Chebyshev's Method: Recursive Algorithm & Polynomial Form

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In summary: Using trigonometric identities (or otherwise), derive the recursive algorithm.This is not a recurrence relation.b) Find a closed polynomial form for .This is not a recurrence relation.c) Compute \int_{-1}^1 T_n\,dxThis is not a recurrence relation.d) Compute \int_{-1}^1 \frac{T_nT_m}{\sqrt{1-x^2}}\,dxThis is not a recurrence relation.
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MarkFL
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Chebyshev's method is a recursive algorithm for computing the th multiple angle formula for the cosine function. If we define:



then the algorithm is given as:



where:



a) Using trigonometric identities (or otherwise), derive the recursive algorithm.

b) Find a closed polynomial form for .

c) Compute

d) Compute
 
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  • #2
a)



So Or
b)



Let



If is odd,

If is even
 
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  • #3
d)

Let

If , If ,
 
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  • #4
I skipped (b) because I initially thought we were dealing with a recurrence relation with a variable coefficient. (Doh) has the associated characteristic polynomial

which has roots .So the general solution is .

So just by inspection,

And which is evidently a polynomial in for all values of
 
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  • #5
Bravo, Random Variable! (Clapping)

Thank you for taking the time to post such lucid explanations. (Yes)

Here are my solutions:

a) Using trigonometric identities (or otherwise), derive the recursive algorithm.

We may begin with the identity:



Add to the right side:



To the first two terms on the right, apply the following sum to product identity:



and we have:



And so we may write:



b) Find a closed polynomial form for .

This is a homogeneous recurrence whose associated auxiliary equation is:



Application of the quadratic formula yields:



Thus, the solution is of the form:



Use initial conditions to determine constants:





Substituting from the first into the second:









Thus, the closed form for is:



Using the binomial theorem, we may state:









c) Compute

From the recurrence relation, we may compute:





From the definition, we see that and , thus:

For even :



For odd :



d) Compute





For :



For :



Let





For :





 

FAQ: Chebyshev's Method: Recursive Algorithm & Polynomial Form

What is Chebyshev's method and how does it work?

Chebyshev's method is a recursive algorithm used for finding the roots of a polynomial. It works by iteratively improving an initial approximation of the root until a desired level of accuracy is reached.

What are the advantages of using Chebyshev's method over other root-finding methods?

Chebyshev's method is known for its ability to quickly converge to the roots of a polynomial, even for polynomials with complex or multiple roots. It also has a high level of accuracy and stability compared to other methods.

Can Chebyshev's method be used for polynomials of any degree?

Yes, Chebyshev's method can be used to find the roots of polynomials of any degree. However, the algorithm may become more computationally intensive for higher degree polynomials.

Are there any limitations or drawbacks to using Chebyshev's method?

One limitation of Chebyshev's method is that it may not converge for polynomials with roots that are very close together. In addition, the initial approximation of the root must be chosen carefully to avoid convergence issues.

How is Chebyshev's method applied in real-world problems?

Chebyshev's method has various applications in fields such as engineering, physics, and economics. It can be used to solve equations that arise in these fields, such as finding the maximum or minimum of a function. It is also used in numerical analysis for interpolation and approximation of functions.

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