I Newton Divided Difference Interpolation Polynomial

AI Thread Summary
The discussion focuses on understanding Newton's Divided Difference Interpolation Polynomial, particularly the intuition behind the multiplication of terms like (x-x(1))(x-x(0)) by coefficients in the polynomial. It explains that divided differences serve as finite approximations to derivatives, with each divided difference representing an approximation of higher-order derivatives. The first divided difference approximates the first derivative, while the second and third divided differences approximate the second and third derivatives, respectively. An important property highlighted is that the n-th divided difference of an n-th degree polynomial remains constant, regardless of the choice of points. This invariance of divided differences is crucial for understanding polynomial behavior in interpolation.
JaredPM
Messages
20
Reaction score
0
f(x)= a(0) + a1(x-x(0)) + a2(x-x(1))(x-x(0))

I am having a hard time understanding the intuition of (x-x(1))(x-x(0)) being multiplied by the coefficient a(2). For example, if I added a(3) to the equation, I would have had to multiply a(3) by (x-x(0))(x-x(1))(x-x(2)). I've researched the Mean Value Theorem and can't make the connection. I don't understand how the multiplications keep increasing. I understand that it needs 4 points to be able to compute the 3rd divided difference. I can't seem to grasp how it works. I am a visual learner, but can slowly reason through arithmetic. Thanks for you help!
 
Mathematics news on Phys.org
The best explanation, I've found on the web is https://mat.iitm.ac.in/home/sryedida/public_html/caimna/interpolation/nddf.html . However, the typography on that page is horribly mangled when I view it.

There are two concepts involve in this topic. The first concept is "divided difference". The second concept is how Newtons polynomial formula is a consequence of the properties of a "divided difference". (The formula for Newtons polynomial is not the formula that defines a "divided difference". )

Let's devote this post just to the topic of "divided differences".

You can think of "divided differences" as finite approximations to derivatives.

The "first divided difference of ##f## evaluated at ##x_0,x_1## is denoted by ##f[x_0,x_1]## It is defined as ##f[x_0,x_1] = \frac{ f(x_1) - f(x_0)} {x_1 - x_0} ##.

To make the analogy with a derivative clear, think of "##x_1##" as "##x_0 + h##" and "##x_1 - x_0##" as "##h##". Then ##f[x_0,x_1] = \frac{ f(x_0+h) - f(x_0)}{h} ##. If we took ##lim_{h\rightarrow 0} ## of that algebraic expression, we would have ##f'(x_0)##.

The second divided difference ##f[x_0, x_1, x_2]## can be regarded as an approximation for the second derivative ##f''(x_0)## computed as the approximate rate of change between two approximations for the first derivative.

##f[x_0, x_1, x_2] = \big{(} \frac{ f(x_2)- f(x_1) } {x2- x_1} - \frac{f(x_1) - f(x_0)}{x_1 - x_0} \big{)} / (x_2 - x_0) ##

It's more concise to denote that expression as ##f[x_0, x_1,x_2] = \frac{ f[x_2,x_1] - f[x_1,x_0] } {x_2 - x_0}##The third divided difference can regarded as an approximation for a third derivative. If we wrote out the algebra it would be the average rate of change between two approximations for second derivatives and the approximations for each of the second derivatives would involve two approximations for first derivatives. So it's more humane to define the third divided difference using the notation ##f[x_0,x_1,x_2,x_3] = \frac{ f[x_1,x_2,x_3] - f[x_0,x_1,x_2]}{x3 - x_0}##.

As you recall, the first derivative of a first degree polynomial is a constant, the second derivative of a second degree polynomial is a constant, and, in general, the n-th derivative of a n-degree polynomial is a constant.

It's an amazing fact that an analogous statement holds true for divided differences of polynomial functions, even though the divided differences do not involve taking any limits. The n-th divided difference of an n-th degree polynomial ##f(x)## is a constant. This is true even if we choose the points ##x_0,x_1,x_2,...,x_n## to be widely separated.

Another way of asserting the same claim is to say that the n-th divided difference of an n-th degree polynomial is an "invariant" of the polynomial. If we have two sets of points ##\{a_1,a_2,...a_n\}## and ##\{b_1,b_2,...b_n\}## then ##f[a_0,a_1,a_2,...,a_n] = f[b_0,b_1,b_2,...,b_n]## for an nth-degree polynomial ##f(x)## because both sides of the equation are equal to the same constant.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
Back
Top