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Evaluating number of iteration with a certain map with While
The 2019 Stack Overflow Developer Survey Results Are InWhy should I avoid the For loop in Mathematica?Out of memory in a Do loopRepeating Calculations/Iterations without a specific functionWhile loop with changing variable , NDSolve and an IntegralUsing Map function with NDSolveHow do I repeat the number of times a nested for loop does an iteration?
$begingroup$
Beeing used to programming in C-like languages I am struggling with iterations and loops with mathematica. I am trying not to use the For command, as already lots of people recommended.
I am trying to solve the following problem:
Given the map $z_i+1= z_i^2 +c$ with $z_i, c in mathbbC$ and $z_0 = 0$ evaluate the contours that represents given the parameter $c$ the number of iterations $i$ that I have to perform in order to have $|z_i|> 2$. Perform the computation with $-0.6 leq Re(z_i)leq -0.4 $ and $0.6 leq Im(z_i)leq 0.4$ and 100 points per axis.
Given the condition I though I could use a While Loop to perform the task
i=0; (*init counter*)
z[i]=0; (*init z[i]*)
g[c_]:= While[
Abs[z[i]]<= 2, (*condition*)
z[i+1]= z[i]^2 +c; (*process*)
i++; (*increment*)
]
Print[i]
g[0.2 + 0.2 I]
This computation with the input, say, $(-0.2 +0.2 i)$ (and with many others) takes all the memory of the machine I am using (Wolfram online). I don't understand whether I am just missing something or the amount of computation I can perform on the server just isn't enough (which seems really unlikely)
Furthermore I would like the function to return an integer (i - the number of iterations) but I really struggle with how to correctly use the synthax of Mathematica to do that.
Thanks in advance to everyone who is so keen to stop by and help :)
procedural-programming
New contributor
$endgroup$
add a comment |
$begingroup$
Beeing used to programming in C-like languages I am struggling with iterations and loops with mathematica. I am trying not to use the For command, as already lots of people recommended.
I am trying to solve the following problem:
Given the map $z_i+1= z_i^2 +c$ with $z_i, c in mathbbC$ and $z_0 = 0$ evaluate the contours that represents given the parameter $c$ the number of iterations $i$ that I have to perform in order to have $|z_i|> 2$. Perform the computation with $-0.6 leq Re(z_i)leq -0.4 $ and $0.6 leq Im(z_i)leq 0.4$ and 100 points per axis.
Given the condition I though I could use a While Loop to perform the task
i=0; (*init counter*)
z[i]=0; (*init z[i]*)
g[c_]:= While[
Abs[z[i]]<= 2, (*condition*)
z[i+1]= z[i]^2 +c; (*process*)
i++; (*increment*)
]
Print[i]
g[0.2 + 0.2 I]
This computation with the input, say, $(-0.2 +0.2 i)$ (and with many others) takes all the memory of the machine I am using (Wolfram online). I don't understand whether I am just missing something or the amount of computation I can perform on the server just isn't enough (which seems really unlikely)
Furthermore I would like the function to return an integer (i - the number of iterations) but I really struggle with how to correctly use the synthax of Mathematica to do that.
Thanks in advance to everyone who is so keen to stop by and help :)
procedural-programming
New contributor
$endgroup$
add a comment |
$begingroup$
Beeing used to programming in C-like languages I am struggling with iterations and loops with mathematica. I am trying not to use the For command, as already lots of people recommended.
I am trying to solve the following problem:
Given the map $z_i+1= z_i^2 +c$ with $z_i, c in mathbbC$ and $z_0 = 0$ evaluate the contours that represents given the parameter $c$ the number of iterations $i$ that I have to perform in order to have $|z_i|> 2$. Perform the computation with $-0.6 leq Re(z_i)leq -0.4 $ and $0.6 leq Im(z_i)leq 0.4$ and 100 points per axis.
Given the condition I though I could use a While Loop to perform the task
i=0; (*init counter*)
z[i]=0; (*init z[i]*)
g[c_]:= While[
Abs[z[i]]<= 2, (*condition*)
z[i+1]= z[i]^2 +c; (*process*)
i++; (*increment*)
]
Print[i]
g[0.2 + 0.2 I]
This computation with the input, say, $(-0.2 +0.2 i)$ (and with many others) takes all the memory of the machine I am using (Wolfram online). I don't understand whether I am just missing something or the amount of computation I can perform on the server just isn't enough (which seems really unlikely)
Furthermore I would like the function to return an integer (i - the number of iterations) but I really struggle with how to correctly use the synthax of Mathematica to do that.
Thanks in advance to everyone who is so keen to stop by and help :)
procedural-programming
New contributor
$endgroup$
Beeing used to programming in C-like languages I am struggling with iterations and loops with mathematica. I am trying not to use the For command, as already lots of people recommended.
I am trying to solve the following problem:
Given the map $z_i+1= z_i^2 +c$ with $z_i, c in mathbbC$ and $z_0 = 0$ evaluate the contours that represents given the parameter $c$ the number of iterations $i$ that I have to perform in order to have $|z_i|> 2$. Perform the computation with $-0.6 leq Re(z_i)leq -0.4 $ and $0.6 leq Im(z_i)leq 0.4$ and 100 points per axis.
Given the condition I though I could use a While Loop to perform the task
i=0; (*init counter*)
z[i]=0; (*init z[i]*)
g[c_]:= While[
Abs[z[i]]<= 2, (*condition*)
z[i+1]= z[i]^2 +c; (*process*)
i++; (*increment*)
]
Print[i]
g[0.2 + 0.2 I]
This computation with the input, say, $(-0.2 +0.2 i)$ (and with many others) takes all the memory of the machine I am using (Wolfram online). I don't understand whether I am just missing something or the amount of computation I can perform on the server just isn't enough (which seems really unlikely)
Furthermore I would like the function to return an integer (i - the number of iterations) but I really struggle with how to correctly use the synthax of Mathematica to do that.
Thanks in advance to everyone who is so keen to stop by and help :)
procedural-programming
procedural-programming
New contributor
New contributor
New contributor
asked 11 hours ago
JacquesLeenJacquesLeen
303
303
New contributor
New contributor
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
The problem with your code is that for some values of c
, Abs[z]
will never become larger than 2. You need to cap the number of iterations.
For this type of iteration, the typical function to use is Nest
and related functions.
countIter[c_] :=
Length@NestWhileList[
#^2 + c &,
0.0,
Abs[#] <= 2 &,
1,
100 (* limit number of iterations to 100 *)
]
result =
Table[
countIter[re + im I],
re, -0.6, -0.4, 0.2/100,
im, 0.4, 0.6, 0.2/100
];
ArrayPlot[result, ColorFunction -> "Rainbow"]
However, this type of problem is quite amenable to compilation with Compile
. When using Compile
, the usual advice does not apply: a procedural style is still the best. (This does not mean that For
is good, I'd still argue against that. But there are many other procedural constructs such as Do
and While
).
countIterCompiled = Compile[c, _Complex,
Block[z = 0.0 + 0.0 I, i = 0,
While[i <= 100 && Abs[z] <= 2,
z = z^2 + c;
i++
];
i
]
]
Using countIterCompiled
will be much faster than countIter
.
$endgroup$
$begingroup$
thank u very much for the suggestion... I previously had a similar idea using Module instead of Block, and the problem was that the exercise did not specify that for many values the map was converging so I had to cap the number of iterations.
$endgroup$
– JacquesLeen
10 hours ago
$begingroup$
@JacquesLeen Maybe that was part of the exercise: will you discover it on your own? InsideCompile
,Module
andBlock
are the same, I think. (Not outside of it.)
$endgroup$
– Szabolcs
9 hours ago
add a comment |
$begingroup$
For iterated function systems like this, Nest
and relatives are the preferred tools. Just exploring your (rather famous) map:
f[z_, c_] := z^2 + c
Abs[NestList[f[#, 0.2 + 0.2 I] &, 0, 30]]
(* 0, 0.282843, 0.344093, 0.351367, 0.327239, 0.304778, 0.303605,
0.311545, 0.316158, 0.315818, 0.313773, 0.312729, 0.31295, 0.313482,
0.313697, 0.313611, 0.313477, 0.313435, 0.313464, 0.313497, 0.313504,
0.313495, 0.313487, 0.313486, 0.313489, 0.313491, 0.313491, 0.31349,
0.31349, 0.31349, 0.31349 *)
As you can see, it converges to a value inside your radius. That's why your function doesn't terminate.
$endgroup$
add a comment |
$begingroup$
You could also use MandelbrotSetPlot
to create Szabolcs' graphic:
MandelbrotSetPlot[-0.6 + 0.4 I, -0.4 + 0.6 I, PlotLegends -> Automatic]
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The problem with your code is that for some values of c
, Abs[z]
will never become larger than 2. You need to cap the number of iterations.
For this type of iteration, the typical function to use is Nest
and related functions.
countIter[c_] :=
Length@NestWhileList[
#^2 + c &,
0.0,
Abs[#] <= 2 &,
1,
100 (* limit number of iterations to 100 *)
]
result =
Table[
countIter[re + im I],
re, -0.6, -0.4, 0.2/100,
im, 0.4, 0.6, 0.2/100
];
ArrayPlot[result, ColorFunction -> "Rainbow"]
However, this type of problem is quite amenable to compilation with Compile
. When using Compile
, the usual advice does not apply: a procedural style is still the best. (This does not mean that For
is good, I'd still argue against that. But there are many other procedural constructs such as Do
and While
).
countIterCompiled = Compile[c, _Complex,
Block[z = 0.0 + 0.0 I, i = 0,
While[i <= 100 && Abs[z] <= 2,
z = z^2 + c;
i++
];
i
]
]
Using countIterCompiled
will be much faster than countIter
.
$endgroup$
$begingroup$
thank u very much for the suggestion... I previously had a similar idea using Module instead of Block, and the problem was that the exercise did not specify that for many values the map was converging so I had to cap the number of iterations.
$endgroup$
– JacquesLeen
10 hours ago
$begingroup$
@JacquesLeen Maybe that was part of the exercise: will you discover it on your own? InsideCompile
,Module
andBlock
are the same, I think. (Not outside of it.)
$endgroup$
– Szabolcs
9 hours ago
add a comment |
$begingroup$
The problem with your code is that for some values of c
, Abs[z]
will never become larger than 2. You need to cap the number of iterations.
For this type of iteration, the typical function to use is Nest
and related functions.
countIter[c_] :=
Length@NestWhileList[
#^2 + c &,
0.0,
Abs[#] <= 2 &,
1,
100 (* limit number of iterations to 100 *)
]
result =
Table[
countIter[re + im I],
re, -0.6, -0.4, 0.2/100,
im, 0.4, 0.6, 0.2/100
];
ArrayPlot[result, ColorFunction -> "Rainbow"]
However, this type of problem is quite amenable to compilation with Compile
. When using Compile
, the usual advice does not apply: a procedural style is still the best. (This does not mean that For
is good, I'd still argue against that. But there are many other procedural constructs such as Do
and While
).
countIterCompiled = Compile[c, _Complex,
Block[z = 0.0 + 0.0 I, i = 0,
While[i <= 100 && Abs[z] <= 2,
z = z^2 + c;
i++
];
i
]
]
Using countIterCompiled
will be much faster than countIter
.
$endgroup$
$begingroup$
thank u very much for the suggestion... I previously had a similar idea using Module instead of Block, and the problem was that the exercise did not specify that for many values the map was converging so I had to cap the number of iterations.
$endgroup$
– JacquesLeen
10 hours ago
$begingroup$
@JacquesLeen Maybe that was part of the exercise: will you discover it on your own? InsideCompile
,Module
andBlock
are the same, I think. (Not outside of it.)
$endgroup$
– Szabolcs
9 hours ago
add a comment |
$begingroup$
The problem with your code is that for some values of c
, Abs[z]
will never become larger than 2. You need to cap the number of iterations.
For this type of iteration, the typical function to use is Nest
and related functions.
countIter[c_] :=
Length@NestWhileList[
#^2 + c &,
0.0,
Abs[#] <= 2 &,
1,
100 (* limit number of iterations to 100 *)
]
result =
Table[
countIter[re + im I],
re, -0.6, -0.4, 0.2/100,
im, 0.4, 0.6, 0.2/100
];
ArrayPlot[result, ColorFunction -> "Rainbow"]
However, this type of problem is quite amenable to compilation with Compile
. When using Compile
, the usual advice does not apply: a procedural style is still the best. (This does not mean that For
is good, I'd still argue against that. But there are many other procedural constructs such as Do
and While
).
countIterCompiled = Compile[c, _Complex,
Block[z = 0.0 + 0.0 I, i = 0,
While[i <= 100 && Abs[z] <= 2,
z = z^2 + c;
i++
];
i
]
]
Using countIterCompiled
will be much faster than countIter
.
$endgroup$
The problem with your code is that for some values of c
, Abs[z]
will never become larger than 2. You need to cap the number of iterations.
For this type of iteration, the typical function to use is Nest
and related functions.
countIter[c_] :=
Length@NestWhileList[
#^2 + c &,
0.0,
Abs[#] <= 2 &,
1,
100 (* limit number of iterations to 100 *)
]
result =
Table[
countIter[re + im I],
re, -0.6, -0.4, 0.2/100,
im, 0.4, 0.6, 0.2/100
];
ArrayPlot[result, ColorFunction -> "Rainbow"]
However, this type of problem is quite amenable to compilation with Compile
. When using Compile
, the usual advice does not apply: a procedural style is still the best. (This does not mean that For
is good, I'd still argue against that. But there are many other procedural constructs such as Do
and While
).
countIterCompiled = Compile[c, _Complex,
Block[z = 0.0 + 0.0 I, i = 0,
While[i <= 100 && Abs[z] <= 2,
z = z^2 + c;
i++
];
i
]
]
Using countIterCompiled
will be much faster than countIter
.
edited 10 hours ago
answered 11 hours ago
SzabolcsSzabolcs
163k14448945
163k14448945
$begingroup$
thank u very much for the suggestion... I previously had a similar idea using Module instead of Block, and the problem was that the exercise did not specify that for many values the map was converging so I had to cap the number of iterations.
$endgroup$
– JacquesLeen
10 hours ago
$begingroup$
@JacquesLeen Maybe that was part of the exercise: will you discover it on your own? InsideCompile
,Module
andBlock
are the same, I think. (Not outside of it.)
$endgroup$
– Szabolcs
9 hours ago
add a comment |
$begingroup$
thank u very much for the suggestion... I previously had a similar idea using Module instead of Block, and the problem was that the exercise did not specify that for many values the map was converging so I had to cap the number of iterations.
$endgroup$
– JacquesLeen
10 hours ago
$begingroup$
@JacquesLeen Maybe that was part of the exercise: will you discover it on your own? InsideCompile
,Module
andBlock
are the same, I think. (Not outside of it.)
$endgroup$
– Szabolcs
9 hours ago
$begingroup$
thank u very much for the suggestion... I previously had a similar idea using Module instead of Block, and the problem was that the exercise did not specify that for many values the map was converging so I had to cap the number of iterations.
$endgroup$
– JacquesLeen
10 hours ago
$begingroup$
thank u very much for the suggestion... I previously had a similar idea using Module instead of Block, and the problem was that the exercise did not specify that for many values the map was converging so I had to cap the number of iterations.
$endgroup$
– JacquesLeen
10 hours ago
$begingroup$
@JacquesLeen Maybe that was part of the exercise: will you discover it on your own? Inside
Compile
, Module
and Block
are the same, I think. (Not outside of it.)$endgroup$
– Szabolcs
9 hours ago
$begingroup$
@JacquesLeen Maybe that was part of the exercise: will you discover it on your own? Inside
Compile
, Module
and Block
are the same, I think. (Not outside of it.)$endgroup$
– Szabolcs
9 hours ago
add a comment |
$begingroup$
For iterated function systems like this, Nest
and relatives are the preferred tools. Just exploring your (rather famous) map:
f[z_, c_] := z^2 + c
Abs[NestList[f[#, 0.2 + 0.2 I] &, 0, 30]]
(* 0, 0.282843, 0.344093, 0.351367, 0.327239, 0.304778, 0.303605,
0.311545, 0.316158, 0.315818, 0.313773, 0.312729, 0.31295, 0.313482,
0.313697, 0.313611, 0.313477, 0.313435, 0.313464, 0.313497, 0.313504,
0.313495, 0.313487, 0.313486, 0.313489, 0.313491, 0.313491, 0.31349,
0.31349, 0.31349, 0.31349 *)
As you can see, it converges to a value inside your radius. That's why your function doesn't terminate.
$endgroup$
add a comment |
$begingroup$
For iterated function systems like this, Nest
and relatives are the preferred tools. Just exploring your (rather famous) map:
f[z_, c_] := z^2 + c
Abs[NestList[f[#, 0.2 + 0.2 I] &, 0, 30]]
(* 0, 0.282843, 0.344093, 0.351367, 0.327239, 0.304778, 0.303605,
0.311545, 0.316158, 0.315818, 0.313773, 0.312729, 0.31295, 0.313482,
0.313697, 0.313611, 0.313477, 0.313435, 0.313464, 0.313497, 0.313504,
0.313495, 0.313487, 0.313486, 0.313489, 0.313491, 0.313491, 0.31349,
0.31349, 0.31349, 0.31349 *)
As you can see, it converges to a value inside your radius. That's why your function doesn't terminate.
$endgroup$
add a comment |
$begingroup$
For iterated function systems like this, Nest
and relatives are the preferred tools. Just exploring your (rather famous) map:
f[z_, c_] := z^2 + c
Abs[NestList[f[#, 0.2 + 0.2 I] &, 0, 30]]
(* 0, 0.282843, 0.344093, 0.351367, 0.327239, 0.304778, 0.303605,
0.311545, 0.316158, 0.315818, 0.313773, 0.312729, 0.31295, 0.313482,
0.313697, 0.313611, 0.313477, 0.313435, 0.313464, 0.313497, 0.313504,
0.313495, 0.313487, 0.313486, 0.313489, 0.313491, 0.313491, 0.31349,
0.31349, 0.31349, 0.31349 *)
As you can see, it converges to a value inside your radius. That's why your function doesn't terminate.
$endgroup$
For iterated function systems like this, Nest
and relatives are the preferred tools. Just exploring your (rather famous) map:
f[z_, c_] := z^2 + c
Abs[NestList[f[#, 0.2 + 0.2 I] &, 0, 30]]
(* 0, 0.282843, 0.344093, 0.351367, 0.327239, 0.304778, 0.303605,
0.311545, 0.316158, 0.315818, 0.313773, 0.312729, 0.31295, 0.313482,
0.313697, 0.313611, 0.313477, 0.313435, 0.313464, 0.313497, 0.313504,
0.313495, 0.313487, 0.313486, 0.313489, 0.313491, 0.313491, 0.31349,
0.31349, 0.31349, 0.31349 *)
As you can see, it converges to a value inside your radius. That's why your function doesn't terminate.
answered 10 hours ago
John DotyJohn Doty
7,57811124
7,57811124
add a comment |
add a comment |
$begingroup$
You could also use MandelbrotSetPlot
to create Szabolcs' graphic:
MandelbrotSetPlot[-0.6 + 0.4 I, -0.4 + 0.6 I, PlotLegends -> Automatic]
$endgroup$
add a comment |
$begingroup$
You could also use MandelbrotSetPlot
to create Szabolcs' graphic:
MandelbrotSetPlot[-0.6 + 0.4 I, -0.4 + 0.6 I, PlotLegends -> Automatic]
$endgroup$
add a comment |
$begingroup$
You could also use MandelbrotSetPlot
to create Szabolcs' graphic:
MandelbrotSetPlot[-0.6 + 0.4 I, -0.4 + 0.6 I, PlotLegends -> Automatic]
$endgroup$
You could also use MandelbrotSetPlot
to create Szabolcs' graphic:
MandelbrotSetPlot[-0.6 + 0.4 I, -0.4 + 0.6 I, PlotLegends -> Automatic]
answered 9 hours ago
Carl WollCarl Woll
73.2k397191
73.2k397191
add a comment |
add a comment |
JacquesLeen is a new contributor. Be nice, and check out our Code of Conduct.
JacquesLeen is a new contributor. Be nice, and check out our Code of Conduct.
JacquesLeen is a new contributor. Be nice, and check out our Code of Conduct.
JacquesLeen is a new contributor. Be nice, and check out our Code of Conduct.
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