Determines the parameters of a function that interpolates the two-dimensional gridded data [x(i),y(j),fcn(i,j)] for i=1,..,nx and j=1,..,ny The interpolating function and its derivatives may subsequently be evaluated by the function db2val.
The interpolating function is a piecewise polynomial function represented as a tensor product of one-dimensional b-splines. the form of this function is
s(x,y)=nx∑i=1ny∑j=1aijui(x)vj(y)
where the functions ui and vj are one-dimensional b-spline basis functions. the coefficients aij are chosen so that
s(x(i),y(j))=fcn(i,j) for i=1,..,nx and j=1,..,ny
Note that for each fixed value of y, s(x,y) is a piecewise polynomial function of x alone, and for each fixed value of x, s(x,y) is a piecewise polynomial function of y 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 ui and vj above is a piecewise polynomial.
Users of db2ink choose the order (degree+1) of the polynomial
pieces used to define the piecewise polynomial in each of the x and
y directions (kx and ky). users also may define their own knot
sequence in x and y separately (tx and ty). if iflag=0, however,
db2ink will choose sequences of knots that result in a piecewise
polynomial interpolant with kx-2 continuous partial derivatives in
x and ky-2 continuous partial derivatives in y. (kx knots are taken
near each endpoint in the x direction, 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 y
direction is treated similarly.)
After a call to db2ink, all information necessary to define the
interpolating function are contained in the parameters nx, ny, kx,
ky, tx, ty, and bcoef. These quantities should not be altered until
after the last call of the evaluation routine db2val.
| Type | Intent | Optional | Attributes | Name | ||
|---|---|---|---|---|---|---|
| real(kind=wp), | intent(in), | dimension(:) | :: | x |
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| integer(kind=ip), | intent(in) | :: | nx |
Number of x abcissae |
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| real(kind=wp), | intent(in), | dimension(:) | :: | y |
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| integer(kind=ip), | intent(in) | :: | ny |
Number of y 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 x ( 2≤kx<nx ) (order = polynomial degree + 1) |
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| integer(kind=ip), | intent(in) | :: | ky |
The order of spline pieces in y ( 2≤ky<ny ) (order = polynomial degree + 1) |
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| integer(kind=ip), | intent(in) | :: | iknot |
knot sequence flag:
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| real(kind=wp), | intent(inout), | dimension(:) | :: | tx |
The
Must be non-decreasing. |
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| real(kind=wp), | intent(inout), | dimension(:) | :: | ty |
The
Must be non-decreasing. |
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| real(kind=wp), | intent(out), | dimension(:,:) | :: | bcoef |
|
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| integer(kind=ip), | intent(out) | :: | iflag |
|
pure subroutine db2ink(x,nx,y,ny,fcn,kx,ky,iknot,tx,ty,bcoef,iflag) implicit none integer(ip),intent(in) :: nx !! Number of \(x\) abcissae integer(ip),intent(in) :: ny !! Number of \(y\) abcissae 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) 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) :: fcn !! `(nx,ny)` matrix of function values to interpolate. !! `fcn(i,j)` should contain the function value at the !! point (`x(i)`,`y(j)`) integer(ip),intent(in) :: iknot !! knot sequence flag: !! !! * 0 = knot sequence chosen by [[db1ink]]. !! * 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 [[db2ink]]. !! * 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 [[db2ink]]. !! * If `iknot=1` these are specified by the user. !! !! Must be non-decreasing. real(wp),dimension(:,:),intent(out) :: bcoef !! `(nx,ny)` 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. !! * 700 = `size(x)` \( \ne \) `size(fcn,1)` !! * 701 = `size(y)` \( \ne \) `size(fcn,2)` !! * 706 = `size(x)` \( \ne \) `nx` !! * 707 = `size(y)` \( \ne \) `ny` !! * 712 = `size(tx)` \( \ne \) `nx+kx` !! * 713 = `size(ty)` \( \ne \) `ny+ky` !! * 800 = `size(x)` \( \ne \) `size(bcoef,1)` !! * 801 = `size(y)` \( \ne \) `size(bcoef,2)` logical :: status_ok real(wp),dimension(:),allocatable :: temp !! work array of length `nx*ny` real(wp),dimension(:),allocatable :: work !! work array of length `max(2*kx*(nx+1),2*ky*(ny+1))` !check validity of inputs call check_inputs( iknot,& iflag,& nx=nx,ny=ny,& kx=kx,ky=ky,& x=x,y=y,& tx=tx,ty=ty,& f2=fcn,& bcoef2=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) end if allocate(temp(nx*ny)) allocate(work(max(2_ip*kx*(nx+1_ip),2_ip*ky*(ny+1_ip)))) !construct b-spline coefficients call dbtpcf(x,nx,fcn, nx,ny,tx,kx,temp, work,iflag) if (iflag==0_ip) call dbtpcf(y,ny,temp,ny,nx,ty,ky,bcoef,work,iflag) deallocate(temp) deallocate(work) end if end subroutine db2ink