PZLATTRS(l) ) PZLATTRS(l)NAME
PZLATTRS - solve one of the triangular systems A * x = s*b, A**T * x =
s*b, or A**H * x = s*b,
SYNOPSIS
SUBROUTINE PZLATTRS( 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
DOUBLE PRECISION SCALE
INTEGER DESCA( * ), DESCX( * )
DOUBLE PRECISION CNORM( * )
COMPLEX*16 A( * ), X( * )
PURPOSE
PZLATTRS 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 n-element 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
PZTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j)
then s is set to 0 and a non-trivial solution to A*x = 0 is returned.
This is very slow relative to PZTRSV. 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 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 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 col‐
umn.
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
ARGUMENTS
UPLO (global input) CHARACTER*1
Specifies whether the matrix A is upper or lower triangular. =
'U': Upper triangular
= 'L': Lower triangular
TRANS (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': Non-unit triangular
= 'U': Unit triangular
NORMIN (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*16 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 trian‐
gular matrix, and the strictly lower triangular part of A is
not referenced. If UPLO = 'L', the leading n by n lower trian‐
gular 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 refer‐
enced 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*16 array,
dimension (DESCX(LLD_),*) On entry, the right hand side b of
the triangular system. On exit, X is overwritten by the solu‐
tion 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) DOUBLE PRECISION
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) DOUBLE PRECISION array,
dimension (N) If NORMIN = 'Y', CNORM is an input argument and
CNORM(j) contains the norm of the off-diagonal part of the j-th
column of A. If TRANS = 'N', CNORM(j) must be greater than or
equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
must be greater than or equal to the 1-norm.
If NORMIN = 'N', CNORM is an output argument and CNORM(j)
returns the 1-norm of the offdiagonal part of the j-th column
of A.
INFO (global output) INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
FURTHER DETAILS
A rough bound on x is computed; if that is less than overflow, PZTRSV
is called, otherwise, specific code is used which checks for possible
overflow or divide-by-zero 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]
end
Define 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 infinity-norm of col‐
umn j+1 of A, not counting the diagonal. Hence
G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
1<=i<=j
and
|x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
1<=i< j
Since |x(j)| <= M(j), we use the Level 2 PBLAS routine PZTRSV 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 column‐
wise method can be performed without fear of overflow. If the computed
bound is greater than a large constant, x is scaled to prevent over‐
flow, but if the bound overflows, x is set to 0, x(j) to 1, and scale
to 0, and a non-trivial solution to A*x = 0 is found.
Similarly, a row-wise 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:j-1,j]' * x[1:j-1] ) / A(j,j)
end
We simultaneously compute two bounds
G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
M(j) = bound on x(i), 1<=i<=j
The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add
the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the
bound on x(j) is
M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
<= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
1<=i<=j
and we can safely call PZTRSV 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 13 August 2001 PZLATTRS(l)
