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


SYNOPSIS

       SUBROUTINE DGESVX( 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

           DOUBLE         PRECISION RCOND

           INTEGER        IPIV( * ), IWORK( * )

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


PURPOSE

       DGESVX 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) DOUBLE PRECISION 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)  DOUBLE  PRECISION  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 DGETRF.  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 DGETRF; 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


               If FACT = 'E', then IPIV is an output argument and
               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) DOUBLE PRECISION 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) DOUBLE PRECISION 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)   DOUBLE   PRECISION   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) DOUBLE PRECISION 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

               = '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 >=
               max(1,N).

       RCOND   (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  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) DOUBLE PRECISION 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) DOUBLE PRECISION 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)  DOUBLE  PRECISION  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

               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 num­
               ber of situations where the computed solution  can
               be  more  accurate  than  the value of RCOND would
               suggest.


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