Question

write down the conditions for applying newton's advancing difference method of interpolation.​

Answer 1

Answer:

There are no sudden change in the values of dependent variable from one period to another. There is a sort of uniformity in the rise or fall of the values of the dependent variable. There will be no Consecutive missing values in the series.

Answer 2

Answer: The conditions for applying Newton's advancing difference method of interpolation are to find a tabulated value near the beginning of the table, use Newton’s forward formula, and find a value near the end of the table, use Newton’s backward formula.

Step-by-step explanation:
Interpolation is a method of estimating a function's value for any intermediate value of the independent variable, whereas extrapolation is the process of calculating the function's value outside of the specified range.

Conditions for using Newton's advancing difference method of interpolation.

1. Use Newton's forward formula to locate a tabulated value close to the beginning of the table.

A finite difference identity, Newton's forward difference formula provides an interpolated value between tabulated points ${f_p}$ in terms of the first value $f_0$ and the powers of the forward difference $\Delta$.
For a ∈[0,1], the formula states

$f_a=f_0+a\Delta+\frac{1}{2!} a(a-1)\Delta^2+\frac{1}{3!} a(a-1)(a-2)\Delta^3+....$

When written in the form

$f(x+a)=\sum_{(n=0)} ^{\infty} \frac{(a)_n\Delta^nf(x)}{n!}$

with $(a)_n$  the falling factorial, the formula resembles a finite version of a Taylor series expansion suspiciously. One of the driving motivations behind the development of umbral calculus was this connection.

An alternate form of this equation using binomial coefficients is

$f(x+a)=\sum_{(n=0)}^{\infty} {(a;n)} \Delta^nf(x)$

where the binomial coefficient (a; n) represents a polynomial of degree n in a.
This formula is used for interpolating the values of y near the beginning of a set of tabulated values and extrapolating values of y a little backward (i.e., to the left) of y₀.

The first two terms of this formula give linear interpolation while the first three terms give parabolic interpolation and so on.

2. Use Newton's backward formula to locate a tabulated value near the end of the table
$f(a+nh+uh)=f(a+nh)+u\nabla f(a+nh)+\frac{u\left ( u+1 \right )}{2!}\nabla ^{2}f(a+nh)+...+\frac{u\left ( u+1 \right )...\left ( u+\overline{n-1} \right )}{n!}\nabla ^{n}f(a+nh)$

This formula is used for interpolating the values of y near the end of a set of tabulated values and also for extrapolating values of y a little ahead (to the right) of y

This formula is useful when the value of f(x) is required near the end of the table. h is called the interval of difference and u = $x-(a)_n$/ h, Here  $(a)_n$  is the last term.




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