PCLANSY - return the value of the one norm, or the Frobenius norm,

REAL FUNCTION

PCLANSY( NORM, UPLO, N, A, IA, JA, DESCA, WORK )

CHARACTER NORM, UPLO

INTEGER IA, JA, N

INTEGER DESCA( * )

REAL WORK( * )

COMPLEX A( * )

PCLANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1).

PCLANSY returns the value

( max(abs(A(i,j))), NORM = 'M' or 'm' with IA <= i <= IA+N-1, ( and JA <= j <= JA+N-1, (

( norm1( sub( A ) ), NORM = '1', 'O' or 'o'

(

( normI( sub( A ) ), NORM = 'I' or 'i'

(

( normF( sub( A ) ), NORM = 'F', 'f', 'E' or 'e'

where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a matrix norm.

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

NORM (global input) CHARACTER

Specifies the value to be returned in PCLANSY as described above.

UPLO (global input) CHARACTER

Specifies whether the upper or lower triangular part of the symmetric matrix sub( A ) is to be referenced. = 'U': Upper triangular part of sub( A ) is referenced,

= 'L': Lower triangular part of sub( A ) is referenced.

N (global input) INTEGER

The number of rows and columns to be operated on i.e the number of rows and columns of the distributed submatrix sub( A ). When N = 0, PCLANSY is set to zero. N >= 0.

A (local input) COMPLEX pointer into the local memory

to an array of dimension (LLD_A, LOCc(JA+N-1)) containing the local pieces of the symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular matrix which norm is to be computed, and the strictly lower triangular part of this matrix is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular matrix which norm is to be computed, and the strictly upper triangular part of sub( A ) is not referenced.

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.

WORK (local workspace) REAL array dimension (LWORK)

LWORK >= 0 if NORM = 'M' or 'm' (not referenced), 2*Nq0+Np0+LDW if NORM = '1', 'O', 'o', 'I' or 'i', where LDW is given by: IF( NPROW.NE.NPCOL ) THEN LDW = MB_A*CEIL(CEIL(Np0/MB_A)/(LCM/NPROW)) ELSE LDW = 0 END IF 0 if NORM = 'F', 'f', 'E' or 'e' (not referenced),

where LCM is the least common multiple of NPROW and NPCOL LCM = ILCM( NPROW, NPCOL ) and CEIL denotes the ceiling operation (ICEIL).

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 ), Np0 = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFFA, NB_A, MYCOL, IACOL, NPCOL ),

ICEIL, ILCM, INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.