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We can also make use of the undocumented syntax of FindRoot:
Clear@myfun14;
myfun14[list_] :=
Module[{b, A, B},
b = Transpose@{{1/2, 1/4, 0, 1/8, 0, 0, 0, 1/16, 0, 0, 0, 0, 0, 0}};
A = DiagonalMatrix[ConstantArray[1., 13], -1]; A[[All, -1]] = list;
B = A + Sum[MatrixPower[A, (2^n - 15)]/2.^(n - 4), {n, 5, 50}];
(list - b - 1/32 B.list)]
...
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