# FreeBSD Manual Pages

PZLABRD(l) ) PZLABRD(l)NAMEPZLABRD - 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 re- turns the matrices X and Y which are needed to apply the transfor- ma- tion to the unreduced part of sub( A )SYNOPSISSUBROUTINE PZLABRD( M, N, NB, A, IA, JA, DESCA, D, E, TAUQ, TAUP, X, IX, JX, DESCX, Y, IY, JY, DESCY, WORK ) INTEGER IA, IX, IY, JA, JX, JY, M, N, NB INTEGER DESCA( * ), DESCX( * ), DESCY( * ) DOUBLE PRECISION D( * ), E( * ) COMPLEX*16 A( * ), TAUP( * ), TAUQ( * ), X( * ), Y( * ), WORK( * )PURPOSEPZLABRD 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 re- turns the matrices X and Y which are needed to apply the transfor- ma- tion 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. This is an auxiliary routine called by PZGEBRD. Notes ===== Each global data object is described by an associated description vec- tor. This vector stores the information required to establish the map- ping 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 as- sume that its process grid has dimension p x q. LOCr( K ) denotes the number of elements of K that a process would re- ceive 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_AARGUMENTSM (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 re- duced. 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 dis- tributed 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 ar- ray TAUQ, represent the unitary matrix Q as a product of ele- mentary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the unitary ma- trix P as a product of elementary reflectors. If m < n, ele- ments below the diagonal in the first NB columns, with the ar- ray TAUQ, represent the unitary matrix Q as a product of ele- mentary 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 dis- tributed 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 re- flectors 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) re- quired 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 DETAILSThe matrices Q and P are represented as products of elementary reflec- tors: 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 vec- tors. If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(ia+i-1:ia+m-1,ja+i-1); u(1:i) = 0, u(i+1) = 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). If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1: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: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 elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U' which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block up- date of the form: sub( A ) := sub( A ) - V*Y' - X*U'. The contents of sub( A ) on exit are illustrated by the following exam- ples with nb = 2: m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( 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 ) where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). ScaLAPACK version 1.7 13 August 2001 PZLABRD(l)

NAME | SYNOPSIS | PURPOSE | ARGUMENTS | FURTHER DETAILS

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