This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.
There were 5 solutions found in 1.653 seconds (with a Bezout bound of 120).
Note: This system is ill-conditioned. There are 4 complex and 1 real solution.
Reference: "The construction and application of wavelets in numerical analysis" by Wim Sweldens.
See also: http://homepages.math.uic.edu/~jan/Demo/quadgrid.html.
i1 : quadgrid(RR_53)
o1 = {w + w + w + w - 1, w b + w b + w b + w b + .5w + w + 1.5w -
0 1 2 3 0 1 2 3 1 2 3
------------------------------------------------------------------------
2 2 2 2
.633975, w b + w b + w b + w b + w b + 2w b + 3w b + .25w + w +
0 1 2 3 1 2 3 1 2
------------------------------------------------------------------------
3 3 3 3 2 2 2
4.5w - .401924, w b + w b + w b + w b + 1.5w b + 3w b + 4.5w b +
3 0 1 2 3 1 2 3
------------------------------------------------------------------------
4 4
.75w b + 3w b + 6.75w b + .125w + w + 3.375w - .131092, w b + w b +
1 2 3 1 2 3 0 1
------------------------------------------------------------------------
4 4 3 3 3 2 2 2
w b + w b + 2w b + 4w b + 6w b + 1.5w b + 6w b + 13.5w b + .5w b
2 3 1 2 3 1 2 3 1
------------------------------------------------------------------------
+ 4w b + 13.5w b + .0625w + w + 5.0625w + .302193}
2 3 1 2 3
o1 : List
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The object quadgrid is a method function.