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# splin3d

spline gridded 3d interpolation

### Syntax

tl = splin3d(x, y, z, v, [order])

### Arguments

- x,y,z
strictly increasing row vectors (each with at least 3 components) defining the 3d interpolation grid

- v
nx x ny x nz hypermatrix (nx, ny, nz being the length of

`x`

,`y`

and`z`

)- order
(optional) a 1x3 vector [kx,ky,kz] given the order of the tensor spline in each direction (default [4,4,4], i.e. tricubic spline)

- tl
a tlist of type splin3d defining the spline

### Description

This function computes a 3d tensor spline *s*
which interpolates the *(xi,yj,zk,vijk)* points, ie, we
have *s(xi,yj,zk)=vijk* for all
*i=1,..,nx*, *j=1,..,ny* and
*k=1,..,nz*. The resulting spline
*s* is defined by `tl`

which consists
in a B-spline-tensor representation of *s*. The
evaluation of *s* at some points must be done by the
interp3d function (to compute
*s* and its first derivatives) or by the bsplin3val function (to compute an arbitrary
derivative of *s*) . Several kind of splines may be
computed by selecting the order of the spline in each direction
`order=[kx,ky,kz]`

.

### Remark

This function works under the conditions:

an error being issued when they are not respected.

### Examples

// example 1 // ============================================================================= func = "v=cos(2*%pi*x).*sin(2*%pi*y).*cos(2*%pi*z)"; deff("v=f(x,y,z)",func); n = 10; // n x n x n interpolation points x = linspace(0,1,n); y=x; z=x; // interpolation grid [X,Y,Z] = ndgrid(x,y,z); V = f(X,Y,Z); tl = splin3d(x,y,z,V,[5 5 5]); m = 10000; // compute an approximated error xp = grand(m,1,"def"); yp = grand(m,1,"def"); zp = grand(m,1,"def"); vp_exact = f(xp,yp,zp); vp_interp = interp3d(xp,yp,zp, tl); er = max(abs(vp_exact - vp_interp)) // now retry with n=20 and see the error

// example 2 (see linear_interpn help page which have the // same example with trilinear interpolation) // ============================================================================= exec("SCI/modules/interpolation/demos/interp_demo.sci") func = "v=(x-0.5).^2 + (y-0.5).^3 + (z-0.5).^2"; deff("v=f(x,y,z)",func); n = 5; x = linspace(0,1,n); y=x; z=x; [X,Y,Z] = ndgrid(x,y,z); V = f(X,Y,Z); tl = splin3d(x,y,z,V); // compute (and display) the 3d spline interpolant on some slices m = 41; direction = ["z=" "z=" "z=" "x=" "y="]; val = [ 0.1 0.5 0.9 0.5 0.5]; ebox = [0 1 0 1 0 1]; XF=[]; YF=[]; ZF=[]; VF=[]; for i = 1:length(val) [Xm,Xp,Ym,Yp,Zm,Zp] = slice_parallelepiped(direction(i), val(i), ebox, m, m, m); Vm = interp3d(Xm,Ym,Zm, tl); [xf,yf,zf,vf] = nf3dq(Xm,Ym,Zm,Vm,1); XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf]; VF = [VF vf]; Vp = interp3d(Xp,Yp,Zp, tl); [xf,yf,zf,vf] = nf3dq(Xp,Yp,Zp,Vp,1); XF = [XF xf]; YF = [YF yf]; ZF = [ZF zf]; VF = [VF vf]; end nb_col = 128; vmin = min(VF); vmax = max(VF); color_example = dsearch(VF,linspace(vmin,vmax,nb_col+1)); gcf().color_map = jetcolormap(nb_col); clf(); gca().hiddencolor = gca().background; colorbar(vmin,vmax) plot3d(XF, YF, list(ZF,color_example), flag=[-1 6 4]) xtitle("3d spline interpolation of "+func) show_window()

### See also

- linear_interpn — n dimensional linear interpolation
- interp3d — 3d spline evaluation function
- bsplin3val — 3d spline arbitrary derivative evaluation function

### History

Version | Description |

5.4.0 | previously, imaginary part of input arguments were implicitly ignored. |

## Comments

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