PCTRCON(3)    ScaLAPACK routine of NEC Numeric Library Collection   PCTRCON(3)



NAME
       PCTRCON - estimate the reciprocal of the condition number of a triangu-
       lar 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  triangu-
       lar  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  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
       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 col-
       umn.
       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 trian-
               gular part of this ditributed matrix contains the lower  trian-
               gular  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.



ScaLAPACK routine               31 October 2017                     PCTRCON(3)