PCGESVX(3) ScaLAPACK routine of NEC Numeric Library Collection PCGESVX(3)
NAME
PCGESVX - use the LU factorization to compute the solution to a complex
system of linear equations A(IA:IA+N-1,JA:JA+N-1) * X =
B(IB:IB+N-1,JB:JB+NRHS-1),
SYNOPSIS
SUBROUTINE PCGESVX( FACT, TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF,
JAF, DESCAF, IPIV, EQUED, R, C, B, IB, JB, DESCB,
X, IX, JX, DESCX, RCOND, FERR, BERR, WORK, LWORK,
RWORK, LRWORK, INFO )
CHARACTER EQUED, FACT, TRANS
INTEGER IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LRWORK,
LWORK, N, NRHS
REAL RCOND
INTEGER DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * ),
IPIV( * )
REAL BERR( * ), C( * ), FERR( * ), R( * ), RWORK( * )
COMPLEX A( * ), AF( * ), B( * ), WORK( * ), X( * )
PURPOSE
PCGESVX uses the LU factorization to compute the solution to a complex
system of linear equations A(IA:IA+N-1,JA:JA+N-1) * X =
B(IB:IB+N-1,JB:JB+NRHS-1), where A(IA:IA+N-1,JA:JA+N-1) is an N-by-N
matrix and X and B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS matrices.
Error bounds on the solution and a condition estimate are also pro-
vided.
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 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 col-
umn.
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
DESCRIPTION
In the following description, A denotes A(IA:IA+N-1,JA:JA+N-1), B
denotes B(IB:IB+N-1,JB:JB+NRHS-1) and X denotes
X(IX:IX+N-1,JX:JX+NRHS-1).
The following steps are performed:
1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
or diag(C)*B (if TRANS = 'T' or 'C').
2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
matrix A (after equilibration if FACT = 'E') as
A = P * L * U,
where P is a permutation matrix, L is a unit lower triangular
matrix, and U is upper triangular.
3. The factored form of A is used to estimate the condition number
of the matrix A. If the reciprocal of the condition number is
less than machine precision, steps 4-6 are skipped.
4. The system of equations is solved for X using the factored form
of A.
5. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
6. If FACT = 'E' and equilibration was used, the matrix X is
premultiplied by diag(C) (if TRANS = 'N') or diag(R) (if
TRANS = 'T' or 'C') so that it solves the original system
before equilibration.
ARGUMENTS
FACT (global input) CHARACTER
Specifies whether or not the factored form of the matrix
A(IA:IA+N-1,JA:JA+N-1) is supplied on entry, and if not,
whether the matrix A(IA:IA+N-1,JA:JA+N-1) should be equili-
brated before it is factored. = 'F': On entry,
AF(IAF:IAF+N-1,JAF:JAF+N-1) and IPIV con-
tain the factored form of A(IA:IA+N-1,JA:JA+N-1). If EQUED is
not 'N', the matrix A(IA:IA+N-1,JA:JA+N-1) has been equili-
brated with scaling factors given by R and C.
A(IA:IA+N-1,JA:JA+N-1), AF(IAF:IAF+N-1,JAF:JAF+N-1), and IPIV
are not modified. = 'N': The matrix A(IA:IA+N-1,JA:JA+N-1)
will be copied to
AF(IAF:IAF+N-1,JAF:JAF+N-1) and factored.
= 'E': The matrix A(IA:IA+N-1,JA:JA+N-1) will be equili-
brated if necessary, then copied to AF(IAF:IAF+N-1,JAF:JAF+N-1)
and factored.
TRANS (global input) CHARACTER
Specifies the form of the system of equations:
= 'N': A(IA:IA+N-1,JA:JA+N-1) * X(IX:IX+N-1,JX:JX+NRHS-1)
= B(IB:IB+N-1,JB:JB+NRHS-1) (No transpose)
= 'T': A(IA:IA+N-1,JA:JA+N-1)**T * X(IX:IX+N-1,JX:JX+NRHS-1)
= B(IB:IB+N-1,JB:JB+NRHS-1) (Transpose)
= 'C': A(IA:IA+N-1,JA:JA+N-1)**H * X(IX:IX+N-1,JX:JX+NRHS-1)
= B(IB:IB+N-1,JB:JB+NRHS-1) (Conjugate transpose)
N (global input) INTEGER
The number of rows and columns to be operated on, i.e. the
order of the distributed submatrix A(IA:IA+N-1,JA:JA+N-1). N
>= 0.
NRHS (global input) INTEGER
The number of right-hand sides, i.e., the number of columns of
the distributed submatrices B(IB:IB+N-1,JB:JB+NRHS-1) and
X(IX:IX+N-1,JX:JX+NRHS-1). NRHS >= 0.
A (local input/local output) COMPLEX pointer into
the local memory to an array of local dimension
(LLD_A,LOCc(JA+N-1)). On entry, the N-by-N matrix
A(IA:IA+N-1,JA:JA+N-1). If FACT = 'F' and EQUED is not 'N',
then A(IA:IA+N-1,JA:JA+N-1) must have been equilibrated by
the scaling factors in R and/or C. A(IA:IA+N-1,JA:JA+N-1) is
not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED
= 'N' on exit.
On exit, if EQUED .ne. 'N', A(IA:IA+N-1,JA:JA+N-1) is scaled as
follows:
EQUED = 'R': A(IA:IA+N-1,JA:JA+N-1) :=
diag(R) * A(IA:IA+N-1,JA:JA+N-1)
EQUED = 'C': A(IA:IA+N-1,JA:JA+N-1) :=
A(IA:IA+N-1,JA:JA+N-1) * diag(C)
EQUED = 'B': A(IA:IA+N-1,JA:JA+N-1) :=
diag(R) * A(IA:IA+N-1,JA:JA+N-1) * diag(C).
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.
AF (local input or local output) COMPLEX pointer
into the local memory to an array of local dimension
(LLD_AF,LOCc(JA+N-1)). If FACT = 'F', then
AF(IAF:IAF+N-1,JAF:JAF+N-1) is an input argument and on entry
contains the factors L and U from the factorization
A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PCGETRF. If
EQUED .ne. 'N', then AF is the factored form of the equili-
brated matrix A(IA:IA+N-1,JA:JA+N-1).
If FACT = 'N', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output
argument and on exit returns the factors L and U from the fac-
torization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the original
matrix A(IA:IA+N-1,JA:JA+N-1).
If FACT = 'E', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output
argument and on exit returns the factors L and U from the fac-
torization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the equili-
brated matrix A(IA:IA+N-1,JA:JA+N-1) (see the description of
A(IA:IA+N-1,JA:JA+N-1) for the form of the equilibrated
matrix).
IAF (global input) INTEGER
The row index in the global array AF indicating the first row
of sub( AF ).
JAF (global input) INTEGER
The column index in the global array AF indicating the first
column of sub( AF ).
DESCAF (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix AF.
IPIV (local input or local output) INTEGER array, dimension
LOCr(M_A)+MB_A. If FACT = 'F', then IPIV is an input argu- ment
and on entry contains the pivot indices from the fac- toriza-
tion A(IA:IA+N-1,JA:JA+N-1) = P*L*U as computed by PCGETRF;
IPIV(i) -> The global row local row i was swapped with. This
array must be aligned with A( IA:IA+N-1, * ).
If FACT = 'N', then IPIV is an output argument and on exit con-
tains the pivot indices from the factorization
A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the original matrix
A(IA:IA+N-1,JA:JA+N-1).
If FACT = 'E', then IPIV is an output argument and on exit con-
tains the pivot indices from the factorization
A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the equilibrated matrix
A(IA:IA+N-1,JA:JA+N-1).
EQUED (global input or global output) CHARACTER
Specifies the form of equilibration that was done. = 'N': No
equilibration (always true if FACT = 'N').
= 'R': Row equilibration, i.e., A(IA:IA+N-1,JA:JA+N-1) has
been premultiplied by diag(R). = 'C': Column equilibration,
i.e., A(IA:IA+N-1,JA:JA+N-1) has been postmultiplied by
diag(C). = 'B': Both row and column equilibration, i.e.,
A(IA:IA+N-1,JA:JA+N-1) has been replaced by
diag(R) * A(IA:IA+N-1,JA:JA+N-1) * diag(C). EQUED is an input
variable if FACT = 'F'; otherwise, it is an output variable.
R (local input or local output) REAL array,
dimension LOCr(M_A). The row scale factors for
A(IA:IA+N-1,JA:JA+N-1).
If EQUED = 'R' or 'B', A(IA:IA+N-1,JA:JA+N-1) is multiplied on
the left by diag(R); if EQUED='N' or 'C', R is not acces- sed.
R is an input variable if FACT = 'F'; otherwise, R is an output
variable. If FACT = 'F' and EQUED = 'R' or 'B', each element
of R must be positive. R is replicated in every process
column, and is aligned with the distributed matrix A.
C (local input or local output) REAL array,
dimension LOCc(N_A). The column scale factors for
A(IA:IA+N-1,JA:JA+N-1).
If EQUED = 'C' or 'B', A(IA:IA+N-1,JA:JA+N-1) is multiplied on
the right by diag(C); if EQUED = 'N' or 'R', C is not accessed.
C is an input variable if FACT = 'F'; otherwise, C is an output
variable. If FACT = 'F' and EQUED = 'C' or C is replicated in
every process row, and is aligned with the distributed matrix
A.
B (local input/local output) COMPLEX pointer
into the local memory to an array of local dimension
(LLD_B,LOCc(JB+NRHS-1) ). On entry, the N-by-NRHS right-hand
side matrix B(IB:IB+N-1,JB:JB+NRHS-1). On exit, if
EQUED = 'N', B(IB:IB+N-1,JB:JB+NRHS-1) is not modified; if
TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
diag(R)*B(IB:IB+N-1,JB:JB+NRHS-1); if TRANS = 'T' or 'C'
and EQUED = 'C' or 'B', B(IB:IB+N-1,JB:JB+NRHS-1) is over-
written by diag(C)*B(IB:IB+N-1,JB:JB+NRHS-1).
IB (global input) INTEGER
The row index in the global array B indicating the first row of
sub( B ).
JB (global input) INTEGER
The column index in the global array B indicating the first
column of sub( B ).
DESCB (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix B.
X (local input/local output) COMPLEX pointer
into the local memory to an array of local dimension (LLD_X,
LOCc(JX+NRHS-1)). If INFO = 0, the N-by-NRHS solution matrix
X(IX:IX+N-1,JX:JX+NRHS-1) to the original
system of equations. Note that A(IA:IA+N-1,JA:JA+N-1) and
B(IB:IB+N-1,JB:JB+NRHS-1) are modified on exit if EQUED .ne.
'N', and the solution to the equilibrated system is
inv(diag(C))*X(IX:IX+N-1,JX:JX+NRHS-1) if TRANS = 'N' and EQUED
= 'C' or 'B', or inv(diag(R))*X(IX:IX+N-1,JX:JX+NRHS-1) if
TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
IX (global input) INTEGER
The row index in the global array X indicating the first row of
sub( X ).
JX (global input) INTEGER
The column index in the global array X indicating the first
column of sub( X ).
DESCX (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix X.
RCOND (global output) REAL
The estimate of the reciprocal condition number of the matrix
A(IA:IA+N-1,JA:JA+N-1) after equilibration (if done). If RCOND
is less than the machine precision (in particular, if RCOND =
0), the matrix is singular to working precision. This condi-
tion is indicated by a return code of INFO > 0.
FERR (local output) REAL array, dimension LOCc(N_B)
The estimated forward error bounds for each solution vector
X(j) (the j-th column of the solution matrix
X(IX:IX+N-1,JX:JX+NRHS-1). If XTRUE is the true solution,
FERR(j) bounds the magnitude of the largest entry in (X(j) -
XTRUE) divided by the magnitude of the largest entry in X(j).
The estimate is as reliable as the estimate for RCOND, and is
almost always a slight overestimate of the true error. FERR is
replicated in every process row, and is aligned with the matri-
ces B and X.
BERR (local output) REAL array, dimension LOCc(N_B).
The componentwise relative backward error of each solution vec-
tor X(j) (i.e., the smallest relative change in any entry of
A(IA:IA+N-1,JA:JA+N-1) or
B(IB:IB+N-1,JB:JB+NRHS-1) that makes X(j) an exact solution).
BERR is replicated in every process row, and is aligned with
the matrices B and X.
WORK (local workspace/local output) COMPLEX 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 = MAX( PCGECON( LWORK ), PCGERFS( LWORK ) ) +
LOCr( N_A ).
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.
RWORK (local workspace/local output) REAL array,
dimension (LRWORK) On exit, RWORK(1) returns the minimal and
optimal LRWORK.
LRWORK (local or global input) INTEGER
The dimension of the array RWORK. LRWORK is local input and
must be at least LRWORK = 2*LOCc(N_A).
If LRWORK = -1, then LRWORK 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 INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= N: U(IA+I-1,IA+I-1) is exactly zero. The factorization has
been completed, but the factor U is exactly singular, so the
solution and error bounds could not be computed. = N+1: RCOND
is less than machine precision. The factorization has been
completed, but the matrix is singular to working precision, and
the solution and error bounds have not been computed.
ScaLAPACK routine 31 October 2017 PCGESVX(3)