c
c     Sample_Code_8
c     Solves a linear BVP using Chebyshev Collocation Method
c 
      IMPLICIT DOUBLE PRECISION (a-h,o-z) 
      parameter (N=10,N1=N+1)
      DOUBLE PRECISION A(N1,N1),B(N1)
      INTEGER IPVT(N1)
      COMMON /CHEB/ z1,z2,PI,Alpha,Beta
      COMMON /TRANS/ C(N1),CBAR(N1),CT(N1,N1),CTINV(N1,N1),Z(N1)
      COMMON /COLLEC/ D(N1,N1),DD(N1,N1)
      OPEN(11,FILE='sample_code_8_output.txt')
c
      PI    = 4.d0*ATAN(1.d0)
      z1    = 0.d0
      z2    = 1.d0
c
      call Chebyshev
c
      call MATRIX(A)
c
      call ludcmp(A,N1,N1,ipvt,dummy)
c
      do i = 1, N1
         B(i) = 0.d0
      enddo
      B(N1) = 1.d0
c
      call lubksb(A,N1,N1,ipvt,B)
c
      do i = 1, N1
         ye = 1.d0/(dexp(-1.d0)-dexp(-4.d0))*
     !        (dexp(-Z(i))-dexp(-4.d0*Z(i)))
         write(11,'(3f16.6)') Z(i),B(i),ye
         write(* ,'(3f16.6)') Z(i),B(i),ye
      enddo
c
1000  stop 
      end
c
c
c
      SUBROUTINE MATRIX(A)
      IMPLICIT DOUBLE PRECISION (a-h,o-z) 
      parameter (N=10,N1=N+1)
      DOUBLE PRECISION A(N1,N1)
      COMMON /CHEB/ z1,z2,PI,Alpha,Beta
      COMMON /TRANS/ C(N1),CBAR(N1),CT(N1,N1),CTINV(N1,N1),Z(N1)
      COMMON /COLLEC/ D(N1,N1),DD(N1,N1)
c
      do i = 1, N1
      do j = 1, N1
        A(i,j) = 0.d0
      enddo
      enddo
c
c     Construct Residuals using Chebyshev Collocation
c
      do i = 2, N1-1
         A(i,i) = 4.d0
         do j = 1, N1
           A(i,j) = A(i,j)+DD(i,j)+5.d0*D(i,j)
         enddo
      enddo
c
c     Implement boundary conditions
c
      A( 1, 1) = 1.d0 
      A(N1,N1) = 1.d0 
c
      return   
      end 
c
c
c
      SUBROUTINE Chebyshev
      IMPLICIT DOUBLE PRECISION (a-h,o-z) 
      parameter (N=10,N1=N+1)
      DOUBLE PRECISION XCHEB(N1)
      COMMON /CHEB/ z1,z2,PI,Alpha,Beta
      COMMON /TRANS/ C(N1),CBAR(N1),CT(N1,N1),CTINV(N1,N1),Z(N1)
      COMMON /COLLEC/ D(N1,N1),DD(N1,N1)
c
c     Vector XCHEB (1,-1) contains the Gauss-Lobatto grid. 
      do i = 1, N1 
        XCHEB(i) = dcos((i-1)*pi/N)
        CBAR(i)  = 1.d0  
        C(i)     = 1.d0  
      enddo
c
      C(1)     = 2.d0  
      CBAR(1)  = 2.d0  
      CBAR(N1) = 2.d0  
c
c     Vector X maps XCHEB to the interval (z1,z2)	==> X(i) = Alpha + Beta*XCHEB(i)
      Alpha = (z1+z2)/2.d0
      Beta  = (z1-z2)/2.d0
c
      do i = 1, N1 
        Z(i) = Alpha + Beta*XCHEB(i)
      enddo
c
      do i = 1, N1
      do j = 1, N1
        CT(i,j)    = 2.d0/N/CBAR(i)/CBAR(j)*dcos(pi*(i-1)*(j-1)/N)  
        CTINV(i,j) = dcos(pi*(i-1)*(j-1)/N)   
      enddo
      enddo
c
      do i = 1, N1
         ii = i-1
         do j = 1, N1
            jj = j-1
            if (ii .ne. jj) then
              D(i,j) = CBAR(i)/CBAR(j)*(-1)**(ii+jj)/(XCHEB(i)-XCHEB(j))
            else
              if (ii .ne. 0 .and. ii .ne. N) then
              D(i,j) = -XCHEB(j)/2.d0/(1.d0-XCHEB(j)**2)
              endif
            endif
         enddo
      enddo
      D( 1, 1) = (2.d0*dble(N)**2+1.d0)/6.d0
      D(N1,N1) =-(2.d0*dble(N)**2+1.d0)/6.d0
c
      do i=1,N1
        do j=1,N1 
          DD(i,j) = 0.d0
          do k=1,N1
            DD(i,j) = DD(i,j) + D(i,k)*D(k,j)
          enddo
        enddo
      enddo
c
      do i = 1, N1
         do j = 1, N1
             D(i,j) =  D(i,j)/BETA
            DD(i,j) = DD(i,j)/BETA/BETA
         enddo
      enddo
c
      return   
      end 
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      subroutine RESP(AR,AS) 
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      implicit double precision (a-h,o-z)
      parameter (N=10,N1=N+1)
      DOUBLE PRECISION AR(N1),AS(N1)
      COMMON /TRANS/ C(N1),CBAR(N1),CT(N1,N1),CTINV(N1,N1),Z(N1)
c
c     invert chebychev
c
      do i = 1, N1 
        AS(i) = 0.d0 
        do j = 1, N1 
          AS(i) = AS(i) + CT(i,j) * AR(j)
        enddo
      enddo
c
      return  
      end 
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      subroutine SPRE(AS,AR) 
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      implicit double precision (a-h,o-z)
      parameter (N=10,N1=N+1)
      DOUBLE PRECISION AR(N1),AS(N1)
      COMMON /TRANS/ C(N1),CBAR(N1),CT(N1,N1),CTINV(N1,N1),Z(N1)
c
c     invert chebychev
c
      do i = 1, N1 
        AR(i) = 0.d0 
        do j = 1, N1 
          AR(i) = AR(i) + CTINV(i,j) * AS(j)
        enddo
      enddo
c
      return  
      end 
c
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      SUBROUTINE ludcmp(a,n,np,indx,d)
      IMPLICIT DOUBLE PRECISION (A-H,O-Z)
      INTEGER n,np,indx(n),NMAX
      DOUBLE PRECISION d,a(np,np),TINY
      PARAMETER (NMAX=500,TINY=1.0e-20)
      INTEGER i,imax,j,k
      DOUBLE PRECISION aamax,dum,sum,vv(NMAX)
      d=1.
      do 12 i=1,n
        aamax=0.d0
        do 11 j=1,n
          if (abs(a(i,j)).gt.aamax) aamax=abs(a(i,j))
11      continue
        if (aamax.eq.0.d0) pause 'singular matrix in ludcmp'
        vv(i)=1./aamax
12    continue
      do 19 j=1,n
        do 14 i=1,j-1
          sum=a(i,j)
          do 13 k=1,i-1
            sum=sum-a(i,k)*a(k,j)
13        continue
          a(i,j)=sum
14      continue
        aamax=0.d0
        do 16 i=j,n

          sum=a(i,j)
          do 15 k=1,j-1
            sum=sum-a(i,k)*a(k,j)
15        continue
          a(i,j)=sum
          dum=vv(i)*abs(sum)
          if (dum.ge.aamax) then
            imax=i
            aamax=dum
          endif
16      continue
        if (j.ne.imax)then
          do 17 k=1,n
            dum=a(imax,k)
            a(imax,k)=a(j,k)
            a(j,k)=dum
17        continue
          d=-d
          vv(imax)=vv(j)
        endif
        indx(j)=imax
        if(a(j,j).eq.0.d0)a(j,j)=TINY
        if(j.ne.n)then
          dum=1./a(j,j)

          do 18 i=j+1,n
            a(i,j)=a(i,j)*dum
18        continue
        endif
19    continue
      return
      END
c
      SUBROUTINE lubksb(a,n,np,indx,b)
      IMPLICIT DOUBLE PRECISION (A-H,O-Z)
      INTEGER n,np,indx(n)
      DOUBLE PRECISION a(np,np),b(n)
      INTEGER i,ii,j,ll
      DOUBLE PRECISION sum
      ii=0
      do 12 i=1,n
        ll=indx(i)
        sum=b(ll)
        b(ll)=b(i)
        if (ii.ne.0)then
          do 11 j=ii,i-1
            sum=sum-a(i,j)*b(j)
11        continue
        else if (sum.ne.0.d0) then
          ii=i
        endif
        b(i)=sum
12    continue
      do 14 i=n,1,-1
        sum=b(i)
        do 13 j=i+1,n
          sum=sum-a(i,j)*b(j)
13      continue
        b(i)=sum/a(i,i)
14    continue
      return
      END