Question

How to find coefficients of interpolating polynomial?

Answer 1

Answer:

Step-by-step explanation:

First of all, sorry for my bad math terminology as it's not my native language and I may misuse some terms in English. I've been tasked with writing ...

Answer 2

Answer:

3.6 Interpolation in Two or More Dimensions

In multidimensional interpolation, we seek an estimate of y(x1, x2,...,xn)

from an n-dimensional grid of tabulated values y and n one-dimensional vec-

tors giving the tabulated values of each of the independent variables x1, x2,...,

xn. We will not here consider the problem of interpolating on a mesh that is not

Cartesian, i.e., has tabulated function values at “random” points in n-dimensional

space rather than at the vertices of a rectangular array. For clarity, we will consider

explicitly only the case of two dimensions, the cases of three or more dimensions

being analogous in every way.

In two dimensions, we imagine that we are given a matrix of functional values

ya[1..m][1..n]. We are also given an array x1a[1..m], and an array x2a[1..n].

The relation of these input quantities to an underlying function y(x1, x2) is

ya[j][k] = y(x1a[j], x2a[k]) (3.6.1)

We want to estimate, by interpolation, the function y at some untabulated point

(x1, x2).

An important concept is that of the grid square in which the point (x1, x2)

falls, that is, the four tabulated points that surround the desired interior point. For

convenience, we will number these points from 1 to 4, counterclockwise starting

from the lower left (see Figure 3.6.1). More precisely, if

x1a[j] ≤ x1 ≤ x1a[j+1]

x2a[k] ≤ x2 ≤ x2a[k+1] (3.6.2)

defines j and k, then

y1 ≡ ya[j][k]

y2 ≡ ya[j+1][k]

y3 ≡ ya[j+1][k+1]

y4 ≡ ya[j][k+1]

(3.6.3)

The simplest interpolation in two dimensions is bilinear interpolation on the

grid square. Its formulas are:

t ≡ (x1 − x1a[j])/(x1a[j+1] − x1a[j])

u ≡ (x2 − x2a[k])/(x2a[k+1] − x2a[k]) (3.6.4)

(so that t and u each lie between 0 and 1), and

y(x1, x2) = (1 − t)(1 − u)y1 + t(1 − u)y2 + tuy3 + (1 − t)uy4 (3.6.5)

Bilinear interpolation is frequently “close enough for government work.” As

the interpolating point wanders from grid square to grid square, the interpolated