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text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"><f family="Arial" charset="0"><u><inlineAttr line-through="false"><c val="#808000"><link href="./regression.xmcd" title="Links to Section Contents" popup="false"><sp count="6" />REGRESSION</link></c></inlineAttr></u></f></p></text></region><region region-id="1321" left="36" top="90.75" width="156" height="13.5" align-x="36" align-y="102" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Sous-titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Rational Function Regression </p></text><indexes><phrase>rational function regression</phrase><phrase>regression, rational function regression</phrase></indexes></region><region region-id="2118" left="66" top="115.5" width="297" height="12.75" align-x="75" align-y="126" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">rationalfit(vx, vy, conf, m, n, [Stdy], [LBUB], [Acc],["noscale"]) </p></text><indexes><phrase>rationalfit</phrase><phrase>functions, rationalfit</phrase></indexes></region><region region-id="2121" left="66" top="133.5" width="308.25" height="12.75" align-x="75" align-y="144" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">rationalfitnp(vx, vy, conf, m, n, [Stdy], [LBUB], [Acc],["noscale"]) </p></text><indexes><phrase>rationalfitnp</phrase><phrase>functions, rationalfitnp</phrase></indexes></region><region region-id="2713" left="66" top="151.5" width="295.5" height="12.75" align-x="66" align-y="162" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">LeastSquaresFit(vx, vy, F(x), guess, conf, [Stdy], [LBUB], [Acc])</p></text><indexes><phrase>LeastSquareFit</phrase><phrase>functions, LeastSquareFit</phrase></indexes></region><region region-id="2093" left="36" top="182.25" width="316.5" height="120" align-x="36" align-y="192" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Rational functions can be useful in fitting data with asymptotes, or other intractable shapes, much like <b><u><link href="./ratint.xmcd" title="Links to section" popup="false">rational function interpolation</link></u></b>. The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">rationalfit</c></inlineAttr></b></f> function will fit a rational polynomial to data, estimating a guess value by doing a linear fit for the parameters in the numerator, assuming a denominator of 1. It requires you to provide the order of the numerator and denominator, m and n, respectively. The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">rationalfitnp</c></inlineAttr></b></f> function has an additional calculation to avoid poles in the denominator within the data range, so the function will be well behaved everywhere in the interval.</p></text></region><region region-id="1998" left="36" top="314.25" width="315" height="48" align-x="36" align-y="324" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The following data are the result of a NIST study involving semiconductor electron mobility. The response variable is a measure of electron mobility, and the predictor variable is the natural log of the density.</p></text></region><region region-id="1999" left="66" top="375" width="110.25" height="46.5" align-x="90" align-y="384" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><component clsid-buddy="577d93c8-ea65-4d05-b631-401168b146e4" item-idref="4" disable-calc="false"><inputs hide="false" /><outputs hide="false"><ml:id>data</ml:id></outputs></component><rendering item-idref="5" /></region><region region-id="2385" left="216" top="380.25" width="162" height="12" align-x="217.5" align-y="390" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">1 Response variable, y (mobility)</p></text></region><region region-id="2386" left="216" top="404.25" width="157.5" height="12" align-x="217.5" align-y="414" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">1 Predictor variable, x (density)</p></text></region><region region-id="2002" left="66" top="429.75" width="52.5" height="18" align-x="82.5" align-y="444" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>vy</ml:id><ml:apply><ml:matcol /><ml:id>data</ml:id><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="6" /></region><region region-id="2382" left="132" top="429.75" width="51" height="18" align-x="147" align-y="444" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>vx</ml:id><ml:apply><ml:matcol /><ml:id>data</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="7" /></region><region region-id="2362" left="36" top="469.5" width="123.75" height="12.75" align-x="36" align-y="480" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solving for the parameters</p></text><indexes><phrase>solving for the parameters</phrase><phrase>parameters, solving for the</phrase></indexes></region><region region-id="2426" left="36" top="494.25" width="321" height="24" align-x="36" align-y="504" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Specify an order for the numerator and denominator of the rational function:</p></text></region><region region-id="2706" left="66" top="531" width="24.75" height="12.75" align-x="78.75" align-y="540" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>m</ml:id><ml:real>3</ml:real></ml:define></math><rendering item-idref="8" /></region><region region-id="2707" left="144" top="531" width="23.25" height="12.75" align-x="155.25" align-y="540" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>n</ml:id><ml:real>3</ml:real></ml:define></math><rendering item-idref="9" /></region><region region-id="2430" left="36" top="560.25" width="86.25" height="12" align-x="36" align-y="570" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Fitting function:</p></text></region><region region-id="2555" left="66" top="582" width="167.25" height="44.25" align-x="115.5" align-y="606" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:function><ml:id>NIST</ml:id><ml:boundVars><ml:id>x</ml:id><ml:id>β</ml:id></ml:boundVars></ml:function><ml:apply><ml:div /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:indexer /><ml:id>β</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>β</ml:id><ml:real>1</ml:real></ml:apply><ml:id>x</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>β</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:pow /><ml:id>x</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>β</ml:id><ml:real>3</ml:real></ml:apply><ml:apply><ml:pow /><ml:id>x</ml:id><ml:real>3</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:real>1</ml:real><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>β</ml:id><ml:real>4</ml:real></ml:apply><ml:id>x</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer 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list-style-type="inherit" tabs="inherit">Apply the <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">rationalfit</c></inlineAttr></b></f> function:</p></text></region><region region-id="2233" left="66" top="687" width="185.25" height="12.75" align-x="87" align-y="696" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Res</ml:id><ml:apply><ml:id>rationalfit</ml:id><ml:sequence><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:real>0.95</ml:real><ml:id>m</ml:id><ml:id>n</ml:id><ml:str>noscale</ml:str></ml:sequence></ml:apply></ml:define></math><rendering item-idref="11" /></region><region region-id="2153" left="36" top="716.25" width="309" height="36" align-x="42.75" align-y="726" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The first column of output contains the values for the parameters. The second and third column contain outputs for the lower and upper confidence limits, respectively.</p></text></region><region region-id="2155" left="36" top="764.25" width="195.75" height="12" align-x="36" align-y="774" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Correct values from the NIST web site:</p></text></region><region region-id="2156" left="36" top="777.75" width="422.25" height="24" align-x="69" align-y="798" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>NISTβ</ml:id><ml:apply><ml:transpose /><ml:matrix rows="1" cols="7"><ml:apply><ml:mult style="auto-select" /><ml:real>1.2881397</ml:real><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>3</ml:real></ml:apply></ml:apply><ml:apply><ml:mult style="auto-select" /><ml:real>1.4910793</ml:real><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>3</ml:real></ml:apply></ml:apply><ml:real>583.2383688</ml:real><ml:real>75.4166443</ml:real><ml:real>0.966295</ml:real><ml:real>0.3979729</ml:real><ml:real>0.0497273</ml:real></ml:matrix></ml:apply></ml:define></math><rendering item-idref="12" /></region><region region-id="2157" left="36" top="812.25" width="78" height="12" align-x="36" align-y="822" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Compare values</p></text></region><region region-id="2498" left="6" top="829.5" width="113.25" height="123.75" align-x="38.25" align-y="894" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>NISTβ</ml:id></ml:eval></math><rendering item-idref="13" /></region><region region-id="2506" left="120" top="829.5" width="222.75" height="123.75" align-x="140.25" align-y="894" show-border="false" show-highlight="true" is-protected="true" background-color="#80ffff" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>Res</ml:id></ml:eval></math><rendering item-idref="14" /></region><region region-id="2334" left="66" top="963.75" width="96.75" height="18" align-x="91.5" align-y="978" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:function><ml:id>fit</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>NIST</ml:id><ml:sequence><ml:id>x</ml:id><ml:apply><ml:matcol /><ml:id>Res</ml:id><ml:real>0</ml:real></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="15" /></region><region region-id="2335" left="222" top="968.25" width="119.25" height="13.5" align-x="269.25" align-y="978" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:function><ml:id>NISTfit</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>NIST</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>NISTβ</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="16" /></region><region region-id="2185" left="66" top="993" width="93.75" height="12.75" align-x="75.75" align-y="1002" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>x</ml:id><ml:range><ml:sequence><ml:real>-3.2</ml:real><ml:real>-3.1</ml:real></ml:sequence><ml:apply><ml:id>max</ml:id><ml:id>vx</ml:id></ml:apply></ml:range></ml:define></math><rendering item-idref="17" /></region><region region-id="2186" left="18" top="1014" width="332.25" height="167.25" align-x="18" align-y="1014" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="18" /><rendering item-idref="19" /></region><region region-id="2450" left="66" top="1179" width="110.25" height="18.75" align-x="134.25" align-y="1194" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>corr</ml:id><ml:sequence><ml:apply><ml:vectorize /><ml:apply><ml:id>fit</ml:id><ml:id>vx</ml:id></ml:apply></ml:apply><ml:id>vy</ml:id></ml:sequence></ml:apply></ml:eval></math><rendering item-idref="20" /></region><region region-id="2451" left="0" top="1212" width="6000" height="6" align-x="0" align-y="1212" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2456" left="36" top="1219.5" width="138.75" height="12.75" align-x="36" align-y="1230" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Comparison with NIST results</p></text><indexes><phrase>comparison with NIST results</phrase><phrase>NIST results, comparision with</phrase></indexes></region><region region-id="2452" left="66" top="1245" width="133.5" height="12.75" align-x="107.25" align-y="1254" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>degofree</ml:id><ml:apply><ml:minus /><ml:apply><ml:id>length</ml:id><ml:id>vx</ml:id></ml:apply><ml:parens><ml:apply><ml:plus /><ml:id>m</ml:id><ml:id>n</ml:id></ml:apply></ml:parens></ml:apply></ml:define></math><rendering item-idref="21" /></region><region region-id="2463" left="222" top="1245" width="62.25" height="12.75" align-x="262.5" align-y="1254" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>degofree</ml:id></ml:eval><resultFormat><general radix="dec" complex-threshold="10" precision="15" show-trailing-zeros="false" exponential-threshold="2" /><matrix display-style="auto" expand-nested-arrays="false" /><unit format-units="true" simplify-units="true" /></resultFormat></math><rendering item-idref="22" /></region><region region-id="2454" left="66" top="1269" width="106.5" height="18.75" align-x="103.5" align-y="1284" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>residual</ml:id><ml:parens><ml:apply><ml:vectorize /><ml:parens><ml:apply><ml:minus /><ml:id>vy</ml:id><ml:apply><ml:id>fit</ml:id><ml:id>vx</ml:id></ml:apply></ml:apply></ml:parens></ml:apply></ml:parens></ml:define></math><rendering item-idref="23" /></region><region region-id="2464" left="222" top="1269" width="123" height="18.75" align-x="261.75" align-y="1284" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>NISTres</ml:id><ml:apply><ml:vectorize /><ml:parens><ml:apply><ml:minus /><ml:id>vy</ml:id><ml:apply><ml:id>NISTfit</ml:id><ml:id>vx</ml:id></ml:apply></ml:apply></ml:parens></ml:apply></ml:define></math><rendering item-idref="24" /></region><region region-id="2299" left="36" top="1304.25" width="130.5" height="12" align-x="36" align-y="1314" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Residual sum of squares</p></text></region><region region-id="2300" left="66" top="1323" width="69" height="30" align-x="77.25" align-y="1338" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>b</ml:id><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:id>residual</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="25" /></region><region region-id="2465" left="222" top="1323" width="93" height="30" align-x="255" align-y="1338" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>NISTb</ml:id><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:id>NISTres</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="26" /></region><region region-id="2302" left="66" top="1362.75" width="183" height="33.75" align-x="112.5" align-y="1380" show-border="false" show-highlight="true" is-protected="true" background-color="#80ffff" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:matrix rows="2" cols="1"><ml:id>b</ml:id><ml:id>NISTb</ml:id></ml:matrix></ml:eval><resultFormat><general radix="dec" complex-threshold="10" precision="15" show-trailing-zeros="true" exponential-threshold="3" /><matrix display-style="table" expand-nested-arrays="false" /><unit format-units="true" simplify-units="true" /><table item-idref="27" /></resultFormat></math><rendering item-idref="28" /></region><region region-id="2303" left="36" top="1406.25" width="93.75" height="12" align-x="36" align-y="1416" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Standard deviation</p></text></region><region region-id="2304" left="66" top="1429.5" width="79.5" height="46.5" align-x="78" align-y="1464" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>σ</ml:id><ml:apply><ml:sqrt /><ml:apply><ml:div /><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:id>residual</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:id>degofree</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="29" /></region><region region-id="2703" left="222" top="1429.5" width="102.75" height="46.5" align-x="255" align-y="1464" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>NISTσ</ml:id><ml:apply><ml:sqrt /><ml:apply><ml:div /><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:id>NISTres</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:id>degofree</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="30" /></region><region region-id="2306" left="66" top="1494" width="170.25" height="34.5" align-x="112.5" align-y="1512" show-border="false" show-highlight="true" is-protected="true" background-color="#80ffff" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:matrix rows="2" cols="1"><ml:id>σ</ml:id><ml:id>NISTσ</ml:id></ml:matrix></ml:eval><resultFormat><general radix="dec" complex-threshold="10" precision="15" show-trailing-zeros="true" exponential-threshold="3" /><matrix display-style="table" expand-nested-arrays="false" /><unit format-units="true" simplify-units="true" /><table item-idref="31" /></resultFormat></math><rendering item-idref="32" /></region><region region-id="2307" left="42" top="1536" width="284.25" height="137.25" align-x="42" align-y="1536" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="33" /><rendering item-idref="34" /></region><region region-id="2538" left="0" top="1698" width="6000" height="6" align-x="0" align-y="1698" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2530" left="36" top="1705.5" width="90" height="12.75" align-x="36" align-y="1716" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Further constraints</p></text><indexes><phrase>rationalfit function, further constraints</phrase></indexes></region><region region-id="2512" left="36" top="1730.25" width="304.5" height="36" align-x="36" align-y="1740" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that you can specify lower and upper bounds on fit parameters, a standard deviation on the y values, and a tolerance for this function, just as for <b><u><link href="./nonlin.xmcd" title="Links to section" popup="false">LeastSquaresFit</link></u></b>.</p></text></region><region region-id="2518" left="36" top="1778.25" width="307.5" height="12" align-x="36" align-y="1788" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Create a matrix of lower and upper bounds on the parameters</p></text></region><region region-id="2519" left="66" top="1803" width="59.25" height="12.75" align-x="74.25" align-y="1812" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>i</ml:id><ml:range><ml:real>0</ml:real><ml:parens><ml:apply><ml:plus /><ml:id>m</ml:id><ml:id>n</ml:id></ml:apply></ml:parens></ml:range></ml:define></math><rendering item-idref="35" /></region><region region-id="2520" left="66" top="1827" width="53.25" height="18" align-x="107.25" align-y="1836" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>LBUB</ml:id><ml:sequence><ml:id>i</ml:id><ml:real>0</ml:real></ml:sequence></ml:apply><ml:real>0</ml:real></ml:define></math><rendering item-idref="36" /></region><region region-id="2521" left="222" top="1821" width="62.25" height="24" align-x="263.25" align-y="1836" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>LBUB</ml:id><ml:sequence><ml:id>i</ml:id><ml:real>1</ml:real></ml:sequence></ml:apply><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>4</ml:real></ml:apply></ml:define></math><rendering item-idref="37" /></region><region region-id="2523" left="66" top="1851" width="75.75" height="18" align-x="123" align-y="1860" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>StdY</ml:id><ml:apply><ml:id>last</ml:id><ml:id>vy</ml:id></ml:apply></ml:apply><ml:real>1.0</ml:real></ml:define></math><rendering item-idref="38" /></region><region region-id="2537" left="66" top="1875" width="264" height="18.75" align-x="87" align-y="1890" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Res</ml:id><ml:apply><ml:id>rationalfit</ml:id><ml:sequence><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:real>0.95</ml:real><ml:id>m</ml:id><ml:id>n</ml:id><ml:id>StdY</ml:id><ml:id>LBUB</ml:id><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>-7</ml:real></ml:apply><ml:str>noscale</ml:str></ml:sequence></ml:apply></ml:define></math><rendering item-idref="39" /></region><region region-id="2535" left="12" top="1915.5" width="113.25" height="123.75" align-x="44.25" align-y="1980" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>NISTβ</ml:id></ml:eval></math><rendering item-idref="40" /></region><region region-id="2536" left="126" top="1915.5" width="222.75" height="123.75" align-x="146.25" align-y="1980" show-border="false" show-highlight="true" is-protected="true" background-color="#80ffff" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>Res</ml:id></ml:eval></math><rendering item-idref="41" /></region><region region-id="2546" left="36" top="2060.25" width="301.5" height="36" align-x="36" align-y="2070" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that order of arguments matters here. You are welcome to use any of the optional parameters alone, that is, you can calculate</p></text></region><region region-id="2705" left="66" top="2109" width="169.5" height="18.75" align-x="87" align-y="2124" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Res</ml:id><ml:apply><ml:id>rationalfit</ml:id><ml:sequence><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:real>0.95</ml:real><ml:id>m</ml:id><ml:id>n</ml:id><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>-10</ml:real></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="42" /></region><region region-id="2549" left="36" top="2138.25" width="319.5" height="63" align-x="36" align-y="2148" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">if you just want to adjust the accuracy of the calculation. Default accuracy is 10<f family="Verdana" charset="0"><inlineAttr line-through="false"><c val="#000"><sup>-7</sup></c></inlineAttr></f>. However, if you are using two or all three of the optional parameters, you must first enter the standard deviation vector, then the lower and upper bounds matrix, then the accuracy.</p></text></region><region region-id="2715" left="36" top="2210.25" width="315" height="96" align-x="36" align-y="2220" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Finally, be aware that rationalfit does an automatic scaling on input data, so that the accuracy can remain scale-invariant.<sp count="2" />It then rescales the output parameters to match the original data.<sp count="2" />Most of the time, this ensures that your fit will be achieved regardless of the relative scale of your input data.<sp count="2" />If you are having trouble achieving a good fit, however, you may wish to add the noscale option string at the end of the argument list, to back out this option.</p></text></region><region region-id="2165" left="0" top="2322" width="6000" height="6" align-x="0" align-y="2322" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2187" left="36" top="2335.5" width="212.25" height="12.75" align-x="36" align-y="2346" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solving for the rational function without poles</p></text><indexes><phrase>solving for the rational function without poles</phrase><phrase>rational function, solving for without poles</phrase></indexes></region><region region-id="2198" left="36" top="2354.25" width="319.5" height="72" align-x="43.5" align-y="2364" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">In the case of <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">rationalfitnp</c></inlineAttr></b></f> (no poles), the solution to the traditional least-squares problem is checked for roots of the denominator within the fitting interval. If there are no poles, the generated fit is returned. If there are poles, additional constraints are added to the nonlinear optimization problem. The denominator polynomial is cast in the following form:<sp count="3" /></p></text></region><region region-id="2248" left="66" top="2446.5" width="279" height="73.5" align-x="109.5" align-y="2484" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:id>denom</ml:id><ml:id>x</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:product /><ml:lambda><ml:boundVars><ml:id>i</ml:id></ml:boundVars><ml:parens><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:mult style="auto-select" /><ml:apply><ml:indexer /><ml:id>a</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:pow /><ml:id>x</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:mult style="auto-select" /><ml:apply><ml:indexer /><ml:id>b</ml:id><ml:id>i</ml:id></ml:apply><ml:id>x</ml:id></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>c</ml:id><ml:id>i</ml:id></ml:apply></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:apply><ml:id>floor</ml:id><ml:apply><ml:div /><ml:id>m</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:bounds></ml:apply><ml:apply><ml:product /><ml:lambda><ml:boundVars><ml:id>i</ml:id></ml:boundVars><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="auto-select" /><ml:apply><ml:indexer /><ml:id>b</ml:id><ml:id>i</ml:id></ml:apply><ml:id>x</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>c</ml:id><ml:id>i</ml:id></ml:apply></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:apply><ml:id>floor</ml:id><ml:apply><ml:div /><ml:id>m</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:apply><ml:id>floor</ml:id><ml:apply><ml:div /><ml:id>m</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:id>m</ml:id></ml:apply></ml:bounds></ml:apply></ml:apply></ml:apply></math><rendering item-idref="43" /></region><region region-id="2247" left="66" top="2525.25" width="165.75" height="60" align-x="107.25" align-y="2550" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:apply><ml:equal /><ml:apply><ml:id>g</ml:id><ml:sequence><ml:id>a</ml:id><ml:id>b</ml:id><ml:id>c</ml:id></ml:sequence></ml:apply><ml:apply><ml:integral auto-algorithm="true" /><ml:lambda><ml:boundVars><ml:id>x</ml:id></ml:boundVars><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:pow /><ml:parens><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:id>a</ml:id><ml:apply><ml:pow /><ml:id>x</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:id>b</ml:id><ml:id>x</ml:id></ml:apply></ml:apply><ml:id>c</ml:id></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:apply><ml:id>min</ml:id><ml:id>x</ml:id></ml:apply><ml:apply><ml:id>max</ml:id><ml:id>x</ml:id></ml:apply></ml:bounds></ml:apply></ml:apply></math><rendering item-idref="44" /></region><region region-id="2202" left="36" top="2600.25" width="259.5" height="12" align-x="42.75" align-y="2610" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To avoid poles the following constraints are added: </p></text></region><region region-id="2212" left="66" top="2624.25" width="68.25" height="27.75" align-x="74.25" align-y="2640" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>i</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:id>floor</ml:id><ml:apply><ml:div /><ml:id>m</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:range></ml:define></math><rendering item-idref="45" /></region><region region-id="2219" left="222" top="2630.25" width="62.25" height="18.75" align-x="270" align-y="2640" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:apply><ml:lessOrEqual /><ml:apply><ml:id>g</ml:id><ml:sequence><ml:apply><ml:indexer /><ml:id>a</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>b</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>c</ml:id><ml:id>i</ml:id></ml:apply></ml:sequence></ml:apply><ml:id>∞</ml:id></ml:apply></math><rendering item-idref="46" /></region><region region-id="2245" left="66" top="2660.25" width="81.75" height="27.75" align-x="74.25" align-y="2676" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>i</ml:id><ml:apply><ml:minus /><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:apply><ml:id>floor</ml:id><ml:apply><ml:div /><ml:id>m</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:id>m</ml:id></ml:apply></ml:define></math><rendering item-idref="47" /></region><region region-id="2246" left="222" top="2666.25" width="60" height="18.75" align-x="267.75" align-y="2676" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:apply><ml:lessOrEqual /><ml:apply><ml:id>g</ml:id><ml:sequence><ml:real font="0">0</ml:real><ml:apply><ml:indexer /><ml:id>b</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>c</ml:id><ml:id>i</ml:id></ml:apply></ml:sequence></ml:apply><ml:id>∞</ml:id></ml:apply></math><rendering item-idref="48" /></region><region region-id="2223" left="36" top="2708.25" width="183" height="12" align-x="43.5" align-y="2718" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This modified problem is then solved.</p></text></region><region region-id="2262" left="66" top="2733" width="203.25" height="12.75" align-x="87" align-y="2742" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Res</ml:id><ml:apply><ml:id>rationalfitnp</ml:id><ml:sequence><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:real>0.95</ml:real><ml:id>m</ml:id><ml:id>n</ml:id><ml:id>StdY</ml:id><ml:id>LBUB</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="49" /></region><region region-id="2336" left="66" top="2767.5" width="222.75" height="123.75" align-x="86.25" align-y="2832" show-border="false" show-highlight="true" is-protected="true" background-color="#80ffff" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>Res</ml:id></ml:eval></math><rendering item-idref="50" /></region><region region-id="2469" left="36" top="2912.25" width="315.75" height="24" align-x="36" align-y="2922" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">In this case, no poles are detected in the first fit, so the results are the same.</p></text></region><region region-id="2261" left="0" top="2946" width="6000" height="6" align-x="0" align-y="2946" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2166" left="36" top="2959.5" width="212.25" height="12.75" align-x="36" align-y="2970" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">General nonlinear fitting of rational functions</p></text><indexes><phrase>general nonlinear fitting of rational functions</phrase><phrase>rational functions, General nonlinear fitting of</phrase></indexes></region><region region-id="2137" left="36" top="2984.25" width="324.75" height="72" align-x="36" align-y="2994" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">For comparison, here is the same calculation set up using the <b><u><link href="./nonlin.xmcd" title="Links to section" popup="false">LeastSquaresFit</link></u></b> function, which performs the same algorithm, but requires setup of the fitting function and a guess value. There may be cases where you wish to do this. For example, if you are fitting a function where you desire the constant term in the numerator, rather than the denominator, to be 1. </p></text></region><region region-id="2569" left="36" top="3074.25" width="160.5" height="12" align-x="36" align-y="3084" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here our function is of the form </p></text></region><region region-id="2570" left="222" top="3068.25" width="8.25" height="27.75" align-x="224.25" align-y="3084" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:apply><ml:div /><ml:real>1</ml:real><ml:id>x</ml:id></ml:apply></math><rendering item-idref="51" /></region><region region-id="2571" left="66" top="3099" width="36.75" height="12.75" align-x="74.25" align-y="3108" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>i</ml:id><ml:range><ml:real>1</ml:real><ml:real>15</ml:real></ml:range></ml:define></math><rendering item-idref="52" /></region><region region-id="2680" left="66" top="3123" width="63" height="24" align-x="81.75" align-y="3138" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>X</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:plus /><ml:id>i</ml:id><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>-15</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="53" /></region><region region-id="2681" left="144" top="3122.25" width="84" height="33" align-x="158.25" align-y="3138" show-border="false" show-highlight="false" is-protected="true" background-color="#0ff" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>Y</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:plus /><ml:apply><ml:indexer /><ml:id>X</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>0.5</ml:real><ml:apply><ml:id>rnd</ml:id><ml:id>i</ml:id></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="54" /></region><region region-id="2682" left="246" top="3129" width="35.25" height="12.75" align-x="269.25" align-y="3138" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>num</ml:id><ml:real>3</ml:real></ml:define></math><rendering item-idref="55" /></region><region region-id="2683" left="318" top="3129" width="45" height="12.75" align-x="351" align-y="3138" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>denom</ml:id><ml:real>5</ml:real></ml:define></math><rendering item-idref="56" /></region><region region-id="2593" left="36" top="3164.25" width="101.25" height="12" align-x="36" align-y="3174" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">First try <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">rationalfit</c></inlineAttr></b></f>:</p></text></region><region region-id="2685" left="66" top="3182.25" width="45" height="27.75" align-x="89.25" align-y="3198" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>dev</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:div /><ml:real>1</ml:real><ml:real>0.5</ml:real></ml:apply></ml:define></math><rendering item-idref="57" /></region><region region-id="2686" left="66" top="3213" width="219.75" height="18.75" align-x="96" align-y="3228" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>param</ml:id><ml:apply><ml:id>rationalfit</ml:id><ml:sequence><ml:id>X</ml:id><ml:id>Y</ml:id><ml:real>0.95</ml:real><ml:id>num</ml:id><ml:id>denom</ml:id><ml:id>dev</ml:id><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>-5</ml:real></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="58" /></region><region region-id="2586" left="66" top="3247.5" width="176.25" height="89.25" align-x="96.75" align-y="3294" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:function><ml:id>f</ml:id><ml:boundVars><ml:id>x</ml:id><ml:id>β</ml:id></ml:boundVars></ml:function><ml:apply><ml:div /><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id>n</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>β</ml:id><ml:id>n</ml:id></ml:apply><ml:apply><ml:pow /><ml:id>x</ml:id><ml:id>n</ml:id></ml:apply></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:id>num</ml:id></ml:bounds></ml:apply><ml:apply><ml:plus /><ml:real>1</ml:real><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id>n</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>β</ml:id><ml:id>n</ml:id></ml:apply><ml:apply><ml:pow /><ml:id>x</ml:id><ml:apply><ml:minus /><ml:id>n</ml:id><ml:id>num</ml:id></ml:apply></ml:apply></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:apply><ml:plus /><ml:id>num</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:minus /><ml:apply><ml:id>rows</ml:id><ml:id>param</ml:id></ml:apply><ml:real>1</ml:real></ml:apply></ml:bounds></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="59" /></region><region region-id="2582" left="66" top="3357.75" width="87" height="18" align-x="91.5" align-y="3372" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:function><ml:id>fit</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>f</ml:id><ml:sequence><ml:id>x</ml:id><ml:apply><ml:matcol /><ml:id>param</ml:id><ml:real>0</ml:real></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="60" /></region><region region-id="2583" left="174" top="3363" width="88.5" height="12.75" align-x="183.75" align-y="3372" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>x</ml:id><ml:range><ml:sequence><ml:real>0</ml:real><ml:real>0.001</ml:real></ml:sequence><ml:apply><ml:id>rows</ml:id><ml:id>X</ml:id></ml:apply></ml:range></ml:define></math><rendering item-idref="61" /></region><region region-id="2698" left="30" top="3384" width="316.5" height="173.25" align-x="30" align-y="3384" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="62" /><rendering item-idref="63" /></region><region region-id="2700" left="36" top="3560.25" width="298.5" height="48" align-x="36" align-y="3570" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The no-poles version of the function might provide a nicer fit, once the tolerance was appropriately adjusted, but in general, the functional form doesn't reflect the real nature of the function.</p></text></region><region region-id="2711" left="0" top="3612" width="6000" height="6" align-x="0" align-y="3612" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2661" left="36" top="3626.25" width="117.75" height="12" align-x="36" align-y="3636" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">LeastSquaresFit</c></inlineAttr></b></f>:</p></text></region><region region-id="2658" left="36" top="3650.25" width="82.5" height="12" align-x="36" align-y="3660" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Guess value :</p></text></region><region region-id="2694" left="66" top="3681" width="109.5" height="84.75" align-x="101.25" align-y="3726" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>βguess</ml:id><ml:matrix rows="5" cols="1"><ml:real>-0.198</ml:real><ml:apply><ml:mult style="auto-select" /><ml:real>9.614</ml:real><ml:apply><ml:pow /><ml:real>10</ml:real><ml:real>-3</ml:real></ml:apply></ml:apply><ml:real>1</ml:real><ml:real>0</ml:real><ml:real>0</ml:real></ml:matrix></ml:define></math><rendering item-idref="64" /></region><region region-id="2722" left="216" top="3716.25" width="78.75" height="13.5" align-x="224.25" align-y="3726" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>i</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:id>last</ml:id><ml:id>βguess</ml:id></ml:apply></ml:range></ml:define></math><rendering item-idref="65" /></region><region region-id="2723" left="216" top="3747" width="51.75" height="18" align-x="255.75" align-y="3756" show-border="false" show-highlight="false" 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item-idref="72" /><rendering item-idref="73" /></region><region region-id="2731" left="42" top="4208.25" width="306" height="60" align-x="42" align-y="4218" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Note that any filtering or data <b><u><link href="./scale.xmcd#transforming" title="Links to section" popup="false">transformation</link></u></b> method that biases the data towards a straight line before doing a rational function fit will add considerable convergence speed to the rational function and may have a side benefit of getting rid of unwanted poles.</p></text></region><region region-id="2734" left="6" top="4272" width="427.5" height="5.25" align-x="6" align-y="4272" show-border="false" show-highlight="false" is-protected="true" 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