# sps_fun: Shape Preserving Bi-variate and Uni-variate Interpolation. In spsi: Shape-Preserving Uni-Variate and Bi-Variate Spline Interpolation

## Description

Both functions recognize whether uni-variate or bi-variate interpolation is requested

`sps_fun` calls `sps_prep` to prepare spline object and according to data provided creates uni-variate or bi-variate executable function. Parameters can be set by specifying them as arguments

`sps_interpolate` prepares spline and returns its values on given set of tabulation points

## Usage

 ```1 2 3``` ```sps_fun(x, y, z=NULL, der=0, der.x=der, der.y=der, grid = FALSE, ...) sps_interpolate(x, y, z=NULL, xt, yt=NULL, grid = TRUE, ...) ```

## Arguments

 `x` Vector of x-coordinates of data. `y` Vector of y-coordinates of data. For uni-variate case length should be equal to length of `x` and y[i] = f(x[i]). For bi-variate case length may differ. `z` Bi-variate case only. Matrix of values of function on grid spanned by x and y; z[i,j] = f(x[i],y[j]) OR Vector of values of functions on points (x, y); z[i] = f(x[i],y[i]). In this case length of all three vectors should be equal and it should be possible to transform the point to a gridded form. `xt, yt` Vectors OR Matrices containing coordinates of the tabulation point; yt for bi-variate case only `der, der.x, der.y` Indicate which derivative should be returned; der.y used in bi-variate case only. If length of vector is greater than 1 function will return list of values. For bi-variate case vectors should have equal length. At the moment only 1 and 0 are supported. See examples. `grid` For bi-variate interpolation only. If `TRUE` function will return matrix of values of the spline on grid spanned by vectors of tabulation points. If `FALSE` vector of f(xt[i] , yt[i]), i = 1, 2, ..., length(xt) will be returned. If matrices were given as tabulation points `grid` is meaningless. `...` arguments in tag = value format. The tags must come from the names of parameters of sps_prep and/or sps_eval.

## Details

Following parameters can be specified:

fx, fy, fxy

Matrices with values of the derivatives: see `sps_prep`

maxdeg

Maximum degree of polynomial allowed: see `sps_prep`

smoothness

Smoothness required: see `sps_prep`

tol

Relative tolerance used by program

shape

Vector of shape attributes that must be preserved. Must contain only 'monotonicity' and/or 'curvature'

## Value

`sps_fun`

Function of 1 or 2 variables depending on data provided. Function will accept vector(s) or matrix(es) of tabulation point and return object of the same class. If while calling `sps_fun` length der.x (and possibly der.y) was bigger than 1. Resulting function will be returning list of values of respective derivatives at given tabulation points.

`sps_interpolate`

Vector, matrix or list of vectors/matrices of value of function and/or derivatives on given set of tabulation points

## Author(s)

Szymon Sacher <[email protected]> & Andrew Clausen <[email protected]>

## References

Costantini, P; Fontanella, F; 'Shape Preserving Bi-variate Interpolation' SIAM J NUMER. ANAL. Vol. 27, No.2, pp. 488-506, April 1990

`sps_eval` `sps_prep`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51``` ```## Example 1 # Following example shows usage of sps_fun along with the parameter 'smoothness'. # As you will see if smoothness = 2 then first derivative of function is differentiable # everywhere. x <- c( 1, 2, 3, 4, 5, 6) y <- c(16 ,18, 21, 17, 15, 12) evalK1 <- sps_fun(x, y) derK1 <- sps_fun(x, y, der.x=1) evalK2 <- sps_fun(x, y, smoothness = 2) derK2 <- sps_fun(x, y, smoothness = 2, der.x = 1) xs <- seq(1, 6, 0.01) par(mfrow = c(2, 2)) plot(x, y, col = "red", xlim = c(0, 7), ylim = c(10, 22), main = "Spline, smoothness = 1") grid() lines(xs,evalK1(xs), col="cyan") par(new = TRUE) plot(derK1, from = 1,to = 6, col = "magenta", xlim = c(0,7), ylim = c(-6,5), xaxt = 'n',yaxt='n',ann = FALSE) axis(4, -6:5) plot(x, y, col="red", xlim=c(0,7), ylim=c(10,22), main = "Spline, smoothness = 2") grid() lines(xs,evalK2(xs), col="cyan") par(new = TRUE) plot(derK2, from = 1, to = 6, col = "magenta", xlim = c(0,7), ylim = c(-6,5), xaxt = 'n',yaxt = 'n', ann = FALSE) axis(4, -6:5) plot(derK1, from = 1.5, to = 2.5) plot(derK2, from = 1.5, to = 2.5) ## EXAMPLE 2 par(mfrow = c(1,1)) X <- seq(0, 50, 5) Y <- seq(0, 40, 5) X_ <- seq(0, 50, 0.5) Y_ <- seq(0, 40, 0.5) persp3D(X_, Y_, sps_interpolate(X, Y, akima, X_, Y_, grid = TRUE, shape = 'monotonicity')) ```