Skip site navigation (1)Skip section navigation (2)

FreeBSD Manual Pages

  
 
  

home | help
PZGGQRF(l)			       )			    PZGGQRF(l)

NAME
       PZGGQRF	-  compute  a generalized QR factorization of an N-by-M	matrix
       sub( A )	= A(IA:IA+N-1,JA:JA+M-1) and an	 N-by-P	 matrix	 sub(  B  )  =
       B(IB:IB+N-1,JB:JB+P-1)

SYNOPSIS
       SUBROUTINE PZGGQRF( N,  M, P, A,	IA, JA,	DESCA, TAUA, B,	IB, JB,	DESCB,
			   TAUB, WORK, LWORK, INFO )

	   INTEGER	   IA, IB, INFO, JA, JB, LWORK,	M, N, P

	   INTEGER	   DESCA( * ), DESCB( *	)

	   COMPLEX*16	   A( *	), B( *	), TAUA( * ), TAUB( * ), WORK( * )

PURPOSE
       PZGGQRF computes	a generalized QR factorization	of  an	N-by-M	matrix
       sub(  A	)  =  A(IA:IA+N-1,JA:JA+M-1)  and  an N-by-P matrix sub( B ) =
       B(IB:IB+N-1,JB:JB+P-1):
		   sub(	A ) = Q*R,	  sub( B ) = Q*T*Z,

       where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix,  and
       R and T assume one of the forms:

       if N >= M,  R = ( R11 ) M  ,   or if N <	M,  R =	( R11  R12 ) N,
		       (  0  ) N-M			   N   M-N
			  M

       where R11 is upper triangular, and

       if N <= P,  T = ( 0  T12	) N,   or if N > P,  T = ( T11 ) N-P,
			P-N  N				 ( T21 ) P
							    P

       where T12 or T21	is upper triangular.

       In  particular,	if sub(	B ) is square and nonsingular, the GQR factor-
       ization of sub( A ) and sub( B )	implicitly gives the QR	 factorization
       of inv( sub( B )	)* sub(	A ):

		    inv( sub( B	) )*sub( A )= Z'*(inv(T)*R)

       where  inv(  sub( B ) ) denotes the inverse of the matrix sub( B	), and
       Z' denotes the conjugate	transpose of matrix Z.

       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_A

ARGUMENTS
       N       (global input) INTEGER
	       The number of rows to be	operated on i.e	the number of rows  of
	       the distributed submatrices sub(	A ) and	sub( B ). N >= 0.

       M       (global input) INTEGER
	       The  number of columns to be operated on	i.e the	number of col-
	       umns of the distributed submatrix sub( A	).  M >= 0.

       P       (global input) INTEGER
	       The number of columns to	be operated on i.e the number of  col-
	       umns of the distributed submatrix sub( B	).  P >= 0.

       A       (local input/local output) COMPLEX*16 pointer into the
	       local  memory  to  an array of dimension	(LLD_A,	LOCc(JA+M-1)).
	       On entry, the local pieces of  the  N-by-M  distributed	matrix
	       sub( A )	which is to be factored.  On exit, the elements	on and
	       above the diagonal of sub( A ) contain the min(N,M) by M	 upper
	       trapezoidal matrix R (R is upper	triangular if N	>= M); the el-
	       ements below the	diagonal, with the array TAUA,	represent  the
	       unitary matrix Q	as a product of	min(N,M) 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.

       TAUA    (local output) COMPLEX*16, array, dimension
	       LOCc(JA+MIN(N,M)-1).  This  array  contains  the	scalar factors
	       TAUA of the elementary reflectors which represent  the  unitary
	       matrix  Q.  TAUA	is tied	to the distributed matrix A. (see Fur-
	       ther Details).  B       (local input/local  output)  COMPLEX*16
	       pointer	into the local memory to an array of dimension (LLD_B,
	       LOCc(JB+P-1)).  On entry, the local pieces of the  N-by-P  dis-
	       tributed	matrix sub( B )	which is to be factored. On exit, if N
	       <= P, the upper triangle	of B(IB:IB+N-1,JB+P-N:JB+P-1) contains
	       the N by	N upper	triangular matrix T; if	N > P, the elements on
	       and above the (N-P)-th subdiagonal contain the  N  by  P	 upper
	       trapezoidal  matrix  T;	the remaining elements,	with the array
	       TAUB, represent the unitary matrix Z as a product of elementary
	       reflectors (see Further Details).  IB	  (global input) INTE-
	       GER The row index in the	global array B	indicating  the	 first
	       row of sub( B ).

       JB      (global input) INTEGER
	       The  column  index  in  the global array	B indicating the first
	       column of sub( B	).

       DESCB   (global and local input)	INTEGER	array of dimension DLEN_.
	       The array descriptor for	the distributed	matrix B.

       TAUB    (local output) COMPLEX*16, array, dimension LOCr(IB+N-1)
	       This array contains the scalar factors of  the  elementary  re-
	       flectors	 which represent the unitary matrix Z. TAUB is tied to
	       the distributed matrix B	(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 >= MAX( NB_A	* ( NpA0 + MqA0	+ NB_A ), MAX(
	       (NB_A*(NB_A-1))/2, (PqB0	+ NpB0)*NB_A ) + NB_A *	NB_A, MB_B * (
	       NpB0 + PqB0 + MB_B ) ), where

	       IROFFA  =  MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A ), IAROW
	       = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL  =  INDXG2P(
	       JA,  NB_A,  MYCOL,  CSRC_A, NPCOL ), NpA0   = NUMROC( N+IROFFA,
	       MB_A, MYROW, IAROW, NPROW ), MqA0   = NUMROC(  M+ICOFFA,	 NB_A,
	       MYCOL, IACOL, NPCOL ),

	       IROFFB  =  MOD( IB-1, MB_B ), ICOFFB = MOD( JB-1, NB_B ), IBROW
	       = INDXG2P( IB, MB_B, MYROW, RSRC_B, NPROW ), IBCOL  =  INDXG2P(
	       JB,  NB_B,  MYCOL,  CSRC_B, NPCOL ), NpB0   = NUMROC( N+IROFFB,
	       MB_B, MYROW, IBROW, NPROW ), PqB0   = NUMROC(  P+ICOFFB,	 NB_B,
	       MYCOL, IBCOL, NPCOL ),

	       and NUMROC, INDXG2P 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 matrix Q is represented as a	product	of elementary reflectors

	  Q = H(ja) H(ja+1) . .	. H(ja+k-1), where k = min(n,m).

       Each H(i) has the form

	  H(i) = I - taua * v *	v'

       where taua is a complex scalar, and v is	a complex vector with v(1:i-1)
       = 0 and v(i) = 1; v(i+1:n) is stored on exit in
       A(ia+i:ia+n-1,ja+i-1), and taua in TAUA(ja+i-1).
       To form Q explicitly, use ScaLAPACK subroutine PZUNGQR.
       To use Q	to update another matrix, use ScaLAPACK	subroutine PZUNMQR.

       The matrix Z is represented as a	product	of elementary reflectors

	  Z = H(ib)' H(ib+1)' .	. . H(ib+k-1)',	where k	= min(n,p).

       Each H(i) has the form

	  H(i) = I - taub * v *	v'

       where taub is a complex scalar, and v is	a  complex  vector  with  v(p-
       k+i+1:p)	= 0 and	v(p-k+i) = 1; conjg(v(1:p-k+i-1)) is stored on exit in
       B(ib+n-k+i-1,jb:jb+p-k+i-2), and	taub in	TAUB(ib+n-k+i-1).  To  form  Z
       explicitly, use ScaLAPACK subroutine PZUNGRQ.
       To use Z	to update another matrix, use ScaLAPACK	subroutine PZUNMRQ.

       Alignment requirements
       ======================

       The  distributed	 submatrices  sub(  A  ) and sub( B ) must verify some
       alignment properties, namely the	following expression should be true:

       ( MB_A.EQ.MB_B .AND. IROFFA.EQ.IROFFB .AND. IAROW.EQ.IBROW )

ScaLAPACK version 1.7		13 August 2001			    PZGGQRF(l)

NAME | SYNOPSIS | PURPOSE | ARGUMENTS | FURTHER DETAILS

Want to link to this manual page? Use this URL:
<https://www.freebsd.org/cgi/man.cgi?query=pzggqrf&manpath=FreeBSD+12.1-RELEASE+and+Ports>

home | help