20
votes
Archimedes' Scheme to find $\pi$
This is a "visual proof" of the Archimedean limiting regular polygons. You could implement the recursion and it would progressively approach $\pi$. The proof lies in the "squeezing" argument. $\pi$ ...
- 58.7k
18
votes
Finite difference method not converging to correct steady state or conserving area?
Since a fully NDSolve-based solution is acceptable for you, let me give you one. You simply need the magic of "Pseudospectral" or a dense enough 4th order spatial ...
- 63.5k
15
votes
Paying off an installment
Using Annuity:
pmt /.Solve[TimeValue[Annuity[pmt, 52, 1], .02, 0] == 5000, pmt]
155.545
...
- 10.9k
13
votes
RecurrenceTable iteration variable not localized? Bug or user error?
As Bob Hanlon's answer points out, RecurrenceTable does not hold its arguments, but most especially, it does not hold its iterator arguments. This must surely be ...
- 68.4k
13
votes
Accepted
How do I recursively calculate this equation and generate a list of iteration?
You were correct. NestList is exactly the function you want to use.
...
- 4,644
12
votes
Accepted
Finite difference method not converging to correct steady state or conserving area?
Introduction
A lack of time to write up an answer ironically provided time to reflect on the problem, and some nagging uncertainties about some issues contributed to the delay.
The slowness of the ...
- 233k
11
votes
Accepted
11
votes
Solving Recurrence Equation $a_{n+1}=a_n(1-a_n)$
Your recurrence is an instance of the logistic map $$x_{n+1} = r x_n(1 - x_n).$$
The first sentence of the Solution in some cases section in the link above says:
The special case of r = 4 can in ...
- 35.5k
11
votes
RecurrenceTable iteration variable not localized? Bug or user error?
Unlike many other functions that use an iterator, RecurrenceTable does not have the attribute HoldAll. Presumably, this ...
- 151k
11
votes
Accepted
Finding Hofstadter's Q-Sequence very slow
Proper memoization is the key:
Q[1] = 1;
Q[2] = 1;
Q[n_] := Q[n] = Q[n - Q[n - 1]] + Q[n - Q[n - 2]]
data = Table[Q[n]/n, {n, 10000}]; // AbsoluteTiming
{0....
- 23.7k
11
votes
Accepted
Stability classification code for dynamic system
The code in the article you cited was poorly formatted, poorly indented, and generally not at all readable. However, what it did was relatively simple, so I rewrote it to be more readable and perhaps ...
- 65.8k
10
votes
Accepted
10
votes
Archimedes' Scheme to find $\pi$
You can use Mathematica to prove the induction step in a proof that $a_n, b_n \rightarrow L$ for some limit $L$ by showing that $a_n$ is decreasing and $b_n$ increasing toward each other and that the ...
- 233k
10
votes
How do I recursively calculate this equation and generate a list of iteration?
Your can use MatrixPower for this example:
f[n_] := MatrixPower[{{.5, -.6}, {.75, 1.1}}, n].{2, 0}
f /@ Range[0, 5]
yields:
<...
- 58.7k
10
votes
Find a stable 3-cycle of the sine map
This is how you define the function $g$ in Mathematica:
g[a_, x_] = a*Sin[π*x];
Making a color-plot of the function $g(g(g(x)))-x$ in the $a$-$x$ plane and ...
- 45.9k
10
votes
Accepted
10
votes
Accepted
Plotting an image of a discrete dynamical system
Use the sliders to modify matrix entries. Click and drag locators (small disks) to modify initial points; ALT+Click to add/remove locators.
...
- 384k
10
votes
How to Generate a Phase Portrait for a Jacobian Evaluated at Fixed Points
First, we can plot phase portrait for 3D system as follows (see comment @I.M.)
...
- 41.9k
10
votes
How to find roots of general polynomials as functions in Mathematica?
Outline:
Code
Discussion on polynomial roots that do not involve radicals
What is the point of getting an explicit function ?
Explanation of code
Examples
Discussion about factoring and ...
- 5,727
10
votes
Accepted
Recurrences related to Ramanujan's 1/pi formulas for level 10?
The question is asking to find linear recurrences with polynomial
coefficients for number sequences. As far as I know, Mathematica
still does not have this functionality builtin. That is the context
...
- 4,897
9
votes
How to replace the symbol n in DifferenceRoot
There is a hidden option to change the variables used by DifferenceRoot[]:
...
- 123k
9
votes
How to find roots of general polynomials as functions in Mathematica?
Clear["Global`*"]
poly6 = x^6 - x - z;
The exact solutions as Root expressions
...
- 151k
8
votes
Accepted
Evolution of a trait in an asexual population (performance tuning)
Here is a modified version of your code. On my PC it completes your example run in about 5 seconds.
I won't try to describe every change but will point out the major features. Some of the changes are ...
- 84.6k
8
votes
8
votes
Accepted
Calculating the Feigenbaum Constants
Several changes are required to obtain the desired results. First, the syntax error mu[n] == mu must be replaced by mu[n] = mu. ...
- 60.7k
8
votes
Accepted
Using Total inside the RecurrenceTable
First, as I noted in the comment, a much simpler approach is to use NestList:
NestList[f, 12, 3]
{12, 9, 729, 1080}
Let's ...
- 23.7k
8
votes
Accepted
Using dynamic programming to compute an integer sequence
This is a good chance to use #0:)
If[#1 == 1, 1, # + 1 - #0[EulerPhi @ #]] & /@ Range[20]
...
- 384k
8
votes
How to draw discrete dynamic systems in mathematica
Play with the following example:
$$
\begin{eqnarray}
x(n+1)&=&rx(n)(1-y(n))\\
y(n)&=&x(n),
\end{eqnarray}
$$
with $r=\displaystyle\frac{2005}{1000}$ and three orbits.
The phase ...
- 12.1k
7
votes
Recursion question
Started as a comment to the OP. Turning this into an answer was requested. The OP asked for Mathematica help solving the recurrence. I suggested recognizing that the recurrence was the sum of ...
- 2,980
7
votes
Evolution of a trait in an asexual population (performance tuning)
When building a simulation like yours you should test the performance of the individual components before incorporating them into the simulation. That is, you should know the cost of the components as ...
- 107k
Only top scored, non community-wiki answers of a minimum length are eligible