Help with interpreting an interpolation problem

[PhysicsKid0123]

Homework Statement

I don't know if this is the appropriate place to ask this, but I really do need some help. I am doing a homework problem and I don't understand what is being asked. It goes as follows:

> Write a MATLAB function to evaluate the trigonometric interpolant $p_n(x)$ for a given set of samples, $\bf{y}$:

Here it follows with some comments about making such a function, I'll leave that out since it's not relevant, but it goes on to say:

> To test your program use $f(x) = 10cos(x)+5cos(3x)$ and plot the maximum error $max |p_n(x)-f(x)|$ for $n= 4,8,16,32,64.$ Verify that the maximum error is close to machine precision for $n=32,64.$ What is the reason behind this?

So I was able to find the interpolating function $p_n(x)$, but I don't know what the next part is asking. Am I supposed to find the maximum value of $g_n(x) = |p_n(x)-f(x)|$ for each $n$, say it is $g_n(x_0)$, then plot $g_i(x_0)$ vs $i$ for $i= 1...n$? So effectively, each value $g_i(x_0)$ is plotted on an $y$- axis and $i$ on a corresponding $x$-axis? Or am I simply supposed to make a plot of $g_n(x) = |p_n(x)-f(x)|$ for each n?

Also, how do I know when I am close to machine precision? When I start getting cancellation errors?

Thank in advance for the help.

By the way, I am using Julia to do this, not MATLAB.

Homework Equations

None are relevant to my question.

The Attempt at a Solution

No solution yet, I wouldn't be asking this question.

 

[PhysicsKid0123]

Sorry, I don't know how to enable LaTex.

 

[Ray Vickson]

Sorry, I don't know how to enable LaTex.

Instead of $p_n(x)$, replace each starting and ending $ by #nospace# (two # signs with no space between them) for in-line formulas; that gives you $p_n(x)$. If you want a displayed equation or formula, replace each starting and ending $ by $nospace$, that is, write two $ signs with no space between them. That gives you $$p_n(x)$$

 

[Mark44]

Fixed the LaTeX in post #1.

 

[PhysicsKid0123]

Fixed the LaTeX in post #1.

Yes, I got it. Thank you.

 

[fresh_42]

Fixed the volume in post #1.