Solve Newton Interpolating Polynomial for Error

[Lilia]

Homework Statement

y=1/(x^1/4). I'm given 5 x's and 5y's. I need to write Newton interpolating polynomial and find the error.

Homework Equations

Ln-1(x)=f(x1)+f(x1,x2)(x-x1)+...

The Attempt at a Solution

With the formula above I wrote the Newton interpolating polynomial but I can't find the error since I don't have the formula. Can anyone help?

 

[Ray Vickson]

Homework Statement

y=1/(x^1/4). I'm given 5 x's and 5y's. I need to write Newton interpolating polynomial and find the error.

Homework Equations

Ln-1(x)=f(x1)+f(x1,x2)(x-x1)+...

The Attempt at a Solution

With the formula above I wrote the Newton interpolating polynomial but I can't find the error since I don't have the formula. Can anyone help?

First show your work.

When you say you "don't have the formula..", what do you mean? Are you unsure how to find the Newton polynomial, or unsure how to find the error? Have you consulted your textbook and/or course notes? Have you looked on-line for topics related to "Newton interpolation"?

 

[Mark44]

Homework Statement

y=1/(x^1/4). I'm given 5 x's and 5y's. I need to write Newton interpolating polynomial and find the error.

Homework Equations

Ln-1(x)=f(x1)+f(x1,x2)(x-x1)+...

What does this mean?
Is "Ln-1(x)" supposed to be "ln^-1(x)" (i.e., e^x)? How is what you wrote relevant to your problem?
Also, you wrote f(x1) and f(x1, x2). Is f a function of one variable or is it a function of two variables? It can't be both.

The Attempt at a Solution

With the formula above I wrote the Newton interpolating polynomial but I can't find the error since I don't have the formula. Can anyone help?

The formula for what? It looks like you're supposed to approximate $f(x) = \frac 1 {x^{1/4}}$ using a term or two from the Taylor series.

 

[Lilia]

x | 1 | 2 | 5 | 16 | 39
------------------------------
y | 1 | 0.8 | 0.6 | 0.5 | 0.4

I found the Newton interpolating polynomial:
P₄(x)=1+(-0.2)*(x-1) + (0.04)*(x-1)(x-2) + (-0.003)*(x-1)(x-2)(x-5) + A(x-1)(x-2)(x-5)(x-16)

I wrote this looking at an example we wrote in class. But we haven't written the error in any example nor I have the formula. But in this exercise I need to find the error.

f(x1,x2) etc. these are the divided differences which are used to make up the Newton Interpolating polynomial, -0.2, 0.04 etc in this case, which I calculated too

Should I calculate the error with this formula? R(x) = f(x)-L_n-1(x)≤ M_n/n! * (x-x1)(x-x2)*...*(x-xn)?

 

[lurflurf]

The error term is
f(x,x1,x2,x3,x4,,x5)(x-x1)(x-x2)(x-x3)(x-x4)(x-x5)
often we use a mean value argument to replace f(x,x1,x2,x3,x4,,x5) with
$$\frac{f^{(5)}(\xi)}{5!}$$
where the derivative is evaluated at an unknown point between the largest and smallest of
x,x1,x2,x3,x4,x5

 

[Lilia]

The error term is
f(x,x1,x2,x3,x4,,x5)(x-x1)(x-x2)(x-x3)(x-x4)(x-x5)
often we use a mean value argument to replace f(x,x1,x2,x3,x4,,x5) with
$$\frac{f^{(5)}(\xi)}{5!}$$
where the derivative is evaluated at an unknown point between the largest and smallest of
x,x1,x2,x3,x4,x5

Isn't that the function deviation in the range x=x1÷x5? I need to find that?

 

[lurflurf]

The deviation and error are the same right? This is an interesting question. If I calculated correctly M is 9945/1024~10 but that is quite pessimistic as 0.0002 models the error well. Overall that is a quite bad approximation especially around 30.