comphy (version 1.0.5)

decidepoly_n: Degree of best-interpolating polynomial

Description

The degree is chosen making use of divided differences. As more and more points are used for the interpolation, the components of columns of divided differences corresponding to higher orders, are smaller and smaller. They are exactly zero when the function to interpolate is a polynomial of degree, say, \(k\). More specifically, all divided differences of order \(k+1\) and above are exactly zero. The criterion used suggests a value \(k\) if the average of the absolute value of all divided differences of order \(k+1\) is less than a given threshold thr (default 1e-6).

Usage

decidepoly_n(x, f, thr = 1e-06, ntrial = 30)

Value

An integer corresponding to the best interpolating polynomial's degree.

Arguments

x

A vector of real numbers. Grid points corresponding to the tabulated (known) values of the function.

f

A vector of real numbers. Tabulated (known) values of the function, corresponding to the grid x.

thr

A real number. This is the threshold to decide when a column in the triangular matrix of divided differences has a small-enough average value (small than thr). Default is thr=1e-6.

ntrial

A positive integer to decide how many random selections of the provided known (tabulated) points have to be carried out. Default is ntrial=30.

Details

The divided differences depend on the specific points selected to calculate the interpolated curve. To avoid potential bias that might occur when the tabulated points used are not distributed uniformly, several random selections of tabulated points are performed (default ntrial=30) and the highest \(k\) is returned.

Examples

# Tabulated grid points for function cos(x)
x <- seq(0,3*pi/2,length=20)
f <- cos(x)

# Suggested polynomial degree (default ntrial)
k <- decidepoly_n(x,f)
print(k)

# Increase number of random selections (ntrial=50)
k <- decidepoly_n(x,f,ntrial=50)
print(k)