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align-x="6" align-y="6" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><png item-idref="2" /><link href="./dataTOC.xmcd" title="Links to Table of Contents" popup="false" /><rendering item-idref="3" /></region><region region-id="513" left="90" top="44.25" width="153" height="12" align-x="90" align-y="54" 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="Titre" margin-left="inherit" margin-right="inherit" 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="./nonparam.xmcd" title="Links to Section Contents" popup="false"><sp count="6" />NONPARAMETRIC FITTING</link></c></inlineAttr></u></f></p></text></region><region region-id="114" left="36" top="72.75" width="156" height="13.5" align-x="36" align-y="84" 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 Interpolation</p></text><indexes><phrase>rational function interpolation</phrase><phrase>interpolation, rational function</phrase><phrase>nonparametric fitting, rational function interpolation</phrase></indexes></region><region region-id="583" left="66" top="97.5" width="88.5" height="12.75" align-x="66" align-y="108" 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">rationalint(vx,vy,x)</p></text></region><region region-id="599" left="36" top="134.25" width="297" height="72" align-x="36" 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="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">rationalint</c></inlineAttr></b></f> function performs interpolation at a requested point, x, using a rational function approximation (quotient of polynomials) to the data. This type of interpolation is useful for data with sharp discontinuities, which are handled by the poles in the denominator of the rational function. </p></text></region><region region-id="748" left="36" top="218.25" width="93" height="12" align-x="36" align-y="228" 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">Consider the data:</p></text></region><region region-id="723" left="66" top="243" width="45" height="144.75" align-x="75.75" align-y="318" 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:matrix rows="9" cols="1"><ml:real>8</ml:real><ml:real>13</ml:real><ml:real>17</ml:real><ml:real>24</ml:real><ml:real>66</ml:real><ml:real>89</ml:real><ml:real>120</ml:real><ml:real>187</ml:real><ml:real>200</ml:real></ml:matrix></ml:define></math><rendering item-idref="4" /></region><region region-id="751" left="144" top="243" width="53.25" height="144.75" align-x="155.25" align-y="318" 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>y</ml:id><ml:matrix rows="9" cols="1"><ml:real>.08</ml:real><ml:real>.131</ml:real><ml:real>.172</ml:real><ml:real>.245</ml:real><ml:real>.766</ml:real><ml:real>1.235</ml:real><ml:real>2.9</ml:real><ml:real>-6.7</ml:real><ml:real>-4.2</ml:real></ml:matrix></ml:define></math><rendering item-idref="5" /></region><region region-id="1032" left="228" top="309" width="75" height="12.75" align-x="237" align-y="318" 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>j</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:minus /><ml:apply><ml:id>rows</ml:id><ml:id>x</ml:id></ml:apply><ml:real>1</ml:real></ml:apply></ml:range></ml:define></math><rendering item-idref="6" /></region><region region-id="746" left="36" top="404.25" width="304.5" height="48" align-x="36" 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">A transition of sorts occurs between x = 120 and x = 187. We would like to estimate where this happens, and also approximate y values for any values of x in the range of the tabulated points.</p></text></region><region region-id="1064" left="66" top="465.75" width="193.5" height="18" align-x="79.5" align-y="480" 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:apply><ml:transpose /><ml:matrix rows="1" cols="9"><ml:real>10</ml:real><ml:real>3</ml:real><ml:real>050</ml:real><ml:real>80</ml:real><ml:real>110</ml:real><ml:real>140</ml:real><ml:real>180</ml:real><ml:real>220</ml:real><ml:real>250</ml:real></ml:matrix></ml:apply></ml:define></math><rendering item-idref="7" /></region><region region-id="744" left="66" top="507" width="103.5" height="18" align-x="81" align-y="516" 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>Y</ml:id><ml:id>j</ml:id></ml:apply><ml:apply><ml:indexer /><ml:apply><ml:id>rationalint</ml:id><ml:id>x</ml:id><ml:id>y</ml:id><ml:apply><ml:indexer /><ml:id>X</ml:id><ml:id>j</ml:id></ml:apply></ml:apply><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="8" /></region><region region-id="1035" left="228" top="507" width="102.75" height="18" align-x="242.25" align-y="516" 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>E</ml:id><ml:id>j</ml:id></ml:apply><ml:apply><ml:indexer /><ml:apply><ml:id>rationalint</ml:id><ml:id>x</ml:id><ml:id>y</ml:id><ml:apply><ml:indexer /><ml:id>X</ml:id><ml:id>j</ml:id></ml:apply></ml:apply><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="9" /></region><region region-id="757" left="36" top="536.25" width="318.75" height="36" align-x="36" align-y="546" 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 output of <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">rationalint </c></inlineAttr></b></f>is a vector of length 2 containing the value of the rational function evaluated at X and an estimate of the error. </p></text></region><region region-id="761" left="0" top="588" width="6000" height="6" align-x="0" align-y="588" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="763" left="36" top="601.5" width="102.75" height="12.75" align-x="36" align-y="612" 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">Finding the asymptote</p></text><indexes><phrase>finding the asymptote</phrase><phrase>asymptote, finding</phrase></indexes></region><region region-id="758" left="36" top="626.25" width="315.75" height="36" align-x="36" align-y="636" 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">To find the value of x at which the approximation has an asymptote, we can solve for the value at which <b>rationalint(x,y,z)</b> has a zero denominator:</p></text></region><region region-id="1066" left="36" top="668.25" width="204.75" height="12" align-x="36" align-y="678" 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">A guess at the location of the transition:</p></text></region><region region-id="1065" left="258" top="669" width="30.75" height="12.75" align-x="267.75" align-y="678" 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>z</ml:id><ml:real>150</ml:real></ml:define></math><rendering item-idref="10" /></region><region region-id="1037" left="66" top="692.25" width="133.5" height="33" align-x="78" align-y="708" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_argument	(Matrix (Value {}))-&gt;I4-&gt;Value {}	(Value {})-&gt;(Value a)-&gt;b	I4	Value a" optimize="false" export="false" disable-calc="false"><ml:define><ml:id>R</ml:id><ml:apply><ml:id>root</ml:id><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:indexer /><ml:apply><ml:id>rationalint</ml:id><ml:id>x</ml:id><ml:id>y</ml:id><ml:id>z</ml:id></ml:apply><ml:real>0</ml:real></ml:apply></ml:apply><ml:id>z</ml:id></ml:apply></ml:define></math><rendering item-idref="11" /></region><region region-id="1068" left="36" top="740.25" width="197.25" height="12" align-x="36" align-y="750" 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">Approximate position of the asymptote:</p></text></region><region region-id="1067" left="258" top="741" width="24.75" height="12.75" align-x="269.25" align-y="750" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math error="bad_argument	(Matrix (Value {}))-&gt;I4-&gt;Value {}	(Value {})-&gt;(Value a)-&gt;b	I4	Value a" optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>R</ml:id></ml:eval></math><rendering item-idref="12" /></region><region region-id="768" left="12" top="768" width="284.25" height="168" align-x="12" align-y="768" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="13" /><rendering item-idref="14" /></region><region region-id="774" left="36" top="968.25" width="312" height="60" align-x="36" align-y="978" 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">The polynomials in the numerator and denominator of the quotient have the same degree if the number of tabulated points is odd. The degree of the denominator is one larger than the degree of the numerator if the number of points is even.</p></text></region><region region-id="872" left="36" top="1045.5" width="144.75" height="12.75" align-x="36" align-y="1056" 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">Poles in the interpolated values</p></text><indexes><phrase>poles in the interpolated values</phrase><phrase>interpolated values, poles in the</phrase></indexes></region><region region-id="873" left="36" top="1064.25" width="313.5" height="96" align-x="36" align-y="1074" 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 the very feature which makes rational function interpolation useful, the existence of poles which result in asymptotic behavior, can be a liability. The algorithm may construct unwanted poles in an effort to fit the data better throughout the rest of the range. The following example compares the error in <f family="Verdana" charset="0"><b><u><inlineAttr line-through="false"><c val="#000"><link href="./polint.xmcd" title="Links to section" popup="false">polyint</link></c></inlineAttr></u></b></f> and <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">rationalint</c></inlineAttr></b></f>, and shows this unwanted effect, as opposed to polynomial interpolation, which has no poles.</p></text></region><region region-id="874" left="0" top="1176" width="6000" height="6" align-x="0" align-y="1176" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1092" left="204" top="1202.25" width="149.25" height="12" align-x="204" align-y="1212" 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">which approximates the curve</p></text></region><region region-id="1093" left="204" top="1225.5" width="38.25" height="14.25" align-x="224.25" align-y="1236" 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:boundVars></ml:function><ml:apply><ml:sqrt /><ml:id>x</ml:id></ml:apply></ml:define></math><rendering item-idref="15" /></region><region region-id="1094" left="204" top="1251" width="64.5" height="12.75" align-x="235.5" align-y="1260" 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>f</ml:id><ml:real>0.35</ml:real></ml:apply></ml:eval></math><rendering item-idref="16" /></region><region region-id="831" left="66" top="1194.75" width="47.25" height="161.25" align-x="80.25" align-y="1278" 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>x1</ml:id><ml:matrix rows="10" cols="1"><ml:real>0.1</ml:real><ml:real>0.2</ml:real><ml:real>0.3</ml:real><ml:real>0.4</ml:real><ml:real>0.5</ml:real><ml:real>0.6</ml:real><ml:real>0.7</ml:real><ml:real>0.8</ml:real><ml:real>0.9</ml:real><ml:real>1</ml:real></ml:matrix></ml:define></math><rendering item-idref="17" /></region><region region-id="1095" left="114" top="1194.75" width="71.25" height="161.25" align-x="129.75" align-y="1278" 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>y1</ml:id><ml:matrix rows="10" cols="1"><ml:real>0.316228</ml:real><ml:real>0.447214</ml:real><ml:real>0.547723</ml:real><ml:real>0.632456</ml:real><ml:real>0.707107</ml:real><ml:real>0.774597</ml:real><ml:real>0.83666</ml:real><ml:real>0.894427</ml:real><ml:real>0.948683</ml:real><ml:real>1</ml:real></ml:matrix></ml:define></math><rendering item-idref="18" /></region><region region-id="1096" left="204" top="1269.75" width="180.75" height="35.25" align-x="300" align-y="1290" 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>rationalint</ml:id><ml:id>x1</ml:id><ml:id>y1</ml:id><ml:real>0.35</ml:real></ml:apply></ml:eval></math><rendering item-idref="19" /></region><region region-id="1097" left="204" top="1311.75" width="163.5" height="35.25" align-x="288" align-y="1332" 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>polyint</ml:id><ml:id>x1</ml:id><ml:id>y1</ml:id><ml:real>0.35</ml:real></ml:apply></ml:eval></math><rendering item-idref="20" /></region><region region-id="937" left="36" top="1370.25" width="170.25" height="12" align-x="36" align-y="1380" 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">Calculate the interpolated vectors</p></text></region><region region-id="940" left="66" top="1395" width="53.25" height="12.75" align-x="93.75" align-y="1404" 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>steps</ml:id><ml:real>1000</ml:real></ml:define></math><rendering item-idref="21" /></region><region region-id="953" left="144" top="1395" width="49.5" height="12.75" align-x="152.25" align-y="1404" 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:id>steps</ml:id></ml:range></ml:define></math><rendering item-idref="22" /></region><region region-id="1053" left="228" top="1388.25" width="94.5" height="27.75" align-x="241.5" align-y="1404" 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>u</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:plus /><ml:parens><ml:apply><ml:div /><ml:apply><ml:mult style="default" /><ml:id>i</ml:id><ml:parens><ml:apply><ml:minus /><ml:real>1</ml:real><ml:real>0.1</ml:real></ml:apply></ml:parens></ml:apply><ml:id>steps</ml:id></ml:apply></ml:parens><ml:real>0.1</ml:real></ml:apply></ml:define></math><rendering item-idref="23" /></region><region region-id="963" left="66" top="1431" width="105.75" height="18" align-x="77.25" align-y="1440" 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>r</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:indexer /><ml:apply><ml:id>rationalint</ml:id><ml:id>x1</ml:id><ml:id>y1</ml:id><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:id>i</ml:id></ml:apply></ml:apply><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="24" /></region><region region-id="1052" left="228" top="1431" width="96" height="18" align-x="241.5" align-y="1440" 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>p</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:indexer /><ml:apply><ml:id>polyint</ml:id><ml:id>x1</ml:id><ml:id>y1</ml:id><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:id>i</ml:id></ml:apply></ml:apply><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="25" /></region><region region-id="948" left="36" top="1460.25" width="127.5" height="12" align-x="36" align-y="1470" 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">and the associated errors</p></text></region><region region-id="949" left="66" top="1482" width="78.75" height="39" align-x="83.25" align-y="1512" 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>Δr</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:minus /><ml:real>1</ml:real><ml:apply><ml:mult style="default" /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:id>i</ml:id></ml:apply></ml:parens><ml:apply><ml:div /><ml:real>-1</ml:real><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>r</ml:id><ml:id>i</ml:id></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="26" /></region><region region-id="1051" left="228" top="1482" width="83.25" height="39" align-x="247.5" align-y="1512" 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>Δp</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:minus /><ml:real>1</ml:real><ml:apply><ml:mult style="default" /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:id>i</ml:id></ml:apply></ml:parens><ml:apply><ml:div /><ml:real>-1</ml:real><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>p</ml:id><ml:id>i</ml:id></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="27" /></region><region region-id="971" left="36" top="1532.25" width="346.5" height="12" align-x="36" align-y="1542" 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 interpolated values from ratint and polint seem virtually identical, </p></text></region><region region-id="969" left="60" top="1566" width="283.5" height="147.75" align-x="60" align-y="1566" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="28" /><rendering item-idref="29" /></region><region region-id="978" left="0" top="1716" width="6000" height="6" align-x="0" align-y="1716" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="973" left="36" top="1736.25" width="111" height="12" align-x="36" align-y="1746" 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">but look at the errors:</p></text></region><region region-id="1013" left="6" top="1770" width="199.5" height="138" align-x="6" align-y="1770" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="30" /><rendering item-idref="31" /></region><region region-id="1098" left="198" top="1770" width="199.5" height="138" align-x="198" align-y="1770" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="32" /><rendering item-idref="33" /></region><region region-id="979" left="36" top="1916.25" width="318.75" height="36" align-x="36" align-y="1926" 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">There is a sharp spike in the error for the rational function interpolation, that is due to a pole in the chosen denominator. We can zoom in on this portion:</p></text></region><region region-id="990" left="66" top="1977" width="53.25" height="12.75" align-x="93.75" align-y="1986" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id>steps</ml:id><ml:real>1000</ml:real></ml:define></math><rendering item-idref="34" /></region><region region-id="991" left="144" top="1977" width="49.5" height="12.75" align-x="152.25" align-y="1986" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id>i</ml:id><ml:range><ml:real>0</ml:real><ml:id>steps</ml:id></ml:range></ml:define></math><rendering item-idref="35" /></region><region region-id="1007" left="228" top="1970.25" width="114.75" height="27.75" align-x="241.5" align-y="1986" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:plus /><ml:parens><ml:apply><ml:div /><ml:apply><ml:mult style="default" /><ml:id>i</ml:id><ml:parens><ml:apply><ml:minus /><ml:real>0.55</ml:real><ml:real>0.52</ml:real></ml:apply></ml:parens></ml:apply><ml:id>steps</ml:id></ml:apply></ml:parens><ml:real>0.52</ml:real></ml:apply></ml:define></math><rendering item-idref="36" /></region><region region-id="1001" left="66" top="2031" width="105.75" height="18" align-x="77.25" align-y="2040" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:apply><ml:indexer /><ml:id>r</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:indexer /><ml:apply><ml:id>rationalint</ml:id><ml:id>x1</ml:id><ml:id>y1</ml:id><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:id>i</ml:id></ml:apply></ml:apply><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="37" /></region><region region-id="1008" left="228" top="2010" width="78.75" height="39" align-x="245.25" align-y="2040" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:apply><ml:indexer /><ml:id>Δr</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:minus /><ml:real>1</ml:real><ml:apply><ml:mult style="default" /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:id>i</ml:id></ml:apply></ml:parens><ml:apply><ml:div /><ml:real>-1</ml:real><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>r</ml:id><ml:id>i</ml:id></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="38" /></region><region region-id="1011" left="48" top="2070" width="201.75" height="138" align-x="48" align-y="2070" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="39" /><rendering item-idref="40" /></region><region region-id="1012" left="36" top="2210.25" width="315.75" height="72" 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">In practice, you may not notice these poles if your step size is too large, but the interpolated value does go to infinity if you are directly on the pole, which may cause problems, for example, if you are integrating these results with a small convergence tolerance. You may find that <b><u><link href="./thiele.xmcd" title="Links to section" popup="false">Thiele interpolation</link></u></b> provides better results in these cases.</p></text></region><region region-id="644" left="36" top="2294.25" width="302.25" height="60" align-x="36" align-y="2304" 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<f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000"> rationalint</c></inlineAttr></b></f> function is based on RATINT (p.112) from the book "Numerical Recipes in C, The Art of Scientific Computing" (Cambridge University Press),<sp count="2" />Copyright (C) 1987, 1988 Numerical Recipes Software, and is used under license. </p></text></region><region region-id="576" left="6" top="2364" width="427.5" height="5.25" align-x="6" align-y="2364" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><png item-idref="41" /><rendering item-idref="42" /></region></regions><binaryContent><item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4zMuw6CMBgF4P5tuYoyOf8mLjZh4hmIk4mPQErFW0I0yODb66HgbpvTy9fm
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