What's the most "functional" way to do Cholesky decomposition?

For reference there is a built-in function CholeskyDecomposition.

For improving your existing code Array may be a minor subjective improvement:

HalfFunctionalCholesky2[matrin_List?PositiveDefiniteMatrixQ] := 
 Module[{dimens, uu},
  dimens = Length[matrin]; 
  uu = ConstantArray[0, {dimens, dimens}]; 
  Array[(uu[[#]] = makerow[matrin, #, uu, dimens]) &, dimens];
  uu
 ]

---

After a look through this function, following your instincts I believe Nest may be used:

HalfFunctionalCholesky3[matrin_List?PositiveDefiniteMatrixQ] :=
  Module[{len = Length[matrin], i = 1},
    Nest[Append[#, makerow[matrin, i++, #, len]] &, {}, len]
  ]

---

Upon further examination I believe that the change to Nest allows a further simplification as follows (within Part):

diagonalelement[matrin_, uu_, rowindex_] := 
 Sqrt[ matrin[[rowindex, rowindex]] - Conjugate[#].# & @ uu[[All, rowindex]] ]

offdiagonalelements[matrin_, uu_, rowindex_, len_, diag_] :=
 (matrin[[rowindex, rowindex - len ;;]] - 
    Conjugate[uu[[All, rowindex]]].uu[[All, rowindex - len ;;]]) / diag

This is not a functional implementation (as it stands, it's rather MATLAB-ish), but I'll leave this snippet around and hope somebody could make something purely functional out of this outer product form of Cholesky decomposition:

m = Array[Min, {4, 4}]; (* example matrix *)

Do[
  m[[k, k]] = b = Sqrt[a = m[[k, k]]];
  m[[k, k + 1 ;;]] = (v = m[[k, k + 1 ;;]])/b;
  m[[k + 1 ;;, k + 1 ;;]] -= Outer[Times, v, v]/a;
  , {k, Length[m]}];
UpperTriangularize[m]

{{1, 1, 1, 1}, {0, 1, 1, 1}, {0, 0, 1, 1}, {0, 0, 0, 1}}

---

(added 6/12/2012)

As expected, the method given above can be made purely functional, but to me the functional version looks a lot less elegant:

k = 0; n = Length[m];
MapAt[Sqrt,
 Nest[Function[m, ++k;
   Block[{a = m[[k, k]], v = m[[k, k + 1 ;;]], s},
    s = Sqrt[m[[k, k]]];
    ArrayFlatten[{{ReplacePart[m[[1 ;; k, 1 ;; k]], {k, k} -> s], 
       DiagonalMatrix[Append[ConstantArray[1, k - 1], 1/s]].m[[1 ;; k, k + 1 ;;]]},
                 {0, m[[k + 1 ;;, k + 1 ;;]] - Outer[Times, v, v]/a}}]]],
  m, n - 1],
 {n, n}]