Gröbner basis design example

Length 8, 1 zero wavelet moment
(N=8, K=1)

The first step of the design procedure is to generate the design equations. The Maple program setup generates the design equations and writes them to the file eqs.

The set of nonlinear design equations is first converted into a grevlex Gröbner basis (gb.dp). Then that is converted in turn into a lexical Gröbner basis (gb.lp) of degree 64. The lexical lexical Gröbner basis is then factored into four Gröbner bases (gb.lp.1, gb.lp.2, gb.lp.3, gb.lp.4). This is all done with the Singular program sfile which reads the design equations and computes the Gröbner bases.

gb.lp.1 yields 6 real solutions.
gb.lp.2 yields 10 real solutions.
gb.lp.3 yields 14 solutions.
gb.lp.4 yields 4 solutions.
The Maple programs result.1, result.2, result.3, result.4 numerically solve the Gröbner bases gb.lp.1, gb.lp.2, gb.lp.3, gb.lp.4 and write the solutions to respective Matlab files.

The solutions are rather poor in this example, as the wavelets each have only 1 zero moment.