05/05/2008, 05:26 AM
What is the flow matrix?
The flow matrix (although it could also be called iterational matrix ... see here) is the matrix of coefficients \( A_{jk} \) where \( f^{\circ t}(x) = \sum_{j=0}^{\infty}\sum_{k=0}^{\infty} A_{jk} t^j x^k \) which are obtained from any method (usually a special case of regular iteration), for parabolic iteration there will only be a finite number of t's, but for hyperbolic iteration (yes, the flow matrix would apply to that as well) this matrix is not triangular (as it is with parabolic iteration). For parabolic iteration the "flow series" is:
Is it the Carleman matrix of parabolic iteration?
No, it is the first row of the Carleman matrix^t, or the first column of the Bell matrix^t, meaning both have been raised to the t. The flow matrix is simply a different expression for the flow series.
Why is it important to know the diagonals?
Well, it seems important to know the asymptotic behavior of the coefficients, and it is very difficult to know the asymptotic behavior of a sequence you only have the first few members of, so my goal with these diagonals is to provide a formula that we can use for root-tests and other convergence tests, so the world will stop calling them "formal power series" and start calling them functions. That is my ultimate goal.
But then again, how am I helping when I only have 3 diagonals, and we don't need to find asymptotic behavior of \( t^k x^k \) but of \( t^k \)? Well, it is obvious that the diagonals follow more of a pattern than the columns, so it seems easier to interpolate the diagonals than the columns of the flow matrix. My hope is that after enough diagonals are found, then a pattern will be found in the columns, as well.
Are these the same as Jabotinsky's double-binomial formula?
Yes. However, that formula is expressed in terms of the coefficients of the n-th iterate of a function. The only way I've seen Daniel Geisler write the flow series has been in terms of the coefficients of the 1-st iterate only, which is a major distinction between Jabotinsky's, and the flow series.
Andrew Robbins
The flow matrix (although it could also be called iterational matrix ... see here) is the matrix of coefficients \( A_{jk} \) where \( f^{\circ t}(x) = \sum_{j=0}^{\infty}\sum_{k=0}^{\infty} A_{jk} t^j x^k \) which are obtained from any method (usually a special case of regular iteration), for parabolic iteration there will only be a finite number of t's, but for hyperbolic iteration (yes, the flow matrix would apply to that as well) this matrix is not triangular (as it is with parabolic iteration). For parabolic iteration the "flow series" is:
\(
\begin{tabular}{rl}
f^{\circ t}(x)
& = x \\
& + x^2 \left( tf_2 \right) \\
& + x^3 \left( t(f_3-f_2^2) + t^2f_2^2 \right) \\
& + x^4 \left( t\left(\frac{f_2}{2}(3f_2^2 - 5f_3) + f_4\right) + t^2\left(\frac{5f_2}{2}(f_3-f_2^2) \right) + t^3f_2^3 \right) \\
& + \cdots
\end{tabular}
\)
this corresponds to the "flow matrix":\begin{tabular}{rl}
f^{\circ t}(x)
& = x \\
& + x^2 \left( tf_2 \right) \\
& + x^3 \left( t(f_3-f_2^2) + t^2f_2^2 \right) \\
& + x^4 \left( t\left(\frac{f_2}{2}(3f_2^2 - 5f_3) + f_4\right) + t^2\left(\frac{5f_2}{2}(f_3-f_2^2) \right) + t^3f_2^3 \right) \\
& + \cdots
\end{tabular}
\)
\(
\left[\begin{tabular}{cccccc}
0 & 0 & 0 & 0 & 0 & \cdots \\
1 & 0 & 0 & 0 & 0 & \cdots \\
0 & f_2 & 0 & 0 & 0 & \cdots \\
0 & (f_3-f_2^2) & f_2^2 & 0 & 0 & \cdots \\
0 & \left(\frac{f_2}{2}(3f_2^2 - 5f_3) + f_4\right)
& \left(\frac{5f_2}{2}(f_3-f_2^2) \right)
& f_2^3 & 0 & \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots
\end{tabular}\right]
\)
However, the flow matrix is not limited to parabolic iteration, but applies to hyperbolic iteration as well. Since the coefficients of hyperbolic iteration are not polynomials, there is a big difference between the series and the matrix, which may serve to illustrate the need for an flow matrix. For hyperbolic iteration the "flow series" is:\left[\begin{tabular}{cccccc}
0 & 0 & 0 & 0 & 0 & \cdots \\
1 & 0 & 0 & 0 & 0 & \cdots \\
0 & f_2 & 0 & 0 & 0 & \cdots \\
0 & (f_3-f_2^2) & f_2^2 & 0 & 0 & \cdots \\
0 & \left(\frac{f_2}{2}(3f_2^2 - 5f_3) + f_4\right)
& \left(\frac{5f_2}{2}(f_3-f_2^2) \right)
& f_2^3 & 0 & \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots
\end{tabular}\right]
\)
\(
\begin{tabular}{rl}
f^{\circ t}(x)
& = x f_1^t \\
& + x^2 \frac{f_1^{t-1}(f_1^{t} - 1)f_2}{(f_1 - 1)} \\
& + x^3 \frac{f_1^{t-2}(f_1^t-1)(f_1((f_1-1)f_3-2f_2^2) + f_1^t(2f_2^2 + (f_1-1)f_1f_3))}{(f_1-1)^2(f_1+1)} \\
& + \cdots
\end{tabular}
\)
which corresponds to the "flow matrix":\begin{tabular}{rl}
f^{\circ t}(x)
& = x f_1^t \\
& + x^2 \frac{f_1^{t-1}(f_1^{t} - 1)f_2}{(f_1 - 1)} \\
& + x^3 \frac{f_1^{t-2}(f_1^t-1)(f_1((f_1-1)f_3-2f_2^2) + f_1^t(2f_2^2 + (f_1-1)f_1f_3))}{(f_1-1)^2(f_1+1)} \\
& + \cdots
\end{tabular}
\)
\(
\left[\begin{tabular}{cccc}
0 & 0 & 0 & \cdots \\
1 & \log(f_1) & \log(f_1)^2/2 & \cdots \\
0 &
\frac{f_2\log(f_1)}{(f_1-1)f_1} &
\frac{3f_2\log(f_1)^2}{2(f_1-1)f_1} & \cdots \\
\vdots & \vdots & \vdots & \ddots
\end{tabular}\right]
\)
\left[\begin{tabular}{cccc}
0 & 0 & 0 & \cdots \\
1 & \log(f_1) & \log(f_1)^2/2 & \cdots \\
0 &
\frac{f_2\log(f_1)}{(f_1-1)f_1} &
\frac{3f_2\log(f_1)^2}{2(f_1-1)f_1} & \cdots \\
\vdots & \vdots & \vdots & \ddots
\end{tabular}\right]
\)
Is it the Carleman matrix of parabolic iteration?
No, it is the first row of the Carleman matrix^t, or the first column of the Bell matrix^t, meaning both have been raised to the t. The flow matrix is simply a different expression for the flow series.
Why is it important to know the diagonals?
Well, it seems important to know the asymptotic behavior of the coefficients, and it is very difficult to know the asymptotic behavior of a sequence you only have the first few members of, so my goal with these diagonals is to provide a formula that we can use for root-tests and other convergence tests, so the world will stop calling them "formal power series" and start calling them functions. That is my ultimate goal.
But then again, how am I helping when I only have 3 diagonals, and we don't need to find asymptotic behavior of \( t^k x^k \) but of \( t^k \)? Well, it is obvious that the diagonals follow more of a pattern than the columns, so it seems easier to interpolate the diagonals than the columns of the flow matrix. My hope is that after enough diagonals are found, then a pattern will be found in the columns, as well.
Are these the same as Jabotinsky's double-binomial formula?
Yes. However, that formula is expressed in terms of the coefficients of the n-th iterate of a function. The only way I've seen Daniel Geisler write the flow series has been in terms of the coefficients of the 1-st iterate only, which is a major distinction between Jabotinsky's, and the flow series.
Andrew Robbins