kroghInterpolate
Syntax
kroghInterpolate(Xi, Yi, X, [der=0])
Arguments
Xi is a numeric vector indicating the x-coordinates. It must be sorted in increasing order with no NULL values contained.
Yi is a numeric vector of the same length as Xi, indicating the y-coordinates. It cannot contain NULL values.
X is a numeric vector specifying the points at which to evaluate the derivatives.
der (optional) is a non-negative integer indicating how many derivatives to evaluate. The default value is 0, meaning the function value is used as the 0th derivative.
Details
Interpolating polynomial for a set of points. The polynomial passes through all the
pairs (Xi, Yi) and returns the derivative interpolated at the
x-points.
One may additionally specify a number of derivatives at each point Xi; this is done by repeating the value Xi and specifying the derivatives as successive Yi values.
-
When the vector of Xi contains only distinct values, Yi represents the function value.
-
When an element of Xi occurs two or more times in a row, the corresponding Yi represents derivative values. For example, if Xi = [0,0,1,1] and Yi= [1,0,2,3], then Yi[0]=f(0), Yi[1]=f'(0), Yi[2]=f(1) and Yi[3]=f'(1).
Examples
sin as an example to interpolate the value and first derivative
at the point of
xx.def linspace(start, end, num, endpoint=true){
if(endpoint) return end$DOUBLE\(num-1), start + end$DOUBLE\(num-1)*0..(num-1)
else return start + end$DOUBLE\(num-1)*0..(num-1)
}
x = 0 1 2 3 4 5
y = sin(x)
xx = linspace(0.0, 5.0, 10)[1]
yy=kroghInterpolate(x,y,xx)
yy;
// output: [0,0.515119011157387,0.898231239576709,0.998548648650381,0.793484053410063,0.354287125066207,-0.188319604452395,-0.678504737959061,-0.969692008469677,-0.958924274663139]
yy1=kroghInterpolate(x,y,xx,1)
yy1;
// output: [0.885486080979582,0.875967413938641,0.459031117252456,-0.103633680213926,-0.612193041424271,-0.92866822117116,-0.976935666075988,-0.742727014588963,-0.273629096989106,0.320916064615744]