Exercise 5.1.11

I just cannot seem to construct the polynomial given at the back of the book. Can anyone figure it out?

My calculation is:

P₂(x) = f(x₂) + (x - x₂)f[x₂,x₃] + (x - x₂)(x - x₃)f[x₂,x₃,x₄]
P₂(x) = 0.3617 + (x - 0.23)(1.7409) + (x - 0.23)(x - 0.27)(-2.9464)

Simplify from there but you can already see its not the same polynomial as whats given.

I think your solution will be to read the middle part of page 261 more slowly.

Then remember you want to calculate a derivative, and not just f.

Then try setting your first term as f[x₃,x₂] (hint)

My apologies for making you kick yourself like this. Bet you enjoy it though! (Only weird people do COS233)

Yo. I've hit a wall it seems. You mentioned I should use f[x₃, x₂]. Did you mean f[x₂, x₃]?

Anyways, at the back of the book they give f(x), and then calculate f'(x). Using their way and equation 5.5 results in failure as I cannot seem to get the right values.

After a bit of work on it I have to tell you my face is every bit as red as I thought yours was. Probably redder. Nice colour, red. Am quite used to it in my face. We're meant to get P(x) before fiddling with P'(x).

I see that if one starts by just solving for P(x) using the divided difference approximation one ends up with a result completely different to that in the answers.

I see also that the coefficient of x² can be found in the answer to question 13, in exactly the right part of that table. No need to recalculate. Just look at the coefficient you have (which should appear "raw" in the table given for the question) and then look at the q13 table in exactly the same position.

This leads me to suspect that the answer to question 11 has been constructed from the table that's the answer to q13.

From what I've read about our text book online it would appear that it's quite riddled with errors, so perhaps you've just chanced on one. In other words you're right. That's your problem. You're right and the text book is wrong.

So there's a chance that the text book is the one that has hit a brick wall, and not you.

You ask good questions.

OK it very much looks like the problem here is simply that the answer was based on the more correct table you're asked to produce in q13. Maybe that's just assuming that we're all very alert numerical folk and immediately would've noticed that the table given for the problem was purportedly accurate to more digits than the f values had in them, and so would've corrected the table before plugging things into formulas.

So then you were actually perfectly right all along. You get a different polynomial if you solve for P₂(x) using the incorrect table supplied, but the method for starting it off that you gave was right.

I had a merry old time removing rust from what I remember about differentiation when checking out their P'(x) expression. The note * at the bottom of page 261 helps. As you get higher order, so it becomes a full on "Product Rule" differentiation, which is a Sum. I think I may have treated it as a product in the last exams. For some reason it took ages for me to log onto the fact that f[..tis..const] is a constant, too. Ja, anyway...

«edit» You mention [x₂,x₃] vs [x₃,x₂] up there. I think the order is unimportant? I'm not sure. Best I check that, eh? I've been using the order of x's very loosely in my rough work until now.

»Another edit« (with the German quotation marks the right way round now, I think)
The order "doesn't matter" says the very middle of page 168. However, in fact you are specifying an order of subtraction there, and an ascending ordering is just neater, so I think I'll follow you in using the f[x_o,x₁,x₂] style.

OK I've finally gotten round to confirming that one. According to my calculations I can almost say for sure that you were dead right. Use the answer to q13 and you should get pretty close to the P(x) they give as an answer to q11. (I would like to say you Do get the exact answer, but my constant term is out by the last two digits. I've made a calculation error -- which I am never ever going to correct ... ONward!!)

I'd better go do P'(x) just to thump through the skull bone the fact that there's a sum of (x-xi) there, not a product.

Hey Edy. Thanks for all the feedback. I figured the table I was using was either wrong or I had the completely wrong end of the stick (heart sank). Anyways let me run through it all again in the AM. I've also asked Dr Rapoo to take a look at this for us.

Dr Rapoo advises that my polynomial first posted is correct. You then calculate the derivative. The books seems to have a typo or something evidently.

Yes. If you plug in values from the other table you'll get the answer you got first.

They've asked you to use TableX, have forgotten that, and given an answer in terms of TableY. It's more than just a typo.