
c---------------------------------------------------------------------
c---------------------------------------------------------------------

       subroutine x_solve

c---------------------------------------------------------------------
c---------------------------------------------------------------------

c---------------------------------------------------------------------
c this function performs the solution of the approximate factorization
c step in the x-direction for all five matrix components
c simultaneously. The Thomas algorithm is employed to solve the
c systems for the x-lines. Boundary conditions are non-periodic
c---------------------------------------------------------------------

       include 'header.h'

       integer i, j, k, i1, i2, m
       double precision  ru1, fac1, fac2


c---------------------------------------------------------------------
c---------------------------------------------------------------------

       if (timeron) call timer_start(t_xsolve)
!$omp parallel do default(shared) private(i,j,k,i1,i2,m,
!$omp&    ru1,fac1,fac2)
       do  k = 1, nz2

          call lhsinit(nx2+1, ny2)

c---------------------------------------------------------------------
c Computes the left hand side for the three x-factors  
c---------------------------------------------------------------------

c---------------------------------------------------------------------
c      first fill the lhs for the u-eigenvalue                   
c---------------------------------------------------------------------
          do  j = 1, ny2
             do  i = 0, grid_points(1)-1
                ru1 = c3c4*rho_i(i,j,k)
                cv(i) = us(i,j,k)
                rhon(i) = dmax1(dx2+con43*ru1, 
     >                          dx5+c1c5*ru1,
     >                          dxmax+ru1,
     >                          dx1)
             end do

             do  i = 1, nx2
                lhs(1,i,j) =   0.0d0
                lhs(2,i,j) = - dttx2 * cv(i-1) - dttx1 * rhon(i-1)
                lhs(3,i,j) =   1.0d0 + c2dttx1 * rhon(i)
                lhs(4,i,j) =   dttx2 * cv(i+1) - dttx1 * rhon(i+1)
                lhs(5,i,j) =   0.0d0
             end do
          end do

c---------------------------------------------------------------------
c      add fourth order dissipation                             
c---------------------------------------------------------------------

          do  j = 1, ny2
             i = 1
             lhs(3,i,j) = lhs(3,i,j) + comz5
             lhs(4,i,j) = lhs(4,i,j) - comz4
             lhs(5,i,j) = lhs(5,i,j) + comz1
  
             lhs(2,i+1,j) = lhs(2,i+1,j) - comz4
             lhs(3,i+1,j) = lhs(3,i+1,j) + comz6
             lhs(4,i+1,j) = lhs(4,i+1,j) - comz4
             lhs(5,i+1,j) = lhs(5,i+1,j) + comz1
          end do

          do  j = 1, ny2
             do   i=3, grid_points(1)-4
                lhs(1,i,j) = lhs(1,i,j) + comz1
                lhs(2,i,j) = lhs(2,i,j) - comz4
                lhs(3,i,j) = lhs(3,i,j) + comz6
                lhs(4,i,j) = lhs(4,i,j) - comz4
                lhs(5,i,j) = lhs(5,i,j) + comz1
             end do
          end do


          do  j = 1, ny2
             i = grid_points(1)-3
             lhs(1,i,j) = lhs(1,i,j) + comz1
             lhs(2,i,j) = lhs(2,i,j) - comz4
             lhs(3,i,j) = lhs(3,i,j) + comz6
             lhs(4,i,j) = lhs(4,i,j) - comz4

             lhs(1,i+1,j) = lhs(1,i+1,j) + comz1
             lhs(2,i+1,j) = lhs(2,i+1,j) - comz4
             lhs(3,i+1,j) = lhs(3,i+1,j) + comz5
          end do

c---------------------------------------------------------------------
c      subsequently, fill the other factors (u+c), (u-c) by adding to 
c      the first  
c---------------------------------------------------------------------
          do  j = 1, ny2
             do   i = 1, nx2
                lhsp(1,i,j) = lhs(1,i,j)
                lhsp(2,i,j) = lhs(2,i,j) - 
     >                            dttx2 * speed(i-1,j,k)
                lhsp(3,i,j) = lhs(3,i,j)
                lhsp(4,i,j) = lhs(4,i,j) + 
     >                            dttx2 * speed(i+1,j,k)
                lhsp(5,i,j) = lhs(5,i,j)
                lhsm(1,i,j) = lhs(1,i,j)
                lhsm(2,i,j) = lhs(2,i,j) + 
     >                            dttx2 * speed(i-1,j,k)
                lhsm(3,i,j) = lhs(3,i,j)
                lhsm(4,i,j) = lhs(4,i,j) - 
     >                            dttx2 * speed(i+1,j,k)
                lhsm(5,i,j) = lhs(5,i,j)
             end do
          end do

c---------------------------------------------------------------------
c                          FORWARD ELIMINATION  
c---------------------------------------------------------------------

c---------------------------------------------------------------------
c      perform the Thomas algorithm; first, FORWARD ELIMINATION     
c---------------------------------------------------------------------

          do  j = 1, ny2
             do    i = 0, grid_points(1)-3
                i1 = i  + 1
                i2 = i  + 2
                fac1      = 1.d0/lhs(3,i,j)
                lhs(4,i,j)  = fac1*lhs(4,i,j)
                lhs(5,i,j)  = fac1*lhs(5,i,j)
                do    m = 1, 3
                   rhs(m,i,j,k) = fac1*rhs(m,i,j,k)
                end do
                lhs(3,i1,j) = lhs(3,i1,j) -
     >                         lhs(2,i1,j)*lhs(4,i,j)
                lhs(4,i1,j) = lhs(4,i1,j) -
     >                         lhs(2,i1,j)*lhs(5,i,j)
                do    m = 1, 3
                   rhs(m,i1,j,k) = rhs(m,i1,j,k) -
     >                         lhs(2,i1,j)*rhs(m,i,j,k)
                end do
                lhs(2,i2,j) = lhs(2,i2,j) -
     >                         lhs(1,i2,j)*lhs(4,i,j)
                lhs(3,i2,j) = lhs(3,i2,j) -
     >                         lhs(1,i2,j)*lhs(5,i,j)
                do    m = 1, 3
                   rhs(m,i2,j,k) = rhs(m,i2,j,k) -
     >                         lhs(1,i2,j)*rhs(m,i,j,k)
                end do
             end do
          end do

c---------------------------------------------------------------------
c      The last two rows in this grid block are a bit different, 
c      since they do not have two more rows available for the
c      elimination of off-diagonal entries
c---------------------------------------------------------------------

          do  j = 1, ny2
             i  = grid_points(1)-2
             i1 = grid_points(1)-1
             fac1      = 1.d0/lhs(3,i,j)
             lhs(4,i,j)  = fac1*lhs(4,i,j)
             lhs(5,i,j)  = fac1*lhs(5,i,j)
             do    m = 1, 3
                rhs(m,i,j,k) = fac1*rhs(m,i,j,k)
             end do
             lhs(3,i1,j) = lhs(3,i1,j) -
     >                      lhs(2,i1,j)*lhs(4,i,j)
             lhs(4,i1,j) = lhs(4,i1,j) -
     >                      lhs(2,i1,j)*lhs(5,i,j)
             do    m = 1, 3
                rhs(m,i1,j,k) = rhs(m,i1,j,k) -
     >                      lhs(2,i1,j)*rhs(m,i,j,k)
             end do
c---------------------------------------------------------------------
c            scale the last row immediately 
c---------------------------------------------------------------------
             fac2             = 1.d0/lhs(3,i1,j)
             do    m = 1, 3
                rhs(m,i1,j,k) = fac2*rhs(m,i1,j,k)
             end do
          end do

c---------------------------------------------------------------------
c      do the u+c and the u-c factors                 
c---------------------------------------------------------------------

          do  j = 1, ny2
             do    i = 0, grid_points(1)-3
                i1 = i  + 1
                i2 = i  + 2
                m = 4
                fac1       = 1.d0/lhsp(3,i,j)
                lhsp(4,i,j)  = fac1*lhsp(4,i,j)
                lhsp(5,i,j)  = fac1*lhsp(5,i,j)
                rhs(m,i,j,k) = fac1*rhs(m,i,j,k)
                lhsp(3,i1,j) = lhsp(3,i1,j) -
     >                        lhsp(2,i1,j)*lhsp(4,i,j)
                lhsp(4,i1,j) = lhsp(4,i1,j) -
     >                        lhsp(2,i1,j)*lhsp(5,i,j)
                rhs(m,i1,j,k) = rhs(m,i1,j,k) -
     >                        lhsp(2,i1,j)*rhs(m,i,j,k)
                lhsp(2,i2,j) = lhsp(2,i2,j) -
     >                        lhsp(1,i2,j)*lhsp(4,i,j)
                lhsp(3,i2,j) = lhsp(3,i2,j) -
     >                        lhsp(1,i2,j)*lhsp(5,i,j)
                rhs(m,i2,j,k) = rhs(m,i2,j,k) -
     >                        lhsp(1,i2,j)*rhs(m,i,j,k)
                m = 5
                fac1       = 1.d0/lhsm(3,i,j)
                lhsm(4,i,j)  = fac1*lhsm(4,i,j)
                lhsm(5,i,j)  = fac1*lhsm(5,i,j)
                rhs(m,i,j,k) = fac1*rhs(m,i,j,k)
                lhsm(3,i1,j) = lhsm(3,i1,j) -
     >                        lhsm(2,i1,j)*lhsm(4,i,j)
                lhsm(4,i1,j) = lhsm(4,i1,j) -
     >                        lhsm(2,i1,j)*lhsm(5,i,j)
                rhs(m,i1,j,k) = rhs(m,i1,j,k) -
     >                        lhsm(2,i1,j)*rhs(m,i,j,k)
                lhsm(2,i2,j) = lhsm(2,i2,j) -
     >                        lhsm(1,i2,j)*lhsm(4,i,j)
                lhsm(3,i2,j) = lhsm(3,i2,j) -
     >                        lhsm(1,i2,j)*lhsm(5,i,j)
                rhs(m,i2,j,k) = rhs(m,i2,j,k) -
     >                        lhsm(1,i2,j)*rhs(m,i,j,k)
             end do
          end do

c---------------------------------------------------------------------
c         And again the last two rows separately
c---------------------------------------------------------------------
          do  j = 1, ny2
             i  = grid_points(1)-2
             i1 = grid_points(1)-1
             m = 4
             fac1       = 1.d0/lhsp(3,i,j)
             lhsp(4,i,j)  = fac1*lhsp(4,i,j)
             lhsp(5,i,j)  = fac1*lhsp(5,i,j)
             rhs(m,i,j,k) = fac1*rhs(m,i,j,k)
             lhsp(3,i1,j) = lhsp(3,i1,j) -
     >                      lhsp(2,i1,j)*lhsp(4,i,j)
             lhsp(4,i1,j) = lhsp(4,i1,j) -
     >                      lhsp(2,i1,j)*lhsp(5,i,j)
             rhs(m,i1,j,k) = rhs(m,i1,j,k) -
     >                      lhsp(2,i1,j)*rhs(m,i,j,k)
             m = 5
             fac1       = 1.d0/lhsm(3,i,j)
             lhsm(4,i,j)  = fac1*lhsm(4,i,j)
             lhsm(5,i,j)  = fac1*lhsm(5,i,j)
             rhs(m,i,j,k) = fac1*rhs(m,i,j,k)
             lhsm(3,i1,j) = lhsm(3,i1,j) -
     >                      lhsm(2,i1,j)*lhsm(4,i,j)
             lhsm(4,i1,j) = lhsm(4,i1,j) -
     >                      lhsm(2,i1,j)*lhsm(5,i,j)
             rhs(m,i1,j,k) = rhs(m,i1,j,k) -
     >                      lhsm(2,i1,j)*rhs(m,i,j,k)
c---------------------------------------------------------------------
c               Scale the last row immediately
c---------------------------------------------------------------------
             rhs(4,i1,j,k) = rhs(4,i1,j,k)/lhsp(3,i1,j)
             rhs(5,i1,j,k) = rhs(5,i1,j,k)/lhsm(3,i1,j)
          end do


c---------------------------------------------------------------------
c                         BACKSUBSTITUTION 
c---------------------------------------------------------------------


          do  j = 1, ny2
             i  = grid_points(1)-2
             i1 = grid_points(1)-1
             do   m = 1, 3
                rhs(m,i,j,k) = rhs(m,i,j,k) -
     >                             lhs(4,i,j)*rhs(m,i1,j,k)
             end do

             rhs(4,i,j,k) = rhs(4,i,j,k) -
     >                          lhsp(4,i,j)*rhs(4,i1,j,k)
             rhs(5,i,j,k) = rhs(5,i,j,k) -
     >                          lhsm(4,i,j)*rhs(5,i1,j,k)
          end do

c---------------------------------------------------------------------
c      The first three factors
c---------------------------------------------------------------------
          do  j = 1, ny2
             do    i = grid_points(1)-3, 0, -1
                i1 = i  + 1
                i2 = i  + 2
                do   m = 1, 3
                   rhs(m,i,j,k) = rhs(m,i,j,k) - 
     >                          lhs(4,i,j)*rhs(m,i1,j,k) -
     >                          lhs(5,i,j)*rhs(m,i2,j,k)
                end do

c---------------------------------------------------------------------
c      And the remaining two
c---------------------------------------------------------------------
                rhs(4,i,j,k) = rhs(4,i,j,k) - 
     >                          lhsp(4,i,j)*rhs(4,i1,j,k) -
     >                          lhsp(5,i,j)*rhs(4,i2,j,k)
                rhs(5,i,j,k) = rhs(5,i,j,k) - 
     >                          lhsm(4,i,j)*rhs(5,i1,j,k) -
     >                          lhsm(5,i,j)*rhs(5,i2,j,k)
             end do
          end do

       end do
       if (timeron) call timer_stop(t_xsolve)

c---------------------------------------------------------------------
c      Do the block-diagonal inversion          
c---------------------------------------------------------------------
       call ninvr

       return
       end