NAME
PCTRCON - estimate the reciprocal of the condition number of a
triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the
1-norm or the infinity-norm
SYNOPSIS
SUBROUTINE PCTRCON( NORM, UPLO, DIAG, N, A, IA, JA, DESCA, RCOND, WORK,
LWORK, RWORK, LRWORK, INFO )
CHARACTER DIAG, NORM, UPLO
INTEGER IA, JA, INFO, LRWORK, LWORK, N
REAL RCOND
INTEGER DESCA( * )
REAL RWORK( * )
COMPLEX A( * ), WORK( * )
PURPOSE
PCTRCON estimates the reciprocal of the condition number of a
triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the
1-norm or the infinity-norm.
The norm of A(IA:IA+N-1,JA:JA+N-1) is computed and an estimate is
obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), then the reciprocal of
the condition number is computed as
RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) *
norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ).
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
NORM (global input) CHARACTER
Specifies whether the 1-norm condition number or the infinity-
norm condition number is required:
= ’1’ or ’O’: 1-norm;
= ’I’: Infinity-norm.
UPLO (global input) CHARACTER
= ’U’: A(IA:IA+N-1,JA:JA+N-1) is upper triangular;
= ’L’: A(IA:IA+N-1,JA:JA+N-1) is lower triangular.
DIAG (global input) CHARACTER
= ’N’: A(IA:IA+N-1,JA:JA+N-1) is non-unit triangular;
= ’U’: A(IA:IA+N-1,JA:JA+N-1) is unit triangular.
N (global input) INTEGER
The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N
>= 0.
A (local input) COMPLEX pointer into the local memory
to an array of dimension ( LLD_A, LOCc(JA+N-1) ). This array
contains the local pieces of the triangular distributed matrix
A(IA:IA+N-1,JA:JA+N-1). If UPLO = ’U’, the leading N-by-N upper
triangular part of this distributed matrix con- tains the upper
triangular matrix, and its strictly lower triangular part is
not referenced. If UPLO = ’L’, the leading N-by-N lower
triangular part of this ditributed matrix contains the lower
triangular matrix, and the strictly upper triangular part is
not referenced. If DIAG = ’U’, the diagonal elements of
A(IA:IA+N-1,JA:JA+N-1) are also not referenced and are assumed
to be 1.
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.
RCOND (global output) REAL
The reciprocal of the condition number of the distributed
matrix A(IA:IA+N-1,JA:JA+N-1), computed as
RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) *
norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ).
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 >= 2*LOCr(N+MOD(IA-1,MB_A)) + MAX( 2,
MAX(NB_A*CEIL(P-1,Q),LOCc(N+MOD(JA-1,NB_A)) + NB_A*CEIL(Q-1,P))
).
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(N+MOD(JA-1,NB_A)).
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.