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Suppose I have the following three lists:
x = {i, j};
y = {a, b, c};
z = {{1, 2, 3}, {4, 5, 6}};
Where z is the data corresponding to the x and y coordinates, z(x_i,y_j) = z_ij. I need to create an array of the following form:
zz = {{i, a, 1}, {i, b, 2}, {i, c, 3}, {j, a, 4}, {j, b, 5}, {j, c, 6}}
How can I do it efficiently in Mathematica 10.0?
This was my attempt so far:
zz = Table[{x[[ii]], y[[jj]], z[[ii, jj]]}, {ii, 1, Length[x]}, {jj, 1, Length[y]}]~Flatten~1
My ultimate goal is to plot (ListPlot3D[zz]) or interpolate this data (Interpolation[zz]), and x and y may be non-uniformly sampled.
Simplifying Bill's answer
x = {i, j};
y = {a, b, c};
z = {{1, 2, 3}, {4, 5, 6}};
MapThread[Append, {Flatten[Outer[List, x, y], 1], Flatten#z}]
{{i, a, 1}, {i, b, 2}, {i, c, 3}, {j, a, 4}, {j, b, 5}, {j, c, 6}}
Also
Transpose#Append[Transpose#Tuples#{x, y}, Flatten#z]
{{i, a, 1}, {i, b, 2}, {i, c, 3}, {j, a, 4}, {j, b, 5}, {j, c, 6}}
Perhaps
x = {i, j}; y = {a, b, c}; z = {{1, 2, 3}, {4, 5, 6}};
zz=MapThread[Flatten[List[#1,#2]] &, {Flatten[Outer[List,x,y],1], Flatten[z]}]
which returns
{{i, a, 1}, {i, b, 2}, {i, c, 3}, {j, a, 4}, {j, b, 5}, {j, c, 6}}
It seems like there should be a simpler way of doing this.
Join ## (Thread[{x[[#]], y, z[[#]]}, List] & /# {1, 2})
{{i, a, 1}, {i, b, 2}, {i, c, 3}, {j, a, 4}, {j, b, 5}, {j, c, 6}}
Let
n=2^10 3^7 5^4...31^2...59^2 61...97
be the factorization of an integer such that the powers of primes are non-increasing.
I would like to write a code in Mathematica to find Min and Max of prime factor of n such that they have the same power.
for example I want a function which take r(the power) and give (at most two) primes in general. A specific answer for the above sample is
minwithpower[7]=3
maxwithpower[7]=3
minwithpower[2]=31
maxwithpower[2]=59
Any idea please.
Let n = 91065388654697452410240000 then
FactorInteger[n]
returns
{{2, 10}, {3, 7}, {5, 4}, {7, 4}, {31, 2}, {37, 2}, {59, 2}, {61, 1}, {97, 1}}
and the expression
Cases[FactorInteger[n], {_, 2}]
returns only those elements from the list of factors and coefficients where the coefficient is 2, ie
{{31, 2}, {37, 2}, {59, 2}}
Next, the expression
Cases[FactorInteger[n], {_, 2}] /. {{min_, _}, ___, {max_, _}} -> {min, max}
returns
{31, 59}
Note that this approach fails if the power you are interested in only occurs once in the output from FactorInteger, for example
Cases[FactorInteger[n], {_, 7}] /. {{min_, _}, ___, {max_, _}} -> {min, max}
returns
{{3, 7}}
but you should be able to fix that deficiency quite easily.
One solution is :
getSamePower[exp_, n_] := With[{powers =
Select[ReleaseHold[n /. {Times -> List, Power[a_, b_] -> {a, b}}], #[[2]] ==
exp &]},
If[Length[powers] == 1, {powers[[1, 1]], powers[[1, 1]]}, {Min[powers[[All, 1]]], Max[powers[[All, 1]]]}]]
to be used as :
getSamePower[7, 2^10 3^7 5^4 \[Pi]^1 31^2 E^1 59^2 61^1 I^1 97^1 // HoldForm]
(* {3, 3} *)
getSamePower[2, 2^10 3^7 5^4 \[Pi]^1 31^2 E^1 59^2 61^1 I^1 97^1 // HoldForm]
(* {31, 59} *)
Given pairs of coordinates
data = {{1, 0}, {2, 0}, {3, 1}, {4, 2}, {5, 1},
{6, 2}, {7, 3}, {8, 4}, {9, 3}, {10, 2}}
I'd like to extract peaks and valleys, thus:
{{4, 2}, {5, 1}, {8, 4}}
My current solution is this clumsiness:
Cases[
Partition[data, 3, 1],
{{ta_, a_}, {tb_, b_}, {tc_, c_}} /; Or[a < b > c, a > b < c] :> {tb, b}
]
which you can see starts out by tripling the size of the data set using Partition. I think it's possible to use Cases and PatternSequence to extract this information, but this attempt doesn't work:
Cases[
data,
({___, PatternSequence[{_, a_}, {t_, b_}, {_, c_}], ___}
/; Or[a < b > c, a > b < c]) :> {t, b}
]
That yields {}.
I don't think anything is wrong with the pattern because it works with ReplaceAll:
data /. ({___, PatternSequence[{_, a_}, {t_, b_}, {_, c_}], ___}
/; Or[a < b > c, a > b < c]) :> {t, b}
That gives the correct first peak, {4, 2}. What's going on here?
One of the reasons why your failed attempt doesn't work is that Cases by default looks for matches on level 1 of your expression. Since your looking for matches on level 0 you would need to do something like
Cases[
data,
{___, {_, a_}, {t_, b_}, {_, c_}, ___} /; Or[a < b > c, a > b < c] :> {t, b},
{0}
]
However, this only returns {4,2} as a solution so it's still not what you're looking for.
To find all matches without partitioning you could do something like
ReplaceList[data, ({___, {_, a_}, {t_, b_}, {_, c_}, ___} /;
Or[a < b > c, a > b < c]) :> {t, b}]
which returns
{{4, 2}, {5, 1}, {8, 4}}
Your "clumsy" solution is fairly fast, because it heavily restricts what gets looked at.
Here is an example.
m = 10^4;
n = 10^6;
ll = Transpose[{Range[n], RandomInteger[m, n]}];
In[266]:=
Timing[extrema =
Cases[Partition[ll, 3,
1], {{ta_, a_}, {tb_, b_}, {tc_, c_}} /;
Or[a < b > c, a > b < c] :> {tb, b}];][[1]]
Out[266]= 3.88
In[267]:= Length[extrema]
Out[267]= 666463
This seems to be faster than using replacement rules.
Faster still is to create a sign table of products of differences. Then pick entries not on the ends of the list that correspond to sign products of 1.
In[268]:= Timing[ordinates = ll[[All, 2]];
signs =
Table[Sign[(ordinates[[j + 1]] -
ordinates[[j]])*(ordinates[[j - 1]] - ordinates[[j]])], {j, 2,
Length[ll] - 1}];
extrema2 = Pick[ll[[2 ;; -2]], signs, 1];][[1]]
Out[268]= 0.23
In[269]:= extrema2 === extrema
Out[269]= True
Handling of consecutive equal ordinates is not considered in these methods. Doing that would take more work since one must consider neighborhoods larger than three consecutive elements. (My spell checker wants me to add a 'u' to the middle syllable of "neighborhoods". My spell checker must think we are in Canada.)
Daniel Lichtblau
Another alternative:
Part[#,Flatten[Position[Differences[Sign[Differences[#[[All, 2]]]]], 2|-2] + 1]] &#data
(* ==> {{4, 2}, {5, 1}, {8, 4}} *)
Extract[#, Position[Differences[Sign[Differences[#]]], {_, 2} | {_, -2}] + 1] &#data
(* ==> {{4, 2}, {5, 1}, {8, 4}} *)
This may be not exactly the implementation you ask, but along those lines:
ClearAll[localMaxPositions];
localMaxPositions[lst : {___?NumericQ}] :=
Part[#, All, 2] &#
ReplaceList[
MapIndexed[List, lst],
{___, {x_, _}, y : {t_, _} .., {z_, _}, ___} /; x < t && z < t :> y];
Example:
In[2]:= test = RandomInteger[{1,20},30]
Out[2]= {13,9,5,9,3,20,2,5,18,13,2,20,13,12,4,7,16,14,8,16,19,20,5,18,3,15,8,8,12,9}
In[3]:= localMaxPositions[test]
Out[3]= {{4},{6},{9},{12},{17},{22},{24},{26},{29}}
Once you have positions, you may extract the elements:
In[4]:= Extract[test,%]
Out[4]= {9,20,18,20,16,20,18,15,12}
Note that this will also work for plateau-s where you have more than one same maximal element in a row. To get minima, one needs to trivially change the code. I actually think that ReplaceList is a better choice than Cases here.
To use it with your data:
In[7]:= Extract[data,localMaxPositions[data[[All,2]]]]
Out[7]= {{4,2},{8,4}}
and the same for the minima. If you want to combine, the change in the above rule is also trivial.
Since one of your primary concerns about your "clumsy" method is the data expansion that takes place with Partition, you may care to know about the Developer` function PartitionMap, which does not partition all the data at once. I use Sequence[] to delete the elements that I don't want.
Developer`PartitionMap[
# /. {{{_, a_}, x : {_, b_}, {_, c_}} /; a < b > c || a > b < c :> x,
_ :> Sequence[]} &,
data, 3, 1
]
Suppose I have a series of strips of paper placed along an infinite ruler, with start and end points specified by pairs of numbers. I would like to create a list representing the number of layers of paper at points along the ruler.
For example:
strips =
{{-27, 20},
{ -2, -1},
{-47, -28},
{-41, 32},
{ 22, 31},
{ 2, 37},
{-28, 30},
{ -7, 39}}
Should output:
-47 -41 -27 -7 -2 -1 2 20 22 30 31 32 37 39
1 2 3 4 5 4 5 4 5 4 3 2 1 0
What is the most efficient, clean, or terse way to do this, accommodating Real and Rational strip positions?
Here's one approach:
Clear[hasPaper,nStrips]
hasPaper[y_, z_] := Piecewise[{{1, x <= z && x >= y}}, 0];
nStrips[y_, strip___] := Total#(hasPaper ### strip) /. x -> y
You can get the number of strips at any value.
Table[nStrips[i, strips], {i, Sort#Flatten#strips}]
{1, 2, 3, 3, 3, 4, 5, 5, 5, 5, 5, 5, 4, 3, 2, 1}
Also, plot it
Plot[nStrips[x, strips], {x, Min#Flatten#strips, Max#Flatten#strips}]
Here is one solution:
In[305]:=
strips = {{-27, 20}, {-2, -1}, {-47, -28}, {-41, 32}, {22, 31}, {2,
37}, {-28, 30}, {-7, 39}};
In[313]:= int = Interval /# strips;
In[317]:= Thread[{Union[Flatten[strips]],
Join[Count[int, x_ /; IntervalMemberQ[x, #]] & /# (Mean /#
Partition[Union[Flatten[strips]], 2, 1]), {0}]}]
Out[317]= {{-47, 1}, {-41, 2}, {-28, 2}, {-27, 3}, {-7, 4}, {-2,
5}, {-1, 4}, {2, 5}, {20, 4}, {22, 5}, {30, 4}, {31, 3}, {32,
2}, {37, 1}, {39, 0}}
EDIT Using SplitBy and postprocessing the following code gets the shortest list:
In[329]:=
strips = {{-27, 20}, {-2, -1}, {-47, -28}, {-41, 32}, {22, 31}, {2,
37}, {-28, 30}, {-7, 39}};
In[330]:= int = Interval /# strips;
In[339]:=
SplitBy[Thread[{Union[Flatten[strips]],
Join[Count[int, x_ /; IntervalMemberQ[x, #]] & /# (Mean /#
Partition[Union[Flatten[strips]], 2, 1]), {0}]}],
Last] /. {b : {{_, co_} ..} :> First[b]}
Out[339]= {{-47, 1}, {-41, 2}, {-27, 3}, {-7, 4}, {-2, 5}, {-1,
4}, {2, 5}, {20, 4}, {22, 5}, {30, 4}, {31, 3}, {32, 2}, {37,
1}, {39, 0}}
You may regard this as a silly approach, but I'll offer it anyway:
f[x_]:=Sum[UnitStep[x-strips[[k,1]]]-UnitStep[x-strips[[k,2]]],{k,Length[strips]}]
f/#Union[Flatten[strips]]
f[u_, s_] := Total[Piecewise#{{1, #1 <= x < #2}} & ### s /. x -> u]
Usage
f[#, strips] & /# {-47, -41, -27, -7, -2, -1, 2, 20, 22, 30, 31, 32, 37, 39}
->
{1, 2, 3, 4, 5, 4, 5, 4, 5, 4, 3, 2, 1, 0}
For Open/Closed ends, just use <= or <
Here's my approach, similar to belisarius':
strips = {{-27, 20}, {-2, -1}, {-47, -28}, {-41, 32}, {22, 31}, {2,
37}, {-28, 30}, {-7, 39}};
pw = PiecewiseExpand[Total[Boole[# <= x < #2] & ### strips]]
Grid[Transpose[
SplitBy[SortBy[Table[{x, pw}, {x, Flatten[strips]}], First],
Last][[All, 1]]], Alignment -> "."]
Here's my attempt - it works on integers, rationals and reals, but makes no claim to being terribly efficient. (I made the same mistake as Sasha, my original version did not return the shortest list. So I stole the SplitBy fix!)
layers[strips_?MatrixQ] := Module[{equals, points},
points = Union#Flatten#strips;
equals = Function[x, Evaluate[(#1 <= x < #2) & ### strips]];
points = {points, Total /# Boole /# equals /# points}\[Transpose];
SplitBy[points, Last] /. {b:{{_, co_}..} :> First[b]}]
strips = {{-27, 20}, {-2, -1}, {-47, -28}, {-41, 32}, {22, 31},
{2, 37}, {-28, 30}, {-7, 39}};
In[3]:= layers[strips]
Out[3]= {{-47, 1}, {-41, 2}, {-27, 3}, {-7, 4}, {-2, 5}, {-1, 4}, {2, 5},
{20, 4}, {22, 5}, {30, 4}, {31, 3}, {32, 2}, {37, 1}, {39, 0}}
In[4]:= layers[strips/2]
Out[4]:= {{-(47/2), 1}, {-(41/2), 2}, {-(27/2), 3}, {-(7/2), 4},
{-1, 5}, {-(1/2), 4}, {1, 5}, {10, 4}, {11, 5}, {15, 4}, {31/2, 3},
{16, 2}, {37/2, 1}, {39/2, 0}}
In[5]:= layers[strips/3.]
Out[5]= {{-15.6667, 1}, {-13.6667, 2}, {-9., 3}, {-2.33333, 4}, {-0.666667, 5},
{-0.333333, 4}, {0.666667, 5}, {6.66667, 4}, {7.33333, 5}, {10.,4},
{10.3333, 3}, {10.6667, 2}, {12.3333, 1}, {13., 0}}
Splice together abutting strips, determine key points where number of layers
changes, and calculate how many strips each key point inhabits:
splice[s_, {}] := s
splice[s_, vals_] := Module[{h = First[vals]},
splice[(s /. {{x___, {k_, h}, w___, {h, j_}, z___} :> {x, {k, j},
w, z}, {x___, {k_, h}, w___, {h, j_}, z___} :> {x, {k, j}, w,
z}}), Rest[vals]]]
splicedStrips = splice[strips, Union#Flatten#strips];
keyPoints = Union#Flatten#splicedStrips;
({#, Total#(splicedStrips /. {a_, b_} :> Boole[a <= # < b])} & /# keyPoints)
// Transpose // TableForm
EDIT
After some struggling I was able to remove splice and more directly eliminate points that did not need checking (-28, in the strips data we've been using) :
keyPoints = Complement[pts = Union#Flatten#strips,
Cases[pts, x_ /; MemberQ[strips, {x, _}] && MemberQ[strips, {_, x}]]];
({#, Total#(strips /. {a_, b_} :> Boole[a <= # < b])} & /# keyPoints)
One approach of solving this is converting the strips
strips = {{-27, 20}, {-2, -1}, {-47, -28}, {-41, 32}
,{ 22, 31}, { 2, 37}, {-28, 30}, {-7, 39}}
to a list of Delimiters, marking the beginning or end of a strip and sort them by position
StripToLimiters[{start_, end_}] := Sequence[BeginStrip[start], EndStrip[end]]
limiterlist = SortBy[StripToLimiters /# strips, First]
Now we can map the sorted limiters to increments/decrements
LimiterToDiff[BeginStrip[_]] := 1
LimiterToDiff[EndStrip[_]] := -1
and use Accumulate to get the intermediate totals of intersected strips:
In[6]:= Transpose[{First/##,Accumulate[LimiterToDiff/##]}]&[limiterlist]
Out[6]= {{-47,1},{-41,2},{-28,3},{-28,2},{-27,3},{-7,4},{-2,5},{-1,4}
,{2,5},{20,4},{22,5},{30,4},{31,3},{32,2},{37,1},{39,0}}
Or without the intermediate limiterlist:
In[7]:= StripListToCountList[strips_]:=
Transpose[{First/##,Accumulate[LimiterToDiff/##]}]&[
SortBy[StripToLimiters/#strips,First]
]
StripListToCountList[strips]
Out[8]= {{-47,1},{-41,2},{-28,3},{-28,2},{-27,3},{-7,4},{-2,5},{-1,4}
,{2,5},{20,4},{22,5},{30,4},{31,3},{32,2},{37,1},{39,0}}
The following solution assumes that the layer count function will be called a large number of times. It uses layer precomputation and Nearest in order to greatly reduce the amount of time required to compute the layer count at any given point:
layers[strips:{__}] :=
Module[{pred, changes, count}
, changes = Union # Flatten # strips /. {c_, r___} :> {c-1, c, r}
; Evaluate[pred /# changes] = {changes[[1]]} ~Join~ Drop[changes, -1]
; Do[count[x] = Total[(Boole[#[[1]] <= x < #[[2]]]) & /# strips], {x, changes}]
; With[{n = Nearest[changes]}
, (n[#] /. {m_, ___} :> count[If[m > #, pred[m], m]])&
]
]
The following example uses layers to define a new function f that will compute the layer count for the provided sample strips:
$strips={{-27,20},{-2,-1},{-47,-28},{-41,32},{22,31},{2,37},{-28,30},{-7,39}};
f = layers[$strips];
f can now be used to compute the number of layers at a point:
Union # Flatten # $strips /. s_ :> {s, f /# s} // TableForm
Plot[f[x], {x, -50, 50}, PlotPoints -> 1000]
For 1,000 layers and 10,000 points, the precomputation stage can take quite a bit of time, but individual point computation is relatively quick:
I'm writing a game of life program in mathematica however there is a caveat in that I need to be able to apply the reproduction rules to some percentage of the cells, I want to try a new method using MapAt but liveNeighbors doesn't work elementwise, and I can't think of a way of fixing it without doing exactly what I did before (lots of messy indexing), does anyone have any suggestions? (I am assuming this will be more efficient then the old method, which is listed below, if not please let me know, I am just a beginner!).
What I am trying to do:
Map[ArrayPlot,FixedPointList[MapAt[update[#,liveNeighbors[#]]&,#,coords]&,Board, 1]]
What I have done already:
LifeGame[ n_Integer?Positive, steps_] := Module [{Board, liveNeighbors, update},
Board = Table [Random [Integer], {n}, {n}];
liveNeighbors[ mat_] :=
Apply[Plus,Map[RotateRight[mat,#]&,{{-1,-1},{-1, 0},{-1,1}, {0, -1}, {0, 1}, {1, -1}, {1, 0}, {1, 1}}]];
update[1, 2] := 1;
update[_, 3] := 1;
update[ _, _] := 0;
SetAttributes[update, Listable];
Seed = RandomVariate[ProbabilityDistribution[0.7 UnitStep[x] + 0.3 UnitStep[x - 1], {x, 0, 1, 1}], {n, n}];
FixedPointList[Table[If[Seed[[i, j]] == 1,update[#[[i, j]], liveNeighbors[#][[i, j]]],#[[i, j]]], {i, n}, {j, n}]&, Board, steps]]]
Thanks!
In[156]:=
LifeGame2[n_Integer?Positive, steps_] :=
Module[{Board, liveNeighbors, update},
Board = RandomInteger[1, {n, n}];
liveNeighbors[mat_] :=
ListConvolve[{{1, 1, 1}, {1, 0, 1}, {1, 1, 1}},
ArrayPad[mat, 1, "Periodic"]];
SetAttributes[update, Listable];
Seed = RandomVariate[BernoulliDistribution[0.3], {n, n}];
update[0, el_, nei_] := el;
update[1, 1, 2] := 1;
update[1, _, 3] := 1;
update[1, _, _] := 0;
FixedPointList[MapThread[update, {Seed, #, liveNeighbors[#]}, 2] &,
Board, steps]
]
This implementation does the same as yours, except is quite a lot faster:
In[162]:= AbsoluteTiming[
res1 = BlockRandom[SeedRandom[11]; LifeGame[20, 100]];]
Out[162]= {6.3476347, Null}
In[163]:= Timing[BlockRandom[Seed[11]; LifeGame2[20, 100]] == res1]
Out[163]= {0.047, True}
Assuming you don't have to roll your own code for a homework problem, have you considered just using the built-in CellularAutomaton function?
Straight from the documentation, the 2D CA rule:
GameOfLife = {224, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}};
And iterate over a 100x100 grid for 100 steps:
ArrayPlot[CellularAutomaton[GameOfLife, RandomInteger[1, {100, 100}], {{{100}}}]]
It would at least give you a baseline for a speed comparison.
Instead of MapAt, you could use Part with the Span syntax to replace a whole subarray at once:
a = ConstantArray[0, {5, 5}];
a[[2 ;; 4, 2 ;; 4]] = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}
HTH!
Here you have my golfed version.