Determines the parameters of a function that interpolates the three-dimensional gridded data The interpolating function and its derivatives may subsequently be evaluated by the function db3val.
The interpolating function is a piecewise polynomial function represented as a tensor product of one-dimensional b-splines. the form of this function is
where the functions , , and are one-dimensional b- spline basis functions. the coefficients are chosen so that:
Note that for fixed values of and is a piecewise polynomial function of alone, for fixed values of and is a piecewise polynomial function of alone, and for fixed values of and is a function of alone. in one dimension a piecewise polynomial may be created by partitioning a given interval into subintervals and defining a distinct polynomial piece on each one. the points where adjacent subintervals meet are called knots. each of the functions , , and above is a piecewise polynomial.
Users of db3ink choose the order (degree+1) of the polynomial
pieces used to define the piecewise polynomial in each of the , ,
and directions (kx, ky, and kz). users also may define their own
knot sequence in , , separately (tx, ty, and tz). if iflag=0,
however, db3ink will choose sequences of knots that result in a
piecewise polynomial interpolant with kx-2 continuous partial
derivatives in , ky-2 continuous partial derivatives in , and kz-2
continuous partial derivatives in . (kx knots are taken near
each endpoint in , not-a-knot end conditions are used, and the
remaining knots are placed at data points if kx is even or at
midpoints between data points if kx is odd. the and directions
are treated similarly.)
After a call to db3ink, all information necessary to define the
interpolating function are contained in the parameters nx, ny, nz,
kx, ky, kz, tx, ty, tz, and bcoef. these quantities should not be
altered until after the last call of the evaluation routine db3val.
| Type | Intent | Optional | Attributes | Name | ||
|---|---|---|---|---|---|---|
| real(kind=wp), | intent(in), | dimension(:) | :: | x |
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| integer(kind=ip), | intent(in) | :: | nx |
number of abcissae ( ) |
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| real(kind=wp), | intent(in), | dimension(:) | :: | y |
|
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| integer(kind=ip), | intent(in) | :: | ny |
number of abcissae ( ) |
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| real(kind=wp), | intent(in), | dimension(:) | :: | z |
|
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| integer(kind=ip), | intent(in) | :: | nz |
number of abcissae ( ) |
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| real(kind=wp), | intent(in), | dimension(:,:,:) | :: | fcn |
|
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| integer(kind=ip), | intent(in) | :: | kx |
The order of spline pieces in ( ) (order = polynomial degree + 1) |
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| integer(kind=ip), | intent(in) | :: | ky |
The order of spline pieces in ( ) (order = polynomial degree + 1) |
||
| integer(kind=ip), | intent(in) | :: | kz |
the order of spline pieces in ( ) (order = polynomial degree + 1) |
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| integer(kind=ip), | intent(in) | :: | iknot |
knot sequence flag:
|
||
| real(kind=wp), | intent(inout), | dimension(:) | :: | tx |
The
Must be non-decreasing. |
|
| real(kind=wp), | intent(inout), | dimension(:) | :: | ty |
The
Must be non-decreasing. |
|
| real(kind=wp), | intent(inout), | dimension(:) | :: | tz |
The
Must be non-decreasing. |
|
| real(kind=wp), | intent(out), | dimension(:,:,:) | :: | bcoef |
|
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| integer(kind=ip), | intent(out) | :: | iflag |
|
pure subroutine db3ink(x,nx,y,ny,z,nz,fcn,kx,ky,kz,iknot,tx,ty,tz,bcoef,iflag) implicit none integer(ip),intent(in) :: nx !! number of \(x\) abcissae ( \( \ge 3 \) ) integer(ip),intent(in) :: ny !! number of \(y\) abcissae ( \( \ge 3 \) ) integer(ip),intent(in) :: nz !! number of \(z\) abcissae ( \( \ge 3 \) ) integer(ip),intent(in) :: kx !! The order of spline pieces in \(x\) !! ( \( 2 \le k_x < n_x \) ) !! (order = polynomial degree + 1) integer(ip),intent(in) :: ky !! The order of spline pieces in \(y\) !! ( \( 2 \le k_y < n_y \) ) !! (order = polynomial degree + 1) integer(ip),intent(in) :: kz !! the order of spline pieces in \(z\) !! ( \( 2 \le k_z < n_z \) ) !! (order = polynomial degree + 1) real(wp),dimension(:),intent(in) :: x !! `(nx)` array of \(x\) abcissae. must be strictly increasing. real(wp),dimension(:),intent(in) :: y !! `(ny)` array of \(y\) abcissae. must be strictly increasing. real(wp),dimension(:),intent(in) :: z !! `(nz)` array of \(z\) abcissae. must be strictly increasing. real(wp),dimension(:,:,:),intent(in) :: fcn !! `(nx,ny,nz)` matrix of function values to interpolate. `fcn(i,j,k)` should !! contain the function value at the point (`x(i)`,`y(j)`,`z(k)`) integer(ip),intent(in) :: iknot !! knot sequence flag: !! !! * 0 = knot sequence chosen by [[db3ink]]. !! * 1 = knot sequence chosen by user. real(wp),dimension(:),intent(inout) :: tx !! The `(nx+kx)` knots in the \(x\) direction for the spline !! interpolant. !! !! * If `iknot=0` these are chosen by [[db3ink]]. !! * If `iknot=1` these are specified by the user. !! !! Must be non-decreasing. real(wp),dimension(:),intent(inout) :: ty !! The `(ny+ky)` knots in the \(y\) direction for the spline !! interpolant. !! !! * If `iknot=0` these are chosen by [[db3ink]]. !! * If `iknot=1` these are specified by the user. !! !! Must be non-decreasing. real(wp),dimension(:),intent(inout) :: tz !! The `(nz+kz)` knots in the \(z\) direction for the spline !! interpolant. !! !! * If `iknot=0` these are chosen by [[db3ink]]. !! * If `iknot=1` these are specified by the user. !! !! Must be non-decreasing. real(wp),dimension(:,:,:),intent(out) :: bcoef !! `(nx,ny,nz)` matrix of coefficients of the b-spline interpolant. integer(ip),intent(out) :: iflag !! * 0 = successful execution. !! * 2 = `iknot` out of range. !! * 3 = `nx` out of range. !! * 4 = `kx` out of range. !! * 5 = `x` not strictly increasing. !! * 6 = `tx` not non-decreasing. !! * 7 = `ny` out of range. !! * 8 = `ky` out of range. !! * 9 = `y` not strictly increasing. !! * 10 = `ty` not non-decreasing. !! * 11 = `nz` out of range. !! * 12 = `kz` out of range. !! * 13 = `z` not strictly increasing. !! * 14 = `ty` not non-decreasing. !! * 700 = `size(x) ` \(\ne\) `size(fcn,1)` !! * 701 = `size(y) ` \(\ne\) `size(fcn,2)` !! * 702 = `size(z) ` \(\ne\) `size(fcn,3)` !! * 706 = `size(x) ` \(\ne\) `nx` !! * 707 = `size(y) ` \(\ne\) `ny` !! * 708 = `size(z) ` \(\ne\) `nz` !! * 712 = `size(tx)` \(\ne\) `nx+kx` !! * 713 = `size(ty)` \(\ne\) `ny+ky` !! * 714 = `size(tz)` \(\ne\) `nz+kz` !! * 800 = `size(x) ` \(\ne\) `size(bcoef,1)` !! * 801 = `size(y) ` \(\ne\) `size(bcoef,2)` !! * 802 = `size(z) ` \(\ne\) `size(bcoef,3)` logical :: status_ok real(wp),dimension(:),allocatable :: temp !! work array of length `nx*ny*nz` real(wp),dimension(:),allocatable :: work !! work array of length `max(2*kx*(nx+1), !! 2*ky*(ny+1),2*kz*(nz+1))` integer(ip) :: i, j, k, ii !! counter ! check validity of input call check_inputs( iknot,& iflag,& nx=nx,ny=ny,nz=nz,& kx=kx,ky=ky,kz=kz,& x=x,y=y,z=z,& tx=tx,ty=ty,tz=tz,& f3=fcn,& bcoef3=bcoef,& status_ok=status_ok) if (status_ok) then ! choose knots if (iknot == 0_ip) then call dbknot(x,nx,kx,tx) call dbknot(y,ny,ky,ty) call dbknot(z,nz,kz,tz) end if allocate(temp(nx*ny*nz)) allocate(work(max(2_ip*kx*(nx+1_ip),2_ip*ky*(ny+1_ip),2_ip*kz*(nz+1_ip)))) ! copy fcn to work in packed for dbtpcf !temp = reshape( fcn, [nx*ny*nz] ) ! replaced with loops to avoid stack ! overflow for large data set: ii = 0_ip do k = 1_ip, nz do j = 1_ip, ny do i = 1_ip, nx ii = ii + 1_ip temp(ii) = fcn(i,j,k) end do end do end do ! construct b-spline coefficients call dbtpcf(x,nx,temp, nx,ny*nz,tx,kx,bcoef,work,iflag) if (iflag==0_ip) call dbtpcf(y,ny,bcoef,ny,nx*nz,ty,ky,temp, work,iflag) if (iflag==0_ip) call dbtpcf(z,nz,temp, nz,nx*ny,tz,kz,bcoef,work,iflag) deallocate(temp) deallocate(work) end if end subroutine db3ink