(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 249025, 4579]*) (*NotebookOutlinePosition[ 249751, 4604]*) (* CellTagsIndexPosition[ 249707, 4600]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "Here's a function that computes the LuckySeven version of x. It ignores \ underflow/overflow (i.e., it returns a1.a2 a3 a4 a5 a6 a7 x ", Cell[BoxData[ \(TraditionalForm\`10\^c\)]], ", regardless of whether c is less than -40 or greater than 40)." }], "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(LuckySeven[ x_]\ := \ \[IndentingNewLine]Block[{}, \[IndentingNewLine]If[ x\ \[NotEqual] \ 0, \[IndentingNewLine]c = 0; \[IndentingNewLine]y = Abs[x]; \[IndentingNewLine]While[ y < 1, c = c - 1; y = y*10]; \[IndentingNewLine]While[y > 10, c = c + 1; y = y/10]; \[IndentingNewLine]z = Floor[y*1000000`25 + 1/2]/ 1000000`25, \[IndentingNewLine]z\ = \ 0]; \[IndentingNewLine]Sign[x]* z*10^c\[IndentingNewLine]];\)\)], "Input"], Cell[TextData[{ "Now here's a function that computes the error in the LuckySeven version of \ the approximation to the second derivative of ", Cell[BoxData[ \(TraditionalForm\`\@x\)]], " at x=1." }], "Text", Background->GrayLevel[0.900008]], Cell[BoxData[ \(\(f[ h_]\ := \ \[IndentingNewLine]Block[{}, \[IndentingNewLine]a = 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What's going on for \ small h, though? 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Roundoff error is kicking in \ around h=0.05, which makes sense, since the formula for approximating the \ second derivative involves subtracting some nearly equal quantities and then \ dividing by a small number. It seems like we should choose h=0.05, since \ that's where the spread of errors is small and centered around zero. For \ h>0.05, the error is consistently negative and getting larger as h increases. \ For h<0.05, the rapid oscillations in the error get larger and larger as h \ decreases.\ \>", "Text", Background->GrayLevel[0.900008]] }, FrontEndVersion->"4.2 for Microsoft Windows", ScreenRectangle->{{0, 1400}, {0, 947}}, WindowSize->{734, 740}, WindowMargins->{{Automatic, 286}, {Automatic, 33}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, Magnification->2 ] (******************************************************************* Cached data follows. 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