# Difference between FindInstance and NSolve

Assume that

w[n_] :=Expand[Sum[Binomial[n - k - 1, k]*(-1)^k*A^(n - 2*k - 1), {k, 0, n - 1}]]
f[y_, z_]:=PolynomialRemainder[(w[z] - 1)*(w[y] - 1), (w[i] - 1), A]
h[{y_, z_}] :=Quiet[N[FindRoot[Abs[f[y, z]] - w[i] + 1 == 0, {A, 10000000}]]]
For[i = 153, i < 210, i++,Print[{i}, Max[ParallelMap[h, Subsets[Range[i + 1, 210], {2}]]]]]


According to above code, there are lots of roots.

But, when I try to find them with NSolve, there is no root. Why?

• Actually it gives and a lot of them! One question why you have chosen 10000000 as the initial value? For that initial value it seems that the method is diverging ... Apr 10, 2013 at 22:45
• @Spawn1701D,Because i want to find the greatest root for all cases. and i guess only where the root? Why does it give? When i try to solve with NSolve, i could not have any solutions. Apr 10, 2013 at 22:47
• Try just one step, take i = 153 construct the subset Subsets[Range[i + 1, 210], {2}]and take the first pair. if you run h you will get {A -> 5.16816*10^6} substitute this value on Abs[f[y, z]] - w[i] + 1 == 0 and you will get a huge number, in other word for that initial value the method diverges. Apr 10, 2013 at 22:51

Try this for h:

h[{y_, z_}] :=Quiet[NSolve[Abs[f[y, z]] - w[i] + 1 == 0, A,Reals]]


Also Max won't work try instead this

For[i = 153, i < 210, i++,
Print[{i}, ParallelMap[Sort[h[#]][[-1]]&, Subsets[Range[i + 1, 210], {2}]]]]]


or alternatively,

For[i = 153, i < 210, i++,
Print[{i}, ParallelMap[{A->Max[h[#][[All,All,2]]]}&, Subsets[Range[i + 1, 210], {2}]]]]]


(using Sort is faster)