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v:ext="edit" data="1"/> </o:shapelayout></xml><![endif]--> </head> <body lang=EN-US link=blue vlink=purple style='tab-interval:.5in'> <div class=WordSection1> <h1>Welcome</h1> <h1>Interpolation</h1> <p class=MsoNormal style='margin-left:.75in;text-indent:-.5in;mso-list:l0 level1 lfo2; tab-stops:list .75in'><![if !supportLists]><span style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span><![endif]>The best linear interpolator, <a href="Bli.htm"><span style='mso-fareast-font-family:"Times New Roman";mso-fareast-theme-font:major-fareast'>Bli.htm</span></a>, finds the optimal set of points for plotting a function. 8/15/2010  This code has been modified in <a href="bli.for"><span class=SpellE>bli.for</span></a> to better include the first and last points in the region.<span style='mso-spacerun:yes'>  </span>The Pascal version <span class=GramE>in </span><a href="../Fittery/robfit/peakfit/uPolyn.pas">..\<span class=SpellE>Fittery</span>\robfit\<span class=SpellE>peakfit</span>\uPolyn.pas</a> has been tested with these modifications.<br> Making an integral of this is discussed in <a href="BLI3.htm">BLI3.htm</a> -<span style='mso-spacerun:yes'>  </span>This is the integration routine for finding G(r) in <a href="../integration/MonteCarlo/Welcome.htm">integration\<span class=SpellE>MonteCarlo</span>\Welcome.htm</a></p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.5in;mso-list:l0 level1 lfo2; tab-stops:list .75in'><![if !supportLists]><span style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span><![endif]>In order to evaluate f(x) from this set of points, one needs to find J such that <span class=GramE>x(</span>J) <span style='font-family: Symbol;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type: symbol;mso-symbol-font-family:Symbol'>£</span></span> x <span style='font-family: Symbol;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type: symbol;mso-symbol-font-family:Symbol'>£</span></span> x(J+1).<span style='mso-spacerun:yes'>  </span>This is accomplished by the subroutines <a href="for/LOCATE.FOR"><span style='mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:major-fareast'>(LOCATE.FOR</span></a>, and <a href="cpp/LOCATE.C"><span style='mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:major-fareast'>LOCATE.C</span></a>) described in <a href="Locate.htm"><span style='mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:major-fareast'>Locate.htm</span></a>. </p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.5in;mso-list:l0 level1 lfo2; tab-stops:list .75in'><![if !supportLists]><span style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span><![endif]>Conversion of this file into a linear function is covered in <a href="FileFun.doc"><span style='mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:major-fareast'>FileFun.doc</span></a>.<span style='mso-spacerun:yes'>  </span></p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.5in;mso-list:l0 level1 lfo2; tab-stops:list .75in'><![if !supportLists]><span style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span><![endif]>Once these points have been found <a href="Lagrange.htm"><span style='mso-fareast-font-family:"Times New Roman";mso-fareast-theme-font:major-fareast'>Lagrange.htm</span></a> is a good way to interpolate between these points to convert a collection of data points into a function.<span style='mso-spacerun:yes'>  </span></p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.5in;mso-list:l0 level1 lfo2; tab-stops:list .75in'><![if !supportLists]><span style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span><![endif]>Semi infinite range interpolation </p> <h2><span style='mso-fareast-font-family:"Times New Roman"'>Numerical Derivatives<o:p></o:p></span></h2> <p class=MsoNormal style='margin-left:.25in'>Numerical derivatives are discussed <span class=GramE>in </span><a href="../Derivatives"><span style='mso-fareast-font-family:"Times New Roman";mso-fareast-theme-font:major-fareast'>..\Derivatives</span></a>.</p> <p class=MsoNormal><o:p>&nbsp;</o:p></p> <h2><span class=SpellE><span style='mso-fareast-font-family:"Times New Roman"'>Findfun</span></span><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h2> <p class=MsoNormal><span style='mso-tab-count:1'>            </span>Lagrange interpolation skipping a data point can be used to locate the largest error region.<span style='mso-spacerun:yes'>  </span>This can also be used in general error analysis.<span style='mso-spacerun:yes'>  </span>It is a bit more complicated <a href="findfun/Findfun.htm"><span class=SpellE>findfun</span>\Findfun.htm</a></p> <h2><span style='mso-fareast-font-family:"Times New Roman"'>Spline Interpolation<o:p></o:p></span></h2> <p class=MsoNormal><span style='mso-tab-count:1'>            </span>The derivatives of the interpolated function are discontinuous at the data points.<span style='mso-spacerun:yes'>  </span>A spline through these points has continuous first and second derivatives and <span class=SpellE>discontinous</span> third derivatives.</p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.5in;mso-list:l1 level1 lfo4; tab-stops:list .75in'><![if !supportLists]><span style='mso-list:Ignore'>1.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span><![endif]><a href="spline1.htm"><span style='mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:major-fareast'>spline1.htm</span></a></p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.5in;mso-list:l1 level1 lfo4; tab-stops:list .75in'><![if !supportLists]><span style='mso-list:Ignore'>2.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span><![endif]><a href="spline2.htm"><span style='mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:major-fareast'>spline2.htm</span></a></p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.5in;mso-list:l1 level1 lfo4; tab-stops:list .75in'><![if !supportLists]><span style='mso-list:Ignore'>3.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span><![endif]><a href="spline3.htm"><span style='mso-fareast-font-family: "Times New Roman";mso-fareast-theme-font:major-fareast'>spline3.htm</span></a></p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.5in;mso-list:l1 level1 lfo4; tab-stops:list .75in'><![if !supportLists]><span style='mso-list:Ignore'>4.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span><![endif]><a href="Cubic%20Spline%20Interpolation.htm">Cubic Spline Interpolation.htm</a></p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.5in;mso-list:l1 level1 lfo4; tab-stops:list .75in'><![if !supportLists]><span style='mso-list:Ignore'>5.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span><![endif]><a href="SplineFitting/Spline%20vs%20Lagrange.htm"><span class=SpellE>SplineFitting</span>\Spline <span class=SpellE>vs</span> Lagrange.htm</a></p> <p class=MsoNormal><o:p>&nbsp;</o:p></p> <p class=MsoNormal><a href="../wsteve/FDERS/The%20Lagrange%20Polynomial.htm">..\<span class=SpellE>wsteve</span>\FDERS\The Lagrange Polynomial.htm</a><span style='mso-spacerun:yes'>  </span>Finds high derivatives for H-H potential.</p> <p class=MsoNormal><a href="LinearInterp.htm">LinearInterp.htm</a>  very simple </p> </div> </body> </html>