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# splin2d

bicubic spline gridded 2d interpolation

### Calling Sequence

C = splin2d(x, y, z, [,spline_type])

### Arguments

- x,y
strictly increasing row vectors (with at least 2 components) defining the interpolation grid

- z
nx x ny matrix (nx being the length of

`x`

and ny the length of`y`

)- spline_type
(optional) a string selecting the kind of bicubic spline to compute

- C
a big vector with the coefficients of the bicubic patches (see details in Remarks)

### Description

This function computes a bicubic spline or sub-spline
*s* which interpolates the
*(xi,yj,zij)* points, ie, we have
*s(xi,yj)=zij* for all *i=1,..,nx*
and *j=1,..,ny*. The resulting spline
*s* is defined by the triplet
`(x,y,C)`

where `C`

is the vector (of
length 16(nx-1)(ny-1)) with the coefficients of each of the (nx-1)(ny-1)
bicubic patches : on *[x(i) x(i+1)]x[y(j) y(j+1)]*,
*s* is defined by :

The evaluation of *s* at some points must be done
by the interp2d function. Several kind of
splines may be computed by selecting the appropriate
`spline_type`

parameter. The method used to compute the
bicubic spline (or sub-spline) is the old fashionned one 's, i.e. to
compute on each grid point *(xi,yj)* an approximation
of the first derivatives *ds/dx(xi,yj)* and
*ds/dy(xi,yj)* and of the cross derivative
*d2s/dxdy(xi,yj)*. Those derivatives are computed by
the mean of 1d spline schemes leading to a C2 function
(*s* is twice continuously differentiable) or by the
mean of a local approximation scheme leading to a C1 function only. This
scheme is selected with the `spline_type`

parameter (see
splin for details) :

- "not_a_knot"
this is the default case.

- "periodic"
to use if the underlying function is periodic : you must have

*z(1,j) = z(nx,j) for all j in [1,ny] and z(i,1) = z(i,ny) for i in [1,nx]*but this is not verified by the interface.

### Remarks

From an accuracy point of view use essentially the **not_a_knot** type or **periodic** type if the underlying interpolated
function is periodic.

The **natural**, **monotone**, **fast** (or
**fast_periodic**) type may be useful in some
cases, for instance to limit oscillations (**monotone** being the most powerfull for that).

To get the coefficients of the bi-cubic patches in a more friendly
way you can use `c = hypermat([4,4,nx-1,ny-1],C)`

then
the coefficient *(k,l)* of the patch
*(i,j)* (see equation here before) is stored at
`c(k,l,i,j)`

. Nevertheless the interp2d function wait for the big vector
`C`

and not for the hypermatrix `c`

(note that one can easily retrieve `C`

from
`c`

with `C=c(:)`

).

### Examples

// example 1 : interpolation of cos(x)cos(y) n = 7; // a regular grid with n x n interpolation points // will be used x = linspace(0,2*%pi,n); y = x; z = cos(x')*cos(y); C = splin2d(x, y, z, "periodic"); m = 50; // discretisation parameter of the evaluation grid xx = linspace(0,2*%pi,m); yy = xx; [XX,YY] = ndgrid(xx,yy); zz = interp2d(XX,YY, x, y, C); emax = max(abs(zz - cos(xx')*cos(yy))); clf() plot3d(xx, yy, zz, flag=[2 4 4]) [X,Y] = ndgrid(x,y); param3d1(X,Y,list(z,-9*ones(1,n)), flag=[0 0]) str = msprintf(" with %d x %d interpolation points. ermax = %g",n,n,emax) xtitle("spline interpolation of cos(x)cos(y)"+str) // example 2 : different interpolation functions on random datas n = 6; x = linspace(0,1,n); y = x; z = rand(n,n); np = 50; xp = linspace(0,1,np); yp = xp; [XP, YP] = ndgrid(xp,yp); ZP1 = interp2d(XP, YP, x, y, splin2d(x, y, z, "not_a_knot")); ZP2 = linear_interpn(XP, YP, x, y, z); ZP3 = interp2d(XP, YP, x, y, splin2d(x, y, z, "natural")); ZP4 = interp2d(XP, YP, x, y, splin2d(x, y, z, "monotone")); xset("colormap", jetcolormap(64)) clf() subplot(2,2,1) plot3d1(xp, yp, ZP1, flag=[2 2 4]) xtitle("not_a_knot") subplot(2,2,2) plot3d1(xp, yp, ZP2, flag=[2 2 4]) xtitle("bilinear interpolation") subplot(2,2,3) plot3d1(xp, yp, ZP3, flag=[2 2 4]) xtitle("natural") subplot(2,2,4) plot3d1(xp, yp, ZP4, flag=[2 2 4]) xtitle("monotone") show_window() // example 3 : not_a_knot spline and monotone sub-spline // on a step function a = 0; b = 1; c = 0.25; d = 0.75; // create interpolation grid n = 11; x = linspace(a,b,n); ind = find(c <= x & x <= d); z = zeros(n,n); z(ind,ind) = 1; // a step inside a square // create evaluation grid np = 220; xp = linspace(a,b, np); [XP, YP] = ndgrid(xp, xp); zp1 = interp2d(XP, YP, x, x, splin2d(x,x,z)); zp2 = interp2d(XP, YP, x, x, splin2d(x,x,z,"monotone")); // plot clf() xset("colormap",jetcolormap(128)) subplot(1,2,1) plot3d1(xp, xp, zp1, flag=[-2 6 4]) xtitle("spline (not_a_knot)") subplot(1,2,2) plot3d1(xp, xp, zp2, flag=[-2 6 4]) xtitle("subspline (monotone)")

### See Also

- cshep2d — bidimensional cubic shepard (scattered) interpolation
- linear_interpn — n dimensional linear interpolation
- interp2d — bicubic spline (2d) evaluation function

### Authors

B. Pincon

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