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$$\frac{\sqrt{\int_0^m\int_0^n (\left[interp2(1:m,1:n,\mbox{data table here},x,y)\right](y,x))^2dxdy}}{\int_0^m\int_0^n \left[interp2(1:m,1:n,\mbox{data table here},x,y)\right](y,x)dxdy}$$

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Block Cholesky

Block Cholesky Decomposition

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Iterated exponentials are an example of an iterated function system based on $$x^x$$. Such systems have induced some interesting mathematical constants and interesting fractal properties based on its generalization to the complex plane. Finch, S. R. (2003). Mathematical constants. Cambridge university press. pp.448-<452/ref>

= Background =

Minimum and Injectivity
We note that $$f(x)=x^x$$ is concave up with minimum

$$(x,f(x))=(\frac{1}{e},(\frac{1}{e})^{\frac{1}{e}})$$

or

=(0.3678794412...,0.6922006276...)

Inverse
In fact, $$f(x)=x^x$$ does have an inverse

$$x=f^{-1}(y)=y^{(\frac{1}{y})^{(\frac{1}{y})^{\dots}}}$$

which is well-defined for $$y\in[(\frac{1}{e})^{\frac{1}{e}},e^e]$$

This has induced interest in the function $$x^{\frac{1}{x}}$$, which has similar limiting properties to $$x^x$$.

Convergence
By an old result of Euler, repeated exponentiation convergence for real values inbetween $$e^{-e}$$ and $$e^{\frac{1}{e}}$$.

= Calculation of Iterated Exponential =

In certain situations, one may calculate the iterated exponential, and certain constants remain of mathematical interest.

Connection to Lambert's Function
If one defines

$$h(z)=z^{z^{z^{\dots}}}$$

for such $$z$$ where such a process converges,

Then $$h(z)$$ actually has a closed form expression in terms of a function known as Lambert's function which is defined implicitly via the following equation:

$$W(z)e^{W(z)}=z$$

Namely, that

$$h(z)=\frac{W(-log(z))}{-log(z)}$$

This can be seen by inputting this definition of $$h(z)$$ into the other equation that $$h(z)$$ satisfies, $$h(z)=z^{h(z)}$$.

Iteration on the Complex Plane
The function may also be extended to the complex plane, where such a map tends to display interesting fractal properties.

Of particular interest is evaluation of the constant

$$i^{i^{i^{\dots}}}$$

Which does indeed converge and has been evaluated as

$$~=0.43828+0.36059i$$

Galidakis, I. N. (2004). On an application of Lambert's W function to infinite exponentials. Complex Variables, Theory and Application: An International Journal, 49(11), 759-780.