(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1791651, 42907]*) (*NotebookOutlinePosition[ 1792708, 42942]*) (* CellTagsIndexPosition[ 1792631, 42936]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "GNGA: ", Cell[BoxData[ \(TraditionalForm\`y'' + y\^3 = 0, \ \ y(0) = \(0 = \(\(y(1)\)\(.\)\)\)\)]] }], "Text", FontSize->24], Cell[BoxData[ RowBox[{\( (*\ grid\ size, \ \ number\ of\ Fourier\ Modes\ *) \), "\[IndentingNewLine]", StyleBox[\(n\ = \ 100; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m\ = \ 20; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[IndentingNewLine]\[IndentingNewLine]dx\ = \ 1/n // N; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ delta\ x\ *) \n x\ = \ Table[i/n, \ {i, \ 0, \ n}]\ // N; \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ Grid\ *) \[IndentingNewLine]\[IndentingNewLine] (*\ nonlinearity\ *) \[IndentingNewLine]\(f[u_]\ \ \ := u^3;\)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \n \(fp[u_]\ := \ 3 u^2;\)\[IndentingNewLine] \(F[u_]\ \ \ := u^4/4;\)\n\n (*\ Fourier\ Basis, \ eigenfunctions\ of\ \ - y'' = \ \[Lambda]y, \ normalized\ in\ L\^2\ *) \n \(p[k_, \ x_] := \ Sqrt[2] Sin[k\ Pi\ x]\ // N;\)\ \n \(lam\ = \ Table[k^2\ Pi^2, \ {k, \ 1, \ m}] // N;\)\ \n\n (*\ Variational\ Functional\ J\ *) \n \(J[u_, a_] := Sum[a[\([j]\)]^2 lam[\([j]\)]/2, {j, 1, m}] - Sum[\(F[u]\)[\([i]\)], {i, 1, n + 1}]* dx;\)\[IndentingNewLine]\[IndentingNewLine] (*\ gradient\ of\ J\ *) \[IndentingNewLine] \(g[u_, \ a_] := \ Module[{v\ = \ f[u], \ j\ = \ 0}, \[IndentingNewLine]Table[ a[\([j]\)]\ lam[\([j]\)]\ - \ \((p[j, \ x]\ . \ v)\)* dx, \ {j, \ 1, \ m}]];\)\[IndentingNewLine]\n (*\ Hessian\ of\ J\ *) \n \(kron[b1_, b2_]\ := \ If[b1 \[Equal] b2, \ 1, \ 0];\)\ \[IndentingNewLine] \(h[u_, \ a_] := \ Module[{v\ = \ fp[u], \ j\ = \ 0, \ k\ = \ 0}, \[IndentingNewLine]Table[ kron[j, \ k]\ lam[\([j]\)]\ - \ \((\((p[k, \ x]* p[j, \ x]\ )\) . \ v)\)*dx, \ {j, \ 1, \ m}, \ {k, \ 1, \ m}]];\)\n\n (*\ Number\ of\ negative\ e - vals\ of\ Hessian\ *) \n \(sig[elist_] := Module[{cnt = 0, i = 0}, \[IndentingNewLine]For[i = 1, i \[LessEqual] Length[elist], \(i++\), \[IndentingNewLine]\(If[ elist[\([i]\)] < \(- .01\), \(cnt++\), Null];\)\[IndentingNewLine]]; \[IndentingNewLine]cnt\ \[IndentingNewLine]];\)\), FontSize->14]}]], "Input", FontSize->14], Cell[BoxData[ \(\(\( (*\ initial\ guess\ of\ Fourier\ coeffs\ *) \)\(\n\)\(\(a\ = \ Table[0, \ {k, \ 1, \ m}];\)\n \(a[\([3]\)]\ = \ 7.5;\)\[IndentingNewLine] \(u\ = \ Sum[a[\([k]\)] p[k, \ x], \ {k, \ 1, \ m}]\ // N;\)\[IndentingNewLine]\[IndentingNewLine] (*\ Gradient\ and\ Hessian\ for\ first\ iteration\ *) \[IndentingNewLine] \(grad\ = \ g[u, \ a];\)\[IndentingNewLine] \(hess\ = \ h[u, \ a];\)\)\)\)], "Input", FontSize->14], Cell[BoxData[{ \(go = \ 1; \ tol\ = \ 10^\((\(-10\))\); \ num\ = \ 0; \ maxits\ = \ 100; delta\ = \ 1.0;\n\[IndentingNewLine] (*\ do\ Newton\ iterations\ until\ grad\ small\ *) \), "\n", \(\(While[ go \[Equal] 1, \[IndentingNewLine]\(num++\); \[IndentingNewLine]\ \[IndentingNewLine] (*\ take\ step\ in\ search\ direction\ \ *) \[IndentingNewLine]a\ = \ a\ - \ delta\ *\ PseudoInverse[hess] . grad; \n\t\n\ \ \ \ (*\ new\ Fourier\ expan\ with\ new\ coeffs\ *) \[IndentingNewLine]u\ = \ \ Sum[a[\([k]\)] p[k, \ x], \ {k, \ 1, \ m}]\ // N; \[IndentingNewLine]\n\ \ \ \ (*\ calc\ Gradient\ and\ Hessian, \ evals\ of\ Hessian\ *) \[IndentingNewLine]grad\ = \ g[u, \ a]; \[IndentingNewLine]hess\ = \ h[u, \ a]; \[IndentingNewLine]evals\ = \ Eigenvalues[hess]; \[IndentingNewLine]\t\n\ \ \ \ (*\ calc\ || \\n abla\ J || _ 2, \ test\ for\ convergence, \ output\ info\ *) \[IndentingNewLine]nrm\ = \ Norm[grad]/ n; \[IndentingNewLine]If[\((nrm\ < \ tol)\)\ || \ \((num\ \[GreaterEqual] \ maxits)\), go = 0]; \[IndentingNewLine]Print["\", num, \ \ \ \ \ \ \ \ "\<, Norm = \>", nrm, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ "\<, J = \>", \ \ \ \ \ \ \ \ \ \ J[u, a], \*"\"\<, ||u\!\(\( || \_\[Infinity]\)\) = \>\"", Max[Abs[u]], "\<, MI = \>", sig[evals]];\[IndentingNewLine]\[IndentingNewLine]];\)\)}], "Input",\ FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(\(a[\([3]\)]\ = \ 7.5;\)\)], "Input", FontSize->24], Cell[BoxData[ InterpretationBox[\("iteration "\[InvisibleSpace]1\[InvisibleSpace]", \ Norm = "\[InvisibleSpace]0.03409691145857005`\[InvisibleSpace]", J = "\ \[InvisibleSpace]1276.2175399682824`\[InvisibleSpace]", ||u\!\(\( || \_\ \[Infinity]\)\) = "\[InvisibleSpace]11.147754093962504`\[InvisibleSpace]", MI \ = "\[InvisibleSpace]3\), SequenceForm[ "iteration ", 1, ", Norm = ", 0.034096911458570051, ", J = ", 1276.2175399682824, ", ||u\!\(\( || 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