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## 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

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
provided.

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

```

## 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
equilibrated   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
equilibrated  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
equilibrated 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
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 AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output
argument and on exit returns the  factors  L  and  U  from  the
factorization 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-
torization  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
contains    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
contains    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
condition 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
matrices B and X.

BERR    (local output) REAL array, dimension LOCc(N_B).
The  componentwise  relative  backward  error  of each solution
vector 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.
```