list manipulation - Best way to create symmetric matrices

From time to time I need to generate symmetric matrices with relatively expensive cost of element evaluation. Most frequently these are Gram matrices where elements are $L_2$ dot products. Here are two ways of efficient implementation which come to mind: memoization and direct procedural generation.

ClearAll[el, elmem];
el[i_, j_] := Integrate[ChebyshevT[i, x] ChebyshevT[j, x], {x, -1, 1}];
elmem[i_, j_] := elmem[j, i] = el[i, j];

n = 30;
ClearSystemCache[];
a1 = Table[el[i, j], {i, n}, {j, n}]; // Timing
ClearSystemCache[];
a2 = Table[elmem[i, j], {i, n}, {j, n}]; // Timing
ClearSystemCache[];

(a3 = ConstantArray[0, {n, n}];
  Do[a3[[i, j]] = a3[[j, i]] = el[i, j], {i, n}, {j, i}];) // Timing
a1 == a2 == a3

---

{34.75, Null}

{18.235, Null}

{18.172, Null}


True

Here a1 is a redundant version for comparison, a2 is using memoization, a3 is a procedural-style one which I don't really like but it beats the built-in function here. The results are quite good but I wonder if there are more elegant ways of generating symmetric matrices?

---

SUMMARY (UPDATED)

Thanks to all participants for their contributions. Now it's time to benchmark. Here is the compilation of all proposed methods with minor modifications.

array[n_, f_] := Array[f, {n, n}];
arraymem[n_, f_] := 
  Block[{mem}, mem[i_, j_] := mem[j, j] = f[i, j]; Array[mem, {n, n}]];

proc[n_, f_] := Block[{res},
  res = ConstantArray[0, {n, n}]; 
  Do[res[[i, j]] = res[[j, i]] = f[i, j], {i, n}, {j, i}];
  res
  ]

acl[size_, fn_] := 
  Module[{rtmp}, rtmp = Table[fn[i, j], {i, 1, size}, {j, 1, i}];
   MapThread[Join, {rtmp, Rest /@ Flatten[rtmp, {{2}, {1}}]}]];


RM1[n_, f_] := 
  SparseArray[{{i_, j_} :> f[i, j] /; i >= j, {i_, j_} :> f[j, i]}, n];
RM2[n_, f_] := 
  Table[{{i, j} -> #, {j, i} -> #} &@f[i, j], {i, n}, {j, i}] // 
    Flatten // SparseArray;

MrWizard1[n_, f_] := 
  Join[#, Rest /@ #~Flatten~{2}, 2] &@Table[i~f~j, {i, n}, {j, i}];
MrWizard2[n_, f_] := Max@##~f~Min@## &~Array~{n, n};


MrWizard3[n_, f_] := Block[{f1, f2},
  f1 = LowerTriangularize[#, -1] + Transpose@LowerTriangularize[#] &@
     ConstantArray[Range@#, #] &;
  f2 = {#, Reverse[(Length@# + 1) - #, {1, 2}]} &;
  f @@ f2@f1@n
  ]


whuber[n_Integer, f_] /; n >= 1 := 
  Module[{data, m, indexes}, 

   data = Flatten[Table[f[i, j], {i, n}, {j, i, n}], 1];
   m = Binomial[n + 1, 2] + 1;
   indexes = 
    Table[m + Abs[j - i] - Binomial[n + 2 - Min[i, j], 2], {i, n}, {j,
       n}];
   Part[data, #] & /@ indexes];

JM[n_Integer, f_, ori_Integer: 1] := 
  Module[{tri = Table[f[i, j], {i, ori, n + ori - 1}, {j, ori, i}]}, 
   Fold[ArrayFlatten[{{#1, Transpose[{Most[#2]}]}, {{Most[#2]}, 

        Last[#2]}}] &, {First[tri]}, Rest[tri]]];

generators = {array, arraymem, proc, acl, RM1, RM2, MrWizard1, 
   MrWizard2, MrWizard3, whuber, JM};

The first three procedures are mine, all other are named after their authors. Let's start from cheap f and (relatively) large dimensions.

fun = Cos[#1 #2] &;
ns = Range[100, 500, 50]
data = Table[ClearSystemCache[]; Timing[gen[n, fun]] // First,
   {n, ns}, {gen, generators}];

Here is a logarithmic diagram for this test:

<< PlotLegends`

ListLogPlot[data // Transpose, PlotRange -> All, Joined -> True, 
 PlotMarkers -> {Automatic, Medium}, DataRange -> {Min@ns, Max@ns}, 
 PlotLegend -> generators, LegendPosition -> {1, -0.5}, 
 LegendSize -> {.5, 1}, ImageSize -> 600, Ticks -> {ns, Automatic}, 
 Frame -> True, FrameLabel -> {"n", "time"}]

Now let's make f numeric:

fun = Cos[N@#1 #2] &;

The result is quite surprising:

As you may guess, the missed quantities are machine zeroes.

The last experiment is old: it doesn't include fresh MrWizard3 and RM's codes are with //Normal. It takes "expensive" f from above and tolerant n:

fun = Integrate[ChebyshevT[#1 , x] ChebyshevT[#2, x], {x, -1, 1}] &;
ns = Range[10, 30, 5]

The result is

As we see, all methods which do not recompute the elements twice behave identically.

Answer

Borrowing liberally from acl's answer:

sim = Join[#, Rest /@ # ~Flatten~ {2}, 2] & @ Table[i ~#~ j, {i, #2}, {j, i}] &;

sim[Subscript[x, ##] &, 5] // Grid

$\begin{array}{ccccc} x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} & x_{5,1} \\ x_{2,1} & x_{2,2} & x_{3,2} & x_{4,2} & x_{5,2} \\ x_{3,1} & x_{3,2} & x_{3,3} & x_{4,3} & x_{5,3} \\ x_{4,1} & x_{4,2} & x_{4,3} & x_{4,4} & x_{5,4} \\ x_{5,1} & x_{5,2} & x_{5,3} & x_{5,4} & x_{5,5} \end{array}$

Trading efficiency for brevity:

sim2[f_, n_] := Max@## ~f~ Min@## & ~Array~ {n, n}

sim2[Subscript[f, ##] &, 5] // Grid

$\begin{array}{ccccc} x_{1,1} & x_{2,1} & x_{3,1} & x_{4,1} & x_{5,1} \\ x_{2,1} & x_{2,2} & x_{3,2} & x_{4,2} & x_{5,2} \\ x_{3,1} & x_{3,2} & x_{3,3} & x_{4,3} & x_{5,3} \\ x_{4,1} & x_{4,2} & x_{4,3} & x_{4,4} & x_{5,4} \\ x_{5,1} & x_{5,2} & x_{5,3} & x_{5,4} & x_{5,5} \end{array}$

---

Just for fun, here's a method for fast vectorized (Listable) functions such as your "cheap f" test, showing what's possible if you keep everything packed. (Cos function given a numeric argument so that it evaluates.)

f1 = LowerTriangularize[#, -1] + Transpose@LowerTriangularize[#] & @

       ConstantArray[Range@#, #] &;

f2 = {#, Reverse[(Length@# + 1) - #, {1, 2}]} &;

f3 = # @@ f2@f1 @ #2 &;

f3[Cos[N@# * #2] &, 500]  // timeAvg

sim[Cos[N@# * #2] &, 500] // timeAvg

0.00712

0.1436