stellinstall/mkbicubw_8f_source.html

      subroutine mkbicubw(x,nx,y,ny,f,nf2,

     >   ibcxmin,bcxmin,ibcxmax,bcxmax,

     >   ibcymin,bcymin,ibcymax,bcymax,

     >   wk,nwk,

     >   ilinx,iliny,ier)

C

C  setup a bicubic spline; store coefficients in compact form

C  (as per suggestion of L. Zakharov, PPPL, Feb. 1999)

C     4 coeffs per grid point:  f,fxx,fyy,fxxyy

C

C

C  input:

      integer nx                        ! length of x vector

      integer ny                        ! length of y vector

      real x(nx)                        ! x vector, strict ascending

      real y(ny)                        ! y vector, strict ascending

C

      integer nf2                       ! 2nd dimension of f, nf2.ge.nx

C  input/output:

C

      real f(4,nf2,ny)                  ! data & spline coefficients

C

C  on input:  f(1,i,j) = f(x(i),y(j))

C  on output:  f(1,i,j) unchanged

C              f(2,i,j) = d2f/dx2(x(i),y(j))

C              f(3,i,j) = d2f/dy2(x(i),y(j))

C              f(4,i,j) = d4f/dx2dy2(x(i),y(j))

C

C  and the interpolation formula for (x,y) in (x(i),x(i+1))^(y(j),y(j+1))

C  is:

C        hx = x(i+1)-x(i)   hy = y(j+1)-y(j)

C        dxp= (x-x(i))/hx   dxm= 1-dxp     dxp,dxm in (0,1)

C        dyp= (x-x(i))/hx   dym= 1-dyp     dyp,dym in (0,1)

C        dx3p = dxp**3-dxp  dx3m = dxm**3-dxm     dxp3,dxm3 in (0,1)

C

C   finterp = dxm*(dym*f(1,i,j)+dyp*f(1,i,j+1))

C            +dxp*(dym*f(1,i+1,j)+dyp*f(1,i+1,j+1))

C     +1/6*hx**2*

C            dx3m*(dym*f(2,i,j)+dyp*f(2,i,j+1))

C           +dx3p*(dym*f(2,i+1,j)+dyp*f(2,i+1,j+1))

C     +1/6*hy**2*

C            dxm*(dy3m*f(3,i,j)+dy3p*f(3,i,j+1))

C           +dxp*(dy3m*f(3,i+1,j)+dy3p*f(3,i+1,j+1))

C     +1/36*hx**2*hy**2*

C            dx3m*(dym*f(4,i,j)+dyp*f(4,i,j+1))

C           +dx3p*(dym*f(4,i+1,j)+dyp*f(4,i+1,j+1))

C

C  where the f(2:4,*,*) are cleverly evaluated to assure

C  (a) finterp is continuous and twice differentiable across all

C      grid cell boundaries, and

C  (b) all boundary conditions are satisfied.

C

C  the boundary conditions specification options are:

C  inputs:

C

      integer ibcxmin                   ! bc flag for x=xmin

      real bcxmin(ny)                   ! bc data vs. y at x=xmin

      integer ibcxmax                   ! bc flag for x=xmax

      real bcxmax(ny)                   ! bc data vs. y at x=xmax

C

      integer ibcymin                   ! bc flag for y=ymin

      real bcymin(nx)                   ! bc data vs. x at y=ymin

      integer ibcymax                   ! bc flag for y=ymax

      real bcymax(nx)                   ! bc data vs. x at y=ymax

C

C  with interpretation:

c   ibcxmin -- indicator for boundary condition at x(1):

c    bcxmin(...) -- boundary condition data

c     =-1 -- periodic boundary condition

c     =0 -- use "not a knot"

c     =1 -- match slope, specified at x(1),th(ith) by bcxmin(ith)

c     =2 -- match 2nd derivative, specified at x(1),th(ith) by bcxmin(ith)

c     =3 -- boundary condition is slope=0 (df/dx=0) at x(1), all th(j)

c     =4 -- boundary condition is d2f/dx2=0 at x(1), all th(j)

c     =5 -- match 1st derivative to 1st divided difference

c     =6 -- match 2nd derivative to 2nd divided difference

c     =7 -- match 3rd derivative to 3rd divided difference

c           (for more detailed definition of BCs 5-7, see the

c           comments of subroutine mkspline)

c   NOTE bcxmin(...) referenced ONLY if ibcxmin=1 or ibcxmin=2

c

c   ibcxmax -- indicator for boundary condition at x(nx):

c    bcxmax(...) -- boundary condition data

c     (interpretation as with ibcxmin, bcxmin)

c   NOTE:  if ibcxmin=-1, ibcxmax is ignored! ...and the BC is periodic.

c

C  and analogous interpretation for ibcymin,bcymin,ibcymax,bcymax

C  (df/dy or d2f/dy2 boundary conditions at y=ymin and y=ymax).

C

C  *** workspace ***

C  This FORTRAN-77 routine requires a workspace of

C    nwk = (AT LEAST) 21*nx*ny

C  *** for dynamic allocation of this workspace use subroutine mkbicub

C    which has the same arguments as mkbicubw, except that the workspace

C    does not have to be provided -- mkbicub is FORTRAN-90.

C

C  input:

      integer nwk                       ! size of workspace

C  modified on output:

      real wk(nwk)                      ! workspace

C

C  and output arguments

C

      integer ilinx                     ! =1: x grid is "nearly" equally spaced

      integer iliny                     ! =1: y grid is "nearly" equally spaced

C

C  ilinx and iliny are set to zero if corresponding grids are not equally

C  spaced

C

C  and completion code

C

      integer ier                       ! =0 on exit if there is no error.

C

C  if there is an error, ier is set and a message is output on unit 6.

C  these are considered programming errors in the calling routine.

C

C  possible errors:

C    x(...) not strict ascending

C    y(...) not strict ascending

C    nx .lt. 4

C    ny .lt. 4

C    invalid boundary condition flag

C    workspace too small

C

C-----------------------------------------------------

C  local arrays

C

      integer iselect(10)

      integer zvalues(10)

C

      data iselect/-1,0,0,0,0,0,0,0,0,0/

C

C-----------------------------------------------------

C

      itest=21*nx*ny

      if(nwk.lt.itest) then

         write(6,9901) nwk,itest

 9901    format(' ?mkbicubw:  workspace too small:'/

     >          '  user supplied:  nwk=',i7,'; need at least:  ',i7/

     >      '  nwk = at least 21*nx*ny is required.')

         ier=1

         return

      endif

C

      iadfp=1

      isiz1=16*nx*ny

      iadfw=iadfp+isiz1

      inwk = nwk-isiz1

C

      call mkbicop(f,nf2,wk(iadfp),nx,ny)

C

C  evaluate 4x4 continuous bicubic spline

C

      call bcspline(x,nx,y,ny,wk(iadfp),nx,

     >   ibcxmin,bcxmin,ibcxmax,bcxmax,

     >   ibcymin,bcymin,ibcymax,bcymax,

     >   wk(iadfw),inwk,

     >   ilinx,iliny,ier)

C

      if(ier.ne.0) return

C

C  convert to 4-coefficient (2nd/4th partials form)

C

      hxlast=x(nx)-x(nx-1)

      hylast=y(ny)-y(ny-1)

      call mkbicon(f,nf2,wk(iadfp),nx,ny,hxlast,hylast)

C

      return

      end

C

C----------------------------------------------------------------

C  mkbicop -- copy spline function input data

C

      subroutine mkbicop(fin,nf2,fwk,nx,ny)

C

      real fin(4,nf2,ny)

      real fwk(4,4,nx,ny)

C

      do iy=1,ny

         do ix=1,nx

            fwk(1,1,ix,iy)=fin(1,ix,iy)

         enddo

      enddo

C

      return

      end

C----------------------------------------------------------------

C  mkbicon -- create compact spline representation from 4x4

C             (bcspline) representation

C

      subroutine mkbicon(fin,nf2,fwk,nx,ny,hxlast,hylast)

C

      real fin(4,nf2,ny)

      real fwk(4,4,nx,ny)

C

C-----------------------------------------------------

C  local arrays

C

      integer iselect(10)

      real zvalues(10)

C

      data iselect/-1,0,0,0,0,0,0,0,0,0/

C

C-----------------------------------------------------

C

      do iy=1,ny

         do ix=1,nx

C

C  copy derivatives from result.  Special treatment needed for end zones

C

            iflag=0

            dxuse=0.0

            dyuse=0.0

            ixuse=ix

            iyuse=iy

            if(ix.eq.nx) then

               iflag=1

               dxuse=hxlast

               ixuse=ix-1

            endif

            if(iy.eq.ny) then

               iflag=1

               dyuse=hylast

               iyuse=iy-1

            endif

C

            if(iflag.eq.1) then

               call bcspevfn(iselect,1,1,zvalues,

     >            ixuse,iyuse,dxuse,dyuse,

     >            fwk,nx,ny)

               do j=2,4

                  fin(j,ix,iy)=zvalues(j)

               enddo

            else

               fin(2,ix,iy)=2.0*fwk(3,1,ix,iy)

               fin(3,ix,iy)=2.0*fwk(1,3,ix,iy)

               fin(4,ix,iy)=4.0*fwk(3,3,ix,iy)

            endif

C

         enddo

      enddo

C

      return

      end