(* Content-type: application/mathematica *)

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

(* CreatedBy='Mathematica 6.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[     24912,        749]
NotebookOptionsPosition[     23769,        705]
NotebookOutlinePosition[     24177,        721]
CellTagsIndexPosition[     24134,        718]
WindowFrame->Normal
ContainsDynamic->False*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["\<\
Math 481A
California State University, Northridge
Cubic Splines\
\>", "Section"],

Cell["\[Copyright]2005 BE Shapiro", "Subsection"]
}, Open  ]],

Cell[CellGroupData[{

Cell["spline: calculate cubic spline polynomials ", "Section"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"<<", "LinearAlgebra`Tridiagonal`"}], ";"}]], "Input"],

Cell[BoxData[
 RowBox[{
  RowBox[{"General", "::", "\<\"obspkg\"\>"}], 
  RowBox[{
  ":", " "}], "\<\"\\!\\(\\\"LinearAlgebra`Tridiagonal`\\\"\\) is now \
obsolete. The legacy version being loaded may conflict with current \
Mathematica functionality. See the Compatibility Guide for updating \
information. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \
ButtonStyle->\\\"Link\\\", ButtonFrame->None, \
ButtonData:>\\\"paclet:Compatibility/Tutorials/LinearAlgebra/Tridiagonal\\\", \
ButtonNote -> \\\"General::obspkg\\\"]\\)\"\>"}]], "Message", "MSG",
 CellChangeTimes->{3.3986018246100616`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"TridiagonalSolve", "::", "\<\"shdw\"\>"}], 
  RowBox[{
  ":", " "}], "\<\"Symbol \\!\\(\\\"TridiagonalSolve\\\"\\) appears in \
multiple contexts \\!\\({\\\"LinearAlgebra`Tridiagonal`\\\", \\\"Global`\\\"}\
\\); definitions in context \\!\\(\\\"LinearAlgebra`Tridiagonal`\\\"\\) may \
shadow or be shadowed by other definitions. \\!\\(\\*ButtonBox[\\\"\
\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", ButtonFrame->None, \
ButtonData:>\\\"paclet:ref/message/General/shdw\\\", ButtonNote -> \
\\\"LinearAlgebra`Tridiagonal`TridiagonalSolve::shdw\\\"]\\)\"\>"}]], \
"Message", "MSG",
 CellChangeTimes->{3.3986018248940163`*^9}]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{"spline", "[", 
   RowBox[{"points__", ",", " ", "x_"}], "]"}], ":=", " ", 
  RowBox[{"Module", "[", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{
     "n", ",", "xi", ",", "f", ",", "a", ",", "b", ",", "c", ",", "d", ",", 
      "h", ",", "r", ",", "  ", "subdiag", ",", "maindiag", ",", "superdiag", 
      ",", " ", "equations", ",", "i", ",", "j", ",", "M", ",", "xmxi"}], 
     "}"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", 
    RowBox[{
     RowBox[{"If", "[", 
      RowBox[{
       RowBox[{"!", 
        RowBox[{"ListQ", "[", "points", "]"}]}], ",", " ", 
       RowBox[{
        RowBox[{"Print", "[", "\"\<First argument must be a list.\>\"", "]"}],
         ";", 
        RowBox[{"Return", "[", "]"}], ";"}]}], "]"}], ";", 
     "\[IndentingNewLine]", 
     RowBox[{"If", "[", 
      RowBox[{
       RowBox[{"!", 
        RowBox[{"And", "@@", 
         RowBox[{"ListQ", "/@", "points"}]}]}], ",", " ", 
       RowBox[{
        RowBox[{
        "Print", "[", "\"\<First argument must be a list of lists.\>\"", 
         "]"}], ";", " ", 
        RowBox[{"Return", "[", "]"}], ";"}]}], "]"}], ";", 
     "\[IndentingNewLine]", 
     RowBox[{"If", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"Union", "[", 
         RowBox[{"Length", "/@", "points"}], "]"}], "\[NotEqual]", " ", 
        RowBox[{"{", "2", "}"}]}], ",", " ", 
       RowBox[{
        RowBox[{"Print", "[", "]"}], ";", 
        RowBox[{
        "Return", "[", 
         "\"\<First argument must be a list of lists of length 2\>\"", "]"}], 
        ";"}]}], "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", 
     "\[IndentingNewLine]", 
     RowBox[{"f", "=", 
      RowBox[{"Last", "/@", "points"}]}], ";", "\[IndentingNewLine]", 
     RowBox[{"a", "=", "f"}], ";", "\[IndentingNewLine]", 
     RowBox[{"n", "=", 
      RowBox[{"Length", "[", "points", "]"}]}], ";", " ", 
     "\[IndentingNewLine]", "\[IndentingNewLine]", 
     RowBox[{"h", "=", 
      RowBox[{"Table", "[", 
       RowBox[{"0", ",", 
        RowBox[{"{", 
         RowBox[{"n", "-", "1"}], "}"}]}], "]"}]}], ";", " ", 
     "\[IndentingNewLine]", 
     RowBox[{"For", "[", 
      RowBox[{
       RowBox[{"i", "=", "1"}], ",", 
       RowBox[{"i", "<", " ", "n"}], ",", " ", 
       RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", 
       RowBox[{
        RowBox[{
         RowBox[{"h", "[", 
          RowBox[{"[", "i", "]"}], "]"}], "=", 
         RowBox[{
          RowBox[{"points", "[", 
           RowBox[{"[", 
            RowBox[{
             RowBox[{"i", "+", "1"}], ",", "1"}], "]"}], "]"}], "-", 
          RowBox[{"points", "[", 
           RowBox[{"[", 
            RowBox[{"i", ",", "1"}], "]"}], "]"}]}]}], ";"}]}], " ", 
      "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", 
     "\[IndentingNewLine]", 
     RowBox[{"subdiag", "=", 
      RowBox[{"Append", "[", 
       RowBox[{
        RowBox[{"Drop", "[", 
         RowBox[{"h", ",", 
          RowBox[{"-", "1"}]}], "]"}], ",", "0"}], "]"}]}], ";", 
     "\[IndentingNewLine]", "\[IndentingNewLine]", 
     RowBox[{"superdiag", "=", 
      RowBox[{"Prepend", "[", 
       RowBox[{
        RowBox[{"Drop", "[", 
         RowBox[{"h", ",", 
          RowBox[{"-", "1"}]}], "]"}], ",", " ", "0"}], "]"}]}], ";", " ", 
     "\[IndentingNewLine]", "\[IndentingNewLine]", 
     RowBox[{"maindiag", "=", 
      RowBox[{"{", "1", "}"}]}], ";", " ", "\[IndentingNewLine]", 
     RowBox[{"For", "[", 
      RowBox[{
       RowBox[{"i", "=", "1"}], ",", 
       RowBox[{"i", "\[LessEqual]", 
        RowBox[{"n", "-", "2"}]}], ",", 
       RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", 
       RowBox[{
        RowBox[{"maindiag", " ", "=", " ", 
         RowBox[{"Append", "[", 
          RowBox[{"maindiag", ",", 
           RowBox[{"2", "*", 
            RowBox[{"(", 
             RowBox[{
              RowBox[{"h", "[", 
               RowBox[{"[", "i", "]"}], "]"}], "+", 
              RowBox[{"h", "[", 
               RowBox[{"[", 
                RowBox[{"i", "+", "1"}], "]"}], "]"}]}], ")"}]}]}], "]"}]}], 
        ";"}]}], " ", "\[IndentingNewLine]", "]"}], ";", 
     "\[IndentingNewLine]", 
     RowBox[{"maindiag", "=", 
      RowBox[{"Append", "[", 
       RowBox[{"maindiag", ",", "1"}], "]"}]}], ";", " ", 
     "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]", 
     RowBox[{"r", "=", 
      RowBox[{"{", "0", "}"}]}], ";", "\[IndentingNewLine]", 
     RowBox[{"For", "[", 
      RowBox[{
       RowBox[{"i", "=", "2"}], ",", " ", 
       RowBox[{"i", "\[LessEqual]", " ", 
        RowBox[{"n", "-", "1"}]}], ",", " ", 
       RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", 
       RowBox[{
        RowBox[{"r", "=", 
         RowBox[{"Append", "[", 
          RowBox[{"r", ",", 
           RowBox[{
            RowBox[{
             RowBox[{"(", 
              RowBox[{"3", "/", 
               RowBox[{"h", "[", 
                RowBox[{"[", "i", "]"}], "]"}]}], ")"}], "*", 
             RowBox[{"(", 
              RowBox[{
               RowBox[{"a", "[", 
                RowBox[{"[", 
                 RowBox[{"i", "+", "1"}], "]"}], "]"}], "-", 
               RowBox[{"a", "[", 
                RowBox[{"[", "i", "]"}], "]"}]}], ")"}]}], "-", 
            RowBox[{
             RowBox[{"(", 
              RowBox[{"3", "/", 
               RowBox[{"h", "[", 
                RowBox[{"[", 
                 RowBox[{"i", "-", "1"}], "]"}], "]"}]}], ")"}], 
             RowBox[{"(", 
              RowBox[{
               RowBox[{"a", "[", 
                RowBox[{"[", "i", "]"}], "]"}], "-", 
               RowBox[{"a", "[", 
                RowBox[{"[", 
                 RowBox[{"i", "-", "1"}], "]"}], "]"}]}], ")"}]}]}]}], 
          "]"}]}], ";"}]}], "\[IndentingNewLine]", "]"}], ";", 
     "\[IndentingNewLine]", 
     RowBox[{"r", "=", 
      RowBox[{"Append", "[", 
       RowBox[{"r", ",", "0"}], "]"}]}], ";", "\[IndentingNewLine]", 
     "\[IndentingNewLine]", 
     RowBox[{"c", "=", 
      RowBox[{"TridiagonalSolve", "[", 
       RowBox[{"subdiag", ",", " ", "maindiag", ",", "superdiag", ",", "r"}], 
       "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", 
     RowBox[{"d", "=", 
      RowBox[{"{", "}"}]}], ";", "\[IndentingNewLine]", 
     RowBox[{"For", "[", 
      RowBox[{
       RowBox[{"j", "=", "1"}], ",", " ", 
       RowBox[{"j", "<", "n"}], ",", " ", 
       RowBox[{"j", "++"}], ",", "\[IndentingNewLine]", 
       RowBox[{
        RowBox[{"d", "=", 
         RowBox[{"Append", "[", 
          RowBox[{"d", ",", " ", 
           RowBox[{
            RowBox[{"(", 
             RowBox[{
              RowBox[{"c", "[", 
               RowBox[{"[", 
                RowBox[{"j", "+", "1"}], "]"}], "]"}], "-", 
              RowBox[{"c", "[", 
               RowBox[{"[", "j", "]"}], "]"}]}], ")"}], "/", 
            RowBox[{"(", 
             RowBox[{"3", "*", 
              RowBox[{"h", "[", 
               RowBox[{"[", "j", "]"}], "]"}]}], ")"}]}]}], "]"}]}], ";"}]}], 
      "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", 
     "\[IndentingNewLine]", 
     RowBox[{"b", "=", 
      RowBox[{"{", "}"}]}], ";", "\[IndentingNewLine]", 
     RowBox[{"For", "[", 
      RowBox[{
       RowBox[{"j", "=", "1"}], ",", 
       RowBox[{"j", "<", "n"}], ",", " ", 
       RowBox[{"j", "++"}], ",", " ", "\[IndentingNewLine]", 
       RowBox[{
        RowBox[{"b", "=", 
         RowBox[{"Append", "[", 
          RowBox[{"b", ",", 
           RowBox[{
            RowBox[{
             RowBox[{"(", 
              RowBox[{
               RowBox[{"a", "[", 
                RowBox[{"[", 
                 RowBox[{"j", "+", "1"}], "]"}], "]"}], "-", 
               RowBox[{"a", "[", 
                RowBox[{"[", "j", "]"}], "]"}]}], ")"}], "/", 
             RowBox[{"h", "[", 
              RowBox[{"[", "j", "]"}], "]"}]}], " ", "-", " ", 
            RowBox[{
             RowBox[{"(", 
              RowBox[{
               RowBox[{"h", "[", 
                RowBox[{"[", "j", "]"}], "]"}], "/", "3"}], ")"}], 
             RowBox[{"(", 
              RowBox[{
               RowBox[{"2", " ", 
                RowBox[{"c", "[", 
                 RowBox[{"[", "j", "]"}], "]"}]}], " ", "+", " ", 
               RowBox[{"c", "[", 
                RowBox[{"[", 
                 RowBox[{"j", "+", "1"}], "]"}], "]"}]}], ")"}]}]}]}], 
          "]"}]}], ";"}]}], "\[IndentingNewLine]", "]"}], ";", 
     "\[IndentingNewLine]", 
     RowBox[{"a", "=", 
      RowBox[{"Drop", "[", 
       RowBox[{"a", ",", 
        RowBox[{"-", "1"}]}], "]"}]}], ";", "\[IndentingNewLine]", 
     RowBox[{"c", "=", 
      RowBox[{"Drop", "[", 
       RowBox[{"c", ",", 
        RowBox[{"-", "1"}]}], "]"}]}], ";", "\[IndentingNewLine]", 
     RowBox[{"xi", "=", 
      RowBox[{"Drop", "[", 
       RowBox[{
        RowBox[{"First", "/@", "points"}], ",", 
        RowBox[{"-", "1"}]}], "]"}]}], ";", "\[IndentingNewLine]", 
     RowBox[{"M", "=", 
      RowBox[{"Transpose", "[", 
       RowBox[{"{", 
        RowBox[{"a", ",", "b", ",", "c", ",", "d"}], "}"}], "]"}]}], ";", 
     "\[IndentingNewLine]", 
     RowBox[{"equations", "=", 
      RowBox[{"{", "}"}]}], ";", "\[IndentingNewLine]", 
     RowBox[{"For", "[", 
      RowBox[{
       RowBox[{"i", "=", "1"}], ",", 
       RowBox[{"i", "<", "n"}], ",", 
       RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", 
       RowBox[{
        RowBox[{"xmxi", "=", 
         RowBox[{"Table", "[", 
          RowBox[{
           RowBox[{
            RowBox[{"(", 
             RowBox[{"x", "-", 
              RowBox[{"xi", "[", 
               RowBox[{"[", "i", "]"}], "]"}]}], ")"}], "^", "j"}], ",", " ", 
           
           RowBox[{"{", 
            RowBox[{"j", ",", "0", ",", 
             RowBox[{"n", "-", "2"}]}], "}"}]}], "]"}]}], ";", " ", 
        "\[IndentingNewLine]", 
        RowBox[{"equations", "=", 
         RowBox[{"Append", "[", 
          RowBox[{"equations", ",", 
           RowBox[{"xmxi", ".", 
            RowBox[{"M", "[", 
             RowBox[{"[", "i", "]"}], "]"}]}]}], "]"}]}], ";"}]}], " ", 
      "\[IndentingNewLine]", "\[IndentingNewLine]", "]"}], ";", 
     "\[IndentingNewLine]", "\[IndentingNewLine]", 
     RowBox[{"Return", "[", "equations", "]"}], ";"}]}], 
   "\[IndentingNewLine]", "]"}]}]], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Example", "Section"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"spline", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"0", ",", "75"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"3", ",", "77"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"6", ",", "80"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"9", ",", "74"}], "}"}], ",", " ", 
     RowBox[{"{", 
      RowBox[{"12", ",", "72"}], "}"}]}], "}"}], ",", "x"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"75", "+", 
    FractionBox[
     RowBox[{"19", " ", "x"}], "56"], "+", 
    FractionBox[
     RowBox[{"55", " ", 
      SuperscriptBox["x", "3"]}], "1512"]}], ",", 
   RowBox[{"77", "+", 
    RowBox[{
     FractionBox["37", "28"], " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"-", "3"}], "+", "x"}], ")"}]}], "+", 
    RowBox[{
     FractionBox["55", "168"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "3"}], "+", "x"}], ")"}], "2"]}], "-", 
    RowBox[{
     FractionBox["73", "504"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "3"}], "+", "x"}], ")"}], "3"]}]}], ",", 
   RowBox[{"80", "-", 
    RowBox[{
     FractionBox["5", "8"], " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"-", "6"}], "+", "x"}], ")"}]}], "-", 
    RowBox[{
     FractionBox["41", "42"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "6"}], "+", "x"}], ")"}], "2"]}], "+", 
    RowBox[{
     FractionBox["29", "168"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "6"}], "+", "x"}], ")"}], "3"]}]}], ",", 
   RowBox[{"74", "-", 
    RowBox[{
     FractionBox["51", "28"], " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"-", "9"}], "+", "x"}], ")"}]}], "+", 
    RowBox[{
     FractionBox["97", "168"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "9"}], "+", "x"}], ")"}], "2"]}], "-", 
    FractionBox[
     RowBox[{"97", " ", 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "9"}], "+", "x"}], ")"}], "3"]}], "1512"]}]}], 
  "}"}]], "Output",
 CellChangeTimes->{
  3.398601733440155*^9, 3.398601774292286*^9, {3.398601830398733*^9, 
   3.398601838006586*^9}}]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["equalIntervalSplines: assumes points are equally spaced", "Section"],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"equalIntervalSplines", "[", 
    RowBox[{"h_", ",", "x0_", ",", " ", "x_", ",", " ", "f__"}], "]"}], ":=", 
   " ", 
   RowBox[{"Module", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
      "n", ",", "a", ",", "A", ",", "i", ",", " ", "w", ",", "c", ",", "b", 
       ",", "M", ",", " ", "equations", ",", "xi"}], "}"}], ",", 
     "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"n", "=", 
       RowBox[{"Length", "[", 
        RowBox[{"{", "f", "}"}], "]"}]}], ";", "\[IndentingNewLine]", 
      RowBox[{"a", "=", 
       RowBox[{"{", "f", "}"}]}], ";", "\[IndentingNewLine]", 
      RowBox[{"A", "=", 
       RowBox[{"Table", "[", 
        RowBox[{"0", ",", 
         RowBox[{"{", "n", "}"}], ",", 
         RowBox[{"{", "n", "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", 
      RowBox[{
       RowBox[{"A", "[", 
        RowBox[{"[", 
         RowBox[{"1", ",", "1"}], "]"}], "]"}], "=", "h"}], ";", 
      "\[IndentingNewLine]", 
      RowBox[{"w", "=", 
       RowBox[{"{", "0", "}"}]}], ";", "\[IndentingNewLine]", 
      RowBox[{"For", "[", 
       RowBox[{
        RowBox[{"i", "=", "2"}], ",", 
        RowBox[{"i", "<", "n"}], ",", 
        RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", 
        RowBox[{
         RowBox[{
          RowBox[{"A", "[", 
           RowBox[{"[", 
            RowBox[{"i", ",", "i"}], "]"}], "]"}], "=", 
          RowBox[{"4", "*", "h"}]}], ";", "\[IndentingNewLine]", 
         RowBox[{
          RowBox[{"A", "[", 
           RowBox[{"[", 
            RowBox[{"i", ",", 
             RowBox[{"i", "-", "1"}]}], "]"}], "]"}], "=", "h"}], ";", 
         "\[IndentingNewLine]", 
         RowBox[{
          RowBox[{"A", "[", 
           RowBox[{"[", 
            RowBox[{"i", ",", 
             RowBox[{"i", "+", "1"}]}], "]"}], "]"}], "=", "h"}], ";", 
         "\[IndentingNewLine]", 
         RowBox[{"w", "=", 
          RowBox[{"Append", "[", 
           RowBox[{"w", ",", 
            RowBox[{
             RowBox[{"a", "[", 
              RowBox[{"[", 
               RowBox[{"i", "-", "1"}], "]"}], "]"}], "-", 
             RowBox[{"2", 
              RowBox[{"a", "[", 
               RowBox[{"[", "i", "]"}], "]"}]}], "+", 
             RowBox[{"a", "[", 
              RowBox[{"[", 
               RowBox[{"i", "+", "1"}], "]"}], "]"}]}]}], "]"}]}], ";"}]}], 
       "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", 
      RowBox[{
       RowBox[{"A", "[", 
        RowBox[{"[", 
         RowBox[{"n", ",", "n"}], "]"}], "]"}], "=", "1"}], ";", 
      "\[IndentingNewLine]", 
      RowBox[{"w", "=", 
       RowBox[{
        RowBox[{"(", 
         RowBox[{"3", "/", "h"}], ")"}], "*", 
        RowBox[{"Append", "[", 
         RowBox[{"w", ",", "0"}], "]"}]}]}], ";", "\[IndentingNewLine]", 
      RowBox[{"c", "=", 
       RowBox[{
        RowBox[{"Inverse", "[", "A", "]"}], ".", "w"}]}], ";", 
      "\[IndentingNewLine]", "\[IndentingNewLine]", 
      RowBox[{"b", "=", 
       RowBox[{"{", "}"}]}], ";", "\[IndentingNewLine]", 
      RowBox[{"d", "=", 
       RowBox[{"{", "}"}]}], ";", "\[IndentingNewLine]", 
      RowBox[{"For", "[", 
       RowBox[{
        RowBox[{"i", "=", "1"}], ",", 
        RowBox[{"i", "<", "n"}], ",", 
        RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", 
        RowBox[{
         RowBox[{"b", "=", 
          RowBox[{"Append", "[", 
           RowBox[{"b", ",", 
            RowBox[{
             RowBox[{
              RowBox[{"(", 
               RowBox[{
                RowBox[{"a", "[", 
                 RowBox[{"[", 
                  RowBox[{"i", "+", "1"}], "]"}], "]"}], "-", 
                RowBox[{"a", "[", 
                 RowBox[{"[", "i", "]"}], "]"}]}], ")"}], "/", "h"}], " ", 
             "-", " ", 
             RowBox[{
              RowBox[{"(", 
               RowBox[{"h", "/", "3"}], ")"}], "*", 
              RowBox[{"(", 
               RowBox[{
                RowBox[{"2", 
                 RowBox[{"c", "[", 
                  RowBox[{"[", "i", "]"}], "]"}]}], "+", 
                RowBox[{"c", "[", 
                 RowBox[{"[", 
                  RowBox[{"i", "+", "1"}], "]"}], "]"}]}], ")"}]}]}]}], 
           "]"}]}], ";", "\[IndentingNewLine]", 
         RowBox[{"d", "=", 
          RowBox[{"Append", "[", 
           RowBox[{"d", ",", 
            RowBox[{
             RowBox[{"(", 
              RowBox[{
               RowBox[{"c", "[", 
                RowBox[{"[", 
                 RowBox[{"i", "+", "1"}], "]"}], "]"}], "-", 
               RowBox[{"c", "[", 
                RowBox[{"[", "i", "]"}], "]"}]}], ")"}], "/", 
             RowBox[{"(", 
              RowBox[{"3", " ", "h"}], ")"}]}]}], "]"}]}], ";"}]}], 
       "\[IndentingNewLine]", "]"}], ";", "\[IndentingNewLine]", 
      RowBox[{"a", "=", 
       RowBox[{"Drop", "[", 
        RowBox[{"a", ",", 
         RowBox[{"-", "1"}]}], "]"}]}], ";", "\[IndentingNewLine]", 
      RowBox[{"c", "=", 
       RowBox[{"Drop", "[", 
        RowBox[{"c", ",", 
         RowBox[{"-", "1"}]}], "]"}]}], ";", "\[IndentingNewLine]", 
      RowBox[{"M", "=", 
       RowBox[{"Transpose", "[", 
        RowBox[{"{", 
         RowBox[{"a", ",", "b", ",", "c", ",", "d"}], "}"}], "]"}]}], ";", 
      "\[IndentingNewLine]", 
      RowBox[{"equations", "=", 
       RowBox[{"{", "}"}]}], ";", "\[IndentingNewLine]", 
      RowBox[{"For", "[", 
       RowBox[{
        RowBox[{"i", "=", "1"}], ",", 
        RowBox[{"i", "<", "n"}], ",", 
        RowBox[{"i", "++"}], ",", "\[IndentingNewLine]", 
        RowBox[{
         RowBox[{"xi", "=", 
          RowBox[{"x0", "+", 
           RowBox[{
            RowBox[{"(", 
             RowBox[{"i", "-", "1"}], ")"}], "*", "h"}]}]}], ";", 
         "\[IndentingNewLine]", 
         RowBox[{"xmxi", "=", 
          RowBox[{"Table", "[", 
           RowBox[{
            RowBox[{
             RowBox[{"(", 
              RowBox[{"x", "-", "xi"}], ")"}], "^", "j"}], ",", " ", 
            RowBox[{"{", 
             RowBox[{"j", ",", "0", ",", 
              RowBox[{"n", "-", "2"}]}], "}"}]}], "]"}]}], ";", " ", 
         "\[IndentingNewLine]", 
         RowBox[{"equations", "=", 
          RowBox[{"Append", "[", 
           RowBox[{"equations", ",", 
            RowBox[{"xmxi", ".", 
             RowBox[{"M", "[", 
              RowBox[{"[", "i", "]"}], "]"}]}]}], "]"}]}], ";"}]}], " ", 
       "\[IndentingNewLine]", "\[IndentingNewLine]", "]"}], ";", 
      "\[IndentingNewLine]", 
      RowBox[{"Return", "[", "equations", "]"}], ";"}]}], 
    "\[IndentingNewLine]", "]"}]}], ";"}]], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Example", "Section"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"equalIntervalSplines", "[", 
  RowBox[{
  "3", ",", "0", ",", "x", ",", "75", ",", "77", ",", "80", ",", "74", ",", 
   "72"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"75", "+", 
    FractionBox[
     RowBox[{"19", " ", "x"}], "56"], "+", 
    FractionBox[
     RowBox[{"55", " ", 
      SuperscriptBox["x", "3"]}], "1512"]}], ",", 
   RowBox[{"77", "+", 
    RowBox[{
     FractionBox["37", "28"], " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"-", "3"}], "+", "x"}], ")"}]}], "+", 
    RowBox[{
     FractionBox["55", "168"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "3"}], "+", "x"}], ")"}], "2"]}], "-", 
    RowBox[{
     FractionBox["73", "504"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "3"}], "+", "x"}], ")"}], "3"]}]}], ",", 
   RowBox[{"80", "-", 
    RowBox[{
     FractionBox["5", "8"], " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"-", "6"}], "+", "x"}], ")"}]}], "-", 
    RowBox[{
     FractionBox["41", "42"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "6"}], "+", "x"}], ")"}], "2"]}], "+", 
    RowBox[{
     FractionBox["29", "168"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "6"}], "+", "x"}], ")"}], "3"]}]}], ",", 
   RowBox[{"74", "-", 
    RowBox[{
     FractionBox["51", "28"], " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"-", "9"}], "+", "x"}], ")"}]}], "+", 
    RowBox[{
     FractionBox["97", "168"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "9"}], "+", "x"}], ")"}], "2"]}], "-", 
    FractionBox[
     RowBox[{"97", " ", 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "9"}], "+", "x"}], ")"}], "3"]}], "1512"]}]}], 
  "}"}]], "Output",
 CellChangeTimes->{3.398601741646119*^9, 3.398601844738155*^9}]
}, Open  ]]
}, Open  ]]
},
WindowSize->{690, 840},
WindowMargins->{{361, Automatic}, {140, Automatic}},
FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (June 19, 2007)",
StyleDefinitions->FrontEnd`FileName[{"Creative"}, "PastelColor.nb", 
  CharacterEncoding -> "UTF-8"]
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[590, 23, 90, 4, 112, "Section"],
Cell[683, 29, 49, 0, 34, "Subsection"]
}, Open  ]],
Cell[CellGroupData[{
Cell[769, 34, 62, 0, 72, "Section"],
Cell[CellGroupData[{
Cell[856, 38, 88, 2, 37, "Input"],
Cell[947, 42, 595, 11, 59, "Message"],
Cell[1545, 55, 661, 12, 59, "Message"]
}, Open  ]],
Cell[2221, 70, 10360, 275, 968, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[12618, 350, 26, 0, 72, "Section"],
Cell[CellGroupData[{
Cell[12669, 354, 437, 14, 37, "Input"],
Cell[13109, 370, 1846, 66, 132, "Output"]
}, Open  ]]
}, Open  ]],
Cell[CellGroupData[{
Cell[15004, 442, 74, 0, 72, "Section"],
Cell[15081, 444, 6608, 178, 548, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[21726, 627, 26, 0, 72, "Section"],
Cell[CellGroupData[{
Cell[21777, 631, 168, 4, 37, "Input"],
Cell[21948, 637, 1793, 64, 132, "Output"]
}, Open  ]]
}, Open  ]]
}
]
*)

(* End of internal cache information *)