Differences in Mathematica's behaviour with identical functions

Question

I have three functions whose identity was verified, but WM doesn't behave itself equally with them.

The code is:

θ[x_] := Boole[x >= 0]
{R, L, C1, Rl, T, Um} := {500, 0.2, 2*10^(-10), 10^3, 10^(-3), 10};
{t1, t2, t3} := {T/2, T/3, T/6};
u1[t_] := Um/t3*(t*θ[t] - (t - t3) θ[t - t3] - (t - t2) θ[t - t2] + (t - t1) θ[t - t1])
U1[s_] := LaplaceTransform[u1[t], t, s]
A10[ω_] := Abs[U1[s]] /. s -> I*ω
Solve[A10[ω] == 0 && ω ∈ Interval[{10000, 20000}], ω]
A11[ω_]:=(1.210^5/ω^2)Sqrt[2(Cos[10^(-3)ω/6])^3-2(Cos[10^(-3)ω/6])^2-2Cos[10^(-3)ω/6]+2]
Solve[A11[ω] == 0 && ω ∈ Interval[{10000,20000}],ω]
A12[ω_]:=Abs[-(610^4/ω^2)(1-E^(-10^(-3)Iω/6)-E^(-10^(-3)Iω/3)+E^(-10^(-3)Iω/2))]
Solve[A12[ω] == 0 && ω ∈ Interval[{10000,20000}],ω]
Plot[A10[ω], {ω, 0, 80000}, PlotRange -> {{0, 80000}, {0, 0.004}}]
Plot[A11[ω], {ω, 0, 80000}, PlotRange -> {{0, 80000}, {0, 0.004}}]
Plot[A12[ω], {ω, 0, 80000}, PlotRange -> {{0, 80000}, {0, 0.004}}]
(Plot performance velocity was compared by isolating each two plots with comment syntax)

First phenomenon is that Solve[] doesn't process A11[ω]==0 equation correctly, returning {{ω -> {18849.6}}} instead of {{ω -> {6000 π}}} and raising such a message:

Solve::ratnz: Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result.

Second phenomenon is that Plot[] for A10[ω] performs unbelievably slow. Thus, I've got two questions...

Why I observe problems with getting A11[ω]==0 roots?
Why A10[ω] plot performance speed is so different from others?

Thanks in advance.

Please review Numbers, esp. "Exact and Approximate Results." — Michael E2, May 15 '21 at 14:23
Possible duplicate: https://mathematica.stackexchange.com/questions/6055/how-to-get-rid-of-warnings-when-using-solve-on-an-equation-with-inexact-coeffici — Michael E2, May 15 '21 at 14:37
Conclusion: the problem was with Set, SetDelayed and floating numbers comprehension. — Cpp Nosavvier, May 23 '21 at 16:52

bill s · Accepted Answer · 2021-05-15T11:37:44.770

6

By typing 1.2*10^5 you are telling Mathematica to do things in floating point. Change this to 120000 and you get the desired exact form. To make it plot faster use Set (shortcut =) rather than SetDelayed (shortcut :=), that is, define

A10[ω_] = Abs[U1[s]] /. s -> I*ω

instead of

A10[ω_] := Abs[U1[s]] /. s -> I*ω

Now it plots almost instantly. (The only difference is the colon before the equals sign).

edited May 15 '21 at 11:37

answered May 12 '21 at 20:54

bill s

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Differences in Mathematica's behaviour with identical functions

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