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October 14th, 2012, 08:08 AM   #1
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explicit formula for chebychev's method?

chebychev's method is a recursion formula:



where:





Thus:











But now i would like to know if there exists a explicit way to get for example without calculating to first
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October 14th, 2012, 12:01 PM   #2
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Re: explicit formula for chebychev's method?

Begin with the recurrence:



This is a homogeneous recurrence whose associated auxiliary equation is:



Application of the quadratic formula to find the characteristic roots:



Thus, the closed form is of the form:



Use initial conditions to determine the parameters :





Substituting from the first into the second:









thus, the closed form for is:



Using the binomial theorem, we may state:







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October 15th, 2012, 11:27 AM   #3
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Re: explicit formula for chebychev's method?

what is a homogeneous recurrence and a associated auxiliary equation?
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October 15th, 2012, 11:36 AM   #4
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Re: explicit formula for chebychev's method?

Read this article for an explanation of the technique I used to get the closed form, before applying the binomial theorem.
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