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 Applying the iteration methods to simpler functions andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 11/12/2007, 09:40 AM You are confusing Bell matrices and Carleman matrices again. The (generalized) Bell matrix of $f(x) = cx+d$ is $\left[\begin{tabular}{ccc} 1 & d & d^2 \\ 0 & c & 2cd \\ 0 & 0 & c^2 \end{tabular}\right]$ whereas the Carleman matrix (is $\left[\begin{tabular}{ccc} 1 & 0 & 0 \\ d & c & 0 \\ d^2 & 2cd & c^2 \end{tabular}\right]$ all the other stuff is right, though, and interesting. I like your notation, but I'm having trouble following your series notation. I think the example of $f(x) = cx+d$ is an interesting one, and I intend on looking into it, but first for the purposes of illustration, your simplest example is the nicest, for example, addition $f(x) = x+d$. Taking the Bell matrix: $\mathbf{B}_x[x + d] = \left[\begin{tabular}{ccc} 1 & d & d^2 & d^3 \\ 0 & 1 & 2d & 3d^2 \\ 0 & 0 & 1 & 3d \\ 0 & 0 & 0 & 1 \end{tabular}\right]$ and subtracting the identity matrix: $\mathbf{B}_x[x + d] - \mathbf{I}= \left[\begin{tabular}{ccc} 0 & d & d^2 & d^3 \\ 0 & 0 & 2d & 3d^2 \\ 0 & 0 & 0 & 3d \\ 0 & 0 & 0 & 0 \end{tabular}\right]$ gives a non-invertible matrix, so doing the choppy thing gives: $\mathbf{C}(\mathbf{B}_x[x + d] - \mathbf{I})\mathbf{D}= \left[\begin{tabular}{ccc} 1 & 0& 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{tabular}\right]\left[\begin{tabular}{ccc} 0 & d & d^2 & d^3 \\ 0 & 0 & 2d & 3d^2 \\ 0 & 0 & 0 & 3d \\ 0 & 0 & 0 & 0 \end{tabular}\right]\left[\begin{tabular}{ccc} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{tabular}\right] = \left[\begin{tabular}{ccc} d & d^2 & d^3 \\ 0 & 2d & 3d^2 \\ 0 & 0 & 3d \end{tabular}\right]$ putting this in the matrix equation gives: $\left[\begin{tabular}{ccc} d & d^2 & d^3 \\ 0 & 2d & 3d^2 \\ 0 & 0 & 3d \end{tabular}\right]\left[\begin{tabular}{c} g_1 \\ g_2 \\ g_3 \end{tabular}\right] == \left[\begin{tabular}{c} 1 \\ 0 \\ 0 \end{tabular}\right]$ and since the matrix is invertible now, we can multiply both sides by its inverse: $\left[\begin{tabular}{c} g_1 \\ g_2 \\ g_3 \end{tabular}\right] == \left[\begin{tabular}{ccc} \frac{1}{d} & -\frac{1}{2} & \frac{d}{6} \\ 0 & \frac{1}{2d} & -\frac{1}{2} \\ 0 & 0 & \frac{1}{3d} \end{tabular}\right]\left[\begin{tabular}{c} 1 \\ 0 \\ 0 \end{tabular}\right] = \left[\begin{tabular}{c} 1/d \\ 0 \\ 0 \end{tabular}\right]$ This gives the natural Abel function $A_f(x) = f^{*}(x) = x/d$ (choosing the constant term to be zero) which satisfies the equation $\frac{x+d}{d} = \frac{x}{d} + 1$. I like that example I know you already talked about it, but I wanted to show a matrix-based way of doing the chopping process. So in general, the matrix equation $(\mathbf{C}(\mathbf{B}[f] - \mathbf{I})\mathbf{D}) \mathbf{a} = \mathbf{b}$ would give the coefficients $\mathbf{a}$ of the natural Abel function of f. Andrew Robbins « Next Oldest | Next Newest »

 Messages In This Thread Applying the iteration methods to simpler functions - by bo198214 - 11/05/2007, 11:39 PM RE: Applying the iteration methods to simpler functions - by Gottfried - 11/06/2007, 12:30 AM RE: Applying the iteration methods to simpler functions - by andydude - 11/06/2007, 03:32 AM RE: Applying the iteration methods to simpler functions - by bo198214 - 11/06/2007, 10:36 AM RE: Applying the iteration methods to simpler functions - by jaydfox - 11/09/2007, 05:34 AM RE: Applying the iteration methods to simpler functions - by bo198214 - 11/09/2007, 08:36 AM RE: Applying the iteration methods to simpler functions - by andydude - 11/12/2007, 09:40 AM RE: Applying the iteration methods to simpler functions - by Gottfried - 11/12/2007, 10:40 AM RE: Applying the iteration methods to simpler functions - by andydude - 11/13/2007, 12:39 AM RE: Applying the iteration methods to simpler functions - by bo198214 - 11/12/2007, 11:23 AM RE: Applying the iteration methods to simpler functions - by jaydfox - 11/12/2007, 04:41 PM

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