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 Iteration basics andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 04/05/2008, 11:35 PM Well, these are really good questions. If you are familiar with Taylor series, then you will know that you can represent any analytic function as a power series: $ f(x) = \sum_{k=0}^{\infty} \frac{x^k D_x^k f (0)}{k!} = f(0) + x f'(0) + \frac{x^2}{2} f''(0) + \frac{x^3}{6} f'''(0) + \cdots$ This is the first iterate of the function. When a function is iterated, its output becomes its next input, so $f^3(x) = f(f(f(x)))$ and in general we write $f^n(x)$ when n is an integer, or $f^t(x)$ when t is non-integer. Finding the derivative of $f^t(x)$ is one of the goals of natural iteration. Continuous iteration can be classified into two major methods, because there are two primary ways that you can turn $f^t(x)$ into a 1-variable power series, because there are two variables: x and t. One could also construct a 2-variable power series, but thats complicated, so I wont do that now. Here is the power series that corresponds to regular iteration: $ f^t(x) \ =\ \sum_{k=0}^{\infty} x^k G_k(t) \ =\ f^t(0) \ +\ x \left[D_x f^t (x)\right]_{x=0} \ +\ \frac{x^2}{2} \left[D_x^2 f^t (x)\right]_{x=0} \ +\ \cdots$ And here is the power series that corresponds to natural iteration: $ f^t(x) \ =\ \sum_{k=0}^{\infty} t^k H_k(x) \ =\ f^0(x) \ +\ t \left[D_t f^t (x)\right]_{t=0} \ +\ \frac{t^2}{2} \left[D_t^2 f^t (x)\right]_{t=0} \ +\ \cdots$ Whats weird about these two methods, is that what they are doing is not really finding derivatives, but finding the coefficients in the power series, but because the coefficients in the power series are related to the derivatives, you can find the derivatives with these methods. For example, if you wanted to find the second derivative of $f^t(x)$ with respect to t, then you could apply natural iteration to find $H_2(x)$, then solve for the derivative to obtain $\left[D_t^2 f^t (x)\right]_{t=0} = 2 H_2(x)$. So if you are interested in derivatives with respect to x, then you should search this forum for "regular" and if you are interested in derivatives with respect to t, then you should search this forum for "natural" and see what you find in your search. Andrew Robbins « Next Oldest | Next Newest »

 Messages In This Thread Iteration basics - by Ivars - 03/20/2008, 10:34 AM RE: Iteration basics - by Ivars - 03/20/2008, 05:24 PM RE: Iteration basics - by Gottfried - 03/21/2008, 08:37 AM RE: Iteration basics - by Ivars - 04/03/2008, 08:05 AM RE: Iteration basics - by Gottfried - 04/03/2008, 08:19 AM RE: Iteration basics - by Ivars - 04/03/2008, 10:13 AM RE: Iteration basics - by andydude - 04/05/2008, 11:35 PM RE: Iteration basics - by Gottfried - 04/06/2008, 07:16 AM RE: Iteration basics - by Ivars - 04/06/2008, 08:55 AM RE: Iteration basics - by andydude - 04/07/2008, 12:01 AM RE: Iteration basics - by Ivars - 04/20/2008, 10:15 PM RE: Iteration basics - by bo198214 - 04/21/2008, 08:21 PM RE: Iteration basics - by andydude - 04/22/2008, 05:30 AM RE: Iteration basics - by Ivars - 04/22/2008, 07:02 AM RE: Iteration basics - by Ivars - 04/09/2008, 07:56 PM RE: Iteration basics - by Ivars - 05/09/2008, 09:45 AM RE: Iteration basics - by bo198214 - 05/09/2008, 02:49 PM RE: Iteration basics - by Ivars - 05/27/2008, 10:33 AM RE: Iteration basics - by Gottfried - 05/27/2008, 07:35 PM RE: Iteration basics - by Ivars - 05/30/2008, 06:08 AM RE: Iteration basics - by Xorter - 01/02/2017, 05:21 PM RE: Imaginary iterates of exponentiation - by Gottfried - 03/20/2008, 12:04 PM RE: Imaginary iterates of exponentiation - by Ivars - 03/20/2008, 12:35 PM RE: Imaginary iterates of exponentiation - by bo198214 - 03/20/2008, 03:10 PM RE: Imaginary iterates of exponentiation - by Ivars - 05/24/2008, 08:35 AM RE: Imaginary iterates of exponentiation - by bo198214 - 05/24/2008, 09:04 AM RE: Imaginary iterates of exponentiation - by Gottfried - 05/24/2008, 09:47 AM RE: Imaginary iterates of exponentiation - by Ivars - 05/24/2008, 07:12 PM

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