Compare some lists, if elements are almost equal

Question

I've got the solutions of three systems on two variables (a and b) that gives results sometimes almost equal. The three list are not of equal length! For example:

sol1={{1.13627*10^-10, 0.517141}, {1822.1623, 0.130514}, {4104.7111, 
0.303997}, {3.89363*10^19, 1.01968}};
sol2={{5.56174*10^-10, 0.517141}, {1822.1612, 0.130514}, {3592.4222, 0.26549}};
sol3={{1.75707*10^-11, 0.517141}, {1822.1723, 0.130514}, {3273.7822, 
0.228122}, {806077., 2.98304}}

As you can notice the second term of sol1, the second term of sol2, and the second term of sol3 are almost equal. I want to extract only one of them, for example {1822.1623, 0.130514}. So, can I extract the couple of solutions that are almost equal? For example if |ai-aj|<0.01 and |bi-bj|<0.0001? Thanks!

Answer 1

You can use Intersection or Union with the option SameTest. For example, solutions for all 3 equations:

Intersection[sol1, sol2, sol3, 
  SameTest -> (Abs[#[[1]] - #2[[1]]] < 0.01 && Abs[#[[2]] - #2[[2]]] < 0.0001 &)]

(* {{1.13627*10^-10, 0.517141}, {1822.16, 0.130514}} *)

Or if you want to gather almost equal solutions:

Gather[Join[sol1, sol2, sol3], 
 Abs[#[[1]] - #2[[1]]] < 0.01 && Abs[#[[2]] - #2[[2]]] < 0.0001 &]

(* {{{1.13627*10^-10, 0.517141}, {5.56174*10^-10, 0.517141}, {1.75707*10^-11, 0.517141}}, 
    {{1822.16, 0.130514}, {1822.16, 0.130514}, {1822.17, 0.130514}},
    {{4104.71, 0.303997}}, 
    {{3.89363*10^19, 1.01968}}, 
    {{3592.42, 0.26549}}, 
    {{3273.78, 0.228122}}, 
    {{806077., 2.98304}}} *)

Or to find all solutions, which are close to sol1[[2]]

With[{s = sol1[[2]]},
 Select[Join[sol1, sol2, sol3], 
  Abs[#[[1]] - s[[1]]] < 0.01 && Abs[#[[2]] - s[[2]]] < 0.0001 &]]
(* {{1822.16, 0.130514}, {1822.16, 0.130514}, {1822.17, 0.130514}} *)

I hope one of these cases helps you.