Skip to main content

list manipulation - Make a huge vector in a wise way


I need to construct a vector similar to:


v[x_]:={0, 0, x, 0, 0, 2*x, 0, 0, 3*x, ....., 0, 0, n*x};

where n=10^9. How can I make such a vector wisely?



Answer



When n is large and x is known, using a PackedArray may be a good option.


ar3=ConstantArray[0,3n];
ar3[[3;;;;3]]=Range[x, n x, x];

ar3

To see that the result is a PackedArray, we see that


<< Developer`;
PackedArrayQ[ar]


True



Whereas for Kuba's last array we would have False (even if x has a value). Note that ConstantArray also produces a PackedArray.



Whether this is useful really depends on what you want to do with the List. PackedArrays take up less space in memory, which is an advantage, but they probably take up more space than a SparseArray (in this case). Computations using PackedArrays can be much faster as well.


Some timings/comparisons


I mainly compare my code with Kuba's MapAt answer. I also compare with my original answer, found further below.


x = 5;
i = 0;
n = 1000000;

(
ar = ConstantArray[0, 3 n];
Do[

ar[[3 i]] = i x
,
{i, 1, n}
];
ar) // Timing // First

(ar2 = MapAt[(++i; i x) &, ConstantArray[0, 3 n], 3 ;; ;; 3]) //
Timing // First

(ar3 = ConstantArray[0, 3 n];

ar3[[3 ;; ;; 3]] = Range[x, n x, x];
ar3
) // Timing // First


3.125590
3.139499
0.048840



So generating the PackedArray can be considerably faster, if done right. ar1 and ar3 must be the same as they are Equal and both PackedArrays, but I added timings/comparisons using both these anyway.



ByteCount[ar]
ByteCount[ar2]
ByteCount[ar3]


24000144
72000080
24000144



which is in favor of the PackedArray. Doing calculations may also be faster using a PackedArray, like in the following example



ar // Total // Timing
ar2 // Total // Timing
ar3 //Total //Timing


{0.010633, 2500002500000}
{0.629586, 2500002500000}
{0.012352, 2500002500000}



Original answer



Note: It now turns out this use of Do is not so nice


This is very similar to Kuba's answer with MapAt. But I think Do works better with PackedArrays. The following gives a "list" of the required form.


n = 500;
ar = ConstantArray[0, 3 n];
Do[
ar[[3 i]] = i x
,
{i, 1, n}
];
ar

Comments

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]