list manipulation - How to use Reap and Sow instead of Append to

I have matrix and and i want to do increment in a do loop and and i want to store in list using append to. It works fine for less values, But when i want to do it for values around 1 million the program is slow.

 x = {{0.1},{0.2}};xm = {{0.4},{0.5}};
Do[datax[i] = {}; dataxm[i] = {};, {i, 1, 2}]


Do[
 x = x + t;
xm = xm + t;
 Do[AppendTo[datax[i], Flatten[{t, x[[i]]}]];, {i, 1, 2}];
Do[AppendTo[dataxm[i], Flatten[{t, xm[[i]]}]];, {i, 1, 2}], {t, 0, 1, 0.1}]


 datax[1]


 {{0., 0.1}, {0.1, 0.2}, {0.2, 0.4}, {0.3, 0.7}, {0.4, 1.1}, {0.5, 1.6}, {0.6, 2.2}, {0.7, 2.9}, {0.8, 3.7}, {0.9, 4.6}, {1., 5.6}}

datax[2]

{{0., 0.2}, {0.1, 0.3}, {0.2, 0.5}, {0.3, 0.8}, {0.4, 1.2}, {0.5, 
  1.7}, {0.6, 2.3}, {0.7, 3.}, {0.8, 3.8}, {0.9, 4.7}, {1., 5.7}}

dataxm[1]

 {{0., 0.4}, {0.1, 0.5}, {0.2, 0.7}, {0.3, 1.}, {0.4, 

  1.4}, {0.5, 1.9}, {0.6, 2.5}, {0.7, 3.2}, {0.8, 4.}, {0.9, 
  4.9}, {1., 5.9}}

 dataxm[2]

 {{0., 0.5}, {0.1, 0.6}, {0.2, 0.8}, {0.3, 1.1}, {0.4, 
  1.5}, {0.5, 2.}, {0.6, 2.6}, {0.7, 3.3}, {0.8, 4.1}, {0.9, 5.}, {1.,
   6.}}

Similar to the above I want to use reap and sow functions to speed it up but i can store only the last value. Why?

 x = {{0.1},{0.2}};xm = {{0.4},{0.5}};


Do[
x = x + t;xm = xm + t;

Do[datax[i] = Reap[Sow[{t, x[[i]]}]][[2, 1]], {i, 1, 2}];

Do[dataxm[i] = Reap[Sow[{t, xm[[i]]}]][[2, 1]], {i, 1, 2}];



, {t, 0, 1, 0.1}]

datax[1]
{{1., {5.6}}}

 datax[2]
{{1., {5.7}}}



datax[1]
{{1., {5.9}}}

datax[2]
{{1., {6.0}}

Can anyone help in fixing this issue? Thanks in advance

Answer

You need olny one Reap for all Sows -- that's primarily the advantage of using them.

Moreover, you may use tags to obtain datax[1] and datax[2]:

x = {{0.1}, {0.2}};
ClearAll[datax];
datax = Association@Reap[
    Do[
     x = x + t;
     Do[
      Sow[
       Flatten[{t, x[[tag]]}], 
       tag
       ],

      {tag, 1, 2}],
     {t, 0, 1, 0.1}],
    _, Rule][[2]]

<|1 -> {{0., 0.1}, {0.1, 0.2}, {0.2, 0.4}, {0.3, 0.7}, {0.4, 1.1}, {0.5, 1.6}, {0.6, 2.2}, {0.7, 2.9}, {0.8, 3.7}, {0.9, 4.6}, {1., 5.6}}, 2 -> {{0., 0.2}, {0.1, 0.3}, {0.2, 0.5}, {0.3, 0.8}, {0.4, 1.2}, {0.5, 1.7}, {0.6, 2.3}, {0.7, 3.}, {0.8, 3.8}, {0.9, 4.7}, {1., 5.7}}|>

Now you can access datax[1] and datax[2] as before.

Because the question changed: This should allow you to assemble more than one list in one go:

data = Merge[
   Reap[
     Do[
      x = x + t;
      xm = xm + t;
      Do[
       Sow[Flatten[{t, x[[i]]}], "datax" -> i];
       Sow[Flatten[{t, xm[[i]]}], "dataxm" -> i]
       ,
       {i, 1, 2}], {t, 0, 1, 0.1}],

     _, #1[[1]] -> Association[#1[[2]] -> #2] &][[2]]
   , Association
   ];

Now data["datax"] and data["dataxm"] should behave like datax and dataxm.

You can obtain essentially the same result much more efficiently with arrays produced by Table:

x = {0.1, 0.2};
xm = {0.4, 0.5};
{datax, dataxm} = Transpose[

   Table[{{{t, t}, x += t}, {{t, t}, xm += t}}, {t, 0, 1, 0.1}], 
   {3, 1, 4, 2}];

Now you have (notice the double brackets):

datax[[1]]
datax[[2]]
dataxm[[1]]
dataxm[[2]]

Even faster for this particular problem is to avoid cursive addition and to use vectorized operations.

The Reap/Sow method from above:

n = 1000000;

x = {0.1, 0.2};
xm = {0.4, 0.5};
First@AbsoluteTiming[
  data = Merge[
     Reap[
       Do[

        x = x + t;
        xm = xm + t;
        Do[
         Sow[Flatten[{t, x[[i]]}], "datax" -> i];
         Sow[Flatten[{t, xm[[i]]}], "dataxm" -> i]
         ,
         {i, 1, 2}], {t, 0., 1., 1./n}],
       _, #1[[1]] -> Association[#1[[2]] -> #2] &][[2]]
     , Association
     ];

  ]

11.4845

The Table method from above:

x = {0.1, 0.2};
xm = {0.4, 0.5};
First@AbsoluteTiming[
  {datax, dataxm} = 

    Transpose[
     Table[{{{t, t}, x += t}, {{t, t}, xm += t}}, {t, 0., 1., 
       1./n}], {3, 1, 4, 2}];
  ]

2.74062

Vectorized version:

x = {0.1, 0.2};

xm = {0.4, 0.5};
First@AbsoluteTiming[
  tlist = Subdivide[0., 1., n];
  slist = Accumulate[tlist];
  datax2 = {
    Transpose[{tlist, x[[1]] + slist}],
    Transpose[{tlist, x[[2]] + slist}]
    };
  dataxm2 = {
    Transpose[{tlist, xm[[1]] + slist}],

    Transpose[{tlist, xm[[2]] + slist}]
    };
  ]

0.061355

So, the latter is 44/187 time faster than the other methods. So you should take these considerations into account when using Reap and Sow.

Errors:

Max[Abs[data["datax"][1] - datax[[1]]]]

Max[Abs[data["datax"][2] - datax[[2]]]]
Max[Abs[data["dataxm"][1] - dataxm[[1]]]]
Max[Abs[data["dataxm"][2] - dataxm[[2]]]]
Max[Abs[datax - datax2]]
Max[Abs[dataxm - dataxm2]]

0.

0.

0.

0.

5.82077*10^-11

5.82077*10^-11