From Random Polygon to Ellipse: An Eigenanalysis.

Suppose x and y are unit 2-norm n-vectors whose components sum to zero. Let P(x, y) be the polygon obtained by connecting (x1, y1),..., (xn, yn), (x1, y1) in order. We say that P̂(x̂, ŷ) is the normalized average of P(x, y) if it is obtained by connecting the midpoints of its edges and then normalizing the resulting vertex vectors x̂ and ŷ so that they have unit 2-norm. If this process is repeated starting with P0 = P(x(0), y(0)), then in the limit the vertices of the polygon iterates P(x(k), y(k)) converge to an ellipse ε that is centered at the origin and whose semiaxes are tilted forty-five degrees from the coordinate axes. An eigenanalysis together with the singular value decomposition is used to explain this phenomenon. The problem and its solution is a metaphor for matrix-based research in computational science and engineering. [ABSTRACT FROM AUTHOR]

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