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See the ", StyleBox["Mathematica ", FontSlant->"Italic"], "Help browser or wolfram.com for more information on using ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell["\<\ This notebook and those for the other seven chapters use a common stylesheet \ that has been imported into the notebook. This style sheet supports \ chapters, section, subsections, numbered equations, numbered figures and \ citations. A separate copy of the stylesheet can be download from the same \ page where this notebook was downloaded. \ \>", "Text"], Cell[TextData[{ StyleBox["COPYRIGHT ISSUES:", FontWeight->"Bold"], " The authors reserve all copyrights associated with this work. Any of \ the material appearing in these notebooks (such as the polyhedral meshes in \ chapter 7) can be used and modified without restriction as long as the use is \ non\[Hyphen]commercial. We simply ask that you acknowledge the authors when \ using material from these notebooks. 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Next, extend eigenvectors until until uniform rules apply" }], "Text"], Cell[BoxData[ RowBox[{"(", GridBox[{ {"1", \(-4\), \(47\/3\), "60", "0"}, {"1", \(-3\), \(26\/3\), "24", "0"}, {"1", \(-2\), \(11\/3\), "6", "0"}, {"1", \(-1\), \(2\/3\), "0", "0"}, {"1", \(1\/3\), \(\(-2\)\/3\), "0", "0"}, {"1", "2", \(8\/3\), "0", "0"}, {"1", "4", \(44\/3\), "0", "48"}, {"1", "6", \(104\/3\), "0", "192"}, {"1", "8", \(188\/3\), "0", "480"} }], ")"}]], "Input"], Cell[TextData[{ "Build a recursive implementation of B\[ODoubleDot]hm's algorithm.. For B\ \[Hyphen]spline of order ", Cell[BoxData[ \(n + 1\)]], ", pass a vector of ", Cell[BoxData[ \(n + 1\)]], " coefficients ", Cell[BoxData[ \(p\)]], " and vector of ", Cell[BoxData[ \(2 n\)]], " knots ", Cell[BoxData[ \(kn\)]], " centered around interval containing ", Cell[BoxData[ \(\[ScriptX]\)]], ". " }], "Text"], Cell[BoxData[ \(bohm[p_, kn_, \[ScriptX]_] := \[IndentingNewLine]If[ Length[p] \[Equal] 1, p\[LeftDoubleBracket]1\[RightDoubleBracket], \ \[IndentingNewLine]With[{n = Length[p] - 1}, \[IndentingNewLine]bohm[\[IndentingNewLine]Table[\(\((kn\ \[LeftDoubleBracket]n + i\[RightDoubleBracket] - \[ScriptX])\) p\ \[LeftDoubleBracket]i\[RightDoubleBracket] + \((\[ScriptX] - kn\ \[LeftDoubleBracket]i\[RightDoubleBracket])\) p\[LeftDoubleBracket]i + 1\ \[RightDoubleBracket]\)\/\(kn\[LeftDoubleBracket]n + i\[RightDoubleBracket] - \ kn\[LeftDoubleBracket]i\[RightDoubleBracket]\), {i, n}], \[IndentingNewLine]Join[ kn\[LeftDoubleBracket]Range[2, n]\[RightDoubleBracket], kn\[LeftDoubleBracket]Range[n + 1, 2 n - 1]\[RightDoubleBracket]], \[IndentingNewLine]\ \[ScriptX]]]]\)], "Input", InitializationCell->True, CellTags->"EQN bohm's algorithm"], Cell[TextData[{ "Evaluate extensions of eigenvectors at ", Cell[BoxData[ \(\[ScriptX] \[Equal] 8\/3\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({bohm[{1, 1, 1, 1}, {0, 1, 2, 3, 4, 5}, 8\/3], \[IndentingNewLine]bohm[{2, 4, 6, 8}, {0, 1, 2, 3, 4, 5}, 8\/3], \[IndentingNewLine]bohm[{8\/3, 44\/3, 104\/3, 188\/3}, {0, 1, 2, 3, 4, 5}, 8\/3], \[IndentingNewLine]bohm[{0, 0, 0, 0}, {0, 1, 2, 3, 4, 5}, 8\/3], \[IndentingNewLine]bohm[{0, 48, 192, 480}, {0, 1, 2, 3, 4, 5}, 8\/3]}\)], "Input"], Cell[BoxData[ \({1, 16\/3, 256\/9, 0, 4096\/27}\)], "Output"] }, Open ]], Cell[TextData[{ "Apply theorem to compute values at ", Cell[BoxData[ \(\[ScriptX] \[Equal] 1\/3\)]], ", i.e. ", Cell[BoxData[ \({1, 2\/3, 4\/9, 0, 8\/27}\)]], ". 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x)\)\ \((2 + x)\)\ \((1 + 2\ x)\)\ \((1 + 3\ x + x\^2)\)\^2\)\), \(\(3\ x\ \((5 \ + 23\ x + 5\ x\^2)\)\)\/\(\((2 + x)\)\ \((1 + 2\ x)\)\ \((1 + 3\ x + x\^2)\)\)\), \(\(x\ \((7 + 55\ x + 87\ x\^2 + 32\ x\^3 + 2\ x\^4)\)\)\/\(\((1 + x)\)\ \((2 + x)\)\ \((1 + 2\ x)\)\ \((1 + 3\ x + x\^2)\)\)\), \(\(x\ \((2 + 32\ x + 87\ x\^2 + 55\ x\^3 + 7\ x\^4)\)\)\/\(\((1 + x)\)\ \((2 + x)\)\ \((1 + 2\ x)\)\ \((1 + 3\ x + x\^2)\)\)\)} }], "\[NoBreak]", ")"}]}]], "Input", InitializationCell->True], Cell[TextData[{ Cell[BoxData[ \(normMesh\)]], " separates ", Cell[BoxData[ \(submesh\)]], " into real and imaginary parts using ", Cell[BoxData[ \(x = \[Omega]\_n\)]], ". Note that ", Cell[BoxData[ \(normMesh\)]], " takes a few seconds to compute!" }], "Text"], Cell[BoxData[ \(\(normMesh[ n_] = \[IndentingNewLine]\(Map[coord, subMesh[Cos[\(2 \[Pi]\)\/n] + \[ImaginaryI]\ Sin[\(2\ \ \[Pi]\)\/n]] // FullSimplify, {2}] // ComplexExpand\) // FullSimplify;\)\)], "Input"], Cell[TextData[{ Cell[BoxData[ \(displaySubmesh\)]], " display the ", Cell[BoxData[ \(4\[Times]4\)]], " submesh" }], "Text"], Cell[BoxData[ \(displaySubMesh[net_] := \[IndentingNewLine]Show[ Graphics[{{GrayLevel[0.5], Polygon[{net\[LeftDoubleBracket]2, 2\[RightDoubleBracket], net\[LeftDoubleBracket]3, 2\[RightDoubleBracket], net\[LeftDoubleBracket]3, 3\[RightDoubleBracket], net\[LeftDoubleBracket]2, 3\[RightDoubleBracket]}]}, \[IndentingNewLine]Table[\ \[IndentingNewLine]Line[{net\[LeftDoubleBracket]i, j\[RightDoubleBracket], net\[LeftDoubleBracket]i + 1, j\[RightDoubleBracket], net\[LeftDoubleBracket]i + 1, j + 1\[RightDoubleBracket], net\[LeftDoubleBracket]i, j\[RightDoubleBracket], 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