table of contents
PZLABRD(l) | LAPACK auxiliary routine (version 1.5) | PZLABRD(l) |
NAME¶
PZLABRD - reduce the first NB rows and columns of 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 by an unitary transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transfor- mation to the unreduced part of sub( A )SYNOPSIS¶
- SUBROUTINE PZLABRD(
- M, N, NB, A, IA, JA, DESCA, D, E, TAUQ, TAUP, X, IX, JX, DESCX, Y, IY, JY, DESCY, WORK )
PURPOSE¶
PZLABRD reduces the first NB rows and columns of 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 by an unitary transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transfor- mation to the unreduced part of sub( A ). If M >= N, sub( A ) is reduced to upper bidiagonal form; if M < N, to lower bidiagonal form.DTYPE_A = 1.
the BLACS process grid A is distribu-
ted over. The context itself is glo-
bal, but the handle (the integer
value) may vary.
array A.
array A.
the rows of the array.
the columns of the array.
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.
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( 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.
- NB (global input) INTEGER
- The number of leading rows and columns of sub( A ) to be reduced.
- 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 ) to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the distributed matrix sub( A ) is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the unitary 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
- LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-1) otherwise. The distributed diagonal elements of the bidiagonal matrix B: D(i) = A(ia+i-1,ja+i-1). 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(ia+i-1,ja+i) for i = 1,2,...,n-1; if m < n, E(i) = A(ia+i,ja+i-1) 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. X (local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_X,NB). On exit, the local pieces of the distributed M-by-NB matrix X(IX:IX+M-1,JX:JX+NB-1) required to update the unreduced part of sub( A ).
- IX (global input) INTEGER
- The row index in the global array X indicating the first row of sub( X ).
- JX (global input) INTEGER
- The column index in the global array X indicating the first column of sub( X ).
- DESCX (global and local input) INTEGER array of dimension DLEN_.
- The array descriptor for the distributed matrix X.
- Y (local output) COMPLEX*16 pointer into the local memory
- to an array of dimension (LLD_Y,NB). On exit, the local pieces of the distributed N-by-NB matrix Y(IY:IY+N-1,JY:JY+NB-1) required to update the unreduced part of sub( A ).
- IY (global input) INTEGER
- The row index in the global array Y indicating the first row of sub( Y ).
- JY (global input) INTEGER
- The column index in the global array Y indicating the first column of sub( Y ).
- DESCY (global and local input) INTEGER array of dimension DLEN_.
- The array descriptor for the distributed matrix Y.
- WORK (local workspace) COMPLEX*16 array, dimension (LWORK)
- LWORK >= NB_A + NQ, with NQ = NUMROC( N+MOD( IA-1, NB_Y ), NB_Y, MYCOL, IACOL, NPCOL ) IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ) INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.
FURTHER DETAILS¶
The matrices Q and P are represented as products of elementary reflectors:Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) 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.
( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
( v1 v2 a a a ) ( v1 1 a a a a )
( v1 v2 a a a ) ( v1 v2 a a a a )
( v1 v2 a a a ) ( v1 v2 a a a a )
( v1 v2 a a a )
12 May 1997 | LAPACK version 1.5 |