SGESVX  - use the LU factorization to compute the solution
       to a real system of linear equations A * X = B,


SYNOPSIS

       SUBROUTINE SGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF,
                          IPIV,  EQUED,  R,  C,  B,  LDB, X, LDX,
                          RCOND, FERR, BERR, WORK, IWORK, INFO )

           CHARACTER      EQUED, FACT, TRANS

           INTEGER        INFO, LDA, LDAF, LDB, LDX, N, NRHS

           REAL           RCOND

           INTEGER        IPIV( * ), IWORK( * )

           REAL           A( LDA, * ), AF( LDAF, * ), B(  LDB,  *
                          ),  BERR(  * ), C( * ), FERR( * ), R( *
                          ), WORK( * ), X( LDX, * )


PURPOSE

       SGESVX uses the LU factorization to compute  the  solution
       to a real system of linear equations A * X = B, where A is
       an N-by-N matrix and X and B are N-by-NRHS matrices.

       Error bounds on the solution and a condition estimate  are
       also provided.


DESCRIPTION

       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,

       angular
          matrix, and U is upper triangular.

       3. If some U(i,i)=0, so that U is exactly  singular,  then
       the routine
          returns  with INFO = i. Otherwise, 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,
          INFO = N+1 is returned as a warning,  but  the  routine
       still goes on
          to  solve  for  X and compute error bounds as described
       below.

       4. The system of equations is solved for X using the  fac­
       tored 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 premulti­
       plied by
          diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T'  or
       'C') so
          that  it  solves  the original system before equilibra­
       tion.


ARGUMENTS

       FACT    (input) CHARACTER*1
               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 and IPIV contain
               the factored form of A.  If EQUED is not 'N',  the
               matrix  A  has been equilibrated with scaling fac­
               tors given by R and C.  A, AF, and  IPIV  are  not
               modified.   = 'N':  The matrix A will be copied to
               AF and factored.
               = 'E':  The matrix A will be equilibrated if  nec­
               essary, then copied to AF and factored.

       TRANS   (input) CHARACTER*1
               Specifies the form of the system of equations:
               = 'N':  A * X = B     (No transpose)
               = 'T':  A**T * X = B  (Transpose)
               = 'C':  A**H * X = B  (Transpose)

               The number of linear equations, i.e., the order of
               the matrix A.  N >= 0.

       NRHS    (input) INTEGER
               The number of right hand sides, i.e.,  the  number
               of columns of the matrices B and X.  NRHS >= 0.

       A       (input/output) REAL array, dimension (LDA,N)
               On  entry, the N-by-N matrix A.  If FACT = 'F' and
               EQUED is not 'N', then A must  have  been  equili­
               brated by the scaling factors in R and/or C.  A is
               not modified if FACT = 'F' or

               On exit, if EQUED .ne. 'N', A is  scaled  as  fol­
               lows: EQUED = 'R':  A := diag(R) * A
               EQUED = 'C':  A := A * diag(C)
               EQUED = 'B':  A := diag(R) * A * diag(C).

       LDA     (input) INTEGER
               The  leading  dimension  of  the  array A.  LDA >=
               max(1,N).

       AF      (input or output) REAL array, dimension (LDAF,N)
               If FACT = 'F', then AF is an input argument and on
               entry  contains  the factors L and U from the fac­
               torization A = P*L*U as computed  by  SGETRF.   If
               EQUED  .ne.  'N',  then AF is the factored form of
               the equilibrated matrix A.

               If FACT = 'N', then AF is an output  argument  and
               on  exit returns the factors L and U from the fac­
               torization A = P*L*U of the original matrix A.

               If FACT = 'E', then AF is an output  argument  and
               on  exit returns the factors L and U from the fac­
               torization A = P*L*U of the equilibrated matrix  A
               (see  the  description  of  A  for the form of the
               equilibrated matrix).

       LDAF    (input) INTEGER
               The leading dimension of the array  AF.   LDAF  >=
               max(1,N).

       IPIV    (input or output) INTEGER array, dimension (N)
               If  FACT = 'F', then IPIV is an input argument and
               on entry contains the pivot indices from the  fac­
               torization  A = P*L*U as computed by SGETRF; row i
               of the matrix was interchanged with row IPIV(i).

               If FACT = 'N', then IPIV is an output argument and
               on  exit  contains the pivot indices from the fac­
               torization A = P*L*U of the original matrix A.

               on  exit  contains the pivot indices from the fac­
               torization A = P*L*U of the equilibrated matrix A.

       EQUED   (input or output) CHARACTER*1
               Specifies the form of equilibration that was done.
               = 'N':  No equilibration (always true  if  FACT  =
               'N').
               =  'R':   Row equilibration, i.e., A has been pre­
               multiplied by diag(R).  = 'C':  Column  equilibra­
               tion,  i.e., A has been postmultiplied by diag(C).
               = 'B':  Both row and column equilibration, i.e., A
               has been replaced by diag(R) * A * diag(C).  EQUED
               is an input argument if FACT = 'F'; otherwise,  it
               is an output argument.

       R       (input or output) REAL array, dimension (N)
               The  row  scale  factors for A.  If EQUED = 'R' or
               'B', A is multiplied on the left  by  diag(R);  if
               EQUED  =  'N'  or 'C', R is not accessed.  R is an
               input argument if FACT = 'F'; otherwise, R  is  an
               output argument.  If FACT = 'F' and EQUED = 'R' or
               'B', each element of R must be positive.

       C       (input or output) REAL array, dimension (N)
               The column scale factors for A.  If EQUED = 'C' or
               'B',  A  is multiplied on the right by diag(C); if
               EQUED = 'N' or 'R', C is not accessed.   C  is  an
               input  argument  if FACT = 'F'; otherwise, C is an
               output argument.  If FACT = 'F' and EQUED = 'C' or
               'B', each element of C must be positive.

       B       (input/output) REAL array, dimension (LDB,NRHS)
               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 overwrit­
               ten by diag(R)*B; if TRANS = 'T' or 'C' and  EQUED
               = 'C' or 'B', B is overwritten by diag(C)*B.

       LDB     (input) INTEGER
               The  leading  dimension  of  the  array B.  LDB >=
               max(1,N).

       X       (output) REAL array, dimension (LDX,NRHS)
               If INFO = 0 or INFO = N+1, 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(C))*X if TRANS = 'N' and  EQUED
               =  'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or
               'C' and EQUED = 'R' or 'B'.

       LDX     (input) INTEGER
               The leading dimension of  the  array  X.   LDX  >=


       RCOND   (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 par­
               ticular, if RCOND = 0), the matrix is singular  to
               working precision.  This condition is indicated by
               a return code of INFO > 0.

       FERR    (output) REAL array, dimension (NRHS)
               The estimated forward error bound for  each  solu­
               tion  vector X(j) (the j-th column of the solution
               matrix X).  If XTRUE is the true  solution  corre­
               sponding  to  X(j),  FERR(j) is an estimated upper
               bound for the magnitude of the largest element  in
               (X(j)  -  XTRUE)  divided  by the magnitude of the
               largest element in X(j).  The estimate is as reli­
               able  as  the  estimate  for  RCOND, and is almost
               always a slight overestimate of the true error.

       BERR    (output) REAL array, dimension (NRHS)
               The componentwise relative backward error of  each
               solution  vector X(j) (i.e., the smallest relative
               change in any element of A or B that makes X(j) an
               exact solution).

       WORK    (workspace/output) REAL array, dimension (4*N)
               On  exit,  WORK(1)  contains  the reciprocal pivot
               growth factor norm(A)/norm(U). The  "max  absolute
               element"  norm  is  used.  If WORK(1) is much less
               than 1, then the stability of the LU factorization
               of the (equilibrated) matrix A could be poor. This
               also means that the solution X, condition  estima­
               tor  RCOND,  and forward error bound FERR could be
               unreliable. If factorization fails with 0<INFO<=N,
               then  WORK(1) contains the reciprocal pivot growth
               factor for the leading INFO columns of A.

       IWORK   (workspace) INTEGER array, dimension (N)

       INFO    (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value
               > 0:  if INFO = i, and i is
               <=  N:  U(i,i) 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. RCOND = 0 is returned.  = N+1:  U
               is  nonsingular,  but  RCOND  is less than machine
               precision, meaning that the matrix is singular  to
               working precision.  Nevertheless, the solution and
               error bounds are  computed  because  there  are  a

               can be more accurate than the value of RCOND would
               suggest.


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