Using While Loops in Mathematica for Iterative Computations

[member 428835]

Hi PF!

I'm trying to compute $$\phi_n(x)=\pi x^2+\int_0^{\pi} 3(0.5 \sin (3x)-tx^2)\phi_{n-1}(t)\,dt$$ given $\phi_0 = 0$. I'd like to iterate this computation until $\sqrt{\int(\phi_{n+1}-\phi_n)^2}< 0.1$. I've never used the while loop in Mathematica, tried reading the documentation but was still uncertain. Any help would be great!

 

[.Scott]

n=0; NotDone = True;
While [NotDone,
  <your computation>;
  NotDone = <your decision>;
  n++]

 

[scottdave]

I'm just a passerby on this question. I don't know Mathematica, but I know other software. What is the way to have multiple operations in (your computation)?

 

[member 428835]

n=0; NotDone = True;
While [NotDone,
  <your computation>;
  NotDone = <your decision>;
  n++]

Could you explain this a little bit please? Why have you defined NotDone as True? For <your computation> and <your decision>, what would I put? I've only used Matlab for while loops, so I'm very lost with Mathematica. Also, what does the n++ mean?

We could simplify the problem and state x = x+1, initializing x = 0, and have this run until x is greater than 10. I just want to understand how to do this in Mathematica.

I'm just a passerby on this question. I don't know Mathematica, but I know other software. What is the way to have multiple operations in (your computation)?

I assume you're asking .Scott this?

 

[DrClaude]

Why have you defined NotDone as True?

Because you want the while loop to execute at least once. While starts by checking the condition, and will execute <your computation> only if the condition is true.

For <your computation> and <your decision>, what would I put?

<your computation>: what you want Mathematica to calculate. In your case, I guess its $\phi_n$. <your decision> must by something that will be false when you want the loop to terminate, so the inverse of $\sqrt{\int(\phi_{n+1}-\phi_n)^2}< 0.1$.

Also, what does the n++ mean?

Increment n by 1. You might not actually need to keep track of n in your case.

We could simplify the problem and state x = x+1, initializing x = 0, and have this run until x is greater than 10. I just want to understand how to do this in Mathematica.

Try the example in the Mathematica help:

n = 1; While[n < 4, Print[n]; n++]

 

[DrClaude]

I'm just a passerby on this question. I don't know Mathematica, but I know other software. What is the way to have multiple operations in (your computation)?

Curly brackets {}.

 

[member 428835]

Because you want the while loop to execute at least once. While starts by checking the condition, and will execute <your computation> only if the condition is true.<your computation>: what you want Mathematica to calculate. In your case, I guess its $\phi_n$. <your decision> must by something that will be false when you want the loop to terminate, so the inverse of $\sqrt{\int(\phi_{n+1}-\phi_n)^2}< 0.1$.Increment n by 1. You might not actually need to keep track of n in your case.Try the example in the Mathematica help:

n = 1; While[n < 4, Print[n]; n++]

Thanks, this was very helpful!

I was able to get the loop to work for some simple iterations, like

n = 0;
NotDone = True;
While[NotDone,
 \[Phi][x_] = x + n;
 NotDone = Integrate[\[Phi][x] , {x, 0, 1}] < 3;
 n++]

but when I try the real thing what I have is

which I know can't work since I have no way of updating $\phi old$. Could you help me here?

 

[scottdave]

After setting the new value for NotDone, can you just say that Φold = Φnew ? Then it will use this on the next iteration.

 

[member 428835]

After setting the new value for NotDone, can you just say that Φold = Φnew ? Then it will use this on the next iteration.

I thought this might work too, but I'm getting a bunch of errors using

 

[scottdave]

Could you explain this a little bit please? Why have you defined NotDone as True? For <your computation> and <your decision>, what would I put? I've only used Matlab for while loops, so I'm very lost with Mathematica. Also, what does the n++ mean?

We could simplify the problem and state x = x+1, initializing x = 0, and have this run until x is greater than 10. I just want to understand how to do this in Mathematica.

I assume you're asking .Scott this?

Anyone who could answer. I wasn't sure how to do multiple statements in Mathematica. From the sample code, I see that it looks like you separate with semicolons.

 

[mfb]

I thought this might work too, but I'm getting a bunch of errors usingView attachment 221100

Missing semicolons to separate the statements?

 

[DrClaude]

Here is the while loop:

While[NotDone, \[Phi]new[x_] = \[Pi] x^2 +
   Integrate[3 (Sin[3 x]/2 - t x^2) \[Phi]old[t], {t, 0, \[Pi]}];
NotDone =
  Sqrt[Integrate[(\[Phi]new[x] - \[Phi]old[x])^2, {x, 0, \[Pi]}]] >
   0.1; \[Phi]old[x_] = \[Phi]new[x]; n++;]

But your problem is flawed: the sequence $\{\phi_n\}$ does not converge.

 

[member 428835]

Gotcha! Ok, I thought this might be the case! Thanks so much for the advise! And I learned something. How do you work with the solution output though? When I run the code I do not receive an output.