NAME
PCGEQPF - compute a QR factorization with column pivoting of a M-by-N
distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1)
SYNOPSIS
SUBROUTINE PCGEQPF( M, N, A, IA, JA, DESCA, IPIV, TAU, WORK, LWORK,
RWORK, LRWORK, INFO )
INTEGER IA, JA, INFO, LRWORK, LWORK, M, N
INTEGER DESCA( * ), IPIV( * )
REAL RWORK( * )
COMPLEX A( * ), TAU( * ), WORK( * )
PURPOSE
PCGEQPF computes a QR factorization with column pivoting of a M-by-N
distributed matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1):
sub( A ) * P = Q * R.
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 pointer into the
local memory to an array of dimension (LLD_A, LOCc(JA+N-1)).
On entry, the local pieces of the M-by-N distributed matrix
sub( A ) which is to be factored. On exit, the elements on and
above the diagonal of sub( A ) contain the min(M,N) by N upper
trapezoidal matrix R (R is upper triangular if M >= N); the
elements below the diagonal, with the array TAU, repre- sent
the unitary matrix Q 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.
IPIV (local output) INTEGER array, dimension LOCc(JA+N-1).
On exit, if IPIV(I) = K, the local i-th column of sub( A )*P
was the global K-th column of sub( A ). IPIV is tied to the
distributed matrix A.
TAU (local output) COMPLEX, array, dimension
LOCc(JA+MIN(M,N)-1). This array contains the scalar factors TAU
of the elementary reflectors. TAU is tied to the distributed
matrix A.
WORK (local workspace/local output) COMPLEX 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(3,Mp0 + Nq0).
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.
RWORK (local workspace/local output) REAL array,
dimension (LRWORK) On exit, RWORK(1) returns the minimal and
optimal LRWORK.
LRWORK (local or global input) INTEGER
The dimension of the array RWORK. LRWORK is local input and
must be at least LRWORK >= LOCc(JA+N-1)+Nq0.
IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW =
INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA,
NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A,
MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL,
IACOL, NPCOL ), LOCc(JA+N-1) = NUMROC( JA+N-1, NB_A, MYCOL,
CSRC_A, NPCOL )
and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL,
NPROW and NPCOL can be determined by calling the subroutine
BLACS_GRIDINFO.
If LRWORK = -1, then LRWORK 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(1) H(2) . . . H(n)
Each H(i) has the form
H = I - tau * v * v’
where tau is a complex scalar, and v is a complex vector with v(1:i-1)
= 0 and v(i) = 1; v(i+1:m) is stored on exit in
A(ia+i-1:ia+m-1,ja+i-1).
The matrix P is represented in jpvt as follows: If
jpvt(j) = i
then the jth column of P is the ith canonical unit vector.