
*
WORK( 1 ) = DCMPLX( DBLE( LWMIN ) )
*
RETURN
*
* End of PZGEBRD
*
END