(* Content-type: application/vnd.wolfram.mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 8.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       157,          7]
NotebookDataLength[      9337,        251]
NotebookOptionsPosition[      9022,        235]
NotebookOutlinePosition[      9366,        250]
CellTagsIndexPosition[      9323,        247]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["Cholesky", "Section",
 CellChangeTimes->{{3.7528526993294487`*^9, 3.7528527021568923`*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{"Cholesky", "[", "A_", "]"}], ":=", 
  RowBox[{"(", "\[IndentingNewLine]", 
   RowBox[{
    RowBox[{"n", "=", 
     RowBox[{"Length", "[", "A", "]"}]}], ";", "\[IndentingNewLine]", 
    RowBox[{"L", "=", 
     RowBox[{"Table", "[", 
      RowBox[{"0", ",", 
       RowBox[{"{", 
        RowBox[{"i", ",", "n"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"j", ",", "n"}], "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", 
    RowBox[{"For", "[", 
     RowBox[{
      RowBox[{"k", "=", "1"}], ",", " ", 
      RowBox[{"k", "\[LessEqual]", "n"}], ",", " ", 
      RowBox[{"k", "++"}], ",", " ", 
      RowBox[{"(*", 
       RowBox[{"Iterate", " ", "over", " ", "the", " ", "columns"}], "*)"}], 
      " ", "\[IndentingNewLine]", 
      RowBox[{"(*", 
       RowBox[{"Find", " ", "the", " ", "diagonal", " ", "element"}], "*)"}], 
      "\[IndentingNewLine]", 
      RowBox[{
       RowBox[{"sum", "=", "0"}], ";", "\[IndentingNewLine]", 
       RowBox[{"For", "[", 
        RowBox[{
         RowBox[{"p", "=", "1"}], ",", 
         RowBox[{"p", "\[LessEqual]", 
          RowBox[{"k", "-", "1"}]}], ",", 
         RowBox[{"p", "++"}], ",", "\[IndentingNewLine]", 
         RowBox[{
          RowBox[{"sum", "+=", 
           RowBox[{
            RowBox[{"L", "[", 
             RowBox[{"[", 
              RowBox[{"k", ",", "p"}], "]"}], "]"}], "^", "2"}]}], ";"}]}], 
        "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", 
       RowBox[{
        RowBox[{"L", "[", 
         RowBox[{"[", 
          RowBox[{"k", ",", "k"}], "]"}], "]"}], "=", 
        RowBox[{"\[Sqrt]", 
         RowBox[{"(", 
          RowBox[{
           RowBox[{"A", "[", 
            RowBox[{"[", 
             RowBox[{"k", ",", "k"}], "]"}], "]"}], "-", "sum"}], ")"}]}]}], 
       ";", "\[IndentingNewLine]", 
       RowBox[{"For", "[", 
        RowBox[{
         RowBox[{"j", "=", 
          RowBox[{"k", "+", "1"}]}], ",", 
         RowBox[{"j", "\[LessEqual]", "n"}], ",", 
         RowBox[{"j", "++"}], ",", " ", 
         RowBox[{"(*", 
          RowBox[{
          "Find", " ", "all", " ", "the", " ", "elements", " ", "below", " ", 
           "the", " ", "main", " ", "diagonal"}], "*)"}], 
         "\[IndentingNewLine]", 
         RowBox[{
          RowBox[{"sum", "=", "0"}], ";", "\[IndentingNewLine]", 
          RowBox[{"For", "[", 
           RowBox[{
            RowBox[{"i", "=", "1"}], ",", 
            RowBox[{"i", "\[LessEqual]", 
             RowBox[{"k", "-", "1"}]}], ",", 
            RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", 
            RowBox[{"sum", "+=", 
             RowBox[{
              RowBox[{"L", "[", 
               RowBox[{"[", 
                RowBox[{"k", ",", "i"}], "]"}], "]"}], 
              RowBox[{"L", "[", 
               RowBox[{"[", 
                RowBox[{"j", ",", "i"}], "]"}], "]"}]}]}]}], 
           "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", 
          RowBox[{
           RowBox[{"L", "[", 
            RowBox[{"[", 
             RowBox[{"j", ",", "k"}], "]"}], "]"}], "=", 
           RowBox[{
            FractionBox["1", 
             RowBox[{"L", "[", 
              RowBox[{"[", 
               RowBox[{"k", ",", "k"}], "]"}], "]"}]], 
            RowBox[{"(", 
             RowBox[{
              RowBox[{"A", "[", 
               RowBox[{"[", 
                RowBox[{"k", ",", "j"}], "]"}], "]"}], "-", "sum"}], 
             ")"}]}]}]}]}], "\[IndentingNewLine]", "]"}]}]}], 
     "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", "L"}], 
   "\[IndentingNewLine]", ")"}]}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"LUSolve", "[", 
   RowBox[{"L_", ",", "U_", ",", "b_"}], "]"}], ":=", 
  RowBox[{"(", "\[IndentingNewLine]", 
   RowBox[{"Module", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"x", ",", "y"}], "}"}], ",", " ", 
     RowBox[{"(*", 
      RowBox[{
      "Define", " ", "x", " ", "and", " ", "y", " ", "to", " ", "be", " ", 
       "local", " ", "variables"}], "*)"}], "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"n", "=", 
       RowBox[{"Length", "[", "b", "]"}]}], ";", "\[IndentingNewLine]", 
      RowBox[{"y", "=", 
       RowBox[{"Table", "[", 
        RowBox[{"0", ",", 
         RowBox[{"{", 
          RowBox[{"i", ",", "n"}], "}"}]}], "]"}]}], " ", ";", 
      "\[IndentingNewLine]", 
      RowBox[{"For", "[", 
       RowBox[{
        RowBox[{"i", "=", "1"}], ",", 
        RowBox[{"i", "\[LessEqual]", "n"}], ",", " ", 
        RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", 
        RowBox[{
         RowBox[{"sum", "=", "0"}], ";", "\[IndentingNewLine]", 
         RowBox[{"For", "[", 
          RowBox[{
           RowBox[{"j", "=", "1"}], ",", 
           RowBox[{"j", "\[LessEqual]", 
            RowBox[{"i", "-", "1"}]}], ",", 
           RowBox[{"j", "++"}], ",", "\[IndentingNewLine]", 
           RowBox[{"sum", "+=", 
            RowBox[{
             RowBox[{"L", "[", 
              RowBox[{"[", 
               RowBox[{"i", ",", "j"}], "]"}], "]"}], 
             RowBox[{"y", "[", 
              RowBox[{"[", "j", "]"}], "]"}]}]}]}], "\[IndentingNewLine]", 
          "]"}], ";", "\[IndentingNewLine]", 
         RowBox[{
          RowBox[{"y", "[", 
           RowBox[{"[", "i", "]"}], "]"}], "=", 
          RowBox[{
           FractionBox["1", 
            RowBox[{"L", "[", 
             RowBox[{"[", 
              RowBox[{"i", ",", "i"}], "]"}], "]"}]], 
           RowBox[{"(", 
            RowBox[{
             RowBox[{"b", "[", 
              RowBox[{"[", "i", "]"}], "]"}], "-", "sum"}], ")"}]}]}]}]}], 
       "\[IndentingNewLine]", "\[IndentingNewLine]", "]"}], ";", 
      "\[IndentingNewLine]", 
      RowBox[{"x", "=", 
       RowBox[{"Table", "[", 
        RowBox[{"0", ",", 
         RowBox[{"{", 
          RowBox[{"i", ",", "n"}], "}"}]}], "]"}]}], " ", ";", 
      "\[IndentingNewLine]", 
      RowBox[{"For", "[", 
       RowBox[{
        RowBox[{"i", "=", "n"}], ",", 
        RowBox[{"i", "\[GreaterEqual]", "1"}], ",", " ", 
        RowBox[{"i", "--"}], ",", "\[IndentingNewLine]", 
        RowBox[{
         RowBox[{"sum", "=", "0"}], ";", "\[IndentingNewLine]", 
         RowBox[{"For", "[", 
          RowBox[{
           RowBox[{"j", "=", 
            RowBox[{"i", "+", "1"}]}], ",", 
           RowBox[{"j", "\[LessEqual]", "n"}], ",", 
           RowBox[{"j", "++"}], ",", "\[IndentingNewLine]", 
           RowBox[{"sum", "+=", 
            RowBox[{
             RowBox[{"U", "[", 
              RowBox[{"[", 
               RowBox[{"i", ",", "j"}], "]"}], "]"}], 
             RowBox[{"x", "[", 
              RowBox[{"[", "j", "]"}], "]"}]}]}]}], "\[IndentingNewLine]", 
          "]"}], ";", "\[IndentingNewLine]", 
         RowBox[{
          RowBox[{"x", "[", 
           RowBox[{"[", "i", "]"}], "]"}], "=", 
          RowBox[{
           FractionBox["1", 
            RowBox[{"U", "[", 
             RowBox[{"[", 
              RowBox[{"i", ",", "i"}], "]"}], "]"}]], 
           RowBox[{"(", 
            RowBox[{
             RowBox[{"y", "[", 
              RowBox[{"[", "i", "]"}], "]"}], "-", "sum"}], ")"}]}]}]}]}], 
       "\[IndentingNewLine]", "\[IndentingNewLine]", "]"}], ";", 
      "\[IndentingNewLine]", "x"}]}], "\[IndentingNewLine]", "]"}], 
   ")"}]}]}], "Input",
 CellChangeTimes->{{3.6557929521573105`*^9, 3.6557929531373124`*^9}, {
   3.655792992582371*^9, 3.655793106106538*^9}, {3.655793138812117*^9, 
   3.655793169457162*^9}, {3.6557932145632286`*^9, 3.6557935316707597`*^9}, {
   3.655793580854902*^9, 3.6557936143739595`*^9}, 3.655793659345024*^9, {
   3.655793741346494*^9, 3.6557937470565023`*^9}, {3.6557939568350487`*^9, 
   3.655794013999318*^9}, {3.655794130158848*^9, 3.655794131032898*^9}, {
   3.6558002898616323`*^9, 3.6558003467928886`*^9}, {3.655801062470823*^9, 
   3.6558010922085238`*^9}, {3.6558012685996127`*^9, 3.655801272430832*^9}, {
   3.65607152594147*^9, 3.6560716019998207`*^9}, {3.656071642224121*^9, 
   3.6560721417216907`*^9}, {3.6560722196281466`*^9, 3.656072224283413*^9}, 
   3.656072399894457*^9, {3.656073073059471*^9, 3.65607307514359*^9}, {
   3.688455147842004*^9, 3.68845518106205*^9}, {3.688455223272109*^9, 
   3.688455233862124*^9}, {3.754031528371827*^9, 3.7540316013002625`*^9}, {
   3.7872855405608215`*^9, 3.7872856342171783`*^9}}]
}, Open  ]]
},
WindowSize->{1366, 685},
WindowMargins->{{-8, Automatic}, {Automatic, -8}},
FrontEndVersion->"10.1 for Microsoft Windows (64-bit) (March 23, 2015)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[579, 22, 97, 1, 70, "Section"],
Cell[679, 25, 8327, 207, 1061, "Input"]
}, Open  ]]
}
]
*)