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For those readers interested in \ commercial use of the material in these notebooks, please contact \ jwarren@cs.rice.edu. 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Open ]], Cell[CellGroupData[{ Cell["Exact evaluation near extraordinary vertices", "Subsection"], Cell[CellGroupData[{ Cell["Evaluation of running example", "Subsubsection", CellTags->"SUBSEC running example evaluation"], Cell[TextData[{ "Coefficients for expansion into eigenvector ", Cell[BoxData[ \(z\&_\_i\)]], " are ", Cell[BoxData[ \({3\/5, 3\/10, \(-\(9\/20\)\), 33\/20, 3\/5}\)]], ". 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[ \({1, 2\/3, 4\/9, 0, 8\/27}\)]], ". Compute final value by multiplying by the ", Cell[BoxData[ \(c\_i\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({3\/5, 3\/10, \(-\(9\/20\)\), 11\/40, 1\/10} . {1, 2\/3, 4\/9, 0, 8\/27}\)], "Input"], Cell[BoxData[ \(17\/27\)], "Output"] }, Open ]], Cell[TextData[{ "Check our answer since ", Cell[BoxData[ \(S\)]], " is subdivision matrix for cubic B\[Hyphen]spline over knot sequence ", Cell[BoxData[ \({\[Ellipsis], \(-1\), \(-\(1\/2\)\), 0, 1, 2, \[Ellipsis]}\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bohm[{0, 1, 0, 0}, {\(-1\), \(-1\)\/2, 0, 1, 2, 3}, 1\/3]\)], "Input"], Cell[BoxData[ \(17\/27\)], "Output"] }, Open ]] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Smoothness analysis at extraordinary vertices", "Section"], Cell[CellGroupData[{ Cell["The characteristic map", "Subsection"], Cell["\<\ Plots of the effect of the characteristic map on the running example\ \>", "Text"], Cell[BoxData[ \(S\&~[p_] := With[{n = Length[p] - 1}, 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