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

```       PSPOSVX  - use the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution
to   a   real   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 PSPOSVX( 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, IWORK, LIWORK, INFO )

CHARACTER       EQUED, FACT, UPLO

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

REAL            RCOND

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

REAL            A( * ), AF( * ), B( * ), BERR( * ), FERR( * ), SC( * ), SR( * ), WORK(
* ), X( * )

```

#### PURPOSE

```       PSPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute  the  solution
to a real 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) REAL pointer into
the  local  memory  to  an  array  of local dimension ( LLD_A, LOCc(JA+N-1) ).  On
entry, the symmetric 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) REAL 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) REAL 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) REAL 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) REAL 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) REAL 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) REAL
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) REAL 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) REAL 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) REAL 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( PSPOCON( LWORK ), PSPORFS( 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.

IWORK   (local workspace/local output) INTEGER array,
dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK.

LIWORK  (local or global input) INTEGER
The dimension of the array IWORK.  LIWORK is local input  and  must  be  at  least
LIWORK = DESCA( LLD_ ) LIWORK = LOCr(N_A).

If  LIWORK = -1, then LIWORK 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.
```