Why am I getting RootSearch::numb:

Question

I'm using Ersek's RootSearch Package to find roots of function Nwes[t] which contains variables β[t], φ[t], γ][t] that were evaluated by NDSolve. However RootSearch fails to return any roots and gives a message:

RootSearch::numb: RootSearch took a number of inital samples and at each sample point the function sampled did not evaluate to a numeric value.

I dont understand why I got this message and how it occured that sample points dont have their numeric values.

My tries were:

• increase InitialSamples variable 10 times to 3000

• All input was treated with N[] function to assure it's numeric (EDIT: however Nwes and eqs had originally machine-precision)

• checked that there exist roots in given range

Whole code:

eqs = {1.` Sin[φ[t]] + 
 0.03504358287544395` Sin[γ[t] - 1.` φ[t]] Derivative[
   1][γ][t]^2 + 
 0.31395122075073245` Cos[β[t] - 
    1.` φ[t]] (β^′′)[t] + 
 0.35557468770802175` (φ^′′)[t] == 
0.31395122075073245` Sin[β[t] - 1.` φ[t]] Derivative[
   1][β][t]^2 + 
 4.303069289595211`*^-18 Sin[γ[t] - 1.` φ[t]] Derivative[
   1][γ][t] Derivative[1][φ][t] + 
 0.03504358287544395` Cos[γ[t] - 
    1.` φ[t]] (γ^′′)[t],4.739130434782608` Sin[β[
    t]] + (1.` Cos[φ[t]] Sin[β[t]] - 
    1.` Cos[β[t]] Sin[φ[t]]) Derivative[1][φ][
   t]^2 + 1.0082125603864733` (β^′′)[t] + 
 1.` Cos[β[t]] Cos[φ[t]] (φ^′′)[
   t] + 1.` Sin[β[t]] Sin[φ[
    t]] (φ^′′)[t] == 
0.`, (1.` Cos[φ[t]] Sin[γ[t]] - 
    1.` Cos[γ[t]] Sin[φ[t]]) Derivative[1][φ][
   t]^2 + 1.` Cos[γ[t]] Cos[φ[
    t]] (φ^′′)[t] + 
 1.` Sin[γ[t]] Sin[φ[t]] (φ^′′)[
   t] == 18.391451068616426` Sin[γ[t]] + 
 1.142857142857143` (γ^′′)[t]};
Subscript[φ, start] = 0.7853981633974483096;

ic1 = {φ[0] == Subscript[φ, start], φ'[0] == 0};

ic2 = {γ[0] == 0, γ'[0] == 0};
ic3 = {β[0] == -(Pi/2), β'[0] == 0};
Nwes[t_] =0.48309178743961356` Csc[φ[
 t]] (39.43705641300001` Sin[φ[t]] - 
 266.5071770107747` Sin[φ[t]] (1.` Cos[φ[t]] + 
    1.` Sin[φ[t]] + 
    14.006711409395976` Sin[β[t] - φ[t]] Derivative[
      1][β][t]^2 - 
    14.006711409395976` Cos[β[t] - φ[
        t]] (β^′′)[t] - 
    13.89261744966443` (φ^′′)[
      t]) (1.` Cos[φ[t]] + 1.` Sin[φ[t]] - 
    0.15903539639826123` (φ^′′)[t]) + 
 0.28451556` Cos[φ[t]] (83.40832395950507` + 
    1.142857142857143` Sin[γ[t]] Derivative[1][γ][t]^2 - 
    1.` Sin[φ[t]] Derivative[1][φ][t]^2 - 
    1.142857142857143` Cos[γ[t]] (γ^′′)[t] + 
    1.` Cos[φ[t]] (φ^′′)[t]) + 
 0.28451556` Sin[φ[t]] (83.40832395950507` - 
    1.142857142857143` Cos[γ[t]] Derivative[1][γ][t]^2 + 
    1.` Cos[φ[t]] Derivative[1][φ][t]^2 - 
    1.142857142857143` Sin[γ[t]] (γ^′′)[t] + 
    1.` Sin[φ[t]] (φ^′′)[t]));

SetDirectory[NotebookDirectory[]];
Get["RootSearch.m"];


sol = NDSolve[{eqs, ic1, ic2, ic3}, {φ, γ, β}, {t, 0, 
2.2}];
Ersek`RootSearch`RootSearch[Evaluate[Nwes[t] /. sol] == 0, {t, 0, 2.2},InitialSamples -> 3000]

• to get RootSearch working on mathematica 9 I set $minPrecision to Infinity as in: http://forums.wolfram.com/mathgroup/archive/2012/Apr/msg00444.html

Answer 1

A few problems here, some of which may have come from copy/pasting code:

1. The derivative should be indicated with φ''[t] and NOT (φ^′′)[t]; the latter does not have a meaning. This should be changed in both your equation and in your Nwes expression.
2. You should define Nwes using SetDelayed (i.e. :=) instead of Set (=), so its value is recalculated each time a different value of t is assigned.
3. NDSolve returns a list of solutions; you just need the first component of that list when substituting back in Nwes, i.e. First[sol].

Once those changes are applied, RootSearch does work, although to a point:

Ersek`RootSearch`RootSearch[Evaluate[Nwes[t] /. First@sol] == 0, {t, 0, 2.2}]

{{t -> 0.685516}, {t -> 1.24982}, {t -> 1.37817}, 
 {t -> 1.52045}, {t -> 1.81702}, {t -> 1.87851}}

Errors are still returned, involving precision issues. You may be better off replacing the machine precision numbers in your equations with arbitrary precision ones, as you probably had before running N on your expressions.