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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="1448" left="174" top="44.25" width="62.25" height="12" align-x="174" 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="./examples.xmcd" title="Links to Section Contents" popup="false"><sp count="2" />EXAMPLES</link></c></inlineAttr></u></f></p></text></region><region region-id="2056" left="36" top="90.75" width="201.75" 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">Optimal Spacing of Interpolation Points</p></text><indexes><phrase>using a rational function to fit optical constants</phrase><phrase>examples, using a rational function to fit optical constants</phrase><phrase>optical constants, using a rational function to fit</phrase><phrase>rational function, using to fit optical constants</phrase></indexes></region><region region-id="1450" left="36" top="104.25" width="76.5" height="12" align-x="36" align-y="114" 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">by Robert Adair</p></text></region><region region-id="2473" left="36" top="128.25" width="311.25" height="168" align-x="36" align-y="138" 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 his seminal book on spline fitting, deBoor says, "Polynomial interpolation at appropriately chosen points (e.g. the Chebyshev points) produces an approximation which ... differs very little from the best possible approximant by polynomials of the same order". That is, if your points are well chosen, a single polynomial of appropriate order will fit all your data well. If they are not, then a <b><u><link href="./nonparam.xmcd" title="Links to section" popup="false">spline fitting</link></u></b> method may be a better choice. In particular, uniformly spaced points can have bad consequences. Further, "...If the function to be approximated is badly behaved anywhere in the interval of approximation, then the approximation is poor everywhere. This global dependence on local properties can be avoided when using <f family="Verdana" charset="0"><i><inlineAttr line-through="false">piecewise</inlineAttr></i></f> polynomial approximants." by which he means cubic splines or bsplines.</p></text></region><region region-id="2478" left="36" top="314.25" width="310.5" height="72" 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="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">What this means is that a polynomial of the same order as the number of the data points will pass through all the points, but can give a terrible approximation to the function elsewhere if the points are badly chosen. For example, consider the following uniformly spaced sample points across a Lorenzian function:</p></text></region><region region-id="2489" left="36" top="404.25" width="52.5" height="12" 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">Endpoints:</p></text></region><region region-id="2515" left="144" top="405" width="30" height="12.75" align-x="154.5" align-y="414" 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>a</ml:id><ml:real>-.9</ml:real></ml:define></math><rendering item-idref="4" /></region><region region-id="2526" left="222" top="405" width="23.25" height="12.75" 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/><ml:id>i</ml:id><ml:parens><ml:apply><ml:minus /><ml:id>b</ml:id><ml:id>a</ml:id></ml:apply></ml:parens></ml:apply><ml:id>n</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="8" /></region><region region-id="2528" left="36" top="458.25" width="99.75" height="12" align-x="36" align-y="468" 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">Sample the function</p></text></region><region region-id="2529" left="66" top="482.25" width="74.25" height="33.75" align-x="93" align-y="498" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" 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align-x="36" align-y="534" 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 global polynomial interpolation using this data spacing</p></text></region><region region-id="2552" left="66" top="549" width="105.75" height="18" align-x="91.5" align-y="558" 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:indexer /><ml:apply><ml:id>polyint</ml:id><ml:id>vx</ml:id><ml:id>vy</ml:id><ml:id>x</ml:id></ml:apply><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="11" /></region><region 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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>j</ml:id></ml:apply><ml:apply><ml:plus /><ml:id>a</ml:id><ml:apply><ml:mult style="default" /><ml:id>j</ml:id><ml:id>step</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="14" /></region><region region-id="2555" left="0" top="618" width="6000" height="-6" align-x="0" align-y="612" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2557" left="36" top="626.25" width="309" 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="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You might expect that as you increase the number of sampled points and the order of the polynomial, n, the approximation would improve, but it does not. Try n = 20.</p></text></region><region region-id="2562" left="66" top="669" width="21.75" height="12.75" align-x="76.5" 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:globalDefine><ml:id>n</ml:id><ml:real>8</ml:real></ml:globalDefine></math><rendering item-idref="15" /></region><region region-id="2560" left="18" top="696" width="308.25" height="185.25" align-x="18" align-y="696" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="16" /><rendering item-idref="17" /></region><region region-id="2563" left="36" top="896.25" width="338.25" height="24" align-x="36" align-y="906" 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">What deBoor recommends is instead choosing the interpolating points as the zeros of the Chebyshev polynomial of degree n,</p></text></region><region region-id="2567" left="66" top="934.5" width="180" height="43.5" align-x="104.25" align-y="966" 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>vxCheb</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:plus /><ml:id>a</ml:id><ml:id>b</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:minus /><ml:id>a</ml:id><ml:id>b</ml:id></ml:apply></ml:parens><ml:apply><ml:id>cos</ml:id><ml:apply><ml:div /><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id>i</ml:id></ml:apply><ml:real>1</ml:real></ml:apply></ml:parens><ml:id>π</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:parens><ml:apply><ml:plus /><ml:id>n</ml:id><ml:real>1</ml:real></ml:apply></ml:parens></ml:apply></ml:apply></ml:apply></ml:apply></ml:apply><ml:real>2</ml:real></ml:apply></ml:define></math><rendering item-idref="18" /></region><region region-id="2566" left="66" top="993" width="96.75" height="12.75" align-x="102" 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 warning="WarnRedefinedUDScalar"><ml:id>vxCheb</ml:id><ml:apply><ml:id>sort</ml:id><ml:id>vxCheb</ml:id></ml:apply></ml:define></math><rendering item-idref="19" /></region><region region-id="2585" left="222" top="993" width="96.75" height="18" align-x="261.75" 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:apply><ml:indexer /><ml:id>vyCheb</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:id>g1</ml:id><ml:apply><ml:indexer /><ml:id>vxCheb</ml:id><ml:id>i</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="20" /></region><region region-id="2570" left="66" top="1023.75" width="72" height="18" align-x="100.5" align-y="1038" 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:matcol /><ml:id>data</ml:id><ml:real>0</ml:real></ml:apply><ml:id>vxCheb</ml:id></ml:define></math><rendering item-idref="21" /></region><region region-id="2586" left="222" top="1023.75" width="73.5" height="18" align-x="256.5" align-y="1038" 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:matcol /><ml:id>data</ml:id><ml:real>1</ml:real></ml:apply><ml:id>vyCheb</ml:id></ml:define></math><rendering item-idref="22" /></region><region region-id="2589" left="36" top="1052.25" width="336" height="36" align-x="36" align-y="1062" 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">Interpolate the same function using these new points. Go back and try a few values of n, to see that these points generally provide a nicer fit everywhere for larger values of n.</p></text></region><region region-id="2640" left="66" top="1095" width="168.75" height="18" align-x="112.5" align-y="1104" 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>fitCheb</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:indexer /><ml:apply><ml:id>polyint</ml:id><ml:id>vxCheb</ml:id><ml:id>vyCheb</ml:id><ml:id>x</ml:id></ml:apply><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="23" /></region><region region-id="2639" left="0" top="1116" width="6000" height="6" align-x="0" align-y="1116" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2592" left="24" top="1122" width="327" height="191.25" align-x="24" align-y="1122" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="24" /><rendering item-idref="25" /></region><region region-id="2671" left="36" top="1328.25" width="308.25" height="84" align-x="36" align-y="1338" 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 Chebyshev point spacing does indeed provide a better interpolation for higher values of n, as we expect. However, it would be nice to know what the optimal spacing of points was so that we could use a lower value of n and still get good results. This can be calculated if we recall that a global polynomial interpolation is the same as a polynomial regression of the same order as the number of data points.</p></text></region><region region-id="2673" left="36" top="1429.5" width="100.5" height="12.75" align-x="36" align-y="1440" 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">Optimal point spacing</p></text></region><region region-id="2767" left="36" top="1448.25" width="328.5" height="84" align-x="36" align-y="1458" 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">Based on the notion that a polynomial of order n can always be found that passes through n points, we can find the polynomial that best approximates the given function g, and find which points on the curve it fits exactly. These points are the optimal interpolation points, that is, the points that, when interpolated with a polynomial, provide the best possible approximation everywhere on the interval. </p></text></region><region region-id="2768" left="36" top="1544.25" width="312.75" height="24" align-x="36" align-y="1554" 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">We need to work with the integral version of the least squares problem to get the optimal solution for a polynomial fit.</p></text></region><region region-id="2645" left="66" top="1582.5" width="165" height="69.75" align-x="91.5" align-y="1620" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="true"><ml:define><ml:id>Error</ml:id><ml:apply><ml:integral auto-algorithm="true" /><ml:lambda><ml:boundVars><ml:id>x</ml:id></ml:boundVars><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><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>c</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>degree</ml:id></ml:bounds></ml:apply><ml:apply><ml:id>g</ml:id><ml:id>x</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:lambda><ml:bounds><ml:id>a</ml:id><ml:id>b</ml:id></ml:bounds></ml:apply></ml:define></math><rendering item-idref="26" /></region><region region-id="2674" left="36" top="1658.25" width="318.75" height="36" align-x="36" align-y="1668" 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 normal equations are generated by taking the derivative of the error with respect to the polynomial coefficients and setting the result equal to 0.</p></text></region><region region-id="2683" left="66" top="1702.5" width="204" height="63.75" align-x="107.25" align-y="1734" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="true"><ml:define><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:apply><ml:indexer /><ml:id>c</ml:id><ml:id>k</ml:id></ml:apply></ml:boundVars><ml:id>Error</ml:id></ml:lambda></ml:apply><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:apply><ml:integral auto-algorithm="true" /><ml:lambda><ml:boundVars><ml:id>x</ml:id></ml:boundVars><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:minus /><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>c</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>degree</ml:id></ml:bounds></ml:apply><ml:apply><ml:id>g</ml:id><ml:id>x</ml:id></ml:apply></ml:apply></ml:parens><ml:apply><ml:pow /><ml:id>x</ml:id><ml:id>k</ml:id></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:id>a</ml:id><ml:id>b</ml:id></ml:bounds></ml:apply></ml:apply></ml:define></math><rendering item-idref="27" /></region><region region-id="2769" left="0" top="1770" width="6000" height="6" align-x="0" align-y="1770" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2826" left="36" top="1784.25" width="309" height="12" align-x="36" align-y="1794" 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 solve the equations, we will need to evaluate the integrals</p></text></region><region region-id="2771" left="66" top="1812" width="90.75" height="40.5" align-x="104.25" 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:function><ml:id>fI0</ml:id><ml:boundVars><ml:id>n</ml:id><ml:id>k</ml:id></ml:boundVars></ml:function><ml:apply><ml:integral auto-algorithm="true" algorithm="adaptive" /><ml:lambda><ml:boundVars><ml:id>x</ml:id></ml:boundVars><ml:apply><ml:pow /><ml:id>x</ml:id><ml:apply><ml:plus /><ml:id>n</ml:id><ml:id>k</ml:id></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:id>a</ml:id><ml:id>b</ml:id></ml:bounds></ml:apply></ml:define></math><rendering item-idref="28" /></region><region region-id="2772" left="222" top="1812" width="94.5" height="40.5" align-x="250.5" 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:function><ml:id>fI1</ml:id><ml:boundVars><ml:id>k</ml:id></ml:boundVars></ml:function><ml:apply><ml:integral auto-algorithm="true" algorithm="adaptive" /><ml:lambda><ml:boundVars><ml:id>x</ml:id></ml:boundVars><ml:apply><ml:mult style="default" /><ml:apply><ml:id>g1</ml:id><ml:id>x</ml:id></ml:apply><ml:apply><ml:pow /><ml:id>x</ml:id><ml:id>k</ml:id></ml:apply></ml:apply></ml:lambda><ml:bounds><ml:id>a</ml:id><ml:id>b</ml:id></ml:bounds></ml:apply></ml:define></math><rendering item-idref="29" /></region><region region-id="2831" left="36" top="1862.25" width="313.5" height="48" align-x="39" align-y="1872" 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 general, the integral fI1 will require numerical evaluation.<sp count="2" />In the case of a Runge function, it could be evaluated symbolically, but the speed of evaluation is good enough that both fI0 and fI1 will be left as numerical integrals.</p></text></region><region region-id="2694" left="36" top="1916.25" width="301.5" height="24" 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="0" text-align="inherit" list-style-type="none" tabs="inherit">The least squares matrix equation for the coefficients then becomes</p></text></region><region region-id="2695" left="66" top="1953" width="64.5" height="15" align-x="92.25" align-y="1962" 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:mult style="default" /><ml:id>M</ml:id><ml:id>c</ml:id></ml:apply><ml:id subscript="integral">y</ml:id></ml:apply></math><rendering item-idref="30" /></region><region region-id="2697" left="36" top="1976.25" width="34.5" height="12" align-x="36" align-y="1986" 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">where:</p></text></region><region region-id="2703" left="66" top="1995" width="45.75" height="12.75" align-x="99" align-y="2004" 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>degree</ml:id><ml:id>n</ml:id></ml:define></math><rendering item-idref="31" /></region><region region-id="2709" left="144" top="1995" width="57" height="12.75" align-x="154.5" align-y="2004" 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>k</ml:id><ml:range><ml:real>0</ml:real><ml:id>degree</ml:id></ml:range></ml:define></math><rendering item-idref="32" /></region><region region-id="2710" left="222" top="1995" width="61.5" height="12.75" align-x="237" align-y="2004" 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>k1</ml:id><ml:range><ml:real>0</ml:real><ml:id>degree</ml:id></ml:range></ml:define></math><rendering item-idref="33" /></region><region region-id="2717" left="66" top="2019" width="75.75" height="21" align-x="111.75" align-y="2028" 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 subscript="integral">y</ml:id><ml:id>k</ml:id></ml:apply><ml:apply><ml:id>fI1</ml:id><ml:id>k</ml:id></ml:apply></ml:define></math><rendering item-idref="34" /></region><region region-id="2713" left="66" top="2049" width="76.5" height="18" align-x="99" align-y="2058" 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>M</ml:id><ml:sequence><ml:id>k</ml:id><ml:id>k1</ml:id></ml:sequence></ml:apply><ml:apply><ml:id>fI0</ml:id><ml:id>k</ml:id><ml:id>k1</ml:id></ml:apply></ml:define></math><rendering item-idref="35" /></region><region region-id="2718" left="36" top="2078.25" width="285.75" height="12" align-x="36" align-y="2088" 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">Solving for the optimal interpolation coefficients, c gives:</p></text></region><region region-id="2719" left="66" top="2103" width="71.25" height="12.75" align-x="77.25" align-y="2112" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:id>L</ml:id><ml:apply><ml:id>cholesky</ml:id><ml:id>M</ml:id></ml:apply></ml:define></math><rendering item-idref="36" /></region><region region-id="2724" left="144" top="2097" width="65.25" height="22.5" align-x="184.5" align-y="2112" 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 subscript="inverse">L</ml:id><ml:apply><ml:pow /><ml:id>L</ml:id><ml:real>-1</ml:real></ml:apply></ml:define></math><rendering item-idref="37" /></region><region region-id="2725" left="66" top="2127.75" width="133.5" height="21.75" align-x="76.5" align-y="2142" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:id>c</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:apply><ml:transpose /><ml:id subscript="inverse">L</ml:id></ml:apply><ml:id subscript="inverse">L</ml:id></ml:apply><ml:id subscript="integral">y</ml:id></ml:apply></ml:define></math><rendering item-idref="38" /></region><region region-id="2730" left="36" top="2156.25" width="343.5" height="24" align-x="36" align-y="2166" 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">Then the optimal fit equation is given by the new polynomial of order n</p></text></region><region region-id="2723" left="66" top="2193" width="95.25" height="36" align-x="106.5" align-y="2208" 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>fitOpt</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id>k</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>c</ml:id><ml:id>k</ml:id></ml:apply><ml:apply><ml:pow /><ml:id>x</ml:id><ml:id>k</ml:id></ml:apply></ml:apply></ml:parens></ml:lambda></ml:apply></ml:define></math><rendering item-idref="39" /></region><region region-id="2783" left="36" top="2234.25" width="314.25" height="12" align-x="36" align-y="2244" 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">Take a look at the interpolated error over the data range:</p></text></region><region region-id="2784" left="66" top="2253" width="129.75" height="12.75" align-x="115.5" align-y="2262" 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>ChebErr</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:minus /><ml:apply><ml:id>g1</ml:id><ml:id>x</ml:id></ml:apply><ml:apply><ml:id>fitCheb</ml:id><ml:id>x</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="40" /></region><region region-id="2785" left="222" top="2253" width="117.75" height="12.75" align-x="265.5" align-y="2262" 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>OptErr</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:minus /><ml:apply><ml:id>g1</ml:id><ml:id>x</ml:id></ml:apply><ml:apply><ml:id>fitOpt</ml:id><ml:id>x</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="41" /></region><region region-id="2735" left="36" top="2280" width="297.75" height="154.5" align-x="36" align-y="2280" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="42" /><rendering item-idref="43" /></region><region region-id="2775" left="0" top="2430" width="6000" height="6" align-x="0" align-y="2430" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2743" left="36" top="2438.25" width="303.75" height="60" align-x="36" align-y="2448" 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">Finding the zeros of the error will give the points at which the optimal fit polynomial is exactly equal to the function, and these are the optimal interpolation points. To do this, let's bracket the zero crossings, then use these brackets as guesses for the root solver.</p></text></region><region region-id="2793" left="36" top="2510.25" width="101.25" height="12" align-x="36" align-y="2520" 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">Bracket all the roots</p></text></region><region region-id="2824" left="66" top="2535" width="210" height="333.75" align-x="121.5" align-y="2544" 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>fGuess</ml:id><ml:boundVars><ml:id>a</ml:id><ml:id>b</ml:id></ml:boundVars></ml:function><ml:program><ml:localDefine><ml:id>npts</ml:id><ml:real>200</ml:real></ml:localDefine><ml:localDefine><ml:id>dx</ml:id><ml:apply><ml:div /><ml:apply><ml:minus /><ml:id>b</ml:id><ml:id>a</ml:id></ml:apply><ml:id>npts</ml:id></ml:apply></ml:localDefine><ml:localDefine><ml:id>x0</ml:id><ml:id>a</ml:id></ml:localDefine><ml:localDefine><ml:id>y0</ml:id><ml:apply><ml:id>OptErr</ml:id><ml:id>x0</ml:id></ml:apply></ml:localDefine><ml:localDefine><ml:id>k</ml:id><ml:real>0</ml:real></ml:localDefine><ml:for><ml:id>i</ml:id><ml:range><ml:real>1</ml:real><ml:id>npts</ml:id></ml:range><ml:program><ml:localDefine><ml:id>x1</ml:id><ml:apply><ml:plus /><ml:id>a</ml:id><ml:apply><ml:mult style="default" /><ml:id>i</ml:id><ml:id>dx</ml:id></ml:apply></ml:apply></ml:localDefine><ml:localDefine><ml:id>y1</ml:id><ml:apply><ml:id>OptErr</ml:id><ml:id>x1</ml:id></ml:apply></ml:localDefine><ml:ifThen><ml:apply><ml:lessOrEqual /><ml:apply><ml:mult style="default" /><ml:id>y0</ml:id><ml:id>y1</ml:id></ml:apply><ml:real>0</ml:real></ml:apply><ml:program><ml:ifThen><ml:apply><ml:equal /><ml:apply><ml:mult style="default" /><ml:id>y0</ml:id><ml:id>y1</ml:id></ml:apply><ml:real>0</ml:real></ml:apply><ml:program><ml:localDefine><ml:id>x1</ml:id><ml:apply><ml:plus /><ml:id>x1</ml:id><ml:apply><ml:div /><ml:id>dx</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:localDefine><ml:localDefine><ml:id>y1</ml:id><ml:apply><ml:id>OptErr</ml:id><ml:id>x1</ml:id></ml:apply></ml:localDefine></ml:program></ml:ifThen><ml:localDefine><ml:apply><ml:indexer /><ml:id>RootBracket</ml:id><ml:sequence><ml:id>k</ml:id><ml:real>0</ml:real></ml:sequence></ml:apply><ml:id>x0</ml:id></ml:localDefine><ml:localDefine><ml:apply><ml:indexer /><ml:id>RootBracket</ml:id><ml:sequence><ml:id>k</ml:id><ml:real>1</ml:real></ml:sequence></ml:apply><ml:id>x1</ml:id></ml:localDefine><ml:localDefine><ml:id>k</ml:id><ml:apply><ml:plus /><ml:id>k</ml:id><ml:real>1</ml:real></ml:apply></ml:localDefine></ml:program></ml:ifThen><ml:localDefine><ml:id>x0</ml:id><ml:id>x1</ml:id></ml:localDefine><ml:localDefine><ml:id>y0</ml:id><ml:id>y1</ml:id></ml:localDefine></ml:program></ml:for><ml:return><ml:id>RootBracket</ml:id></ml:return></ml:program></ml:define></math><rendering item-idref="44" /></region><region region-id="2798" left="36" top="2882.25" width="253.5" height="12" align-x="36" align-y="2892" 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">Refine the roots from the initial bracketing guesses</p></text></region><region region-id="2801" left="66" top="2901" width="240" height="120.75" align-x="90.75" align-y="2910" 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>xOpt</ml:id><ml:program><ml:localDefine><ml:id>guess</ml:id><ml:apply><ml:id>fGuess</ml:id><ml:id>a</ml:id><ml:id>b</ml:id></ml:apply></ml:localDefine><ml:localDefine><ml:id>nRoots</ml:id><ml:apply><ml:id>rows</ml:id><ml:id>guess</ml:id></ml:apply></ml:localDefine><ml:for><ml:id>i</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:minus /><ml:id>nRoots</ml:id><ml:real>1</ml:real></ml:apply></ml:range><ml:program><ml:localDefine><ml:id>x</ml:id><ml:apply><ml:div /><ml:apply><ml:plus /><ml:apply><ml:indexer /><ml:id>guess</ml:id><ml:sequence><ml:id>i</ml:id><ml:real>0</ml:real></ml:sequence></ml:apply><ml:apply><ml:indexer /><ml:id>guess</ml:id><ml:sequence><ml:id>i</ml:id><ml:real>1</ml:real></ml:sequence></ml:apply></ml:apply><ml:real>2</ml:real></ml:apply></ml:localDefine><ml:localDefine><ml:apply><ml:indexer /><ml:id>r</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:id>root</ml:id><ml:apply><ml:id>OptErr</ml:id><ml:id>x</ml:id></ml:apply><ml:id>x</ml:id><ml:apply><ml:indexer /><ml:id>guess</ml:id><ml:sequence><ml:id>i</ml:id><ml:real>0</ml:real></ml:sequence></ml:apply><ml:apply><ml:indexer /><ml:id>guess</ml:id><ml:sequence><ml:id>i</ml:id><ml:real>1</ml:real></ml:sequence></ml:apply></ml:apply></ml:localDefine></ml:program></ml:for><ml:return><ml:id>r</ml:id></ml:return></ml:program></ml:define></math><rendering item-idref="45" /></region><region region-id="2823" left="66" top="3039" width="83.25" height="12.75" align-x="92.25" align-y="3048" 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>yOpt</ml:id><ml:apply><ml:id>fitOpt</ml:id><ml:id>xOpt</ml:id></ml:apply></ml:define></math><rendering item-idref="46" /></region><region region-id="2809" left="0" top="3066" width="6000" height="6" align-x="0" align-y="3066" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="2807" left="36" top="3074.25" width="327" height="84" 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">Compare the optimal fit to the Chebyshev fit.<sp count="2" />Overall the two fits have the same general appearance, and the ripple is much smaller than the equally spaced data points fit.<sp count="2" />Comparing the location of the interpolation points, one finds the Chebyshev points have a similar distribution to the optimal points which helps to reduce the overall ripple. Adjust the value of n to see how these two approximations change with sampling.</p></text></region><region region-id="2816" left="30" top="3168" width="297.75" height="226.5" align-x="30" align-y="3168" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="47" /><rendering item-idref="48" /></region><region region-id="2811" left="36" top="3398.25" width="306.75" height="36" align-x="36" align-y="3408" 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 have the luxury of choosing where your data will be sampled, you may wish to use guidelines of this sort when choosing your data spacing.</p></text></region><region region-id="2420" left="36" top="3445.5" width="47.25" height="12.75" align-x="36" align-y="3456" 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">Reference</p></text></region><region region-id="2434" left="36" top="3464.25" width="306.75" height="24" align-x="36" align-y="3474" 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">de Boor, Carl (1978), <f family="Verdana" charset="0"><i><inlineAttr line-through="false">A Practical Guide to Splines</inlineAttr></i></f>, Springer-Verlag, Chapter 2.</p></text></region><region region-id="2429" left="6" top="3492" width="427.5" height="8.25" align-x="6" align-y="3492" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><png item-idref="49" /><rendering item-idref="50" /></region></regions><binaryContent><item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4zMuw6CMBgF4P5tuYoyOf8mLjZh4hmIk4mPQErFW0I0yODb66HgbpvTy9fm
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