

SUBROUTINE PCLATTRS(  UPLO, TRANS, DIAG, NORMIN, N, A, IA, JA, DESCA, X, IX, JX, DESCX, SCALE, CNORM, INFO ) 
CHARACTER DIAG, NORMIN, TRANS, UPLO  
INTEGER IA, INFO, IX, JA, JX, N  
REAL SCALE  
INTEGER DESCA( * ), DESCX( * )  
REAL CNORM( * )  
COMPLEX A( * ), X( * )  
PCLATTRS solves one of the triangular systems A * x = s*b, A**T * x = s*b, or A**H * x = s*b, with scaling to prevent overflow. Here A is an upper or lower triangular matrix, A**T denotes the transpose of A, A**H denotes the conjugate transpose of A, x and b are nelement vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 PBLAS routine PCTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j) then s is set to 0 and a nontrivial solution to A*x = 0 is returned.This is very slow relative to PCTRSV. This should only be used when scaling is necessary to control overflow, or when it is modified to scale better.
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 r x c.
LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the r 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 c 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
UPLO (global input) CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = ’U’: Upper triangular
= ’L’: Lower triangularTRANS (global input) CHARACTER*1 Specifies the operation applied to A. = ’N’: Solve A * x = s*b (No transpose)
= ’T’: Solve A**T * x = s*b (Transpose)
= ’C’: Solve A**H * x = s*b (Conjugate transpose)DIAG (global input) CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = ’N’: Nonunit triangular
= ’U’: Unit triangularNORMIN (global input) CHARACTER*1 Specifies whether CNORM has been set or not. = ’Y’: CNORM contains the column norms on entry
= ’N’: CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.N (global input) INTEGER The order of the matrix A. N >= 0. A (local input) COMPLEX array, dimension (DESCA(LLD_),*) The triangular matrix A. If UPLO = ’U’, the leading n by n upper triangular part of the array A contains the upper triangular matrix, and the strictly lower triangular part of A is not referenced. If UPLO = ’L’, the leading n by n lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. If DIAG = ’U’, the diagonal elements of A are also not referenced and are assumed to be 1. IA (global input) pointer to INTEGER The global row index of the submatrix of the distributed matrix A to operate on. JA (global input) pointer to INTEGER The global column index of the submatrix of the distributed matrix A to operate on. DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. X (local input/output) COMPLEX array, dimension (DESCX(LLD_),*) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x. IX (global input) pointer to INTEGER The global row index of the submatrix of the distributed matrix X to operate on. JX (global input) pointer to INTEGER The global column index of the submatrix of the distributed matrix X to operate on. DESCX (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix X. SCALE (global output) REAL The scaling factor s for the triangular system A * x = s*b, A**T * x = s*b, or A**H * x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0. CNORM (global input or global output) REAL array, dimension (N) If NORMIN = ’Y’, CNORM is an input argument and CNORM(j) contains the norm of the offdiagonal part of the jth column of A. If TRANS = ’N’, CNORM(j) must be greater than or equal to the infinitynorm, and if TRANS = ’T’ or ’C’, CNORM(j) must be greater than or equal to the 1norm. If NORMIN = ’N’, CNORM is an output argument and CNORM(j) returns the 1norm of the offdiagonal part of the jth column of A.
INFO (global output) INTEGER = 0: successful exit
< 0: if INFO = k, the kth argument had an illegal value
A rough bound on x is computed; if that is less than overflow, PCTRSV is called, otherwise, specific code is used which checks for possible overflow or dividebyzero at every operation.A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is
x[1:n] := b[1:n]
for j = 1, ..., n
x(j) := x(j) / A(j,j)
x[j+1:n] := x[j+1:n]  x(j) * A[j+1:n,j]
endDefine bounds on the components of x after j iterations of the loop:
M(j) = bound on x[1:j]
G(j) = bound on x[j+1:n]
Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.Then for iteration j+1 we have
M(j+1) <= G(j) /  A(j+1,j+1) 
G(j+1) <= G(j) + M(j+1) *  A[j+2:n,j+1] 
<= G(j) ( 1 + CNORM(j+1) /  A(j+1,j+1)  )where CNORM(j+1) is greater than or equal to the infinitynorm of column j+1 of A, not counting the diagonal. Hence
G(j) <= G(0) product ( 1 + CNORM(i) /  A(i,i)  )
1<=i<=j
andx(j) <= ( G(0) / A(j,j) ) product ( 1 + CNORM(i) / A(i,i) )
1<=i< jSince x(j) <= M(j), we use the Level 2 PBLAS routine PCTRSV if the reciprocal of the largest M(j), j=1,..,n, is larger than
max(underflow, 1/overflow).The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a nontrivial solution to A*x = 0 is found.
Similarly, a rowwise scheme is used to solve A**T *x = b or A**H *x = b. The basic algorithm for A upper triangular is
for j = 1, ..., n
x(j) := ( b(j)  A[1:j1,j]’ * x[1:j1] ) / A(j,j)
endWe simultaneously compute two bounds
G(j) = bound on ( b(i)  A[1:i1,i]’ * x[1:i1] ), 1<=i<=j
M(j) = bound on x(i), 1<=i<=jThe initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j1) and M(j) >= M(j1) for j >= 1. Then the bound on x(j) is
M(j) <= M(j1) * ( 1 + CNORM(j) ) /  A(j,j) 
<= M(0) * product ( ( 1 + CNORM(i) ) / A(i,i) )
1<=i<=jand we can safely call PCTRSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).
Last modified by: Mark R. Fahey, August 2000
ScaLAPACK version 1.7  PCLATTRS (l)  13 August 2001 
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