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October 14th, 2012, 08:08 AM  #1 
Senior Member Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11  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 
October 14th, 2012, 12:01 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,206 Thanks: 513 Math Focus: Calculus/ODEs  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: 
October 15th, 2012, 11:27 AM  #3 
Senior Member Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11  Re: explicit formula for chebychev's method?
what is a homogeneous recurrence and a associated auxiliary equation?

October 15th, 2012, 11:36 AM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,206 Thanks: 513 Math Focus: Calculus/ODEs  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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