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style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>MACROBUTTON MTEditEquationSection2 <span class=MTEquationSection><font color=red>Equation Chapter 1 Section 1</font></span><span style='mso-element:field-begin'></span> SEQ MTEqn \r \h \* MERGEFORMAT <span style='mso-element:field-end'></span><span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>SEQ MTSec \r 1 \h \* MERGEFORMAT <span style='mso-element:field-end'></span><span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>SEQ MTChap \r 1 \h \* MERGEFORMAT <span style='mso-element:field-end'></span><![endif]--><!--[if supportFields]><span style='mso-element:field-end'></span><![endif]--><span class=Heading1Char1><b><font size=5 face=Arial><span style='font-size:16.0pt'>Welcome to the nlfit folder<o:p></o:p></span></font></b></span></p> <p class=MsoNormal style='text-indent:.5in'><span class=Heading1Char1><b style='mso-ansi-font-weight:normal'><font size=3 face="Times New Roman"><span style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt; font-family:"Times New Roman","serif";font-weight:normal;mso-bidi-font-weight: bold'>The direction file is described in </span></font></b></span><a href="file:///E:\public_html\Fittery\nlfit\Direction%20file.doc"><span style='mso-font-kerning:16.0pt'>Direction file.doc</span></a><span class=Heading1Char1><b style='mso-ansi-font-weight:normal'><font face="Times New Roman"><span style='mso-ansi-font-size:12.0pt;mso-bidi-font-size: 12.0pt;font-family:"Times New Roman","serif";font-weight:normal;mso-bidi-font-weight: bold'> </span></font></b></span><a href="file:///E:\public_html\Fittery\nlfit\Direction%20file.htm"><span style='mso-font-kerning:16.0pt'>.htm</span></a><span class=Heading1Char1><b style='mso-ansi-font-weight:normal'><font face="Times New Roman"><span style='mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;font-family:"Times New Roman","serif"; font-weight:normal;mso-bidi-font-weight:bold'>.<o:p></o:p></span></font></b></span></p> <p class=MsoNormal style='text-indent:.5in'><span class=Heading1Char1><b style='mso-ansi-font-weight:normal'><font size=3 face="Times New Roman"><span style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt; font-family:"Times New Roman","serif";font-weight:normal;mso-bidi-font-weight: bold'>The code nlfit is designed to minimize </span></font></b></span><span style='mso-tab-count:1'>          </span></p> <p class=MsoNormal><sub><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='position:relative;top:17.0pt;mso-text-raise: -17.0pt'><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"/> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"/> <v:f eqn="sum @0 1 0"/> <v:f eqn="sum 0 0 @1"/> <v:f eqn="prod @2 1 2"/> <v:f eqn="prod @3 21600 pixelWidth"/> <v:f eqn="prod @3 21600 pixelHeight"/> <v:f eqn="sum @0 0 1"/> <v:f eqn="prod @6 1 2"/> <v:f eqn="prod @7 21600 pixelWidth"/> <v:f eqn="sum @8 21600 0"/> <v:f eqn="prod @7 21600 pixelHeight"/> <v:f eqn="sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/> <o:lock v:ext="edit" aspectratio="t"/> </v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:217.5pt; height:42pt' o:ole=""> <v:imagedata src="Welcome_files/image033.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=290 height=56 src="Welcome_files/image002.gif" v:shapes="_x0000_i1025"><![endif]></span><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1420617789"> </o:OLEObject> </xml><![endif]--></span></font></sub><span style='mso-spacerun:yes'> </span><!--[if supportFields]><span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>MACROBUTTON MTPlaceRef \* MERGEFORMAT <span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>SEQ MTEqn \h \* MERGEFORMAT <span style='mso-element:field-end'></span>(<span style='mso-field-code:" SEQ MTEqn \\c \\* Arabic \\* MERGEFORMAT "'><span style='mso-no-proof:yes'>1</span></span>)<![endif]--><!--[if supportFields]><span style='mso-element:field-end'></span><![endif]--></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><o:p>&nbsp;</o:p></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'>The function <i style='mso-bidi-font-style:normal'><span style='font-style:italic;mso-bidi-font-style:normal'>f</span></i><sub>A</sub> is different for every functional form.<span style='mso-spacerun:yes'>  </span>This means that it is necessary to re-compile at least part of nlfit every time this form is changed.<span style='mso-spacerun:yes'>  </span>The optional penalty term is discussed in <a href="file:///E:\public_html\optimization\Penalty.docx">Penalty.docx</a> <a href="file:///E:\public_html\optimization\Penalty.htm">.htm</a>. These codes differ from previous versions of nlfit in that they make more extensive use of unformatted data sets and they have a <b style='mso-bidi-font-weight:normal'><span style='font-weight:bold;mso-bidi-font-weight:normal'>PolySA</span></b> option that allows Pen(c) and <i style='mso-bidi-font-style:normal'><span style='font-style:italic;mso-bidi-font-style:normal'>f</span></i><sub>A</sub>(<b style='mso-bidi-font-weight:normal'><span style='font-weight:bold;mso-bidi-font-weight: normal'>c,x)</span></b> to be written as separate stand alone code, possibly in a different computer language.<span style='mso-spacerun:yes'>  </span></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span><span style='mso-spacerun:yes'> </span><a href="file:///E:\public_html\Fittery\nlfit\nlfitFortran.zip">NlfitFortran.zip</a> is stored using <b style='mso-bidi-font-weight:normal'><span style='font-weight: bold;mso-bidi-font-weight:normal'>bnlfitFortran.bat </span></b>operating on <a href="file:///E:\public_html\Fittery\nlfit\nlfitFortran.txt">nlfitFortran.txt</a>.<span style='mso-spacerun:yes'>    </span>There are 5 slightly different codes in the zip and a 6<sup>th</sup> code in <a href="file:///E:\public_html\Fittery\nlfit\nlfit-Pas.zip">nlfit-Pas.zip</a> (<b style='mso-bidi-font-weight:normal'><span style='font-weight:bold;mso-bidi-font-weight: normal'>bnlfitpas.bat</span></b>). </span></font></p> <ol style='margin-top:0in' start=1 type=1> <li class=MsoNormal style='mso-list:l0 level1 lfo1;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The simplest codes require recompilation each time the approximating function is changed.<span style='mso-spacerun:yes'>  </span>The ide s for these are stored in the folders using them.<span style='mso-spacerun:yes'>  </span>The penalty is<span style='position:relative;top:14.0pt;mso-text-raise: -14.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1026" type="#_x0000_t75" style='width:36pt;height:27pt' o:ole=""> <v:imagedata src="Welcome_files/image034.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=48 height=36 src="Welcome_files/image004.gif" v:shapes="_x0000_i1026"><![endif]></span><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1420617790"> </o:OLEObject> </xml><![endif]-->.</span></font></li> <ol style='margin-top:0in' start=1 type=a> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\GePairpFit\nlfitW1.wpj">GePairpFit\nlfitW1.wpj</a> (<a href="file:///E:\public_html\Fittery\nlfit\nlfitw1.zip">nlfitw1.zip</a>) double precision only, calculates PC and PPCC in the main code and uses RobminW1.for.<span style='mso-spacerun:yes'>  </span><a href="file:///E:\public_html\Fittery\nlfit\GePairpFit\nlfitw1.zip">GePairpFit\nlfitw1.zip</a><span style='mso-spacerun:yes'>  </span>This is the simplest<span style='mso-spacerun:yes'>  </span>to use when precision neede for fit is 8 digits or so, and there are no linear dependencies, <a href="#PadeApproximates">#PadeApproximates</a>)</span></font></li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///C:\public_html\Fittery\nlfit\nlfitmp.wpj">nlfitmp.wpj</a> (Poly is double precision, Minimization is multiple precision) <a href="file:///E:\public_html\Fittery\nlfit\GePairpFit\nlfitmp.zip">GePairpFit\nlfitmp.zip</a> (This is slow, but the array is now relatively immune to Pade problems.)</span></font></li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\GePairpFit\nlfit.vfproj">GePairpFit\nlfit.vfproj</a> (Intel10 Fortran, uses quadruple precision) <a href="file:///E:\public_html\Fittery\nlfit\nlfitq.zip">nlfitq.zip</a> (This is about as accurate as the above for the array and also allows the hope of up to 15 digit accuracy in the fit.</span></font></li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\GePairpFit\nlfitqmp.vfproj">GePairpFit\nlfitqmp.vfproj</a> (Intel10 Fortran, quadruple precision, combines with multiple precision array) <a href="file:///E:\public_html\Fittery\nlfit\GePairpFit\nlfitqmp.zip">GePairpFit\nlfitqmp.zip</a> (This is the biggest hammer in the section.<span style='mso-spacerun:yes'>  </span>When it fails try using higher values of NPOW to isolate the troublesome regions.)</span></font></li> </ol> <li class=MsoNormal style='mso-list:l0 level1 lfo1;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The prototype code is reached through <a href="file:///E:\public_html\Fittery\nlfit\nlfit.vfproj">nlfit.vfproj</a>.<span style='mso-spacerun:yes'>  </span>It uses the quadruple precision extension made by the intel 10 version of Fortran 90.<span style='mso-spacerun:yes'>  </span>All arrays are dynamically allocated in the main code.<span style='mso-spacerun:yes'>  </span></span></font></li> <li class=MsoNormal style='mso-list:l0 level1 lfo1;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The code <a href="file:///E:\public_html\Fittery\nlfit\nlfit.wpj">nlfit.wpj</a> utilizes Watcom s extension of Fortran 77 to include dynamic allocation.<span style='mso-spacerun:yes'>  </span>It has the real*16 parts commented out and a few other changes.<span style='mso-spacerun:yes'>  </span>Thus</span></font></li> <ol style='margin-top:0in' start=1 type=a> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\nlfit.for">nlfit.for</a> </span></font><font face=Wingdings><span style='font-family:Wingdings;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type: symbol;mso-symbol-font-family:Wingdings'><span style='mso-char-type:symbol; mso-symbol-font-family:Wingdings'>à</span></span></font> <a href="file:///E:\public_html\Fittery\nlfit\nlfitW.for">nlfitW.for</a> </li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>nlfsubq.for </span></font><font face=Wingdings><span style='font-family:Wingdings;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type: symbol;mso-symbol-font-family:Wingdings'><span style='mso-char-type:symbol; mso-symbol-font-family:Wingdings'>à</span></span></font> nlfsubW.for </li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>robmin.for </span></font><font face=Wingdings><span style='font-family:Wingdings;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type: symbol;mso-symbol-font-family:Wingdings'><span style='mso-char-type:symbol; mso-symbol-font-family:Wingdings'>à</span></span></font> robminW.for</li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><a name="Numerical_Partial_Derivatives"></a><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>This code is set up to check the partial derivatives.<span style='mso-spacerun:yes'>  </span>If<span style='mso-spacerun:yes'>  </span>ivorf(i) in the direction file is made N &gt; 2, the code calculates both analytical and numerical derivatives for the N th data point and prints them out.<span style='mso-spacerun:yes'>  </span>The delta used to find the numerical derivatives is the error estimate in the direction file.</span></font></li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-spacerun:yes'> </span>The sample code <a href="file:///E:\public_html\Fittery\nlfit\nlfit1.wpj">nlfit1.wpj</a> is still under development  most recently accessed through <a href="file:///E:\public_html\wsteve\h2fit\BobNlfit\ednlfit.bat">..\..\wsteve\h2fit\BobNlfit\ednlfit.bat</a>.<span style='mso-spacerun:yes'>  </span>The function f<sub>A­</sub> is assumed to be a pade approximate. (<a href="file:///E:\public_html\Fittery\nlfit\pade\Welcome.docx">pade\Welcome.docx</a> <a href="file:///E:\public_html\Fittery\nlfit\pade\Welcome.htm">.htm</a>)<span style='mso-spacerun:yes'>  </span>The partials of <i style='mso-bidi-font-style: normal'><span style='font-style:italic;mso-bidi-font-style:normal'>Ç</span></i><sup>2</sup> are calculated in the main code nlfsubq </span></font><font face=Wingdings><span style='font-family:Wingdings;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type: symbol;mso-symbol-font-family:Wingdings'><span style='mso-char-type:symbol; mso-symbol-font-family:Wingdings'>à</span></span></font> main.<span style='mso-spacerun:yes'>  </span>The options for double precision and multiple precision second derivative arrays are available.<span style='mso-spacerun:yes'>  </span>The test code is fitting &lt;H<sub>2</sub>&gt; as a function of the separation distance R of the two nuclei.<span style='mso-spacerun:yes'>  </span>It is normally called as &gt;<b style='mso-bidi-font-weight:normal'><span style='font-weight:bold; mso-bidi-font-weight:normal'>nlfit.bat </span></b>potfit.dir from<b style='mso-bidi-font-weight:normal'><span style='font-weight:bold; mso-bidi-font-weight:normal'><span style='mso-spacerun:yes'>  </span></span></b><a href="file:///E:\public_html\wsteve\h2fit\BobNlfit">..\..\wsteve\h2fit\BobNlfit</a>.</li> </ol> </ol> <p class=MsoNormal style='margin-left:.75in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><o:p>&nbsp;</o:p></span></font></p> <ol style='margin-top:0in' start=3 type=1> <ol style='margin-top:0in' start=6 type=a> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The code negg1d is described in <a href="file:///E:\public_html\WaveFunction\WaveFunctionByMinimization.docx">..\..\WaveFunction\WaveFunctionByMinimization.docx</a> <a href="file:///E:\public_html\WaveFunction\WaveFunctionByMinimization.htm">.htm</a>.<span style='mso-spacerun:yes'>  </span>It uses complex constants and first derivatives with a real second derivative matrix <a href="file:///E:\public_html\optimization\Robmin\CRobmin.doc">../../optimization/Robmin/CRobmin.doc</a> <a href="file:///E:\public_html\optimization\Robmin\CRobmin.htm">.htm</a>. </span></font></li> </ol> <li class=MsoNormal style='mso-list:l0 level1 lfo1;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The third code is listed in makefile.g77.<span style='mso-spacerun:yes'>  </span>This code requires that the user dimension (or most often over dimension) a few parameters.<span style='mso-spacerun:yes'>  </span>As with the first code the bulk of the storage is done using unformatted data sets, and the few that do not can be over dimensioned with little harm.<span style='mso-spacerun:yes'>  </span>As the g77 implies, this set of code works on linux and unix operating systems (Extract all files, copy makefile.g77 over makefile and in the nlfit folder issue the command <b style='mso-bidi-font-weight:normal'><span style='font-weight:bold; mso-bidi-font-weight:normal'>make)</span></b>.<span style='mso-spacerun:yes'>  </span>At present in linux it is best to use gnuplot for looking at the results.</span></font></li> <ol style='margin-top:0in' start=1 type=a> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>nlfitW.for<span style='mso-spacerun:yes'>  </span></span></font><font face=Wingdings><span style='font-family:Wingdings;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-font-family: Wingdings'><span style='mso-char-type:symbol;mso-symbol-font-family:Wingdings'>à</span></span></font> <a href="file:///E:\public_html\Fittery\nlfit\nlfitg77.for">nlfitg77.for</a> </li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>other files are the *W.for files</span></font></li> </ol> <li class=MsoNormal style='mso-list:l0 level1 lfo1;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The fourth code differs from nlfitg77 in that it has a series of includes for watfor.<span style='mso-spacerun:yes'>  </span>Watfor has limitations, owing to the default size of the unformatted files, this code works with only 5 of the 6 parameters variable.<span style='mso-spacerun:yes'>  </span>It does give a valuable check of the coding logic.</span></font></li> <ol style='margin-top:0in' start=1 type=a> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\nlfitg77.for">nlfitg77.for</a> </span></font><font face=Wingdings><span style='font-family:Wingdings; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Wingdings'><span style='mso-char-type:symbol;mso-symbol-font-family:Wingdings'>à</span></span></font> nlfitW4.for</li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>other files are the *W.for files</span></font></li> </ol> <li class=MsoNormal style='mso-list:l0 level1 lfo1;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>A fifth code is in the makefile.<span style='mso-spacerun:yes'>  </span>It uses gfortran which for this code differs from Watcom only in that the Fortran 90 allocatable statement is needed for the dynamical arrays. </span></font></li> <ol style='margin-top:0in' start=1 type=a> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>nlfit.for<span style='mso-spacerun:yes'>  </span></span></font><font face=Wingdings><span style='font-family:Wingdings;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-font-family: Wingdings'><span style='mso-char-type:symbol;mso-symbol-font-family:Wingdings'>à</span></span></font> nlfitGF.for <font face=Wingdings><span style='font-family:Wingdings; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Wingdings'><span style='mso-char-type:symbol;mso-symbol-font-family:Wingdings'>ß</span></span></font> real*16 <font face=Wingdings><span style='font-family:Wingdings; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Wingdings'><span style='mso-char-type:symbol;mso-symbol-font-family:Wingdings'>à</span></span></font> real*8 (later real*10) </li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>other files are the *W.for files</span></font></li> </ol> <li class=MsoNormal style='mso-list:l0 level1 lfo1;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>A sixth code, <a href="file:///E:\public_html\Fittery\nlfit\nlfit-Pas.zip">nlfit-Pas.zip</a>, is in Pascal.<span style='mso-spacerun:yes'>  </span>This uses 10 byte extended precision which is about a factor of 100 better able to resolve a near linear dependence than is the usual 8 byte precision.<span style='mso-spacerun:yes'>  </span>Much of the extra precision is in the exponent which helps even more. <a href="file:///E:\public_html\MatrixInverse\Cholesky\Singular.doc">Singular.doc</a> <a href="file:///E:\public_html\MatrixInverse\Cholesky\Singular.htm">.htm</a>.<span style='mso-spacerun:yes'>  </span>The robmin used is very closely related to that in <b style='mso-bidi-font-weight:normal'><span style='font-weight: bold;mso-bidi-font-weight:normal'>Cambio </span></b><a href="file:///E:\public_html\optimization\Robmin\Cambio\Welcome.htm">Cambio\Welcome.htm</a>.<span style='mso-spacerun:yes'>  </span>Pascal is fast (<a href="file:///E:\public_html\Progdet\SoftWare.html">Progdet\SoftWare.html</a>). The code is compiled using <st1:place w:st="on">Delphi</st1:place> in windows and then in linux using code from <a href="http://freepascal.org">http://freepascal.org</a> (<a href="file:///E:\public_html\Progdet\Progdet.htm">Progdet.htm</a>) with the terminal command.<span style='mso-spacerun:yes'>  </span></span></font></li> </ol> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><font size=3 face="Times New Roman"><span style='font-size:12.0pt; font-weight:bold;mso-bidi-font-weight:normal'>fpc  MDelphi nlfit.dpr<o:p></o:p></span></font></b></p> <ol style='margin-top:0in' start=8 type=1> <li class=MsoNormal style='mso-list:l0 level1 lfo1;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>Nlfit2, <a href="file:///E:\public_html\Fittery\nlfit\nlfit2-Pas.zip">nlfit2-Pas.zip</a>, is a simpler version of the Pascal code.<span style='mso-spacerun:yes'>  </span>The code uRobmin2.pas is a version of Robmin in which all possible arrays appear only in robmin.<span style='mso-spacerun:yes'>  </span>The code nlfitsub2 incorporates the sum that is in chiders above.<span style='mso-spacerun:yes'>  </span>This code also has the option of taking numerical derivatives.<span style='mso-spacerun:yes'>  </span></span></font></li> </ol> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><font size=3 face="Times New Roman"><span style='font-size:12.0pt; font-weight:bold;mso-bidi-font-weight:normal'><br style='mso-special-character: line-break'> <![if !supportLineBreakNewLine]><br style='mso-special-character:line-break'> <![endif]><o:p></o:p></span></font></b></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\NlfitSub\Welcome.htm"><span class=Heading1Char1><b><font size=5 color="#00b0f0" face=Arial><span style='font-size:16.0pt;color:#00B0F0'>NlfitSub\Welcome.htm</span></font></b></span></a></span></font><span class=Heading1Char1><b><font size=5 face=Arial><span style='font-size:16.0pt'> <o:p></o:p></span></font></b></span></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>This folder contains the code that one would call to fit data inside a code (e.g. in Cambio)<span style='mso-spacerun:yes'>  </span>This code is called by the main Nlfit code after handling the input and before the output.</span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\PolySA\Welcome%20to%20POLYSA.htm"><span class=Heading1Char1><b><font size=5 color="#00b0f0" face=Arial><span style='font-size:16.0pt;color:#00B0F0'>PolySA\Welcome.htm</span></font></b></span></a></span></font><span class=Heading1Char1><b><font size=5 face=Arial><span style='font-size:16.0pt'> </span></font></b></span><a href="file:///E:\public_html\Fittery\nlfit\PolySA\Welcome%20to%20POLYSA.doc"><span class=Heading1Char1><b><font size=5 color="#00b0f0" face=Arial><span style='font-size:16.0pt;color:#00B0F0'>.doc</span></font></b></span></a> </p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><font size=3 face="Times New Roman"><span style='font-size:12.0pt;font-weight:bold; mso-bidi-font-weight:normal'><span style='mso-tab-count:1'>            </span></span></font></b>The subroutines <a href="file:///E:\public_html\Fittery\nlfit\PolySA\polysubq.for">PolySA\polysubq.for</a> and <a href="file:///E:\public_html\Fittery\nlfit\PolySA\upolysub.pas">PolySA\upolysub.pas</a> organize the results of the calls to poly and take numerical derivatives when requested.<span style='mso-spacerun:yes'>   </span></p> <p class=MsoNormal style='text-indent:.5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>This folder invokes the <b style='mso-bidi-font-weight: normal'><span style='font-weight:bold;mso-bidi-font-weight:normal'>PolySA </span></b>option.<span style='mso-spacerun:yes'>  </span>The Fortran Nlfit uses CALL FSYSTEM( PolySA.exe ), while the Pascal code uses WinExec( PolySA.exe ) at the location where the various poly subroutines find the values Pen and of F<sub>A</sub> and its partials. </span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>The Pascal code does not wait for PolySA to finish.<span style='mso-spacerun:yes'>  </span>This makes it necessary to look for a file with a series of  are we done yet questions that tie up a small amount of the processor for no good use.<span style='mso-spacerun:yes'>  </span>Details are in <a href="file:///E:\public_html\Progdet\dll\pascal\Welcome.htm">..\..\Progdet\dll\pascal\Welcome.htm</a>.<b style='mso-bidi-font-weight:normal'><span style='font-weight:bold;mso-bidi-font-weight: normal'><span style='mso-spacerun:yes'>    </span><o:p></o:p></span></b></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><o:p>&nbsp;</o:p></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><a href="file:///E:\public_html\Fittery\chiders\Welcome.docx"><span class=Heading1Char1><b><font size=5 color="#00b0f0" face=Arial><span style='font-size:16.0pt;color:#00B0F0'>../chiders/Welcome.docx</span></font></b></span></a></span></font><span class=Heading1Char1><b><font size=5 face=Arial><span style='font-size:16.0pt'> </span></font></b></span><a href="file:///E:\public_html\Fittery\chiders\Welcome.htm"><span class=Heading1Char1><b><font size=5 color="#00b0f0" face=Arial><span style='font-size:16.0pt;color:#00B0F0'>.htm</span></font></b></span></a> </p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>This folder contains the chipxx code for making the partials of Ç<sup>2</sup>(cons) from the partials of f<sub>A</sub>.<span style='mso-spacerun:yes'>  </span>The first X refers to the precision with which f<sub>A</sub> and its partials are calculated while the second refers to the precision of Ç<sup>2</sup> and its partials.</span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><a href="file:///E:\public_html\optimization\Robmin\RobminSA\Welcome.doc"><span class=Heading1Char1><b><font size=5 color="#00b0f0" face=Arial><span style='font-size:16.0pt;color:#00B0F0'>RobminSA\Welcome.doc</span></font></b></span></a></span></font><span class=Heading1Char1><b><font size=5 face=Arial><span style='font-size:16.0pt'> </span></font></b></span><a href="file:///E:\public_html\optimization\Robmin\RobminSA\Welcome.htm"><span class=Heading1Char1><b><font size=5 color="#00b0f0" face=Arial><span style='font-size:16.0pt;color:#00B0F0'>.htm</span></font></b></span></a> </p> <p class=MsoNormal style='text-indent:.5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>This is the test folder for the robust minimization routines used in <b style='mso-bidi-font-weight:normal'><span style='font-weight: bold;mso-bidi-font-weight:normal'>nlfit.</span></b><span style='mso-spacerun:yes'>  </span>These in turn depend on the matrix inversion routines in <a href="file:///E:\public_html\MatrixInverse\Cholesky\MatInvSA">Cholesky\MatInvSA</a>.<span style='mso-spacerun:yes'>  </span>Tests of these are described in <a href="file:///E:\public_html\MatrixInverse\Cholesky\Singular.doc">Cholesky\Singular.doc</a> <a href="file:///E:\public_html\MatrixInverse\Cholesky\Singular.htm">.htm</a>.</span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><a href="file:///E:\public_html\Fittery\robfit\TemplFit\eff\Welcome.doc"><span class=Heading1Char1><b><font size=5 color="#00b0f0" face=Arial><span style='font-size:16.0pt;color:#00B0F0'>robfit\TemplFit\eff\Welcome.doc</span></font></b></span></a></span></font><span class=Heading1Char1><b><font size=5 face=Arial><span style='font-size:16.0pt'> </span></font></b></span><a href="file:///E:\public_html\Fittery\robfit\TemplFit\eff\Welcome.htm"><span class=Heading1Char1><b><font size=5 color="#00b0f0" face=Arial><span style='font-size:16.0pt;color:#00B0F0'>.htm</span></font></b></span></a><span class=Heading1Char1><b><u><font size=5 color="#00b0f0" face=Arial><span style='font-size:16.0pt;color:#00B0F0'><o:p></o:p></span></font></u></b></span></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-spacerun:yes'> </span><span style='mso-tab-count:1'>           </span>A fit to the efficiency of an intrinsic Germanium Gamma Ray detector as a function of energy involves fitting 63 points to a linear Pade approximate multiplying the exponential of a Pade.<span style='mso-spacerun:yes'>  </span>This very fast but somewhat difficult fit is the testing ground for the routines described above.<span style='mso-spacerun:yes'>  </span>The poly routines that give f<sub>A­ </sub>and its derivatives are in this folder.</span></font></p> <h1><b style='mso-bidi-font-weight:normal'><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman";mso-bidi-font-weight: normal'>Stop.in</span></font></b><span style='mso-fareast-font-family:"Times New Roman"'> <a name=stop></a><o:p></o:p></span></h1> <p class=MsoNormal style='margin-left:.25in;text-indent:.5in;page-break-after: avoid'><font size=3 face="Times New Roman"><span style='font-size:12.0pt; mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_11" o:spid="_x0000_i1042" type="#_x0000_t75" alt="P…ÏX…Ïx" style='width:308.25pt;height:120.75pt; visibility:visible;mso-wrap-style:square'> <v:imagedata src="Welcome_files/image005.png" o:title="P…ÏX…Ïx"/> </v:shape><![endif]--><![if !vml]><img border=0 width=411 height=161 src="Welcome_files/image035.jpg" alt="P…ÏX…Ïx" v:shapes="Picture_x0020_11"><![endif]></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'>Figure </span></font><!--[if supportFields]><span style='mso-element: field-begin'></span><span style='mso-spacerun:yes'> </span>SEQ Figure \* ARABIC <span style='mso-element:field-separator'></span><![endif]--><span style='mso-no-proof:yes'>1</span><!--[if supportFields]><span style='mso-element: field-end'></span><![endif]--> Author's view of chi-square in the minimization region.<span style='mso-spacerun:yes'>  </span><br> <span style='mso-tab-count:1'>            </span>Look at figure 1 above.<span style='mso-spacerun:yes'>  </span>Nlfit does not stop easily.<span style='mso-spacerun:yes'>  </span>It requires that fr <font face=Wingdings><span style='font-family:Wingdings;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-font-family: Wingdings'><span style='mso-char-type:symbol;mso-symbol-font-family:Wingdings'>à</span></span></font> <st1:date Month="1" Day="10" Year="2015" w:st="on">1-10<sup>-15</sup></st1:date> in rather small steps.<span style='mso-spacerun:yes'>  </span>Physicists frequently look for very small bumps in noisy data, hence the very reluctant stopping.<span style='mso-spacerun:yes'>  </span>That said, when fr becomes small, the routine is Newton-Raphson with quadratic convergence.<span style='mso-spacerun:yes'>  </span>As a practical matter, it should converge in ~100 steps.<span style='mso-spacerun:yes'>  </span>If it does not the derivatives or the matrix inversion are faulty.<span style='mso-spacerun:yes'>  </span><a href="#_Precision_Options">#Precision Options</a></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>It can happen that nlfit goes into a seemingly infinite loop in which the improvement in </span></font><font face=Symbol><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'>c</span></span></font><sup>2</sup> is at an agonizingly slow pace.<span style='mso-spacerun:yes'>  </span>The file stop.in is tested occasionally. Normally, it is empty.<span style='mso-spacerun:yes'>  </span>Making the first character in this file  s or  S causes the routine to stop normally, saving its best values to date.</p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>Be sure to change the s in first position of <b style='mso-bidi-font-weight:normal'><span style='font-weight:bold;mso-bidi-font-weight:normal'>stop.in</span></b> before the next nlfit run. </span></font></p> <h1><b style='mso-bidi-font-weight:normal'><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman";mso-bidi-font-weight: normal'>Quick Start</span></font></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h1> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>The folder <a href="file:///E:\public_html\Fittery\nlfit\PolySA">PolySA</a> contains the zip <a href="file:///E:\public_html\Fittery\nlfit\PolySA\PolySA.zip">PolySA.zip</a>.<span style='mso-spacerun:yes'>  </span>This zip contains </span></font></p> <ol style='margin-top:0in' start=1 type=1> <li class=MsoNormal style='mso-list:l5 level1 lfo2'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>Nlfit.exe  Fortran calling code compiled by Intel-10 to work in windows</span></font></li> <li class=MsoNormal style='mso-list:l5 level1 lfo2'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>Polysa.wpj, Polysa.tgt, Polysa.for, pademp.for.<span style='mso-spacerun:yes'>  </span>A sample set of the stand alone poly called by <b style='mso-bidi-font-weight: normal'><span style='font-weight:bold;mso-bidi-font-weight:normal'>nlfit.exe</span></b></span></font></li> <li class=MsoNormal style='mso-list:l5 level1 lfo2'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-spacerun:yes'> </span><a href="#NumericalPartials">#Numerical Partials</a> require 2M+1 evaluations of the function and generally lose about 7 digits of accuracy.<span style='mso-spacerun:yes'>  </span>Analytical derivatives approximately double the time to evaluate the function, sometimes less, and usually have full double precision accuracy.</span></font></li> </ol> <h1><b style='mso-bidi-font-weight:normal'><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman";mso-bidi-font-weight: normal'>Powers</span></font></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h1> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>If the fit is to a set of points without statistical error, the error in the fit is that of the largest miss.<span style='mso-spacerun:yes'>  </span>Npow values greater than one in </span></font><!--[if supportFields]><span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>GOTOBUTTON ZEqnNum525236<span style='mso-spacerun:yes'>  </span>\* MERGEFORMAT <span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>REF ZEqnNum525236 \* Charformat \! \* MERGEFORMAT <span style='mso-element:field-separator'></span>(1)<span style='mso-element:field-end'></span><![endif]--><!--[if supportFields]><span style='mso-element:field-end'></span><![endif]--> selectively push down the worst misses.<span style='mso-spacerun:yes'>  </span>As Npow increases, the minimization becomes that of the largest error. A value of Npow greater than one also makes the minimization explicitly non linear, since the second derivative array always depends on the last fit. The precision matters less because those points well fitted cease to dominate the second derivative array.<span style='mso-spacerun:yes'>  </span>The recommended procedure is to minimize first with Npow = 1, then 2, 3, and 4 then to reverse this as 4, 2, 1 if and only if the data contains random errors.</p> <h1><a name="_Precision_Options"></a><b style='mso-bidi-font-weight:normal'><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman"; mso-bidi-font-weight:normal'>Precision Options</span></font></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h1> <ol style='margin-top:0in' start=0 type=1> <li class=MsoNormal style='mso-list:l1 level1 lfo3;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The derivatives of f<sub>A</sub> and of Ç<sup>2</sup> are both double precision - fastest </span></font></li> <li class=MsoNormal style='mso-list:l1 level1 lfo3;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The function f<sub>A</sub> is double precision, but the multiplications that form the second derivative matrix are quadruple precision as are the sums that for the first and second partials of Ç<sup>2</sup>.<span style='mso-spacerun:yes'>  </span>The matrix inversion is in quadruple precision.<span style='mso-spacerun:yes'>  </span>This is fast owing to the Intel-10 version of quadruple precision but requires no coding beyond double precision in the calculation of f<sub>A</sub>.<span style='mso-spacerun:yes'>   </span></span></font></li> <li class=MsoNormal style='mso-list:l1 level1 lfo3;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The function f<sub>A </sub>is double precision.<span style='mso-spacerun:yes'>  </span>The multiplications that form the second derivative parts and their sums are multiple precision (it=19, 250 digits).<span style='mso-spacerun:yes'>  </span>The inversion step uses multiple precision Cholesky matrix inversion<b style='mso-bidi-font-weight:normal'><span style='font-weight:bold;mso-bidi-font-weight:normal'> MPSMINV.</span></b></span></font></li> </ol> <p class=MsoNormal style='margin-left:.5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The inversion steps are slow.<span style='mso-spacerun:yes'>  </span>This should normally be used only after step 0 has finished.<span style='mso-spacerun:yes'>  </span>It can be used simply to get around a hangup and the minimization continued with option 0.<span style='mso-spacerun:yes'>  </span>However, when the minimization seems to take forever with very small steps, this is a solution.</span></font></p> <ol style='margin-top:0in' start=3 type=1> <li class=MsoNormal style='mso-list:l1 level1 lfo3;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The function f<sub>A</sub> and its derivatives are quadruple precision.<span style='mso-spacerun:yes'>  </span>All summations and inversions are also quadruple precision.<span style='mso-spacerun:yes'>  </span>The Intel-10 compiler is needed even for PolySA.<span style='mso-spacerun:yes'>  </span>This is easy to code, simply use real*16 for everything.<span style='mso-spacerun:yes'>  </span>It is fairly fast and greatly extends the fitting range.<span style='mso-spacerun:yes'>    </span></span></font></li> <li class=MsoNormal style='mso-list:l1 level1 lfo3;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The function f<sub>A</sub> and its derivatives are quadruple precision (Intel-10).<span style='mso-spacerun:yes'>  </span>The product of these, the sums and the matrix inversions are multiple precision (it=19, 250 digits) </span></font></li> <li class=MsoNormal style='mso-list:l1 level1 lfo3;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The function f<sub>A</sub> and its derivatives along with all other terms are multiple precision.<span style='mso-spacerun:yes'>  </span><b style='mso-bidi-font-weight: normal'><span style='font-weight:bold;mso-bidi-font-weight:normal'>Nlfit</span></b> is based on the notion that the set of cons is real*8.<span style='mso-spacerun:yes'>  </span>This can limit the final accuracy, though normally the practical limit is linear dependency.</span></font></li> </ol> <p class=MsoNormal style='margin-left:.25in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>With an Intel-10 nlfit.exe and <b style='mso-bidi-font-weight: normal'><span style='font-weight:bold;mso-bidi-font-weight:normal'>PolySA</span></b> without Intel 10, the options available are 0, 1, 2, and 5.<span style='mso-spacerun:yes'>  </span>Without quadruple precision (e.g. in Linux), the options are 0,2, and 5.<br style='mso-special-character:line-break'> <![if !supportLineBreakNewLine]><br style='mso-special-character:line-break'> <![endif]></span></font></p> <ol style='margin-top:0in' start=6 type=1> <li class=MsoNormal style='mso-list:l1 level1 lfo3;tab-stops:list .5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The Pascal version of nlfit does not use PolySA.<span style='mso-spacerun:yes'>  </span>It has f<sub>A</sub> and its derivatives in double precision, but the product and sums that form the derivatives of Ç<sup>2</sup> are in extended precision as are the matrix inversions.<span style='mso-spacerun:yes'>  </span>This falls between 0 and 1 above, with roughly a factor of 100 gain in the ability to approach linear dependencies.<span style='mso-spacerun:yes'>  </span>It is fast.<span style='mso-spacerun:yes'>  </span></span></font></li> </ol> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><o:p>&nbsp;</o:p></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><o:p>&nbsp;</o:p></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'>Sections 3 through 5 are out of date.<span style='mso-spacerun:yes'>  </span>They are left in for reference only. </span></font></p> <p class=MsoNormal style='margin-left:.5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\for\nlfit.wpj">for\nlfit.wpj</a> <a href="file:///E:\public_html\Fittery\nlfit\for\Welcome.htm">Welcome.htm</a> double precision extendable using Bailey s routines<a name=One></a>  Uses Cambio s robmin.</span></font></p> <ol style='margin-top:0in' start=8 type=1> <ol style='margin-top:0in' start=1 type=a> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\Ptcheby\nlfit.wpj">Ptcheby\nlfit.wpj</a>  expands a Pade in Tchebyshev polynomials</span></font></li> </ol> <li class=MsoNormal style='mso-list:l0 level1 lfo1;tab-stops:list .5in'><a name=Two></a><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\formp\nlfit.wpj">formp\nlfit.wpj</a> <a href="file:///E:\public_html\Fittery\nlfit\formp\Welcome.htm">Welcome.htm</a> uses multiple precision when requested</span></font></li> <ol style='margin-top:0in' start=1 type=a> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\2dPoly\nlfit.wpj">2dPoly\nlfit.wpj</a></span></font></li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\EkofZ\nlfit.wpj">EkofZ\nlfit.wpj</a></span></font></li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\FermiFit\nlfit.wpj">FermiFit\nlfit.wpj</a></span></font></li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\fit8\nlfit.wpj">fit8\nlfit.wpj</a></span></font></li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///E:\public_html\Fittery\nlfit\LogAigau\nlfit.wpj">LogAigau\nlfit.wpj</a></span></font></li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><a name=Three></a><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><a href="file:///E:\public_html\Fittery\chiders\Welcome.docx">..\chiders\Welcome.docx</a> <a href="file:///E:\public_html\Fittery\chiders\Welcome.htm">.htm</a> This folder contains the code that evaluates <span style='position:relative; top:16.0pt;mso-text-raise:-16.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1027" type="#_x0000_t75" style='width:93.75pt;height:36.75pt' o:ole=""> <v:imagedata src="Welcome_files/image036.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=125 height=49 src="Welcome_files/image008.gif" v:shapes="_x0000_i1027"><![endif]></span><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1420617791"> </o:OLEObject> </xml><![endif]-->.<span style='mso-spacerun:yes'>  </span>Usually the subroutines in this folder are called directly from <b style='mso-bidi-font-weight: normal'><span style='font-weight:bold;mso-bidi-font-weight:normal'>NlfitSub.</span></b></span></font></li> <li class=MsoNormal style='mso-list:l0 level2 lfo1;tab-stops:list 1.0in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The stand alone codes are tested in <a href="file:///E:\public_html\Fittery\robfit\TemplFit\eff">..\robfit\TemplFit\eff</a> <a href="file:///E:\public_html\Fittery\robfit\TemplFit\eff\TestingNlfit.docx">TestingNlfit.docx</a> <a href="file:///E:\public_html\Fittery\robfit\TemplFit\eff\TestingNlfit.htm">.htm</a><br> <br style='mso-special-character:line-break'> <![if !supportLineBreakNewLine]><br style='mso-special-character:line-break'> <![endif]></span></font></li> </ol> </ol> <h1><b style='mso-bidi-font-weight:normal'><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman";mso-bidi-font-weight: normal'>Discussion</span></font></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h1> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>Nlfit has been a working code for many years.<span style='mso-spacerun:yes'>  </span>The old version is discussed in in <a href="#pre2007_Fortran">#pre2007_Fortran</a>.<span style='mso-spacerun:yes'>  </span></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>The data is a set of N values of {<b style='mso-bidi-font-weight:normal'><span style='font-weight: bold;mso-bidi-font-weight:normal'>x</span></b><sub>i</sub>,<i style='mso-bidi-font-style: normal'><span style='font-style:italic;mso-bidi-font-style:normal'>f</span></i><sub>i</sub>,</span></font><font face=Symbol><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'>e</span></span></font><sub>i</sub>}.<span style='mso-spacerun:yes'>  </span>The code <a href="file:///E:\public_html\Fittery\nlfit\PolySA\Welcome%20to%20POLYSA.htm">POLYSA.htm</a> uses <b style='mso-bidi-font-weight:normal'><span style='font-weight:bold; mso-bidi-font-weight:normal'>c<sub>I</sub></span></b> to extend this set to {<b style='mso-bidi-font-weight:normal'><span style='font-weight:bold;mso-bidi-font-weight: normal'>x</span></b><sub>i</sub>,<i style='mso-bidi-font-style:normal'><span style='font-style:italic;mso-bidi-font-style:normal'>f</span></i><sub>i</sub>,<i style='mso-bidi-font-style:normal'><span style='font-style:italic;mso-bidi-font-style: normal'>f</span></i><sub>A,I</sub>,<font face=Symbol><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'>e</span></span></font><sub>i</sub>}.<span style='mso-spacerun:yes'>  </span>The code nlfitSA calculates the vector <span style='position:relative;top:6.0pt;mso-text-raise:-6.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1028" type="#_x0000_t75" style='width:45pt;height:18.75pt' o:ole=""> <v:imagedata src="Welcome_files/image037.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=60 height=25 src="Welcome_files/image010.gif" v:shapes="_x0000_i1028"><![endif]></span><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1028" DrawAspect="Content" ObjectID="_1420617792"> </o:OLEObject> </xml><![endif]--><span style='mso-spacerun:yes'> </span>and the array <span style='position:relative;top:7.0pt;mso-text-raise:-7.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1029" type="#_x0000_t75" style='width:77.25pt;height:20.25pt' o:ole=""> <v:imagedata src="Welcome_files/image038.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=103 height=27 src="Welcome_files/image012.gif" v:shapes="_x0000_i1029"><![endif]></span><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1029" DrawAspect="Content" ObjectID="_1420617793"> </o:OLEObject> </xml><![endif]-->from {<b style='mso-bidi-font-weight:normal'><span style='font-weight:bold;mso-bidi-font-weight:normal'>x</span></b><sub>i</sub>,<i style='mso-bidi-font-style:normal'><span style='font-style:italic;mso-bidi-font-style: normal'>f</span></i><sub>i</sub>,<i style='mso-bidi-font-style:normal'><span style='font-style:italic;mso-bidi-font-style:normal'>f</span></i><sub>A,I</sub>,<font face=Symbol><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'>e</span></span></font><sub>i</sub>}.<span style='mso-spacerun:yes'>  </span>The code RobminSA uses the sequence of values of Ç<sup>2</sup><sub>Pow,I</sub> to determine f<sub>r</sub>, the desired fractional decrease in <!--[if supportFields]><span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>GOTOBUTTON ZEqnNum525236<span style='mso-spacerun:yes'>  </span>\* MERGEFORMAT <span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>REF ZEqnNum525236 \* Charformat \! \* MERGEFORMAT <span style='mso-element:field-separator'></span>(1)<span style='mso-element:field-end'></span><![endif]--><!--[if supportFields]><span style='mso-element:field-end'></span><![endif]-->.<span style='mso-spacerun:yes'>  </span>Then it uses this value along with <b style='mso-bidi-font-weight:normal'><span style='font-weight:bold;mso-bidi-font-weight: normal'>c<sub>I</sub>, </span></b><span style='position:relative;top:6.0pt; mso-text-raise:-6.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1030" type="#_x0000_t75" style='width:45pt;height:18.75pt' o:ole=""> <v:imagedata src="Welcome_files/image037.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=60 height=25 src="Welcome_files/image010.gif" v:shapes="_x0000_i1030"><![endif]></span><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1030" DrawAspect="Content" ObjectID="_1420617794"> </o:OLEObject> </xml><![endif]-->and <span style='position:relative;top:7.0pt;mso-text-raise: -7.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1031" type="#_x0000_t75" style='width:77.25pt;height:20.25pt' o:ole=""> <v:imagedata src="Welcome_files/image038.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=103 height=27 src="Welcome_files/image012.gif" v:shapes="_x0000_i1031"><![endif]></span><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1031" DrawAspect="Content" ObjectID="_1420617795"> </o:OLEObject> </xml><![endif]--><span style='mso-spacerun:yes'> </span>to predict <b style='mso-bidi-font-weight:normal'><span style='font-weight:bold;mso-bidi-font-weight: normal'>c</span></b><sub>I+1</sub>.<span style='mso-spacerun:yes'>  </span>Thus the nlfit code consists of three separable codes with a bit of overhead.<span style='mso-spacerun:yes'>   </span></p> <p class=MsoNormal style='text-indent:.5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The routine Robmin has a set of Marquardt parameters determined experientially from the steps used to minimize </span></font><!--[if supportFields]><span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>GOTOBUTTON ZEqnNum525236<span style='mso-spacerun:yes'>  </span>\* MERGEFORMAT <span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>REF ZEqnNum525236 \* Charformat \! \* MERGEFORMAT <span style='mso-element:field-separator'></span>(1)<span style='mso-element:field-end'></span><![endif]--><!--[if supportFields]><span style='mso-element:field-end'></span><![endif]-->. The newest robmin starts from <a href="file:///E:\public_html\optimization\solving\Welcome.htm">..\..\optimization\solving\Welcome.htm</a> <font face=Wingdings><span style='font-family:Wingdings;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-char-type:symbol; mso-symbol-font-family:Wingdings'><span style='mso-char-type:symbol;mso-symbol-font-family: Wingdings'>ß</span></span></font> see especially the documents on Bracketing, Bracketi-bli, and <st1:City w:st="on"><st1:place w:st="on">Newton</st1:place></st1:City>. </p> <p class=MsoNormal style='text-indent:.5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>In the fall of 2007, the ability to use <a href="file:///E:\public_html\MultiplePrecision\Welcome.htm">MultiplePrecision</a> for the calculations involved in finding PC and PPCC and to the appropriate steps for inverting PPCC was added to the general code.<span style='mso-spacerun:yes'>  </span><a href="#options">#options</a> Option 0 does not use multiple precision.<span style='mso-spacerun:yes'>  </span>Options 1, and 2 use double precision first derivatives from Poly with multiple precision summands.<span style='mso-spacerun:yes'>  </span>These do not use multiple precision data from poly.<span style='mso-spacerun:yes'>  </span>Options 0, 1, and 2 can be used with <a href="file:///E:\public_html\Fittery\nlfit\PolySA\Welcome%20to%20POLYSA.htm">POLYSA.htm</a> which can be in any computer language and use any compiler. Options 3, 4 and 5 require the user written code, in the subroutine <a href="file:///E:\public_html\Fittery\nlfit\formp\poly.for">poly.for</a> or <a href="file:///E:\public_html\Fittery\nlfit\formp\pademp.for">pademp.for</a> to return multiple precision values of <sub><span style='position:relative; top:7.0pt;mso-text-raise:-7.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1032" type="#_x0000_t75" style='width:70.5pt;height:19.5pt' o:ole=""> <v:imagedata src="Welcome_files/image039.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=94 height=26 src="Welcome_files/image014.gif" v:shapes="_x0000_i1032"><![endif]></span><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1032" DrawAspect="Content" ObjectID="_1420617796"> </o:OLEObject> </xml><![endif]--></sub>.<span style='mso-spacerun:yes'>    </span>These are most easily done using <a href="file:///E:\public_html\Fittery\nlfit\formp\mpbob.for">formp\mpbob.for</a> (<a href="file:///E:\public_html\Fittery\nlfit\formp\Mpbob.htm">Mpbob.htm</a>) Multiple precision testing was done by making fits to <sub><span style='position:relative;top:16.0pt;mso-text-raise:-16.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1033" type="#_x0000_t75" style='width:123pt;height:38.25pt' o:ole=""> <v:imagedata src="Welcome_files/image040.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=164 height=51 src="Welcome_files/image016.gif" v:shapes="_x0000_i1033"><![endif]></span><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1033" DrawAspect="Content" ObjectID="_1420617797"> </o:OLEObject> </xml><![endif]--></sub><span style='mso-spacerun:yes'> </span>for -6</span></font><font face=Symbol><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></font>x<font face=Symbol><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></font>0 <a href="#_Log(AiGau)">#Log(AiGau)</a>.<span style='mso-spacerun:yes'>  </span></p> <p class=MsoNormal style='text-indent:.5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>In calculating the first derivative vector and the second derivative array, multiple precision can keep values that would otherwise be lost to truncation error.<span style='mso-spacerun:yes'>  </span>Then in the second derivative matrix may also need to be multiple precision to keep these values.<span style='mso-spacerun:yes'>  </span>This means that while it is often necessary to have f<sub>A</sub> and its derivatives multiple precision that there is also a region where only the stock routines nlfitSa and robmin need to be multiple precision. </span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><o:p>&nbsp;</o:p></span></font></p> <p class=MsoNormal style='text-indent:.5in'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'>The only coding needed to take full advantage of the nlfit described below is that of the definitions of <span style='position:relative; top:7.0pt;mso-text-raise:-7.0pt'><!--[if gte vml 1]><v:shape id="_x0000_i1034" type="#_x0000_t75" style='width:140.25pt;height:20.25pt' o:ole=""> <v:imagedata src="Welcome_files/image041.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=187 height=27 src="Welcome_files/image018.gif" v:shapes="_x0000_i1034"><![endif]></span><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1034" DrawAspect="Content" ObjectID="_1420617798"> </o:OLEObject> </xml><![endif]--><span style='mso-spacerun:yes'> </span>in <a href="file:///E:\public_html\Fittery\nlfit\PolySA\pademp.for">pademp.for</a>.</span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span><span style='mso-spacerun:yes'>  </span></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>In general the htm s contain links to <a href="file:///E:\public_html\Fittery\nlfit\zips">zips</a> with the data fitted and watcom ide s for recompiling the Fortran.<span style='mso-spacerun:yes'>  </span>When the zip contains Fortran code, the routines are older than those in the <a href="file:///E:\public_html\Fittery\nlfit\formp">formp</a> folder, but they worked for the examples shown.<span style='mso-spacerun:yes'>  </span>Presumably the new ones work better, but they have not been tested. </span></font></p> <h1><b style='mso-bidi-font-weight:normal'><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman";mso-bidi-font-weight: normal'>Details2</span></font></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h1> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>The code nlfit minimizes the chi-square of equation </span></font><!--[if supportFields]><span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>GOTOBUTTON ZEqnNum525236<span style='mso-spacerun:yes'>  </span>\* MERGEFORMAT <span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>REF ZEqnNum525236 \* Charformat \! \* MERGEFORMAT <span style='mso-element:field-separator'></span>(1)<span style='mso-element:field-end'></span><![endif]--><!--[if supportFields]><span style='mso-element:field-end'></span><![endif]-->.<span style='mso-spacerun:yes'>  </span>The user needs to to write a double precision routine similar to <a href="file:///E:\public_html\Fittery\nlfit\formp\poly.for">formp\poly.for</a> to utilize most of <a href="file:///E:\public_html\optimization\Robmin\Robmin.doc">Robmin s</a> (<a href="file:///E:\public_html\optimization\Robmin\Robmin.htm">htm</a>) ability to minimize <font face=Symbol><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'>c</span></span></font><sup>2</sup> or a routine that returns multiple precision partials of f<sub>A</sub> with respect to c<sub>J</sub> similar to <a href="file:///E:\public_html\Fittery\nlfit\formp\pademp.for">formp\pademp.for</a>.<span style='mso-spacerun:yes'>  </span></p> <p class=MsoNormal><font size=1 face="Courier New"><span style='font-size:8.0pt; font-family:"Courier New"'><span style='mso-spacerun:yes'>      </span>SUBROUTINE <st1:stockticker w:st="on">POLY</st1:stockticker>(X,PF,CONS,FA,MPPF,NCONS,NX,MP)<o:p></o:p></span></font></p> <p class=MsoNormal><font size=1 face="Courier New"><span style='font-size:8.0pt; font-family:"Courier New"'><span style='mso-spacerun:yes'>      </span>IMPLICIT <st1:stockticker w:st="on">REAL</st1:stockticker>*8 (A-H,O-Z)<o:p></o:p></span></font></p> <p class=MsoNormal><font size=1 face="Courier New"><span style='font-size:8.0pt; font-family:"Courier New"'><span style='mso-spacerun:yes'>      </span>DIMENSION X(NX),PF(*),CONS(*)<o:p></o:p></span></font></p> <p class=MsoNormal><font size=1 face="Courier New"><span style='font-size:8.0pt; font-family:"Courier New"'><span style='mso-spacerun:yes'>      </span>DIMENSION MPPF(64,*)<o:p></o:p></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>The first argument X is the vector <b style='mso-bidi-font-weight:normal'><span style='font-weight: bold;mso-bidi-font-weight:normal'>x</span></b><sub>i</sub> in </span></font><!--[if supportFields]><span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>GOTOBUTTON ZEqnNum525236<span style='mso-spacerun:yes'>  </span>\* MERGEFORMAT <span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>REF ZEqnNum525236 \* Charformat \! \* MERGEFORMAT <span style='mso-element:field-separator'></span>(1)<span style='mso-element:field-end'></span><![endif]--><!--[if supportFields]><span style='mso-element:field-end'></span><![endif]-->.<span style='mso-spacerun:yes'>  </span>It has NX dimensions.<span style='mso-spacerun:yes'>  </span>The second argument PF is <sub><!--[if gte vml 1]><v:shape id="_x0000_i1035" type="#_x0000_t75" style='width:98.25pt;height:36.75pt' o:ole=""> <v:imagedata src="Welcome_files/image042.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=131 height=49 src="Welcome_files/image020.gif" v:shapes="_x0000_i1035"><![endif]><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1035" DrawAspect="Content" ObjectID="_1420617799"> </o:OLEObject> </xml><![endif]--></sub>.<span style='mso-spacerun:yes'>  </span>The vector CONS is <b style='mso-bidi-font-weight:normal'><span style='font-weight:bold; mso-bidi-font-weight:normal'>c</span></b> in <!--[if supportFields]><span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>GOTOBUTTON ZEqnNum525236<span style='mso-spacerun:yes'>  </span>\* MERGEFORMAT <span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>REF ZEqnNum525236 \* Charformat \! \* MERGEFORMAT <span style='mso-element:field-separator'></span>(1)<span style='mso-element:field-end'></span><![endif]--><!--[if supportFields]><span style='mso-element:field-end'></span><![endif]-->.<span style='mso-spacerun:yes'>  </span>Both PF and CONS have NCONS dimensions.<span style='mso-spacerun:yes'>  </span>The scalar FA is the value of<span style='mso-spacerun:yes'>  </span><sub><!--[if gte vml 1]><v:shape id="_x0000_i1036" type="#_x0000_t75" style='width:47.25pt;height:20.25pt' o:ole=""> <v:imagedata src="Welcome_files/image043.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=63 height=27 src="Welcome_files/image022.gif" v:shapes="_x0000_i1036"><![endif]><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1036" DrawAspect="Content" ObjectID="_1420617800"> </o:OLEObject> </xml><![endif]--></sub><span style='mso-spacerun:yes'> </span>in <!--[if supportFields]><span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>GOTOBUTTON ZEqnNum525236<span style='mso-spacerun:yes'>  </span>\* MERGEFORMAT <span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>REF ZEqnNum525236 \* Charformat \! \* MERGEFORMAT <span style='mso-element:field-separator'></span>(1)<span style='mso-element:field-end'></span><![endif]--><!--[if supportFields]><span style='mso-element:field-end'></span><![endif]-->.<span style='mso-spacerun:yes'>  </span>The array MPPF(64,*) is the multi-precision version of PF.<span style='mso-spacerun:yes'>  </span>Each value is represented by 64 integers from -8192 to 8191 leading to approximately 250 digits of precision.<span style='mso-spacerun:yes'>  </span>(<a href="file:///E:\public_html\MultiplePrecision\Bob\Welcome.htm">../../MultiplePrecision/Bob/Welcome.htm</a>).<span style='mso-spacerun:yes'>  </span>The integer MP is passed to the subroutine to inform it that MPPF is needed if it is geater than 3. (<a href="file:///E:\public_html\Fittery\nlfit\formp\pademp.for">formp\pademp.for</a> is a Poly including multiple precision)<span style='mso-spacerun:yes'>  </span>The routine poly is called for every value of x<sub>i</sub>.<span style='mso-spacerun:yes'>  </span>The returned values of FA enable the calculation of <font face=Symbol><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'>c</span></span></font><sup>2</sup>.<span style='mso-spacerun:yes'>  </span>The values of PF give both PC(J)=<sub><!--[if gte vml 1]><v:shape id="_x0000_i1037" type="#_x0000_t75" style='width:24.75pt;height:36.75pt' o:ole=""> <v:imagedata src="Welcome_files/image044.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=33 height=49 src="Welcome_files/image024.gif" v:shapes="_x0000_i1037"><![endif]><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1037" DrawAspect="Content" ObjectID="_1420617801"> </o:OLEObject> </xml><![endif]--></sub>and a surprisingly accurate estimate of<span style='mso-spacerun:yes'>  </span>PPCC(J,K) = <sub><!--[if gte vml 1]><v:shape id="_x0000_i1038" type="#_x0000_t75" style='width:36.75pt;height:36.75pt' o:ole=""> <v:imagedata src="Welcome_files/image045.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=49 height=49 src="Welcome_files/image026.gif" v:shapes="_x0000_i1038"><![endif]><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1038" DrawAspect="Content" ObjectID="_1420617802"> </o:OLEObject> </xml><![endif]--></sub>.<span style='mso-spacerun:yes'>   </span></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>Once these have been found the next estimate of the constants is given by CONS(J)=<sub><!--[if gte vml 1]><v:shape id="_x0000_i1039" type="#_x0000_t75" style='width:152.25pt;height:42.75pt' o:ole=""> <v:imagedata src="Welcome_files/image046.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=203 height=57 src="Welcome_files/image028.gif" v:shapes="_x0000_i1039"><![endif]><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1039" DrawAspect="Content" ObjectID="_1420617803"> </o:OLEObject> </xml><![endif]--></sub> where </span></font><font face=Symbol><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'>l</span></span></font> is a Marquardt parameter that needs to be found rather precisely and S(j) is a smoothing factor that contains an estimate of the range of validity of the second order expansion of <font face=Symbol><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'>c</span></span></font><sup>2</sup>.<span style='mso-spacerun:yes'>  </span>The details begin in <a href="file:///E:\public_html\Fittery\FitData.doc">FitData.doc</a> <a href="file:///E:\public_html\Fittery\FitData.htm">.htm</a>.<span style='mso-spacerun:yes'>  </span></p> <p class=MsoNormal><a name="_Powers"></a><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><o:p>&nbsp;</o:p></span></font></p> <h1><b style='mso-bidi-font-weight:normal'><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman";mso-bidi-font-weight: normal'>Plotting</span></font></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h1> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>Begin by plotting the data.<span style='mso-spacerun:yes'>  </span>Plot codes are described in <a href="file:///E:\public_html\gplot\WELCOME.htm#Windows">gplot\WELCOME.htm</a>.<span style='mso-spacerun:yes'>  </span>I have recently compiled gplot with the Intel-10 compiler.<span style='mso-spacerun:yes'>  </span>The resulting executable contained in <a href="file:///E:\public_html\gplot\windows\gplot\gplot\Debug\gplot.zip">gplot.zip</a> appears to run properly on all windows computers.<span style='mso-spacerun:yes'>  </span>Nlfit produces fit.dat which is  essentially the fit at each of the data points, res.out which is ((f<sub>i</sub>-f<sub>A</sub>(<b style='mso-bidi-font-weight:normal'><span style='font-weight:bold;mso-bidi-font-weight: normal'>x</span></b><sub>i</sub>))/err<sub>i</sub>)<sup>2*Npow</sup> at each of the data points and err.out which is f<sub>i</sub>-f<sub>A</sub>(<b style='mso-bidi-font-weight:normal'><span style='font-weight:bold;mso-bidi-font-weight: normal'>x</span></b><sub>i</sub>) at each of the data points.<span style='mso-spacerun:yes'>  </span></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>The open source code <a href="http://www.gnuplot.info/">http://www.gnuplot.info/</a> is one of the best ways to plot the data.<span style='mso-spacerun:yes'>  </span>On my fedora linux system, I go to the directory with the data, then enter the command gnuplot.<span style='mso-spacerun:yes'>  </span>Then</span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'>plot  BOBc.txt ,  <st1:stockticker w:st="on">FIT</st1:stockticker>.DAT to get a plot of the data and the fit to the data.</span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><o:p>&nbsp;</o:p></span></font></p> <h1><a name="_Options"></a><a name="_Stop.in"></a><b style='mso-bidi-font-weight: normal'><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family: "Times New Roman";mso-bidi-font-weight:normal'>Standard Deviations of the calculated results</span></font></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h1> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>The error matrix in <a href="file:///E:\public_html\Fittery\ErrorMatrix\StdDev.doc">StdDev.doc</a> <a href="file:///E:\public_html\Fittery\ErrorMatrix\StdDev.htm">htm</a> is shown to be closely related to the matrix used in minimizing the </span></font><font face=Symbol><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'>c</span></span></font><sup>2</sup> of equation <!--[if supportFields]><span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>GOTOBUTTON ZEqnNum525236<span style='mso-spacerun:yes'>  </span>\* MERGEFORMAT <span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>REF ZEqnNum525236 \* Charformat \! \* MERGEFORMAT <span style='mso-element:field-separator'></span>(1)<span style='mso-element:field-end'></span><![endif]--><!--[if supportFields]><span style='mso-element:field-end'></span><![endif]--> .<span style='mso-spacerun:yes'>  </span>After the minimum is reached nlfit makes a final inversion of the second derivative matrix and multiplies this inverse by 2 <font face=Symbol><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'>c</span></span></font><sup>2</sup> divided by the number of degrees of freedom to form the error matrix.<span style='mso-spacerun:yes'>  </span>The diagonal terms of this matrix are the standard deviations of the fitted constants and are written in the normal output file.<span style='mso-spacerun:yes'>  </span>The fitted constants, the ivorf terms that tell which constants vary, and the complete error matrix are written to the file ERRMAT.OUT.<span style='mso-spacerun:yes'>  </span></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>A<span style='mso-spacerun:yes'>  </span>fit is made to the derivative of a generalized Fermi function, <a href="file:///E:\public_html\Fittery\nlfit\FermiFit\dfpoly.for">dfpoly.for</a>, in the folder <a href="file:///E:\public_html\Fittery\nlfit\FermiFit">FermiFit</a> <a href="file:///E:\public_html\Fittery\nlfit\FermiFit\Welcome.htm">FermiFit\Welcome.htm</a>.<span style='mso-spacerun:yes'>  </span>The code <a href="file:///E:\public_html\Fittery\nlfit\FermiFit\polstd.for">polstd.for</a> uses the generalized Fermi function <a href="file:///E:\public_html\Fittery\nlfit\FermiFit\FPOLY.FOR">FPOLY.FOR</a> which is the integral of the fitting polynomial, the constants fitted and the error matrix to calculate the standard deviation of the Fermi function over the range specified in the direction file <a href="file:///E:\public_html\Fittery\nlfit\FermiFit\polstd.dir">polstd.dir</a>.</span></font></p> <h1><b style='mso-bidi-font-weight:normal'><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman";mso-bidi-font-weight: normal'>Compilers</span></font></b><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h1> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>Nlfit uses some extensions to Fortran 77.<span style='mso-spacerun:yes'>  </span>In particular it opens files, including scratch files and it also calls the system in the event that the user needs help.<span style='mso-spacerun:yes'>  </span>The addition of a double precision variable to a multiple precision variable uses ISHFT, IBCLR and IBSET. <span style='mso-spacerun:yes'> </span>Compiler dependent routines in appropriately named files are slightly different for each compiler.<span style='mso-spacerun:yes'>  </span></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>My attempts to use dll s mostly failed due to a need to pass locations from one compiler to another (<a href="file:///E:\public_html\Progdet\Progdet.htm#dll">Progdet.htm#dll</a>).<span style='mso-spacerun:yes'>  </span>The fourth method in the series of tests documented in <a href="file:///E:\public_html\Progdet\dll\Welcome.htm">..\..\Progdet\dll\Welcome.htm</a> is not a dll at all, but rather a stand alone code whose arguments are passed as two binary files.<span style='mso-spacerun:yes'>  </span><a href="file:///E:\public_html\Fittery\nlfit\PolySA\Welcome%20to%20POLYSA.htm">PolySA\Welcome to POLYSA.htm</a>. This means that the code <a href="file:///E:\public_html\Fittery\nlfit\PolySA\polysa.for">PolySA\polysa.for</a> can be used to call <a href="file:///E:\public_html\Fittery\nlfit\PolySA\pademp.for">PolySA\pademp.for</a>.<span style='mso-spacerun:yes'>  </span>The Ide for this is <a href="file:///E:\public_html\Fittery\nlfit\PolySA\polysa.wpj">PolySA\polysa.wpj</a> while the Ide for the rest of the code is <a href="file:///E:\public_html\Fittery\nlfit\PolySA\nlfit.vfproj">PolySA\nlfit.vfproj</a>.<span style='mso-spacerun:yes'>  </span>The bulk of the code is compiled in release form by the Intel-10 compiler while the part specific to a fiven analytical approximation to the function of interest is compiled by <a href="file:///E:\public_html\Progdet\openwatcom.htm">..\..\Progdet\openwatcom.htm</a></span></font></p> <h2 style='margin-left:.5in;text-indent:-.25in;mso-list:l4 level1 lfo4; tab-stops:list .5in'><![if !supportLists]><b><font size=4 color="#4f81bd" face=Cambria><span style='font-size:13.0pt;mso-fareast-font-family:Cambria; mso-bidi-font-family:Cambria'><span style='mso-list:Ignore'>1.<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font></b><![endif]><span style='mso-fareast-font-family: "Times New Roman"'>Watfor<o:p></o:p></span></h2> <p class=MsoNormal style='tab-stops:95.4pt'><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-spacerun:yes'>              </span>Watfor is a dos based Fortran 77 compiler with very few extensions. --<span style='mso-spacerun:yes'>  </span><a href="file:///E:\public_html\Progdet\W4.mht">W4.mht</a> <a href="file:///E:\public_html\Progdet\Watfor.htm">Watfor.htm</a> -- It checks for uninitialized variables at run time better than any other compiler that I have used.<span style='mso-spacerun:yes'>  </span>When it has been installed from one of my CD s, the editor/ide is named e.<span style='mso-spacerun:yes'>  </span>The code using lnigau.for as poly is in <a href="file:///E:\public_html\Fittery\nlfit\formp\formp.zip">formp\formp.zip</a>.<span style='mso-spacerun:yes'>  </span>Unzip to a new folder.<span style='mso-spacerun:yes'>  </span>Switch to Dos.<span style='mso-spacerun:yes'>  </span>Then in Dos issue the command.</span></font></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><font size=3 face="Times New Roman"><span style='font-size:12.0pt;font-weight:bold; mso-bidi-font-weight:normal'>e nlfit</span></font></b> </p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'>F9 toggles between the command line and the editor.<span style='mso-spacerun:yes'>  </span>The command $ takes the editor to the bottom line.<span style='mso-spacerun:yes'>  </span>There are a group of c$includes.<span style='mso-spacerun:yes'>  </span>One of these will need to be a new poly.<span style='mso-spacerun:yes'>  </span>Watfor code in this mode is on the order of a factor of 20 slower than the more modern compilers below.<span style='mso-spacerun:yes'>  </span></span></font></p> <h2 style='margin-left:.5in;text-indent:-.25in;mso-list:l4 level1 lfo4; tab-stops:list .5in'><![if !supportLists]><b><font size=4 color="#4f81bd" face=Cambria><span style='font-size:13.0pt;mso-fareast-font-family:Cambria; mso-bidi-font-family:Cambria'><span style='mso-list:Ignore'>2.<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font></b><![endif]><span style='mso-fareast-font-family: "Times New Roman"'>Watcom<o:p></o:p></span></h2> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>Watcom is in the public domain.<span style='mso-spacerun:yes'>  </span>Users will need to recompile their version of poly  here <a href="file:///E:\public_html\Fittery\nlfit\formp\poly.for">poly.for</a>  For options </span></font><font face=Symbol><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'>³</span></span></font> 3 the entire code will need recompilation.<span style='mso-spacerun:yes'>  </span>For options &lt; 3, the IDE <a href="file:///E:\public_html\Fittery\nlfit\PolySA\polysa.wpj">PolySA\polysa.wpj</a> can be used to recompile the poly function, here named pademp.for along with polysa to make a short stand alone routine that can run with the intel-10 version of nlfit.<span style='mso-spacerun:yes'>  </span>The ide for the full code, needed for options <font face=Symbol><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'>³</span></span></font> 3, is <a href="file:///E:\public_html\Fittery\nlfit\formp\nlfit.wpj">formp\nlfit.wpj</a>. The Watfor version C$ <font face=Wingdings><span style='font-family:Wingdings; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Wingdings'><span style='mso-char-type: symbol;mso-symbol-font-family:Wingdings'>à</span></span></font> C# must be changed to make this work.<span style='mso-spacerun:yes'>  </span>Owing to a common heritage, Watcom attemps to compile the code with C$, but since the code no long runs in dos, the dos specific commands will cause it to fail without this change.<span style='mso-spacerun:yes'>  </span>Details on obtaining and using Watcom begin in <a href="file:///E:\public_html\Progdet\Progdet.htm#OpenWatcom">Progdet.htm#OpenWatcom</a>.<span style='mso-spacerun:yes'>  </span>For those who do not wish to program in Fortran see <a href="file:///E:\public_html\Progdet\Mixing%20Fortran%20and%20C%20with%20Watcom.doc">Mixing Fortran and C with Watcom.doc</a> <a href="file:///E:\public_html\Progdet\Watfor.htm">.htm</a>. </p> <h2 style='margin-left:.5in;text-indent:-.25in;mso-list:l4 level1 lfo4; tab-stops:list .5in'><![if !supportLists]><b><font size=4 color="#4f81bd" face=Cambria><span style='font-size:13.0pt;mso-fareast-font-family:Cambria; mso-bidi-font-family:Cambria'><span style='mso-list:Ignore'>3.<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font></b><![endif]><span style='mso-fareast-font-family: "Times New Roman"'>g77<o:p></o:p></span></h2> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span><a href="file:///E:\public_html\Fittery\nlfit\formp\unix.zip">Formp\unix.zip</a>  (To Be Done) is the same fortran as that in <a href="file:///E:\public_html\Fittery\nlfit\formp">formp</a>.<span style='mso-spacerun:yes'>  </span>The zip also contains a makefile for constructing nlfit.<span style='mso-spacerun:yes'>  </span>The current one uses lnigau.for.<span style='mso-spacerun:yes'>  </span>It also has a sample data file algau.out and finally a code named gplot (for directions see<a href="file:///E:\public_html\gplot\WELCOME.DOC">..\..\gplot\WELCOME.DOC</a><span style='mso-spacerun:yes'>  </span><a href="file:///E:\public_html\gplot\WELCOME.htm">.htm</a>) for plotting the data, the fit, err.out and res.OUT.<span style='mso-spacerun:yes'>  </span>This particular gplot assumes that xterm is available.<span style='mso-spacerun:yes'>  </span>Unix is a natural for mixing Fortran and C, see <a href="file:///E:\public_html\Progdet\Mixing%20FORTRAN%20and%20C.htm">Mixing FORTRAN and C.htm</a>.</span></font></p> <h2 style='margin-left:.5in;text-indent:-.25in;mso-list:l4 level1 lfo4; tab-stops:list .5in'><![if !supportLists]><b><font size=4 color="#4f81bd" face=Cambria><span style='font-size:13.0pt;mso-fareast-font-family:Cambria; mso-bidi-font-family:Cambria'><span style='mso-list:Ignore'>4.<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font></b><![endif]><span style='mso-fareast-font-family: "Times New Roman"'>Intel10<o:p></o:p></span></h2> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>This started originally as the Microsoft power station <a href="file:///E:\public_html\Fittery\nlfit\formp\nlfit.mdp">nlfit.mdp</a> and is currently <a href="file:///E:\public_html\Fittery\nlfit\formp\nlfit.vfproj">formp\nlfit.vfproj</a>.<span style='mso-spacerun:yes'>  </span>This produces code about 40% faster than the best that Watcom does.<span style='mso-spacerun:yes'>  </span>This is probably due to optimization of the  school boy loop in MPxMP.<span style='mso-spacerun:yes'>  </span><a href="file:///E:\public_html\Fittery\nlfit\formp\mpbob.for">formp\mpbob.for</a>. </span></font></p> <h1><b style='mso-bidi-font-weight:normal'><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman";mso-bidi-font-weight: normal'>Pade Approximates</span></font></b><a name=PadeApproximates></a><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></h1> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>As referenced in <a href="file:///E:\public_html\Fittery\FA.doc#PadeApproximates">FA.doc#PadeApproximates</a>, Pade Approximate methods are very powerful and not completely understood.<span style='mso-spacerun:yes'>  </span>The function and its partials are given in <a href="file:///E:\public_html\Fittery\nlfit\LogAigau\Pade.doc#PadeAndPartials">LogAigau/Pade.doc#PadeAndPartials</a>.<span style='mso-spacerun:yes'>   </span>A general discussion of the numerical problem seen in fitting to a Pade is given in <a href="file:///E:\public_html\Fittery\nlfit\LinDep.doc">LinDep.doc</a> <a href="file:///E:\public_html\Fittery\nlfit\LinDep.htm">.htm</a>.<span style='mso-spacerun:yes'>  </span><a name=LinDep></a>The specific problem in fitting the error function conjugate is discussed in <a href="file:///E:\public_html\Fittery\nlfit\LogAigau\TheProblem.doc">LogAigau\TheProblem.doc</a> <a href="file:///E:\public_html\Fittery\nlfit\LogAigau\TheProblem.htm">.htm</a> and a partial solution is given in <a href="file:///E:\public_html\Fittery\nlfit\LogAigau\TheProblem.doc#PartialSolution">LogAigau/TheProblem.doc#PartialSolution</a> <a href="file:///E:\public_html\Fittery\nlfit\LogAigau\TheProblem.htm#PartialSolution">.htm</a>.</span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>A somewhat more general solution is discussed in <a href="file:///E:\public_html\Fittery\nlfit\Ptcheby\Welcome.htm">Ptcheby/Welcome.htm</a><span style='mso-spacerun:yes'>  </span>-<span style='mso-spacerun:yes'>  </span>Ratio of two Chebyshev polynomials.</span></font></p> <h1><b style='mso-bidi-font-weight:normal'><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman";mso-bidi-font-weight: normal'>Positive Definite Matrices</span></font></b><span style='mso-fareast-font-family: "Times New Roman"'><o:p></o:p></span></h1> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><a href="file:///E:\public_html\MatrixInverse\Cholesky\CHOLESKY.htm">Curve fit matrices are almost always positive definite</a>. The section title says it all.<span style='mso-spacerun:yes'>  </span>Unfortunately they are not always invertible.<span style='mso-spacerun:yes'>  </span>Errors in the matrix can cause <a href="#_Matrix_Inversion_Problems">#Matrix Inversion Problems</a>.<span style='mso-spacerun:yes'>  </span>These frequently mean that some approximations to the inverse predict maxima, while most predict saddle points.<span style='mso-spacerun:yes'>  </span>In this work a Marquardt parameter is used in combination with smoothing values (<a href="file:///E:\public_html\optimization\Robmin\Smvals.doc">Smvals.doc</a> <a href="file:///E:\public_html\optimization\Robmin\Smvals.htm">.htm</a>) to keep the matrix always invertible.<span style='mso-spacerun:yes'>  </span>The inability to invert the second derivative matrix is almost always due to approximate linear dependencies in the second derivative array.<span style='mso-spacerun:yes'>  </span>In this work a multiple precision array (<a href="file:///E:\public_html\MultiplePrecision\Welcome.htm">MultiplePrecision\Welcome.htm</a>) was formed from sums of double precision terms at each point.<span style='mso-spacerun:yes'>  </span>It was found that this was needed only for the first half of the Cholesky (<a href="file:///E:\public_html\MatrixInverse\Cholesky\Welcome.htm">Cholesky\Welcome.htm</a>) inversion routine <a href="file:///E:\public_html\optimization\solving\Cholesky\for\sminv.for">sminv.for</a> (also in <a href="file:///E:\public_html\Fittery\nlfit\formp\robmin.for">formp\robmin.for</a>).<span style='mso-spacerun:yes'>  </span>This resulted in the routines<span style='mso-spacerun:yes'>  </span>MPHSMINV.for which is called when <b style='mso-bidi-font-weight:normal'><span style='font-weight:bold;mso-bidi-font-weight: normal'>sminv </span></b>says that it cannot invert the matrix.<span style='mso-spacerun:yes'>  </span>This greatly increases the computation time owing to the fact that the multiple precision multiplications as coded here are rather slow.<span style='mso-spacerun:yes'>    </span></span></font></p> <h1><a name="_Log(AiGau)"></a><b><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman"'><a href="file:///E:\public_html\Fittery\nlfit\LogAigau">Log(AiGau)</a><o:p></o:p></span></font></b></h1> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-tab-count:1'>            </span>This folder is a test of the multiple precision routines. <a href="file:///E:\public_html\Fittery\nlfit\LogAigau\Welcome.htm">Welcome.htm</a>.<span style='mso-spacerun:yes'>  </span></span></font></p> <h1><b style='mso-bidi-font-weight:normal'><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman";mso-bidi-font-weight: normal'>Sub-directories<o:p></o:p></span></font></b></h1> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo5; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>1<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\2dFit">2dFit</a> <a href="file:///E:\public_html\Fittery\nlfit\2dFit\2dFit.htm">2dFit.htm</a><span style='mso-spacerun:yes'>  </span>Uses old robmin, slightly re-written gplot and real data.<span style='mso-spacerun:yes'>  </span> demonstrates a fit to x,y,z data.<span style='mso-spacerun:yes'>  </span>That is z is fitted a z <font face=Symbol><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></font> f<sub>A</sub>(x,y)</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo5; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>2<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\2dPoly">2dPoly</a> <a href="file:///E:\public_html\Fittery\nlfit\2dPoly\Welcome.htm">Welcome.htm</a><span style='mso-spacerun:yes'>  </span>Uses new robmin and original gplot.<span style='mso-spacerun:yes'>  </span>Contains a routine for re-writing data into a proper form for plotting as a series of 1d lines.</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo5; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>3<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\aigau">aigau</a> <a href="file:///E:\public_html\Fittery\nlfit\aigau\Welcome.htm">Welcome.htm</a><span style='mso-spacerun:yes'>  </span>16 digit fit to the complementary error function.<span style='mso-spacerun:yes'>  </span>Uses Bailey s multiple precision <a href="file:///E:\public_html\MultiplePrecision\bailey\Welcome.htm">bailey\Welcome.htm</a>.</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo5; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>4<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\efffit">efffit</a> <a href="file:///E:\public_html\Fittery\nlfit\efffit\EFF2.htm">EFF2.htm</a> This uses an old robmin stored in <a href="file:///E:\public_html\Fittery\nlfit\efffit\effitfor.zip">effitfor.zip</a>.<span style='mso-spacerun:yes'>  </span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo5; tab-stops:list .25in'><a name=fit8></a><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list: Ignore'>5<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\fit8">fit8</a> <a href="file:///E:\public_html\Fittery\nlfit\fit8\Welcome.doc">fit8\Welcome.doc</a> <a href="file:///E:\public_html\Fittery\nlfit\fit8\Welcome.htm">.htm</a>  experiments with a different Pade form <span style='mso-tab-count:1'>            </span><sub><!--[if gte vml 1]><v:shape id="_x0000_i1040" type="#_x0000_t75" style='width:258pt;height:66pt' o:ole=""> <v:imagedata src="Welcome_files/image047.wmz" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=344 height=88 src="Welcome_files/image030.gif" v:shapes="_x0000_i1040"><![endif]><!--[if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1040" DrawAspect="Content" ObjectID="_1420617808"> </o:OLEObject> </xml><![endif]--></sub><br> This speeds the convergence by a factor of 100 to 1000 for small N.<span style='mso-spacerun:yes'>  </span>Eventually the derivatives for a<sub>i+N</sub> begin to be very similar so that about 8-10 digits are all that this method can reach easily with a double precision PPCC array.<span style='mso-spacerun:yes'>  </span>This form is also used in <a href="file:///E:\public_html\Fittery\nlfit\LogAigau\PadeM\Welcome.htm">LogAigau\PadeM\Welcome.htm</a> where it is not a significant improvement over the fit in <a href="file:///E:\public_html\Fittery\nlfit\LogAigau">LogAigau</a> <a href="file:///E:\public_html\Fittery\nlfit\LogAigau\Welcome.htm">Welcome.htm</a>.</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo5; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>6<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\formp">formp</a> <a href="file:///E:\public_html\Fittery\nlfit\formp\Welcome.htm">formp\Welcome.htm</a>  current set of fortran and compiler files</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo5; tab-stops:list .25in'><a name="pre2007_Fortran"></a><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>7<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\for">for</a> <a href="file:///E:\public_html\Fittery\nlfit\for\Welcome.htm">for\Welcome.htm</a>  pre 2007 set of Fortran used with most of the data and examples in <a href="file:///E:\public_html\Fittery\nlfit\TestData">TestData</a>.<span style='mso-spacerun:yes'>  </span>Multiple precision  comment lines appropriate for Bailey s transmp codes are in nlfit and robmin.<span style='mso-spacerun:yes'>   </span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo5; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>8<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\LogAigau">LogAigau</a> <a href="file:///E:\public_html\Fittery\nlfit\LogAigau\Welcome.htm">Welcome.htm</a>  16 digit fit to log of complementary error function.<span style='mso-spacerun:yes'>  </span>(not as accurate as 3 above) Tests the routines in 6.<span style='mso-spacerun:yes'>  </span>The log has up to two extra digits making an exact fit that much less accurate than the one above.</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo5; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>9<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\Minsteps">Minsteps</a> <a href="file:///E:\public_html\Fittery\nlfit\Minsteps\Minimization-in-steps-Real.doc">Minsteps\Minimization-in-steps-Real.doc</a> <a href="file:///E:\public_html\Fittery\nlfit\Minsteps\Minimization-in-steps-Real.htm">.htm</a> <a name=minsteps></a> A somewhat disastrous attempt to iterate to more accuracy.<span style='mso-spacerun:yes'>  </span>Demonstrates that an accurate inverse of an approximate matrix frequently does not predict a minimum and in fact may predict a maximum.<span style='mso-spacerun:yes'>  </span> The conclusion is that the Cholesky matrix inversion fails when the curve fit matrix fails to be positive definite.<span style='mso-spacerun:yes'>  </span>This failure in the test was due to the fact that information was lost in the matrix PPCC owing to its fininte number of digits.<span style='mso-spacerun:yes'>  </span>The solution to the equation PPCC<font face=Symbol><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></font>X = PC in this case is a saddle point or even an extremum.<span style='mso-spacerun:yes'>  </span>This is not the solution desired.<span style='mso-spacerun:yes'>  </span>Minimizing the expectation value of PC<font face=Symbol><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></font>X + ½ PPCC <font face=Symbol><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></font>X<font face=Symbol><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></font>X had the problem that the minimum is for extremely large X and the derivative equation is not even close to a solution.<span style='mso-spacerun:yes'>  </span>Finally the  best solution is to use the non-inversion of the matrix as a flag and to set the value of all X s to zero in the case that the matrix does not invert.<br> <span style='mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_14" o:spid="_x0000_i1041" type="#_x0000_t75" style='width:384pt;height:171pt; visibility:visible;mso-wrap-style:square'> <v:imagedata src="Welcome_files/image031.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=512 height=228 src="Welcome_files/image048.jpg" v:shapes="Picture_x0020_14"><![endif]></span><br> Figure <!--[if supportFields]><span style='mso-element:field-begin'></span><span style='mso-spacerun:yes'> </span>SEQ Figure \* ARABIC <span style='mso-element: field-separator'></span><![endif]--><span style='mso-no-proof:yes'>2</span><!--[if supportFields]><span style='mso-element:field-end'></span><![endif]--> <font face=Symbol><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'>c</span></span></font><sub>p</sub><sup>2</sup> /<font face=Symbol><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'>c</span></span></font><sub>o</sub><sup>2</sup> as a function of <font face=Symbol><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'>l</span></span></font></p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo5; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>10<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\Ptcheby">Ptcheby</a> <a href="file:///E:\public_html\Fittery\nlfit\Ptcheby\Welcome.htm">Ptcheby\Welcome.htm</a> <a name=Ptcheby></a>Substituting Tchebyshev polynomials for the polynomials in the Pade, makes the convergence much better.<span style='mso-spacerun:yes'>  </span>Uses Bailey s multiple precision <a href="file:///E:\public_html\MultiplePrecision\bailey\Welcome.htm">bailey\Welcome.htm</a>.<span style='mso-spacerun:yes'>  </span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo5; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>11<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\TestData">TestData</a> <a href="file:///E:\public_html\Fittery\nlfit\TestData\Welcome.htm">TestData\Welcome.htm</a></p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo5; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>12<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\wsteve\h2fit">..\..\wsteve\h2fit</a> <a href="file:///E:\public_html\wsteve\h2fit\H2fit.htm">H2fit.htm</a> </p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo5; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>13<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\zips">zips</a><span style='mso-spacerun:yes'>  </span>-- see especially <a href="file:///E:\public_html\Fittery\nlfit\zips\fit7.zip">fit7.zip</a> This code started much of the current work when it required ~ 500,000 steps to converge.<span style='mso-spacerun:yes'>  </span><span style='mso-tab-count: 1'>       </span>The derivatives in this case are very similar for the denominator and the numerator expansion resulting in PPCC s with many similar terms.<span style='mso-spacerun:yes'>  </span>The routine requied 10,000 to 50,000 iterations and 1 to 3 restarts to converge.<span style='mso-spacerun:yes'>  </span>The problem and a partial solution are documented in <a href="#fit8">#fit8</a> above.</p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><o:p>&nbsp;</o:p></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><o:p>&nbsp;</o:p></span></font></p> <p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size: 12.0pt'><span style='mso-spacerun:yes'>         </span></span></font></p> <h1><b style='mso-bidi-font-weight:normal'><font size=5 face=Arial><span style='font-size:16.0pt;mso-fareast-font-family:"Times New Roman";mso-bidi-font-weight: normal'>Documents/Folders <o:p></o:p></span></font></b></h1> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>1<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\robfit\TemplFit\eff\Welcome.htm">..\robfit\TemplFit\eff\Welcome.htm</a> Test data for a double Pade fit.</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>2<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\mpsminv1.doc">mpsminv1.doc</a> <a href="file:///E:\public_html\Fittery\nlfit\mpsminv1.htm">.htm</a> <font face=Wingdings><span style='font-family:Wingdings;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-char-type:symbol;mso-symbol-font-family: Wingdings'><span style='mso-char-type:symbol;mso-symbol-font-family:Wingdings'>ß</span></span></font> testing MPHSMINV</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>3<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\NLFIT%20Code.doc">NLFIT Code.doc</a> <a href="file:///E:\public_html\Fittery\nlfit\NLFIT.htm">.htm</a>  Details the code, refers to FofLambda.doc</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>4<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\NumericalPartials.doc">NumericalPartials.doc</a> <a name=NumericalPartials></a><a href="file:///E:\public_html\Fittery\nlfit\NumericalPartials.htm">.htm</a>, <a href="file:///E:\public_html\Fittery\nlfit\Numerical%20derivatives.htm">Numerical derivatives.htm</a><span style='mso-spacerun:yes'>  </span>Sample Poly routines using numerical derivatives.<span style='mso-spacerun:yes'>  </span>The newest code has an option to calculate numerical derivatives internally.<span style='mso-spacerun:yes'>  </span>This multiplies the time required by 4×Ncon and still is not as accurate as the analytical derivatives can be, but when one has lots more computer time than coding and mathematical skills this can get the fit<span style='mso-spacerun:yes'>  </span>a lot faster. </p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>5<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\LogAigau\The%20number%20of%20Multiple%20Precision%20integers.htm">The number of Multiple Precision integers.htm</a></p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>6<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\TestData\green\Alex1.htm">Alex1.htm</a> <a href="file:///E:\public_html\Fittery\nlfit\TestData\green\ALEX1.DOC">.doc</a>  Further generalization of the Fermi function.<span style='mso-spacerun:yes'>  </span>A.E.S. Green introduced me to this function.</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>7<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\TestData\green\DFFIT.htm">TestData\green\DFFIT.htm</a> <a href="file:///E:\public_html\Fittery\nlfit\TestData\green\DFFIT.DOC">.doc</a>  details on finding the derivative of the Fermi function </p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>8<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\README.htm">README.htm</a> <a href="file:///E:\public_html\Fittery\nlfit\TestData\green\README.DOC">.doc</a> </p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>9<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\ErrorMatrix\StdDev.doc">StdDevfc.doc</a> <a href="file:///E:\public_html\Fittery\ErrorMatrix\StdDevfc.htm">htm</a> <a href="file:///E:\public_html\Fittery\ErrorMatrix\Welcome.htm">..\ErrorMatrix\Welcome.htm</a>  Derivation of the relation between the second derivative array and the error matrix.</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><a name="_Matrix_Inversion_Problems"></a><a name=PhotoAbsrption></a><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>10<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]>The following four fits were used to construct <a href="file:///E:\public_html\Fittery\nlfit\PabK\tphoto.wpj">PabK\tphoto.wpj</a> which gives the photoabsorption (cm<sup>2</sup>/mol) as a function of E and Z for gamma rays travelling through matter.<span style='mso-spacerun:yes'>  </span>This is used in the templfit extension of <a href="file:///E:\public_html\Fittery\robfit\Welcome.htm">..\robfit\Welcome.htm</a>.<span style='mso-spacerun:yes'>  </span></p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.25in;mso-list:l3 level1 lfo7'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>a.<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><span style='mso-spacerun:yes'> </span><a href="file:///E:\public_html\Fittery\nlfit\AofZ\Welcome.htm">AofZ\Welcome.htm</a> -<span style='mso-spacerun:yes'>  </span>Fits the atomic weight as a function of the atomic number.</p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.25in;mso-list:l3 level1 lfo7'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>b.<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\RhoOfZ\Welcome.htm">RhoOfZ\Welcome.htm</a>  Fits the number of moles/cm<sup>3</sup> as a function of atomic number Z</p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.25in;mso-list:l3 level1 lfo7'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>c.<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\EkofZ\Welcome.htm">EkofZ\Welcome.htm</a> Fits the K shell energy as a function of atomic number Z</p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.25in;mso-list:l3 level1 lfo7'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>d.<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\Pbel\Welcome.htm">Pbel\Welcome.htm</a> <a href="file:///E:\public_html\Fittery\nlfit\PabK">PabK</a> <a href="file:///C:\public_html\Fittery\nlfit\Pabk2">Pabk2</a> <a href="file:///C:\public_html\Fittery\nlfit\Pbel">Pbel</a> Fits to the photo-attenuation (cm<sup>2</sup>/mole) as a function of E and Z </p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>11<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\FermiFit\Welcome.htm">FermiFit\Welcome.htm</a><span style='mso-spacerun:yes'>  </span>Tests error calculations using artificial data</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>12<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\2dPoly">2dPoly</a> </p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>13<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\aigau">aigau</a><span style='mso-spacerun:yes'>  </span>integral of Gaussian from -<font face="Cambria Math"><span style='font-family:"Cambria Math","serif"'>"</span></font> to x  closely related to the error function. <a href="file:///E:\public_html\Fittery\nlfit\LogAigau">LogAigau</a> alternate fit not as accurate.</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>14<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\amu2">amu2</a> better fit to attenuation.<span style='mso-spacerun:yes'>  </span><a href="file:///E:\public_html\Fittery\nlfit\Compton">Compton</a> fits needed to approximate Compton scattering.<span style='mso-spacerun:yes'>  </span><a href="Pairp">Pairp</a> fits pair production.</p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>15<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\AofZ">AofZ</a> <a href="file:///E:\public_html\Fittery\nlfit\EkofZ">EkofZ</a> <a href="file:///C:\public_html\Fittery\nlfit\RhoOfZ">RhoOfZ</a> </p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>16<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\H2ScattFermi">H2ScattFermi</a> </p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l6 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><font size=3 face="Times New Roman"><span style='font-size:12.0pt'><span style='mso-list:Ignore'>17<font size=1 face="Times New Roman"><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></font></span></span></font><![endif]><a href="file:///E:\public_html\Fittery\nlfit\pade2d">pade2d</a></p> </div> </body> </html>