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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

"A polynomial p(x) of degree two or less which takes values y0 y1 and y2 at three distinct values x0 x1 x2 respectively is given by {the la grange's interpolation polynomial}" use this data to answer

1.) A polynomial of degree 2 which takes values y0 , y0 , y1 at points x0 x0+t x1 t is not equal to 0 is given by?

2.) A polynomial of degree 2 and "a" not equal to 0 or 1 then p(x) is....

3.) in last question if a tends to 0 then p(x) is given by

i don't know what this " la grange's interpolation polynomial is" can somebody tell me about it and explain a way to answer these question. i did not have this in my class so i find it hard to understand

"The man was just too bored so he invented maths for fun"

-some wise guy

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,570

Hi gourish;

It is not a whole lot more than plugging in:

just remembering that j ≠ k in 2).

Are you restricted to only the Lagrange interpolation formula?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

yeah only that... plus i don't know what the Lagrange's interpolation formula is... so can you help me with that

"The man was just too bored so he invented maths for fun"

-some wise guy

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,570

The formula is best demonstrated with a live example.

Want to see one?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

yeah for sure.... quite excited to see it

"The man was just too bored so he invented maths for fun"

-some wise guy

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,570

Actually, it is a primitive idea and I am wondering why they still teach it like they do but here is the example.

You are familiar with a set of xy coordinates that are represented as ordered pairs.

{(0,1),(1,5),(2,18),{3,85)}

We have 4 points so that means we can fit a 3rd order poly through them and get an exact fit. This is much easier to see if you use other methods but for right now follow my signature.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

ok i get it.. but does it mean that it's the other way of saying "look there is a relation between two sets and i can get so and so number of functions out of the relations set"?

"The man was just too bored so he invented maths for fun"

-some wise guy

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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

but can you just show me how to use it with an example

"The man was just too bored so he invented maths for fun"

-some wise guy

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,570

Actually what it is saying is that you will need n simultaneous equations to fit n points and the result will be an (n-1)th order polynomial but lets not get caught up with jargon, lets see how we can determine the answer using Lagrange.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

yeah that's what i want to learn....

"The man was just too bored so he invented maths for fun"

-some wise guy

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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

Compute f(0.3) for the data

x

0

1

3

4

7

f

1

3

49

129

813

using Lagrange's interpolation formula (Analytic value is 1.831)

this is what i found on the net. can you show me how to find answer to this question using Lagrange's formula

reference for the actual question :http://mat.iitm.ac.in/home/sryedida/public_html/caimna/interpolation/lagrange.html

"The man was just too bored so he invented maths for fun"

-some wise guy

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,570

That problem is bigger than the one in post #6 so it is more work but it it is not more informative. It will also require more time to post, I therefore recommend the smaller problem first.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

ok yeah let's go with the smaller one then....

"The man was just too bored so he invented maths for fun"

-some wise guy

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,570

I will need some time to post the latex...

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

okay i can wait.. you take your time and post....

"The man was just too bored so he invented maths for fun"

-some wise guy

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,570

We start with:

Where f(2) = 18 and x1 = 1.

Using the formulas;

we get:

Adding all the l's up using 2) in post #2:

So the Lagrangian polynomial for that data is

Now do you see why I would never use this method?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

i would rather hit my head on the desk then use this formula... it's better to guess on the answer from the choices....

"The man was just too bored so he invented maths for fun"

-some wise guy

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,570

I would think that the whole process could be automated.

But much easier is just solving a linear set of equations.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

yeah it kicks out the mystery of things in finding the solution thanks

"The man was just too bored so he invented maths for fun"

-some wise guy

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,570

You understand how it was derived?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

no i am sorry the signs used are quite complicated for me. i never really came across these before...

"The man was just too bored so he invented maths for fun"

-some wise guy

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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

i will go through them again and try to understand how it was derived....

"The man was just too bored so he invented maths for fun"

-some wise guy

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,570

Hi;

I meant the simultaneous set in post 18.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**gourish****Member**- Registered: 2013-05-28
- Posts: 113

i did not understand the l 2 and l 3 part and sorry i still couldn't understand how it was derived

"The man was just too bored so he invented maths for fun"

-some wise guy

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,570

If you want to fit a cubic to 4 data points perfectly you would start with the general form of a cubic.

and now just substitute the data points for x and y.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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