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

FreeBSD Manual Pages

  
 
  

home | help
PCGEBD2(l)			       )			    PCGEBD2(l)

NAME
       PCGEBD2 - reduce	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  uni-
       tary transformation

SYNOPSIS
       SUBROUTINE PCGEBD2( M,  N,  A,  IA,  JA,	DESCA, D, E, TAUQ, TAUP, WORK,
			   LWORK, INFO )

	   INTEGER	   IA, INFO, JA, LWORK,	M, N

	   INTEGER	   DESCA( * )

	   REAL		   D( *	), E( *	)

	   COMPLEX	   A( *	), TAUP( * ), TAUQ( * ), WORK( * )

PURPOSE
       PCGEBD2 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 uni-
       tary 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 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
       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 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 ). 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	 prod-
	       uct  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	subdi-
	       agonal, 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  ar-
	       ray 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) REAL 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) REAL 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(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 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  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)  COM-
	       PLEX array, dimension (LWORK) On	exit, WORK(1) returns the min-
	       imal 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( MpA0, NqA0 )

	       where  NB  =  MB_A = NB_A, IROFFA = MOD(	IA-1, NB ) IAROW = IN-
	       DXG2P( 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+IROFFA,  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    (local 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 reflec-
       tors:

       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  vec-
       tors;
       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 vec-
       tors;
       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 exam-
       ples:

       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 de-
       notes 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 )

ScaLAPACK version 1.7		13 August 2001			    PCGEBD2(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=pcgebd2&manpath=FreeBSD+12.1-RELEASE+and+Ports>

home | help