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PDTRCON - 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

SUBROUTINE PDTRCON( NORM, UPLO, DIAG, N, A, IA, JA, DESCA, RCOND, WORK, LWORK, IWORK, LIWORK, INFO ) CHARACTER DIAG, NORM, UPLO INTEGER IA, JA, INFO, LIWORK, LWORK, N DOUBLE PRECISION RCOND INTEGER DESCA( * ), IWORK( * ) DOUBLE PRECISION A( * ), WORK( * )

PDTRCON 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

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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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)) + LOCc(N+MOD(JA-1,NB_A)) + MAX( 2, MAX( NB_A*MAX( 1, CEIL(NPROW-1,NPCOL) ), LOCc(N+MOD(JA-1,NB_A)) + NB_A*MAX( 1, CEIL(NPCOL-1,NPROW) ) ). 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. IWORK (local workspace/local output) INTEGER array, dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK. LIWORK (local or global input) INTEGER The dimension of the array IWORK. LIWORK is local input and must be at least LIWORK >= LOCr(N+MOD(IA-1,MB_A)). If LIWORK = -1, then LIWORK 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.