CGBSVX  - use the LU factorization to compute the solution
       to a complex system of linear equations A * X = B, A**T  *
       X = B, or A**H * X = B,


SYNOPSIS

       SUBROUTINE CGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB,
                          AFB, LDAFB, IPIV, EQUED, R, C, B,  LDB,
                          X, LDX, RCOND, FERR, BERR, WORK, RWORK,
                          INFO )

           CHARACTER      EQUED, FACT, TRANS

           INTEGER        INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N,
                          NRHS

           REAL           RCOND

           INTEGER        IPIV( * )

           REAL           BERR(  *  ), C( * ), FERR( * ), R( * ),
                          RWORK( * )

           COMPLEX        AB( LDAB, * ), AFB( LDAFB, * ), B( LDB,
                          * ), WORK( * ), X( LDX, * )


PURPOSE

       CGBSVX  uses  the LU factorization to compute the solution
       to a complex system of linear equations A * X = B, A**T  *
       X  = B, or A**H * X = B, where A is a band matrix of order
       N with KL subdiagonals and KU superdiagonals, 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 by this subroutine:

       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

          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 = L * U,
          where L is a product of permutation and unit lower tri­
       angular
          matrices  with KL subdiagonals, and U is upper triangu­
       lar with
          KL+KU superdiagonals.

       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, AFB 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.  AB, AFB, and IPIV are  not
               modified.   = 'N':  The matrix A will be copied to

               = 'E':  The matrix A will be equilibrated if  nec­
               essary, then copied to AFB 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  (Conjugate transpose)

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

       KL      (input) INTEGER
               The  number  of subdiagonals within the band of A.
               KL >= 0.

       KU      (input) INTEGER
               The number of superdiagonals within the band of A.
               KU >= 0.

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

       AB      (input/output) COMPLEX array, dimension (LDAB,N)
               On entry, the matrix A in band storage, in rows  1
               to KL+KU+1.  The j-th column of A is stored in the
               j-th column of the array AB as follows: AB(KU+1+i-
               j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

               If  FACT  =  'F' and EQUED is not 'N', then A must
               have been equilibrated by the scaling factors in R
               and/or  C.   AB  is  not modified if FACT = 'F' or
               'N', or if FACT = 'E' and EQUED = 'N' on exit.

               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).

       LDAB    (input) INTEGER
               The  leading  dimension  of the array AB.  LDAB >=
               KL+KU+1.

       AFB     (input or output) COMPLEX array, dimension
               (LDAFB,N)
               If  FACT  = 'F', then AFB is an input argument and
               on entry contains details of the LU  factorization
               of the band matrix A, as computed by CGBTRF.  U is
               stored as an upper  triangular  band  matrix  with
               KL+KU superdiagonals in rows 1 to KL+KU+1, and the
               multipliers  used  during  the  factorization  are

               .ne. 'N', then AFB is the  factored  form  of  the
               equilibrated matrix A.

               If  FACT = 'N', then AFB is an output argument and
               on exit returns details of the LU factorization of
               A.

               If  FACT = 'E', then AFB is an output argument and
               on exit returns details of the LU factorization of
               the  equilibrated matrix A (see the description of
               AB for the form of the equilibrated matrix).

       LDAFB   (input) INTEGER
               The leading dimension of the array AFB.  LDAFB  >=
               2*KL+KU+1.

       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 = L*U as computed by CGBTRF; 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 = L*U of the original matrix A.

               If FACT = 'E', then IPIV is an output argument and
               on  exit  contains the pivot indices from the fac­
               torization A = 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

               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) COMPLEX array, dimension (LDB,NRHS)
               On  entry, the 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.

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

       X       (output) COMPLEX 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  >=
               max(1,N).

       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).


       RWORK   (workspace/output) REAL array, dimension (N)
               On  exit,  RWORK(1)  contains the reciprocal pivot
               growth factor norm(A)/norm(U). The  "max  absolute
               element"  norm  is  used. If RWORK(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 RWORK(1) contains the reciprocal pivot growth
               factor for the leading INFO columns of A.

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


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