Remove a list element based on condition

Question

I have a list with many expressions as elements. I want to remove the list elements which do not contain certain variables. For example, I would like to remove the element which does not contain x1 in the list I below:

I = {2x1 + 3x2 - 5x1^2, 7x2 - 3x2^2, 2x1 + 5x2 - 9x3}

to become

I = {2x1 + 3x2 - 5x1^2, 2x1 + 5x2 - 9x3}.

Also, I would like to combine conditions to check if the element does not contain any of a list of variables, and remove it if so. For example if neither x1 nor x2 in I[[i]], then remove I[[i]].

Any help is appreciated.

Answer 1

Removing elements which do not contain x1 (i.e. keeping elements which do contain x1)

expr = {2 x1 + 3 x2 - 5 x1^2, 7 x2 - 3 x2^2, 2 x1 + 5 x2 - 9 x3};
Select[expr, MemberQ[Variables[#], x1] &]

{2 x1 - 5 x1^2 + 3 x2, 2 x1 + 5 x2 - 9 x3}

Keeping elements which contain at least one of the variables from a list

(* changing example list to include a term which contains neither `x1` or x2` *)
expr = {2 x1 + 3 x2 - 5 x1^2, 7 x2 - 3 x2^2, 2 x1 + 5 x2 - 9 x3, x3 + x4};
wanted = {x1, x2};

Select[expr, 
 Table[MemberQ[Variables[#], v], {v, wanted}] /. List -> Or &]

{2 x1 - 5 x1^2 + 3 x2, 7 x2 - 3 x2^2, 2 x1 + 5 x2 - 9 x3}

Answer 2

list = {2 x1 + 3 x2 - 5 x1^2, 7 x2 - 3 x2^2, 2 x1 + 5 x2 - 9 x3, Sin[x1 + x4] + x5};

You can use FreeQ with DeleteCases, Cases, Select and Pick as follows:

DeleteCases[_?(FreeQ[x1])]@list

Cases[_?(Not@*FreeQ[x1])]@list

Select[Not@*FreeQ[x1]]@list

Pick[list, Not @* FreeQ[x1] /@ list]

All methods above return

{2 x1 - 5 x1^2 + 3 x2, 2 x1 + 5 x2 - 9 x3, x5 + Sin[x1 + x4]}

To delete cases that does not contain x2 or x3 (to keep only those case that contain both x2 and x3) use Alternatives (|) in the pattern inside FreeQ:

wanted =  {x2, x3};
alt = Alternatives @@ wanted;

DeleteCases[_?(FreeQ[alt])]@list

Cases[_?(Not@*FreeQ[alt])]@list

Select[Not@*FreeQ[alt]]@list

Pick[list, Not @* FreeQ[alt] /@ list]

All four methods give

{2 x1 - 5 x1^2 + 3 x2, 7 x2 - 3 x2^2, 2 x1 + 5 x2 - 9 x3}

You can also use a combination of Level and ContainsNone or ContainsAny as follows:

DeleteCases[_?(ContainsNone[Level[#, {-1}], wanted] &)]@list

Cases[_?(ContainsAny[Level[#, {-1}], wanted] &)]@list

Select[ContainsAny[Level[#, {-1}], wanted] &]@list

Pick[#, ContainsAny[Level[#, {-1}], wanted] & /@ #] &@list

to get

{2 x1 - 5 x1^2 + 3 x2, 7 x2 - 3 x2^2, 2 x1 + 5 x2 - 9 x3}

Note: Variables "gives a list of all independent variables in a polynomial":

Variables /@ list

{{x1, x2}, {x2}, {x1, x2, x3}, {x5, Sin[x1 + x4]}}