Sciweavers

39
Voted
SIAMREV
2010

From Random Polygon to Ellipse: An Eigenanalysis

13 years 6 months ago
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 P(x, y) 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 x and y 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 E 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 phenomena. The problem and its solution is a metaphor for matrix-based research in computational science and engineering. Key words. power method, eigenvalue analysis, ellipse, polygon AMS subject classifications. 15A18, 65F15
Adam N. Elmachtoub, Charles F. Van Loan
Added 21 May 2011
Updated 21 May 2011
Type Journal
Year 2010
Where SIAMREV
Authors Adam N. Elmachtoub, Charles F. Van Loan
Comments (0)