Lagrange's interpolation polynomial

gourish · 2014-04-19 18:14:10

"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

bobbym · 2014-04-19 20:32:11

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?

gourish · 2014-04-19 21:31:39

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

bobbym · 2014-04-19 22:55:12

The formula is best demonstrated with a live example.

Want to see one?

gourish · 2014-04-20 00:21:54

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

bobbym · 2014-04-20 00:38:29

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.

gourish · 2014-04-20 00:40:52

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"?

gourish · 2014-04-20 00:44:15

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

bobbym · 2014-04-20 00:45:40

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.

gourish · 2014-04-20 00:46:47

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

gourish · 2014-04-20 00:48:22

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

bobbym · 2014-04-20 00:52:44

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.

gourish · 2014-04-20 01:01:23

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

bobbym · 2014-04-20 01:01:55

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

gourish · 2014-04-20 01:05:45

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

bobbym · 2014-04-20 01:42:06

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?

gourish · 2014-04-20 02:53:26

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

bobbym · 2014-04-20 03:01:27

I would think that the whole process could be automated.

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

gourish · 2014-04-20 03:04:36

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

bobbym · 2014-04-20 03:07:41

You understand how it was derived?

gourish · 2014-04-20 03:12:57

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

gourish · 2014-04-20 03:15:10

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

bobbym · 2014-04-20 03:17:08

Hi;

I meant the simultaneous set in post 18.

gourish · 2014-04-20 03:18:41

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

bobbym · 2014-04-20 03:25:47

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.

Math Is Fun Forum

#1 2014-04-19 18:14:10

Lagrange's interpolation polynomial

#2 2014-04-19 20:32:11

Re: Lagrange's interpolation polynomial

#3 2014-04-19 21:31:39

Re: Lagrange's interpolation polynomial

#4 2014-04-19 22:55:12

Re: Lagrange's interpolation polynomial

#5 2014-04-20 00:21:54

Re: Lagrange's interpolation polynomial

#6 2014-04-20 00:38:29

Re: Lagrange's interpolation polynomial

#7 2014-04-20 00:40:52

Re: Lagrange's interpolation polynomial

#8 2014-04-20 00:44:15

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#9 2014-04-20 00:45:40

Re: Lagrange's interpolation polynomial

#10 2014-04-20 00:46:47

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#11 2014-04-20 00:48:22

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#12 2014-04-20 00:52:44

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#13 2014-04-20 01:01:23

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#14 2014-04-20 01:01:55

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#15 2014-04-20 01:05:45

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#16 2014-04-20 01:42:06

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#17 2014-04-20 02:53:26

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#18 2014-04-20 03:01:27

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#19 2014-04-20 03:04:36

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#20 2014-04-20 03:07:41

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#21 2014-04-20 03:12:57

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#22 2014-04-20 03:15:10

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#23 2014-04-20 03:17:08

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#24 2014-04-20 03:18:41

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#25 2014-04-20 03:25:47

Re: Lagrange's interpolation polynomial

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