PZLATTRS(3) ScaLAPACK routine of NEC Numeric Library Collection PZLATTRS(3)
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 routine 31 October 2017 PZLATTRS(3)