(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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[ 8045, 225]*) (*NotebookOutlinePosition[ 32402, 1084]*) (* CellTagsIndexPosition[ 32358, 1080]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Error in numerical methods", "Title"], Cell["By Robin Sue Sanders (Spring 1999)", "Subsubtitle", TextAlignment->Center], Cell["\<\ In Problems 3 and 4 of Lab 3, I've asked you to do some elementary \ error analysis. In other words, I've asked you to start looking at HOW the \ error in the Riemann Sums begins to disapper. This notebook should help you \ better understand what I want in problems 3 and 4.\ \>", "Text"], Cell[CellGroupData[{ Cell[TextData[{ "Problem 3 hints\[LongDash]What is error? And how do we make ", StyleBox["Mathematica", FontSlant->"Italic"], " find it for us?" }], "Section"], Cell["\<\ The crudest way to examine how fast the error disappears is to just \ look at the \"raw error\"\ \>", "Text"], Cell[TextData[{ "Raw error ", StyleBox[ " is just your run-of-the-mill error: How far off is the actual estimate \ from the (known) correct value? To make stuff easier to parse, most people \ use the absolute value of the difference between the computed value and the \ exact value. So in problem 3, you want to look at the difference between the \ Riemann Sum and the true, exact value of the integral. If you've named your \ Left Rectangle estimate function something like LeftSum[n], then create a \ function called LeftError[n] with the syntax", FontWeight->"Plain"] }], "Text", FontWeight->"Bold"], Cell[TextData[StyleBox[ "trueValue = Integrate[function, {x, a, b}];\nLeftError[n_] := Abs[LeftSum[n] \ - trueValue]", FontWeight->"Bold"]], "Text"], Cell[TextData[{ "Abs[ ] is ", StyleBox["Mathematica", FontSlant->"Italic"], "'s name for the Absolute Value function. If you use this syntax, you'll \ have to define trueValue first (fill in the correct function and the correct \ values for a and b in the first line.)" }], "Text"], Cell["\<\ After you've defined the LeftError[n] function, you can use it to \ make Tables of values that show you what the error in using n Left Rectangles \ is. As n gets larger, you should see LeftError[n] get smaller.\ \>", "Text"], Cell["\<\ The error functions for RightSum and MiddleSum can be defined in an \ analogous fashion. Once you've defined them, you can make tables that show \ you the error in using n Right Rectangles or n Middle Rectangles to estimate \ the integral. Again, you should see that as n gets larger, both \ RightError[n] and MiddlerError[n] get smaller.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[ "Problem 3 hints\[LongDash]How small does the error have to be to guarentee k \ decimal places of accuracy?"], "Section"], Cell[TextData[{ "When we say that an estimate is accurate to ", StyleBox["k", FontSlant->"Italic"], " decimal places, that means that if we round both the estimate and the \ true value to ", StyleBox["k", FontSlant->"Italic"], " decimal places, we get the same number. What that really means is that \ the two numbers must differ by less than a 5 in the (", StyleBox["k+", FontSlant->"Italic"], "1) decimal place. What this means is that the error, the difference \ between the two numbers must be smaller than 5 \[Cross] ", Cell[BoxData[ \(TraditionalForm\`10\^\(k + 1\)\)]], ". So if LeftError[500] < .005, we've got 2 decimal places of accuracy in \ our estimate\[LongDash]that means that LeftSum[500] and the correct value for \ the integral agree to 2 decimal places. In other words, if you round both \ LeftSum[500] and the correct value to two decimals, you get the same answer." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[ "Problem 4 hints\[LongDash]More sophisticated error analysis"], "Section"], Cell[TextData[{ "In Problem 4 you're asked to compare the error using 2", StyleBox["n", FontSlant->"Italic"], " rectangles to the error using ", StyleBox["n", FontSlant->"Italic"], " rectangles. On the surface, this may seem like a silly thing to do. 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For example, if\ \>", "Text"], Cell["\t\tLeftError[2n]/LeftError[n] -> 0.40 as n-> infinity", "Text"], Cell[TextData[{ "that tells us that the error caused by using 2n rectangles is about 40% \ the size of the error caused by using n rectangles. In other words, doubling \ the number of rectangles reduces the size of the error by about 60%. Note \ that the smaller this ratio is, the faster we're approaching the true value. \ Methods with small Error[2n]/Error[n] ratios ", StyleBox["converge", FontSlant->"Italic"], " faster than those with large Error[2n]/Error[n] ratios. From a practical \ point of view, if the LeftError ratio is about 50% and the MiddleError ratio \ is about 33%, that means that doubling the number of middle rectangles cuts \ the error by about 66%, but doubling the left rectangles only cuts the error \ by about 50%. Hence it takes more (lots more) left rectangles to get a good \ estimate than it takes to get a good estimate using middle rectangles." }], "Text"], Cell["\<\ You should tackle Problem 4 using this approach: For each of the \ two integrals, make a table of values of the ratios\ \>", "Text"], Cell["\t\tLeftError[2n]/LeftError[n]", "Text"], Cell["\t\tRightError[2n]/RightError[n]", "Text"], Cell["and", "Text"], Cell["\t\tMiddleError[2n]/MiddleError[n]", "Text"], Cell["\<\ and see what the long term behavior is. Which method has the best \ error ratio? 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The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Hyperlink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`NotebookLocate[ #2]}]&), Active->True, ButtonNote->ButtonData}], Cell[StyleData["Hyperlink", "Presentation"]], Cell[StyleData["Hyperlink", "Condensed"]], Cell[StyleData["Hyperlink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["\<\ The following styles are for linking automatically to the on-line \ help system.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["MainBookLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "MainBook", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["MainBookLink", "Presentation"]], Cell[StyleData["MainBookLink", "Condensed"]], Cell[StyleData["MainBookLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["AddOnsLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "AddOns", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["AddOnsLink", "Presentation"]], Cell[StyleData["AddOnsLink", "Condensed"]], Cell[StyleData["AddOnLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["RefGuideLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontFamily->"Courier", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "RefGuideLink", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["RefGuideLink", "Presentation"]], Cell[StyleData["RefGuideLink", "Condensed"]], Cell[StyleData["RefGuideLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["GettingStartedLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "GettingStarted", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["GettingStartedLink", "Presentation"]], Cell[StyleData["GettingStartedLink", "Condensed"]], Cell[StyleData["GettingStartedLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["OtherInformationLink"], StyleMenuListing->None, ButtonStyleMenuListing->Automatic, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}, ButtonBoxOptions->{ButtonFunction:>(FrontEndExecute[ { FrontEnd`HelpBrowserLookup[ "OtherInformation", #]}]&), Active->True, ButtonFrame->"None"}], Cell[StyleData["OtherInformationLink", "Presentation"]], Cell[StyleData["OtherInformationLink", "Condensed"]], Cell[StyleData["OtherInformationLink", "Printout"], FontColor->GrayLevel[0], FontVariations->{"Underline"->False}] }, Closed]], Cell["Placeholder Styles", "Section"], Cell["\<\ The cells below define styles useful for making placeholder \ objects in palette templates.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Placeholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->True, StyleMenuListing->None], Cell[StyleData["Placeholder", "Presentation"]], Cell[StyleData["Placeholder", "Condensed"]], Cell[StyleData["Placeholder", "Printout"]] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SelectionPlaceholder"], Editable->False, Selectable->False, StyleBoxAutoDelete->True, Placeholder->Primary, StyleMenuListing->None, DrawHighlighted->True], Cell[StyleData["SelectionPlaceholder", "Presentation"]], Cell[StyleData["SelectionPlaceholder", "Condensed"]], Cell[StyleData["SelectionPlaceholder", "Printout"]] }, Closed]], Cell["FormatType Styles", "Section"], Cell["\<\ The cells below define styles that are mixed in with the styles \ of most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. This is particularly true of Input and Output.\ \>", "Text"], Cell[StyleData["CellExpression"], PageWidth->Infinity, CellMargins->{{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel->False, ShowSpecialCharacters->False, AllowInlineCells->False, AutoItalicWords->{}, StyleMenuListing->None, FontFamily->"Courier", FontSize->12, Background->GrayLevel[1]], Cell[StyleData["InputForm"], AllowInlineCells->False, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["OutputForm"], PageWidth->Infinity, TextAlignment->Left, LineSpacing->{0.6, 1}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["StandardForm"], LineSpacing->{1.25, 0}, StyleMenuListing->None, FontFamily->"Courier"], Cell[StyleData["TraditionalForm"], LineSpacing->{1.25, 0}, SingleLetterItalics->True, TraditionalFunctionNotation->True, DelimiterMatching->None, StyleMenuListing->None], Cell["\<\ The style defined below is mixed in to any cell that is in an \ inline cell within another.\ \>", "Text"], Cell[StyleData["InlineCell"], TextAlignment->Left, ScriptLevel->1, StyleMenuListing->None], Cell[StyleData["InlineCellEditing"], StyleMenuListing->None, Background->RGBColor[1, 0.749996, 0.8]] }], MacintoshSystemPageSetup->"\<\ 00<0004/0B`000002mT8o?mooh<" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1731, 51, 43, 0, 150, "Title"], Cell[1777, 53, 82, 1, 50, "Subsubtitle"], Cell[1862, 56, 300, 5, 82, "Text"], Cell[CellGroupData[{ Cell[2187, 65, 170, 5, 70, "Section"], Cell[2360, 72, 119, 3, 48, "Text"], Cell[2482, 77, 618, 12, 150, "Text"], Cell[3103, 91, 150, 3, 48, "Text"], Cell[3256, 96, 292, 7, 65, "Text"], Cell[3551, 105, 235, 4, 65, "Text"], Cell[3789, 111, 364, 6, 99, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[4190, 122, 136, 2, 50, "Section"], Cell[4329, 126, 938, 20, 184, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[5304, 151, 89, 1, 31, "Section"], Cell[5396, 154, 484, 13, 99, "Text"], Cell[5883, 169, 493, 9, 133, "Text"], Cell[6379, 180, 46, 0, 31, "Text"], Cell[6428, 182, 149, 3, 48, "Text"], Cell[6580, 187, 70, 0, 31, "Text"], Cell[6653, 189, 906, 14, 201, "Text"], Cell[7562, 205, 143, 3, 48, "Text"], Cell[7708, 210, 46, 0, 31, "Text"], Cell[7757, 212, 48, 0, 31, "Text"], Cell[7808, 214, 19, 0, 31, "Text"], Cell[7830, 216, 50, 0, 31, "Text"], Cell[7883, 218, 134, 3, 48, "Text"] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)