Most of my dissertation (1968) was on numerical solution of potential problems.
One of the parts was a proof that some of the known iterative methods
converged. The argument loosely went something like this. Consider the 2D
Poisson equation on a square. If you use an N x N approximation with the usual
discretization of the Laplacian
u_ij = (u_i(j-1) + u_i(j+1) + u_(i_1)j + i_(j+1))/4
i.e, the average of the surrounding points, the problem reduces to the solution
of a set of N^2 linear equations
Ax = b
where x in a vector of the unknown {u_ij} arranged by rows or columns, b is
determined by the boundary conditions and the right side of the Poisson
equation. The interesting part is A which is block tridiagonal. With only a
small error A can be made block cyclic. You can then diagonalize A with a sine
transform and I was able to use that for proofs.
A few years later when the FFT came about, we realized that we could use the
FFT to do the sine transform and the resulting numerical method was as least as
efficient as any other method people had come up with.
Ed
Here’s a reference from 1986 that I think was based on paper at a Bellman
Continuum
``From Dynamic Programming to Fast Transforms,'' E. Angel, J. Math. Anal.
Appl.,119,1986.
Ed
__________
Ed Angel
Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)
Professor Emeritus of Computer Science, University of New Mexico
1017 Sierra Pinon
Santa Fe, NM 87501
505-984-0136 (home) edward.an...@gmail.com
505-453-4944 (cell) http://www.cs.unm.edu/~angel
> On Apr 28, 2023, at 8:18 AM, Stephen Guerin <stephen.gue...@simtable.com>
> wrote:
>
> Special Unitary Groups and Quaternions
>
> Mostly for Ed from the context of last week's Physical Friam if you're coming
> today.
>
> Discussion was around potential ways of visualizing the dynamics of SU(3),
> SU(2), (SU1) that highlights Special Unitary Groups. (wiki link from Frank
> <https://en.wikipedia.org/wiki/Special_unitary_group>), and can we foreground
> how quaternions are used in this process.
>
> and a related bit on forces, I'm searching for ways to visualize/understand
> how FFTs with Poisson equation
> <https://www.codeproject.com/Articles/5308623/Solving-Poisson-Equation> are
> used to compute the forces from scalar fields (eg gravitational force from
> mass density, electric force from charge, etc) and if there's any relation to
> Special Unitary Groups.
>
> -S
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