Provided by: scalapack-doc_1.5-10_all #### NAME

```       PZPOSVX  - use the Cholesky factorization A = U**H*U or A = L*L**H 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 PZPOSVX( FACT,  UPLO,  N,  NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, EQUED,
SR, SC, B, IB, JB, DESCB, X, IX, JX, DESCX, RCOND, FERR,  BERR,  WORK,
LWORK, RWORK, LRWORK, INFO )

CHARACTER       EQUED, FACT, UPLO

INTEGER         IA, IAF, IB, INFO, IX, JA, JAF, JB, JX, LRWORK, LWORK, N, NRHS

DOUBLE          PRECISION RCOND

INTEGER         DESCA( * ), DESCAF( * ), DESCB( * ), DESCX( * )

DOUBLE          PRECISION BERR( * ), FERR( * ), SC( * ), SR( * ), RWORK( * )

COMPLEX*16      A( * ), AF( * ), B( * ), WORK( * ), X( * )

```

#### PURPOSE

```       PZPOSVX  uses  the Cholesky factorization A = U**H*U or A = L*L**H 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.  In the following
comments Y denotes Y(IY:IY+M-1,JY:JY+K-1) a M-by-K matrix where Y can be A, AF, B and X.

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

```       The following steps are performed:

1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
diag(SR) * A * diag(SC) * inv(diag(SC)) * X = diag(SR) * 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(SR)*A*diag(SC) and B by diag(SR)*B.

2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
factor the matrix A (after equilibration if FACT = 'E') as
A = U**T* U,  if UPLO = 'U', or
A = L * L**T,  if UPLO = 'L',
where U is an upper triangular matrix and L is a lower triangular
matrix.

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 equilibration was used, the matrix X is premultiplied by
diag(SR) 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 is supplied on entry,
and if not, whether the matrix A should be equilibrated before it is factored.   =
'F':   On entry, AF contains the factored form of A.  If EQUED = 'Y', the matrix A
has been equilibrated with scaling factors given by S.   A  and  AF  will  not  be
modified.  = 'N':  The matrix A will be copied to AF and factored.
=  'E':   The  matrix  A  will be equilibrated if necessary, then copied to AF and
factored.

UPLO    (global input) CHARACTER
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.

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 and X.  NRHS >= 0.

A       (local input/local output) COMPLEX*16 pointer into
the local memory to an array of local  dimension  (  LLD_A,  LOCc(JA+N-1)  ).   On
entry,  the  Hermitian matrix A, except if FACT = 'F' and EQUED = 'Y', then A must
contain the equilibrated matrix diag(SR)*A*diag(SC).  If UPLO = 'U',  the  leading
N-by-N upper triangular part of A contains the upper triangular part of the matrix
A, and the strictly lower triangular part of A is not referenced.  If UPLO =  'L',
the  leading  N-by-N lower triangular part of A contains the lower triangular part
of the matrix A, and the strictly upper triangular part of A is not referenced.  A
is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(SR)*A*diag(SC).

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*16 pointer
into  the local memory to an array of local dimension ( LLD_AF, LOCc(JA+N-1)).  If
FACT = 'F', then AF is an input argument and  on  entry  contains  the  triangular
factor  U  or  L  from the Cholesky factorization A = U**T*U or A = L*L**T, in the
same storage format as A.  If EQUED .ne. 'N', then AF is the factored form of  the
equilibrated matrix diag(SR)*A*diag(SC).

If  FACT  =  'N', then AF is an output argument and on exit returns the triangular
factor U or L from the Cholesky factorization A = U**T*U or  A  =  L*L**T  of  the
original matrix A.

If  FACT  =  'E', then AF is an output argument and on exit returns the triangular
factor U or L from the Cholesky factorization A = U**T*U or  A  =  L*L**T  of  the
equilibrated  matrix  A (see the description of A 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.

EQUED   (global input/global output) CHARACTER
Specifies the form of equilibration that  was  done.   =  'N':   No  equilibration
(always true if FACT = 'N').
=  'Y':   Equilibration  was  done,  i.e.,  A  has been replaced by diag(SR) * A *
diag(SC).  EQUED is an input variable if FACT = 'F'; otherwise, it  is  an  output
variable.

SR      (local input/local output) COMPLEX*16 array,
dimension  (LLD_A)  The  scale  factors for A distributed across process rows; not
accessed if EQUED = 'N'.  SR is an input variable if FACT = 'F'; otherwise, SR  is
an  output  variable.   If  FACT = 'F' and EQUED = 'Y', each element of SR must be
positive.

SC      (local input/local output) COMPLEX*16 array,
dimension (LOC(N_A)) The scale factors for A distributed across  process  columns;
not  accessed if EQUED = 'N'. SC is an input variable if FACT = 'F'; otherwise, SC
is an output variable.  If FACT = 'F' and EQUED = 'Y', each element of SC must  be
positive.

B       (local input/local output) COMPLEX*16 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.  On exit, if EQUED = 'N', B is  not
modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B; if
TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B.

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*16 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 to the original system of equations.
Note that A and B are modified on exit if EQUED .ne. 'N', and the solution to  the
equilibrated system is inv(diag(SC))*X if TRANS = 'N' and EQUED = 'C' or 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) DOUBLE PRECISION
The   estimate   of  the  reciprocal  condition  number  of  the  matrix  A  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, and the  solution  and  error
bounds are not computed.

FERR    (local output) DOUBLE PRECISION array, dimension (LOC(N_B))
The  estimated forward error bounds for each solution vector X(j) (the j-th column
of the solution matrix X).  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 quality of the error bound depends on the  quality  of
the estimate of norm(inv(A)) computed in the code; if the estimate of norm(inv(A))
is accurate, the error bound is guaranteed.

BERR    (local output) DOUBLE PRECISION array, dimension (LOC(N_B))
The componentwise relative backward error of each solution vector X(j) (i.e.,  the
smallest  relative  change  in  any  entry  of  A  or  B  that makes X(j) an exact
solution).

WORK    (local workspace/local output) COMPLEX*16 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( PZPOCON( LWORK ), PZPORFS( LWORK ) ) + LOCr( N_A ).  LWORK = 3*DESCA( LLD_
)

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) DOUBLE PRECISION 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: if INFO = i, the leading minor of order i of A is not positive definite, so
the factorization could not be completed, and 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.
```