The Joint Spectral Radius Approach

In the eigenanalysis chapter we used the support analysis to tell us how many control points influenced a neighbourhood of a point of the limit curve, and thus how large the matrix needed to be on which to carry out the eigenanalysis. 

However, this told us only the continuity exactly at one point of the limit curve. 

We can also ask the support analysis how many points influence one span of the limit curve, the piece corresponding to one edge of the control polygon. This turns out to be one fewer. Call it m. The value will be 4 for the cubic B-spline. 

After one refinement, the pieces of limit curve are just half as long, and there are two pieces, each dependent on m new points. 

The left hand half depends on the new points a to d, the right hand half on b to e, all of which depend only on A to D. 

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The joint spectral radius approach appeared in a paper by Ingrid Dau-bechies and Jeffrey Lagarias[DaLa91] in 1991, and also in one by Hartmut Prautzsch and Charles Micchelli[PM87] in 1987. Efficient computation is still a hot topic. A strong competitor for the ideas described in section 18.3 above is the depth first search method developed by the team of Ulrich Reif. 

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