ForEach loop in Mathematica

问题:

I'd like something like this:

each[i_, {1,2,3},
  Print[i]
]

Or, more generally, to destructure arbitrary stuff in the list you're looping over, like:

each[{i_, j_}, {{1,10}, {2,20}, {3,30}},
  Print[i*j]
]

Usually you want to use Map or other purely functional constructs and eschew a non-functional programming style where you use side effects. But here's an example where I think a for-each construct is supremely useful:

Say I have a list of options (rules) that pair symbols with expressions, like

attrVals = {a -> 7, b -> 8, c -> 9}

Now I want to make a hash table where I do the obvious mapping of those symbols to those numbers. I don't think there's a cleaner way to do that than

each[a_ -> v_, attrVals, h[a] = v]

Additional test cases

In this example, we transform a list of variables:

a = 1;
b = 2;
c = 3;
each[i_, {a,b,c}, i = f[i]]

After the above, {a,b,c} should evaluate to {f[1],f[2],f[3]}. Note that that means the second argument to "each" should be held unevaluated if it's a list.

If the unevaluated form is not a list, it should evaluate the second argument. For example:

each[i_, Rest[{a,b,c}], Print[i]]

That should print the values of b and c.

Addendum: To do for-each properly, it should support Break[] and Continue[]. I'm not sure how to implement that. Perhaps it will need to somehow be implemented in terms of For, While, or Do since those are the only loop constructs that support Break[] and Continue[].

And another problem with the answers so far: they eat Return[]s. That is, if you are using a ForEach loop in a function and want to return from the function from within the loop, you can't. Issuing Return inside the ForEach loop seems to work like Continue[]. This just (wait for it) threw me for a loop.

回答1:

Newer versions of Mathematica (6.0+) have generalized versions of Do[] and Table[] that do almost precisely what you want, by taking an alternate form of iterator argument. For instance,

Do[
  Print[i],
  {i, {1, 2, 3}}]

is exactly like your

ForEach[i_, {1, 2, 3,},
  Print[i]]

Alterntatively, if you really like the specific ForEach syntax, you can make a HoldAll function that implements it, like so:

Attributes[ForEach] = {HoldAll};

ForEach[var_Symbol, list_, expr_] :=
  ReleaseHold[
    Hold[
      Scan[
        Block[{var = #},
         expr] &,
      list]]];

ForEach[vars : {__Symbol}, list_, expr_] :=
  ReleaseHold[
    Hold[
      Scan[
        Block[vars,
          vars = #;
          expr] &,
      list]]];

This uses symbols as variable names, not patterns, but that's how the various built-in control structures like Do[] and For[] work.

HoldAll[] functions allow you to put together a pretty wide variety of custom control structures. ReleaseHold[Hold[...]] is usually the easiest way to assemble a bunch of Mathematica code to be evaluated later, and Block[{x = #}, ...]& allows variables in your expression body to be bound to whatever values you want.

In response to dreeves' question below, you can modify this approach to allow for more arbitrary destructuring using the DownValues of a unique symbol.

ForEach[patt_, list_, expr_] := 
  ReleaseHold[Hold[
     Module[{f}, 
       f[patt] := expr; 
       Scan[f, list]]]]

At this point, though, I think you may be better off building something on top of Cases.

ForEach[patt_, list_, expr_] :=
  With[{bound = list},
    ReleaseHold[Hold[
       Cases[bound,
         patt :> expr]; 
       Null]]]

I like making Null explicit when I'm suppressing the return value of a function. EDIT: I fixed the bug pointed out be dreeves below; I always like using With to interpolate evaluated expressions into Hold* forms.

回答2:

I'm years late to the party here, and this is perhaps more an answer to the "meta-question", but something many people initially have a hard time with when programming in Mathematica (or other functional languages) is approaching a problem from a functional rather than structural viewpoint. The Mathematica language has structural constructs, but it's functional at its core.

Consider your first example:

ForEach[i_, {1,2,3},
  Print[i]
]

As several people pointed out, this can be expressed functionally as Scan[Print, {1,2,3}] or Print /@ {1,2,3} (although you should favor Scan over Map when possible, as previously explained, but that can be annoying at times since there is no infix operator for Scan).

In Mathematica, there's usually a dozen ways to do everything, which is sometimes beautiful and sometimes frustrating. With that in mind, consider your second example:

ForEach[{i_, j_}, {{1,10}, {2,20}, {3,30}},
  Print[i*j]
]

... which is more interesting from a functional point of view.

One possible functional solution is to instead use list replacement, e.g.:

In[1]:= {{1,10},{2,20},{3,30}}/.{i_,j_}:>i*j
Out[1]= {10,40,90}

...but if the list was very large, this would be unnecessarily slow since we are doing so-called "pattern matching" (e.g., looking for instances of {a, b} in the list and assigning them to i and j) unnecessarily.

Given a large array of 100,000 pairs, array = RandomInteger[{1, 100}, {10^6, 2}], we can look at some timings:

Rule-replacement is pretty quick:

In[3]:= First[Timing[array /. {i_, j_} :> i*j;]]
Out[3]= 1.13844

... but we can do a little better if we take advantage of the expression structure where each pair is really List[i,j] and apply Times as the head of each pair, turning each {i,j} into Times[i,j]:

In[4]:= (* f@@@list is the infix operator form of Apply[f, list, 1] *)
    First[Timing[Times @@@ array;]]
Out[4]= 0.861267

As used in the implementation of ForEach[...] above, Cases is decidedly suboptimal:

In[5]:= First[Timing[Cases[array, {i_, j_} :> i*j];]]
Out[5]= 2.40212

... since Cases does more work than just the rule replacement, having to build an output of matching elements one-by-one. It turns out we can do a lot better by decomposing the problem differently, and take advantage of the fact that Times is Listable, and supports vectorized operation.

The Listable attribute means that a function f will automatically thread over any list arguments:

In[16]:= SetAttributes[f,Listable]
In[17]:= f[{1,2,3},{4,5,6}]
Out[17]= {f[1,4],f[2,5],f[3,6]}

So, since Times is Listable, if we instead had the pairs of numbers as two separate arrays:

In[6]:= a1 = RandomInteger[{1, 100}, 10^6];
        a2 = RandomInteger[{1, 100}, 10^6];

In[7]:= First[Timing[a1*a2;]]
Out[7]= 0.012661

Wow, quite a bit faster! Even if the input wasn't provided as two separate arrays (or you have more than two elements in each pair,) we can still do something optimal:

In[8]:= First[Timing[Times@@Transpose[array];]]
Out[8]= 0.020391

The moral of this epic is not that ForEach isn't a valuable construct in general, or even in Mathematica, but that you can often obtain the same results more efficiently and more elegantly when you work in a functional mindset, rather than a structural one.

回答3:

The built-in Scan basically does this, though it's uglier:

    Scan[Print[#]&, {1,2,3}]

It's especially ugly when you want to destructure the elements:

    Scan[Print[#[[1]] * #[[2]]]&, {{1,10}, {2,20}, {3,30}}]

The following function avoids the ugliness by converting pattern to body for each element of list.

SetAttributes[ForEach, HoldAll];
ForEach[pat_, lst_, bod_] :=  Scan[Replace[#, pat:>bod]&, Evaluate@lst]

which can be used as in the example in the question.

PS: The accepted answer induced me to switch to this, which is what I've been using ever since and it seems to work great (except for the caveat I appended to the question):

SetAttributes[ForEach, HoldAll];             (* ForEach[pattern, list, body]   *)
ForEach[pat_, lst_, bod_] := ReleaseHold[    (*  converts pattern to body for  *)
  Hold[Cases[Evaluate@lst, pat:>bod];]];     (*   each element of list.        *)

回答4:

The built-in Map function does exactly what you want. It can be used in long form:

Map[Print, {1,2,3}]

or short-hand

Print /@ {1,2,3}

In your second case, you'd use "Print[Times@@#]&/@{{1,10}, {2,20}, {3,30}}"

I'd recommend reading the Mathematica help on Map, MapThread, Apply, and Function. They can take bit of getting used to, but once you are, you'll never want to go back!

回答5:

Here is a slight improvement based on the last answer of dreeves that allows to specify the pattern without Blank (making the syntax similar to other functions like Table or Do) and that uses the level argument of Cases

SetAttributes[ForEach,HoldAll];
ForEach[patt_/; FreeQ[patt, Pattern],list_,expr_,level_:1] :=
   Module[{pattWithBlanks,pattern},
      pattWithBlanks = patt/.(x_Symbol/;!MemberQ[{"System`"},Context[x]] :> pattern[x,Blank[]]);
      pattWithBlanks = pattWithBlanks/.pattern->Pattern;

      Cases[Unevaluated@list, pattWithBlanks :> expr, {level}];
      Null
   ];

Tests:

ForEach[{i, j}, {{1, 10}, {2, 20}, {3, 30}}, Print[i*j]]
ForEach[i, {{1, 10}, {2, 20}, {3, 30}}, Print[i], 2]

回答6:

Mathematica have map functions, so lets say you have a function Functaking one argument. Then just write

Func /@ list

Print /@ {1, 2, 3, 4, 5}

The return value is a list of the function applied to each element in the in-list.

PrimeQ /@ {10, 2, 123, 555}

will return {False,True,False,False}

回答7:

Thanks to Pillsy and Leonid Shifrin, here's what I'm now using:

SetAttributes[each, HoldAll];               (* each[pattern, list, body]      *)
each[pat_, lst_List, bod_] :=               (*  converts pattern to body for  *)
  (Cases[Unevaluated@lst, pat:>bod]; Null); (*   each element of list.        *)
each[p_, l_, b_] := (Cases[l, p:>b]; Null); (* (Break/Continue not supported) *)