Daubechies-Lagarias Algorithm -- A Simplified Approach

Daubechies & Lagarias algorithm gave the first proof regarding the convergence of the iterations producing the scaling and wavelet functions. The proof however uses advance concepts from linear algebra. Incidentally, it also uses a concept called joint spectral radius, which was introduced by Strang in 1960. The concept of joint spectral radius remained dormant for a very long time. It was only Daubechies and Lagarias, who understood the importance of the concept and applied it for solving the key equation in wavelet theory, the refinement equation for the scaling function. The concept was developed in two papers consisting of around 70 pages. The mathematics employed is formidable to a mathematically less sophisticated reader. In this paper, we give a simplified and intuitive explanation for the same, with all the required mathematics in one place, so that the material can be used for classroom teaching. Then it is shown how the algorithm is implemented in Microsoft Excel. Also is presented as to how to generalize the same for the M-band case.