Calculating the relative error of an interpolation polynomial

[bremenfallturm]

Homework Statement

You have created two interpolation polynomials, $p_1$ of degree $1$ and $p_2$ of degree $2$. At the point $x=2$, you get the following values
$$
\begin{cases}
p_1(2)=16.52848966 \\
p_2(2)=14.1764705
\end{cases}
$$
Approximate the absolute and relative errors in y at $x=2$.

Relevant Equations

Interpolation absolute error can be approximated as
$|p_{n}(a)-p_{n+1}(a)|$ where $p_{n}$ represents a polynomial of degree n and $a$ the point where the absolute error is to be calculated.

Hi!

It's been a while since I've done this and I am unsure about the relative error. Should I use $p_2(2)$ as the "true value" for the relative error, that is, be in the denominator?

Or in other words, is this correct?

Absolute error : $|p_1(2)-p_2(2)|=|16.52848966−14.01764705|$
Relative error: $\frac{|16.52848966−14.01764705|}{|14.01764705|}$

 

[BvU]

Hi,

Homework Statement: You have created two interpolation polynomials, $p_1$ of degree $1$ and $p_2$ of degree $2$. At the point $x=2$, you get the following values
$$
\begin{cases}
p_1(2)=16.52848966 \\
p_2(2)=14.1764705
\end{cases}
$$
Approximate the absolute and relative errors in y at $x=2$.
Relevant Equations: Interpolation absolute error can be approximated as
$|p_{n}(a)-p_{n+1}(a)|$ where $p_{n}$ represents a polynomial of degree n and $a$ the point where the absolute error is to be calculated.

It's been a while since I've done this and I am unsure about the relative error. Should I use $p_2(2)$ as the "true value" for the relative error, that is, be in the denominator?

Or in other words, is this correct?

Absolute error : $|p_1(2)-p_2(2)|=|16.52848966−14.01764705|$
Relative error: $\frac{|16.52848966−14.01764705|}{|14.01764705|}$

The question confuses me, since it says 'errors'

Approximate the absolute and relative errors in y at $x=2$.

You only give an estimate of the error in $p_1$, which may well be what the exercise composer intended. But he/she could just as well be asking for expressions in terms of derivatives...

And: what are $p_1$ and $p_2$ based on ? The same set of data points ? Or does $p_2$ have twice as many ?

Finally: what if $p_2(a) = 0$ ?

$\$

 

[bremenfallturm]

Hi,



The question confuses me, since it says 'errors'

You only give an estimate of the error in $p_1$, which may well be what the exercise composer intended. But he/she could just as well be asking for expressions in terms of derivatives...

And: what are $p_1$ and $p_2$ based on ? The same set of data points ? Or does $p_2$ have twice as many ?

Finally: what if $p_2(a) = 0$ ?

$\$

Sorry, the question is not in English so I translated it. It is supposed to say "error". It is also given that $p_1$ and $p_2$ are based on the same data points. Though I assume this means that $p_2$ are based on 3 datapoints since it is a 2nd degree polynomial and $p_1$ are based on 2 of those datapoints.

If $p_2(a)=0$, then we do have a problem indeed as division by zero makes the quotient go to infinity. Actually I am not sure how that would be handled? I have been confused by taht in the past and never sorted it out.

 

[BvU]

It is also given that p1 and p2 are based on the same data points. Though I assume this means that p2 are based on 3 datapoints since it is a 2nd degree polynomial and p1 are based on 2 of those datapoints.

Possible: yes. Fair ? Hmmm... piecewise linear would be fairer

Suppose you have three points (I took $e^x\$, x = 1, 2, 3) then a first attempt would be something like

with deviations from $e^x$ as follows:

So red is the absolute error for the linear interpolation.
| red minus green | would be the error estimate for the absolute error.
And yellowish is that divided by $p_2$ (The estimate for the relative error. No worry about 1/0)

The purple line is the actual relative error (i.e. |(red - $e^x$)| / $e^x$)

Draw your own conclusions ...

(but clearly there is no reason for 10 digit reporting )

$\$

 

[bremenfallturm]

(but clearly there is no reason for 10 digit reporting )

Oh yeah, definitely not. Thanks for the pointers in the right direction? Now I'm just left with a curious bonus question (feel free to fill in if you have time ;))

If , then we do have a problem indeed as division by zero makes the quotient go to infinity. Actually I am not sure how that would be handled? I have been confused by taht in the past and never sorted it out.