Efficient method for Inserting arrays into arrays

Question

This is strongly related to the following Questions:

• Looking for a way to insert multiple elements into multiple positions simultaneously in a list
• Building matrices by attaching vectors (columns) and matrices
• What is the most efficient way to add rows and columns to a matrix?

I am looking for an efficient method to insert an array into an array. Let’s assume that both arrays have similar number of rows. Let’s further assume insertion is by column.

a = {{a1, b1, c1, d1, e1}, {a2, b2, c2, d2, e2}, {a3, b3, c3, d3, e3}}; (*example may be larger*)
b = {{x1, y1, z1}, {x2, y2, z2}, {x3, y3, z3}}; (*example may be larger*)

Q1: What is the fastest method to insert b into a at a position p. Let’s assume that both arrays have similar number of rows. Let’s further assume insertion is by column.

My current method is as follows:

insertarray[a_, b_, pos_] :=
FlattenAt[
Transpose@Insert[Transpose@a, b[[##]], pos ] & [Range[Length@a]], 
Table[{i, pos}, {i, 1, Length@a}]];

which allows me to do

Q2: Some times I have to intersperse columns of b into a at multiple positions. How do I best extend insertarray so that:

insertarray[a,b,{1,2,5}]//MatrixForm

Answer 1

Q1

Join[a[[;; , ;; 1]], b, a[[;; , 2 ;;]], 2]

One may want to create function

insCol[a_, b_, n_] := Join[a[[;; , ;; n - 1]], b, a[[;; , n ;;]], 2]

insRow[a_, b_, n_] := Join[a[[;; n - 1]], b, a[[n ;;]]]

Answer 2

After a few attempts, the following gives a noticeable if unremarkable improvement over insertarray:

columnInsert[a_, b_, pos_] := 
  MapThread[Join[#1[[;; pos - 1]], #2, #1[[pos ;;]]] &, {a, b}];

Your second functionality doensn't actually need anything fancy, just some swaps afterwards:

SetAttributes[columnInsert`swap, HoldFirst];
columnInsert`swap[mat_, first_, list_] :=
 Module[{i = first}, 
  Scan[columnInsert`tmp = mat[[All, i]]; 
     mat[[All, i++]] = mat[[All, #]]; 
     mat[[All, #]] = columnInsert`tmp; &, list]; mat]
columnInsert[a_, b_, pos_List] := 
 Module[{res = columnInsert[a, b, First@pos]},
  columnInsert`swap[res, First@pos, pos]; res]

Answer 3

I haven't had a chance to test if this is faster (probably not), but I find it easier to read:

Block[{x, T = Transpose}, Insert[T@a, x, 2] /. x -> Sequence @@ T@b // T]

or alternately:

MapThread[Flatten[## ~Insert~ 2] &, {a, b}]

Both give the following:

Answer 4

Inserting b at column p :-

a = {{a1, b1, c1, d1, e1}, {a2, b2, c2, d2, e2}, {a3, b3, c3, d3, e3}};
b = {{x1, y1, z1}, {x2, y2, z2}, {x3, y3, z3}};

p = 4;

c = ReleaseHold@MapThread[Insert, {a, Hold[Sequence @@ #] & /@ b, Table[p, {Length@a}]}];

MatrixForm[c]