MATLAB 5.0 MAT-file, Platform: PCWIN, Created on: Tue Jan 04 21:53:19 2005 IM8 hgS_070000 7typehandlepropertieschildrenspecial@ figure8 f@CUnitsColorColormapFileNameIntegerHandleInvertHardcopyMenuBarNameNumberTitlePaperPositionPositionRendererRendererModeResizeHandleVisibilityTagUserDataApplicationDataBehaviorH characters00@ ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????1bC:\MFILES.KEA\Guide_NA.dir\Integrate_GUI_Help.fig8off0on8noneX$Integrate_GUI_Help8offP ?@ @@P 43333Y@NN0@gffff_@ONDF@@painters@ manual8off@callback@figure100 GUIDEOptionslastValidTagactive_htaginfooverridereleaseresizeaccessibilitymfilecallbackssingletonsyscolorfigblockinglastSavedFile8 `g@figuretext0000 8none@callback00000/^C:\MFILES.KEA\Guide_NA.dir\Integrate_GUI_Help.m@figure18% 7typehandlepropertieschildrenspecialH uicontrol8 f@UnitsBackgroundColorFontSizePositionStringStyleTagApplicationDataBehaviorH characters08 +@P hffffC@;;D@D@ON?X$Integrate_GUI Help8text@ text1  lastValidTag@ text18H uicontrol8 .@UnitsBackgroundColorFontSizeHorizontalAlignmentPositionStringStyleTagApplicationDataBehaviorH characters00 8leftP 433333@;;?]@؉XC@ 0i This performs numerical integration of a function f(x) over an interval [a,b]. There are four choices U of composite quadrature rules: trapezoidal, midpoint, Simpson's, and Boole's rules.0^ The user specifies an initial number of subdivisions n0 and a number of times this is to be g doubled (call it 'doubles'). The quadrature is then calculated for n = n0*2^j for j=0,1,...,doubles._ We calculate the successive differences and the ratios by which they decrease. We also give 8p the true errors and the ratios by which they decrease.0P Choice of n0:^ For the midpoint and trapezoidal rules, n0 can be any positive integer. For Simpson's rule, c n0 must be an integral multiple of 2; and for Boole's rule, n0 must be an integral multiple of 4.0Z In a separate output box, we list two forms of error estimates, using Richardsons error7n estimate and the asymptotic error estimation formula.0,X For the Richardson error estimate, we use 08p I I_n approx equals (I_n I_{n/2})/(2^p 1)0S where p denotes the order of the method (p=2 for midpoint and trapezoidal rules, 4h p=4 for Simpsons rule, and p=6 for Booles rule).0J For the asymptotic error estimate, we use the standard lead term in the M Euler-MacLaurin error expansion for the method. We also use these estimatesS to give extrapolated values of the original numerical integral, along with errorsp @ for these extrapolated values.8text@ text2  lastValidTag@ text28@ uimenu8  0@PCallbackLabelTagApplicationDataBehavior>|Integrate_GUI_Help('File_Menu_Callback',gcbo,[],guidata(gcbo))8FileH File_Menu  lastValidTagH File_Menu8 7typehandlepropertieschildrenspecial@ uimenu8  1@PCallbackLabelTagApplicationDataBehavior:tIntegrate_GUI_Help('Close_Callback',gcbo,[],guidata(gcbo))@ Close@ Close  lastValidTag@ Close8H uicontrol8  2@UnitsBackgroundColorPositionStringStyleTagApplicationDataBehaviorH charactersH ??P 433333V@;;E@B@NN?x"DCopyright 2004 by Kendall Atkinson8text@ text3  lastValidTag@ text38