how to extract rows from matrix based on value in first entry? - wolfram-mathematica

Question

This is another simple 'matrix' question in Mathematica. I want to show how I did this, and ask if there is a better answer.
I want to select all 'rows' from matrix based on value in the first column (or any column, I used first column here just as an example).
Say, find all rows where the entry in the first position is <=4 in this example:
list = {{1, 2, 3},
{4, 5, 8},
{7 , 8, 9}}
So, the result should be
{{1,2,3},
{4,5,8}}
Well, the problem is I need to use Position, since the result returned by Position can be used directly by Extract. (but can't be used by Part or [[ ]], so that is why I am just looking at Position[] ).
But I do not know how to tell Position to please restrict the 'search' pattern to only the 'first' column so I can do this in one line.
When I type
pos = Position[list, _?(# <= 4 &)]
it returns position of ALL entries which are <=4.
{{1, 1}, {1, 2}, {1, 3}, {2, 1}}
If I first get the first column, then apply Position on it, it works ofcourse
list = {{1, 2, 3},
{4, 5, 8},
{7 , 8, 9}};
pos = Position[list[[All, 1]], _?(# <= 4 &)]
Extract[list, pos]
--> {{1, 2, 3}, {4, 5, 8}}
Also I tried this:
pos = Position[list, _?(# <= 4 &)];
pos = Select[pos, #[[2]] == 1 &] (*only look at ones in the 'first' column*)
{{1, 1}, {2, 1}}--->
and this gives me the correct positions in the first column. To use that to find all rows, I did
pos = pos[[All, 1]] (* to get list of row positions*)
---> {1, 2}
list[[ pos[[1]] ;; pos[[-1]], All]]
{{1, 2, 3},
{4, 5, 8}}
So, to summarize, putting it all together, this is what I did:
method 1
list = {{1, 2, 3},
{4, 5, 8},
{7 , 8, 9}};
pos = Position[list[[All, 1]], _?(# <= 4 &)]
Extract[list, pos]
--> {{1, 2, 3}, {4, 5, 8}}
method 2
list = {{1, 2, 3},
{4, 5, 8},
{7 , 8, 9}}
pos = Position[list, _?(# <= 4 &)];
pos = Select[pos, #[[2]] == 1 &];
pos = pos[[All, 1]];
list[[ pos[[1]] ;; pos[[-1]], All]]
{{1, 2, 3},
{4, 5, 8}}
The above clearly is not too good.
Is method 1 above the 'correct' functional way to do this?
For reference, this is how I do the above in Matlab:
EDU>> A=[1 2 3;4 5 8;7 8 9]
A =
1 2 3
4 5 8
7 8 9
EDU>> A( A(:,1)<=4 , :)
1 2 3
4 5 8
I am trying to improve my 'functional' handling of working with matrices in Mathematica commands, this is an area I feel I am not good at working with lists. I find working with matrices easier for me.
The question is: Is there is a shorter/more functional way to do this in Mathematica?
thanks

Answer 1

You could use Pick[] as follows:
Pick[list, list[[All, 1]], _?(# <= 4 &)]

Answer 2

How about the following?
In[1]:= list = {{1, 2, 3}, {4, 5, 8}, {7, 8, 9}};
In[2]:= Select[list, First[#] <= 4 &]
Out[2]= {{1, 2, 3}, {4, 5, 8}}
Here's a loose translation of your matlab code:
list[[Flatten[Position[Thread[list[[All, 1]] <= 4], True]]]]
(of course, the Flatten would not be needed if I used Extract instead of Part).

Answer 3

There is a faster method than those already presented, using SparseArray. It is:
list ~Extract~
SparseArray[UnitStep[4 - list[[All, 1]]]]["NonzeroPositions"]
Here are speed comparisons with the other methods. I had to modify WReach's method to handle other position specifications.
f1[list_, x_] := Cases[list, {Sequence ## Table[_, {x - 1}], n_, ___} /; n <= 4]
f2[list_, x_] := Select[list, #[[x]] <= 4 &]
f3[list_, x_] := Pick[list, (#[[x]] <= 4 &) /# list]
f4[list_, x_] := Pick[list, UnitStep[4 - list[[All, x]]], 1]
f5[list_, x_] := Pick[list, Thread[list[[All, x]] <= 4]]
f6[list_, x_] := list ~Extract~
SparseArray[UnitStep[4 - list[[All, x]]]]["NonzeroPositions"]
For a table with few rows and many columns (comparing position 7):
a = RandomInteger[99, {250, 150000}];
timeAvg[#[a, 7]] & /# {f1, f2, f3, f4, f5, f6} // Column
0.02248
0.0262
0.312
0.312
0.2808
0.0009728
For a table with few columns and many rows (comparing position 7):
a = RandomInteger[99, {150000, 12}];
timeAvg[#[a, 7]] & /# {f1, f2, f3, f4, f5, f6} // Column
0.0968
0.1434
0.184
0.0474
0.103
0.002872

Answer 4

If you want the rows that meet the criteria, use Cases:
Cases[list, {n_, __} /; n <= 4]
(* {{1, 2, 3}, {4, 5, 8}} *)
If you want the positions within the list rather than the rows themselves, use Position instead of Cases (restricted to the first level only):
Position[list, {n_, __} /; n <= 4, {1}]
(* {{1}, {2}} *)

Answer 5

If you want to be very clever:
Pick[list, UnitStep[4 - list[[All, 1]]], 1]
This also avoids unpacking, which means it'll be faster and use less memory.