Re: New to Mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg126413] Re: New to Mathematica
- From: Bob Hanlon <hanlonr357 at gmail.com>
- Date: Tue, 8 May 2012 04:12:04 -0400 (EDT)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <jntga4$2c$1@smc.vnet.net>
Build one step at a time.
For a given value of k (say k=2) use exact values (i.e., 1082/10
rather than 108.2) and restrict the domain of Solve to Reals
With[{k = 2}, Solve[(x/Log[x]) (1 + 1/Log[x]) == 1082/10 + k,
x, Reals]]
{{x -> Root[{-551 Log[#1]^2 + 5 #1 + 5 Log[#1] #1 &,
0.91655356819195758220}]}, {x ->
Root[{-551 Log[#1]^2 + 5 #1 + 5 Log[#1] #1 &,
1.11136974029598568569}]}, {x ->
Root[{-551 Log[#1]^2 + 5 #1 + 5 Log[#1] #1 &, 611.73540466624378639}]}}
Use N to convert Root objects to their values (last entry in Root object)
With[{k = 2}, Solve[
(x/Log[x]) (1 + 1/Log[x]) == 1082/10 + k,
x, Reals] // N]
{{x -> 0.916554}, {x -> 1.11137}, {x -> 611.735}}
Use ReplaceAll to extract values from rules
With[{k = 2}, x /. Solve[
(x/Log[x]) (1 + 1/Log[x]) == 1082/10 + k,
x, Reals] // N]
{0.916554, 1.11137, 611.735}
Use Part, Last, or Max to extract largest value
With[{k = 2}, x /. Solve[
(x/Log[x]) (1 + 1/Log[x]) == 1082/10 + k,
x, Reals][[-1]] // N]
611.735
Convert to a function of k with k restricted to Integers
r[k_Integer] := x /. Solve[
(x/Log[x]) (1 + 1/Log[x]) == 1082/10 + k,
x, Reals][[-1]] // N
r[2]
611.735
Plot of r
ListPlot[{#, r[#]} & /@ Range[-20, 20, 4],
Frame -> True, Axes -> False]
Define your Product as a function of kmax
p[kmax_Integer] :=
Product[1 - 1/r[k], {k, 0, kmax}]
p[10]
0.982714
This is quite slow so calculation for large values of kmax will take
considerable time. Plot of p
ListPlot[{#, p[#]} & /@ Range[0, 20, 2],
Frame -> True, Axes -> False]
Bob Hanlon
On Sun, May 6, 2012 at 8:28 PM, J.Jack.J. <jack.j.jepper at googlemail.com> wr=
ote:
> Thanks for replying. Responses embedded:
>
>
> On May 6, 8:24 am, Murray Eisenberg <mur... at math.umass.edu> wrote:
>> First, perhaps folks were reluctant to respond because this looked like
>> it could be a homework exercise.
>>
>> Second, you don't even have proper Mathematica syntax in your equation
>> relating x and k. Did you even try to read the documentation to learn
>> the very basics?
>
> I tried and tried for hours but couldn't so much as find any section
> that would even tell me how to write the condition that k be an
> integer.
>
> For example, proper syntax for the equation would be:
>>
>> (x/Log[x]) (1 + 1/Log[x]) == 108.2 + k
>>
>> Function arguments must be enclosed in brackets, not parentheses, and
>> the equality is a doubled "=" sign. Moreover, your original expression
>> had an unbalanced terminal parenthesis.
>>
>
> Thanks.
>
>> Third, the equation itself looks really nasty. Aside from the fact that
>> it mixes exact formulas with an approximate real (108.2), the left-hand
>> side is transcendental.
>>
>> Fourth, the equation does not seem to uniquely define x as a function of
>> k! For example, form the difference between the two sides...
>>
>> f[x_] := (x/Log[x]) (1 + 1/Log[x]) - 108.2 - k
>>
>> ... and plot f for, say, k = 2:
>>
>> Plot[Evaluate[f[x] /. k -> 2], {x, 0.5, 2}, Exclusions -> {x ==
=
> = 1},
>> AxesOrigin -> {0, 0}, PlotRange -> {-5, 5}]
>>
>> The graph crosses the x-axis twice. And indeed, if you use FindRoot with
>> initial guesses above and below 1, you'll see that this is so:
>>
>> FindRoot[Evaluate[f[x] /. k -> 2], {x, 0.9}]
>> {x -> 0.916554}
>>
>> FindRoot[Evaluate[f[x] /. k -> 2], {x, 1.1}]
>> {x -> 1.11137}
>>
>
> Am sorry, I should have said "let r(k) be the highest x such that..."
>
> Can you give me the required inputs? That would help me immensely.
>