How Are Lagrange Polynomials Computed and Proven?

In summary, Lagrange Polynomials are a type of interpolation polynomial used to approximate a function by passing through a set of points. They are calculated using a formula that involves the given set of points and the function values at those points. The purpose of using Lagrange Polynomials is to approximate a function that may not be easy to evaluate directly, and the degree of a Lagrange Polynomial is equal to the number of points used in the interpolation process minus one. They can be used for any type of continuous function, but the accuracy of the approximation may vary depending on the complexity of the function.
  • #1
e.gedge
7
0
Lagrange Polynomals are defined by:

lj(t)= (t-a0) ...(t-aj-1)(t-aj+1)...(t-an) / (aj-a0)...(aj-aj-1)(aj-aj+1)...(aj-an)

A) compute the lagrange polynomials associated with a0=1, a1=2, a2=3. Evaluate lj(ai).

B) prove that (l0, l1, ... ln) form a basis for R[t] less than or equal to n.

C) Deduce the Lagrange interpolation formula.

Thanks!
 
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  • #2
I suggest that rather than asking for help on my homework assignments in a forum where it specifically states "this is not for homework" that you instead come to the tutorials.
 

Related to How Are Lagrange Polynomials Computed and Proven?

1. What are Lagrange Polynomials?

Lagrange Polynomials are a type of interpolation polynomial used to approximate a function by passing through a set of points. They are named after mathematician Joseph-Louis Lagrange.

2. How are Lagrange Polynomials calculated?

Lagrange Polynomials are calculated using a formula that involves the given set of points and the function values at those points. The formula involves multiplying each point's function value by a weight factor, and then summing all of these terms together.

3. What is the purpose of using Lagrange Polynomials?

The purpose of using Lagrange Polynomials is to approximate a function that may not be easy to evaluate directly. It allows us to create a simpler polynomial function that closely matches the original function at a set of given points.

4. What is the degree of a Lagrange Polynomial?

The degree of a Lagrange Polynomial is equal to the number of points used in the interpolation process minus one. For example, if five points are used, the degree of the polynomial will be four.

5. Can Lagrange Polynomials be used for any type of function?

Yes, Lagrange Polynomials can be used to approximate any continuous function, as long as the given points are chosen appropriately. However, the accuracy of the approximation may vary depending on the complexity of the function.

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