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Quadratic interpolation of tabulated function of 1,2 or 3 variables +! +!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! + +subroutine fvn_s_find_interval(x,i,xdata,n) + implicit none + ! This routine find the indice i where xdata(i) <= x < xdata(i+1) + ! xdata(n) must contains a set of increasingly ordered values + ! if x < xdata(1) i=0 is returned + ! if x > xdata(n) i=n is returned + ! special case is where x=xdata(n) then n-1 is returned so + ! we will not exclude the upper limit + ! a simple dichotomy method is used + + real(kind=4), intent(in) :: x + real(kind=4), intent(in), dimension(n) :: xdata + integer(kind=4), intent(in) :: n + integer(kind=4), intent(out) :: i + + integer(kind=4) :: imin,imax,imoyen + + ! special case is where x=xdata(n) then n-1 is returned so + ! we will not exclude the upper limit + if (x == xdata(n)) then + i=n-1 + return + end if + + ! if x < xdata(1) i=0 is returned + if (x < xdata(1)) then + i=0 + return + end if + + ! if x > xdata(n) i=n is returned + if (x > xdata(n)) then + i=n + return + end if + + ! here xdata(1) <= x <= xdata(n) + imin=0 + imax=n+1 + + do while((imax-imin) > 1) + imoyen=(imax+imin)/2 + if (x >= xdata(imoyen)) then + imin=imoyen + else + imax=imoyen + end if + end do + + i=imin + +end subroutine + + +subroutine fvn_d_find_interval(x,i,xdata,n) + implicit none + ! This routine find the indice i where xdata(i) <= x < xdata(i+1) + ! xdata(n) must contains a set of increasingly ordered values + ! if x < xdata(1) i=0 is returned + ! if x > xdata(n) i=n is returned + ! special case is where x=xdata(n) then n-1 is returned so + ! we will not exclude the upper limit + ! a simple dichotomy method is used + + real(kind=8), intent(in) :: x + real(kind=8), intent(in), dimension(n) :: xdata + integer(kind=4), intent(in) :: n + integer(kind=4), intent(out) :: i + + integer(kind=4) :: imin,imax,imoyen + + ! special case is where x=xdata(n) then n-1 is returned so + ! we will not exclude the upper limit + if (x == xdata(n)) then + i=n-1 + return + end if + + ! if x < xdata(1) i=0 is returned + if (x < xdata(1)) then + i=0 + return + end if + + ! if x > xdata(n) i=n is returned + if (x > xdata(n)) then + i=n + return + end if + + ! here xdata(1) <= x <= xdata(n) + imin=0 + imax=n+1 + + do while((imax-imin) > 1) + imoyen=(imax+imin)/2 + if (x >= xdata(imoyen)) then + imin=imoyen + else + imax=imoyen + end if + end do + + i=imin + +end subroutine + + +function fvn_s_quad_interpol(x,n,xdata,ydata) + implicit none + ! This function evaluate the value of a function defined by a set of points + ! and values, using a quadratic interpolation + ! xdata must be increasingly ordered + ! x must be within xdata(1) and xdata(n) to actually do interpolation + ! otherwise extrapolation is done + integer(kind=4), intent(in) :: n + real(kind=4), intent(in), dimension(n) :: xdata,ydata + real(kind=4), intent(in) :: x + real(kind=4) :: fvn_s_quad_interpol + + integer(kind=4) :: iinf,base,i,j + real(kind=4) :: p + + call fvn_s_find_interval(x,iinf,xdata,n) + + ! Settings for extrapolation + if (iinf==0) then + ! TODO -> Lower bound extrapolation warning + iinf=1 + end if + + if (iinf==n) then + ! TODO -> Higher bound extrapolation warning + iinf=n-1 + end if + + ! The three points we will use are iinf-1,iinf and iinf+1 with the + ! exception of the first interval, where iinf=1 we will use 1,2 and 3 + if (iinf==1) then + base=0 + else + base=iinf-2 + end if + + ! The three points we will use are : + ! xdata/ydata(base+1),xdata/ydata(base+2),xdata/ydata(base+3) + + ! Straight forward Lagrange polynomial + fvn_s_quad_interpol=0. + do i=1,3 + ! polynome i + p=ydata(base+i) + do j=1,3 + if (j /= i) then + p=p*(x-xdata(base+j))/(xdata(base+i)-xdata(base+j)) + end if + end do + fvn_s_quad_interpol=fvn_s_quad_interpol+p + end do + +end function + + +function fvn_d_quad_interpol(x,n,xdata,ydata) + implicit none + ! This function evaluate the value of a function defined by a set of points + ! and values, using a quadratic interpolation + ! xdata must be increasingly ordered + ! x must be within xdata(1) and xdata(n) to actually do interpolation + ! otherwise extrapolation is done + integer(kind=4), intent(in) :: n + real(kind=8), intent(in), dimension(n) :: xdata,ydata + real(kind=8), intent(in) :: x + real(kind=8) :: fvn_d_quad_interpol + + integer(kind=4) :: iinf,base,i,j + real(kind=8) :: p + + call fvn_d_find_interval(x,iinf,xdata,n) + + ! Settings for extrapolation + if (iinf==0) then + ! TODO -> Lower bound extrapolation warning + iinf=1 + end if + + if (iinf==n) then + ! TODO Higher bound extrapolation warning + iinf=n-1 + end if + + ! The three points we will use are iinf-1,iinf and iinf+1 with the + ! exception of the first interval, where iinf=1 we will use 1,2 and 3 + if (iinf==1) then + base=0 + else + base=iinf-2 + end if + + ! The three points we will use are : + ! xdata/ydata(base+1),xdata/ydata(base+2),xdata/ydata(base+3) + + ! Straight forward Lagrange polynomial + fvn_d_quad_interpol=0. + do i=1,3 + ! polynome i + p=ydata(base+i) + do j=1,3 + if (j /= i) then + p=p*(x-xdata(base+j))/(xdata(base+i)-xdata(base+j)) + end if + end do + fvn_d_quad_interpol=fvn_d_quad_interpol+p + end do + +end function + + +function fvn_s_quad_2d_interpol(x,y,nx,xdata,ny,ydata,zdata) + implicit none + ! This function evaluate the value of a two variable function defined by a + ! set of points and values, using a quadratic interpolation + ! xdata and ydata must be increasingly ordered + ! the couple (x,y) must be as x within xdata(1) and xdata(nx) and + ! y within ydata(1) and ydata(ny) to actually do interpolation + ! otherwise extrapolation is done + integer(kind=4), intent(in) :: nx,ny + real(kind=4), intent(in) :: x,y + real(kind=4), intent(in), dimension(nx) :: xdata + real(kind=4), intent(in), dimension(ny) :: ydata + real(kind=4), intent(in), dimension(nx,ny) :: zdata + real(kind=4) :: fvn_s_quad_2d_interpol + + integer(kind=4) :: ixinf,iyinf,basex,basey,i + real(kind=4),dimension(3) :: ztmp + !real(kind=4), external :: fvn_s_quad_interpol + + call fvn_s_find_interval(x,ixinf,xdata,nx) + call fvn_s_find_interval(y,iyinf,ydata,ny) + + ! Settings for extrapolation + if (ixinf==0) then + ! TODO -> Lower x bound extrapolation warning + ixinf=1 + end if + + if (ixinf==nx) then + ! TODO -> Higher x bound extrapolation warning + ixinf=nx-1 + end if + + if (iyinf==0) then + ! TODO -> Lower y bound extrapolation warning + iyinf=1 + end if + + if (iyinf==ny) then + ! TODO -> Higher y bound extrapolation warning + iyinf=ny-1 + end if + + ! The three points we will use are iinf-1,iinf and iinf+1 with the + ! exception of the first interval, where iinf=1 we will use 1,2 and 3 + if (ixinf==1) then + basex=0 + else + basex=ixinf-2 + end if + + if (iyinf==1) then + basey=0 + else + basey=iyinf-2 + end if + + ! First we make 3 interpolations for x at y(base+1),y(base+2),y(base+3) + ! stored in ztmp(1:3) + do i=1,3 + ztmp(i)=fvn_s_quad_interpol(x,nx,xdata,zdata(:,basey+i)) + end do + + ! Then we make an interpolation for y using previous interpolations + fvn_s_quad_2d_interpol=fvn_s_quad_interpol(y,3,ydata(basey+1:basey+3),ztmp) +end function + + +function fvn_d_quad_2d_interpol(x,y,nx,xdata,ny,ydata,zdata) + implicit none + ! This function evaluate the value of a two variable function defined by a + ! set of points and values, using a quadratic interpolation + ! xdata and ydata must be increasingly ordered + ! the couple (x,y) must be as x within xdata(1) and xdata(nx) and + ! y within ydata(1) and ydata(ny) to actually do interpolation + ! otherwise extrapolation is done + integer(kind=4), intent(in) :: nx,ny + real(kind=8), intent(in) :: x,y + real(kind=8), intent(in), dimension(nx) :: xdata + real(kind=8), intent(in), dimension(ny) :: ydata + real(kind=8), intent(in), dimension(nx,ny) :: zdata + real(kind=8) :: fvn_d_quad_2d_interpol + + integer(kind=4) :: ixinf,iyinf,basex,basey,i + real(kind=8),dimension(3) :: ztmp + !real(kind=8), external :: fvn_d_quad_interpol + + call fvn_d_find_interval(x,ixinf,xdata,nx) + call fvn_d_find_interval(y,iyinf,ydata,ny) + + ! Settings for extrapolation + if (ixinf==0) then + ! TODO -> Lower x bound extrapolation warning + ixinf=1 + end if + + if (ixinf==nx) then + ! TODO -> Higher x bound extrapolation warning + ixinf=nx-1 + end if + + if (iyinf==0) then + ! TODO -> Lower y bound extrapolation warning + iyinf=1 + end if + + if (iyinf==ny) then + ! TODO -> Higher y bound extrapolation warning + iyinf=ny-1 + end if + + ! The three points we will use are iinf-1,iinf and iinf+1 with the + ! exception of the first interval, where iinf=1 we will use 1,2 and 3 + if (ixinf==1) then + basex=0 + else + basex=ixinf-2 + end if + + if (iyinf==1) then + basey=0 + else + basey=iyinf-2 + end if + + ! First we make 3 interpolations for x at y(base+1),y(base+2),y(base+3) + ! stored in ztmp(1:3) + do i=1,3 + ztmp(i)=fvn_d_quad_interpol(x,nx,xdata,zdata(:,basey+i)) + end do + + ! Then we make an interpolation for y using previous interpolations + fvn_d_quad_2d_interpol=fvn_d_quad_interpol(y,3,ydata(basey+1:basey+3),ztmp) +end function + + +function fvn_s_quad_3d_interpol(x,y,z,nx,xdata,ny,ydata,nz,zdata,tdata) + implicit none + ! This function evaluate the value of a 3 variables function defined by a + ! set of points and values, using a quadratic interpolation + ! xdata, ydata and zdata must be increasingly ordered + ! The triplet (x,y,z) must be within xdata,ydata and zdata to actually + ! perform an interpolation, otherwise extrapolation is done + integer(kind=4), intent(in) :: nx,ny,nz + real(kind=4), intent(in) :: x,y,z + real(kind=4), intent(in), dimension(nx) :: xdata + real(kind=4), intent(in), dimension(ny) :: ydata + real(kind=4), intent(in), dimension(nz) :: zdata + real(kind=4), intent(in), dimension(nx,ny,nz) :: tdata + real(kind=4) :: fvn_s_quad_3d_interpol + + integer(kind=4) :: ixinf,iyinf,izinf,basex,basey,basez,i,j + !real(kind=4), external :: fvn_s_quad_interpol,fvn_s_quad_2d_interpol + real(kind=4),dimension(3,3) :: ttmp + + call fvn_s_find_interval(x,ixinf,xdata,nx) + call fvn_s_find_interval(y,iyinf,ydata,ny) + call fvn_s_find_interval(z,izinf,zdata,nz) + + ! Settings for extrapolation + if (ixinf==0) then + ! TODO -> Lower x bound extrapolation warning + ixinf=1 + end if + + if (ixinf==nx) then + ! TODO -> Higher x bound extrapolation warning + ixinf=nx-1 + end if + + if (iyinf==0) then + ! TODO -> Lower y bound extrapolation warning + iyinf=1 + end if + + if (iyinf==ny) then + ! TODO -> Higher y bound extrapolation warning + iyinf=ny-1 + end if + + if (izinf==0) then + ! TODO -> Lower z bound extrapolation warning + izinf=1 + end if + + if (izinf==nz) then + ! TODO -> Higher z bound extrapolation warning + izinf=nz-1 + end if + + ! The three points we will use are iinf-1,iinf and iinf+1 with the + ! exception of the first interval, where iinf=1 we will use 1,2 and 3 + if (ixinf==1) then + basex=0 + else + basex=ixinf-2 + end if + + if (iyinf==1) then + basey=0 + else + basey=iyinf-2 + end if + + if (izinf==1) then + basez=0 + else + basez=izinf-2 + end if + + ! We first make 9 one dimensional interpolation on variable x. + ! results are stored in ttmp + do i=1,3 + do j=1,3 + ttmp(i,j)=fvn_s_quad_interpol(x,nx,xdata,tdata(:,basey+i,basez+j)) + end do + end do + + ! We then make a 2 dimensionnal interpolation on variables y and z + fvn_s_quad_3d_interpol=fvn_s_quad_2d_interpol(y,z, & + 3,ydata(basey+1:basey+3),3,zdata(basez+1:basez+3),ttmp) +end function + + +function fvn_d_quad_3d_interpol(x,y,z,nx,xdata,ny,ydata,nz,zdata,tdata) + implicit none + ! This function evaluate the value of a 3 variables function defined by a + ! set of points and values, using a quadratic interpolation + ! xdata, ydata and zdata must be increasingly ordered + ! The triplet (x,y,z) must be within xdata,ydata and zdata to actually + ! perform an interpolation, otherwise extrapolation is done + integer(kind=4), intent(in) :: nx,ny,nz + real(kind=8), intent(in) :: x,y,z + real(kind=8), intent(in), dimension(nx) :: xdata + real(kind=8), intent(in), dimension(ny) :: ydata + real(kind=8), intent(in), dimension(nz) :: zdata + real(kind=8), intent(in), dimension(nx,ny,nz) :: tdata + real(kind=8) :: fvn_d_quad_3d_interpol + + integer(kind=4) :: ixinf,iyinf,izinf,basex,basey,basez,i,j + !real(kind=8), external :: fvn_d_quad_interpol,fvn_d_quad_2d_interpol + real(kind=8),dimension(3,3) :: ttmp + + call fvn_d_find_interval(x,ixinf,xdata,nx) + call fvn_d_find_interval(y,iyinf,ydata,ny) + call fvn_d_find_interval(z,izinf,zdata,nz) + + ! Settings for extrapolation + if (ixinf==0) then + ! TODO -> Lower x bound extrapolation warning + ixinf=1 + end if + + if (ixinf==nx) then + ! TODO -> Higher x bound extrapolation warning + ixinf=nx-1 + end if + + if (iyinf==0) then + ! TODO -> Lower y bound extrapolation warning + iyinf=1 + end if + + if (iyinf==ny) then + ! TODO -> Higher y bound extrapolation warning + iyinf=ny-1 + end if + + if (izinf==0) then + ! TODO -> Lower z bound extrapolation warning + izinf=1 + end if + + if (izinf==nz) then + ! TODO -> Higher z bound extrapolation warning + izinf=nz-1 + end if + + ! The three points we will use are iinf-1,iinf and iinf+1 with the + ! exception of the first interval, where iinf=1 we will use 1,2 and 3 + if (ixinf==1) then + basex=0 + else + basex=ixinf-2 + end if + + if (iyinf==1) then + basey=0 + else + basey=iyinf-2 + end if + + if (izinf==1) then + basez=0 + else + basez=izinf-2 + end if + + ! We first make 9 one dimensional interpolation on variable x. + ! results are stored in ttmp + do i=1,3 + do j=1,3 + ttmp(i,j)=fvn_d_quad_interpol(x,nx,xdata,tdata(:,basey+i,basez+j)) + end do + end do + + ! We then make a 2 dimensionnal interpolation on variables y and z + fvn_d_quad_3d_interpol=fvn_d_quad_2d_interpol(y,z, & + 3,ydata(basey+1:basey+3),3,zdata(basez+1:basez+3),ttmp) +end function + + + + !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! Akima spline interpolation and spline evaluation