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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" display-width="235.5" display-height="33" /><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="78.75" width="209.25" height="13.5" align-x="36" align-y="90" 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">About Splines and Interpolation Methods</p></text><indexes><phrase>about Splines and Spline methods</phrase><phrase>nonparametric fitting, about Splines and Spline methods</phrase><phrase>Splines, methods</phrase></indexes></region><region region-id="1002" left="36" top="116.25" width="306.75" height="144" align-x="36" 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="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Splines are used to describe a curve through a data set without a parametric function.<sp count="2" />Spline interpolation can also be used to approximate a curve which has an exceptionally complicated function associated with it. In the case of <b><u><link href="./statspline.xmcd" title="Links to section" popup="false">B-splines</link></u></b>, where a string of knots is used to represent the data instead of the original x-values, significant data compression can be realized as well. The utility of polynomial splines is that, although they do not represent a closed-form expression, they are well-behaved, that is, they are differentiable, integrable, etc., and so are useful in many of the same kinds of mathematical analyses as a closed-form expression.<sp count="2" /></p></text></region><region region-id="1000" left="36" top="272.25" width="302.25" height="84" align-x="56.25" align-y="282" 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">Parametric fit functions, or regressions, are described in the <b><u><link href="./regression.xmcd" title="Links to Section" popup="false">next section</link></u></b>.<sp count="2" />It may seem more natural to find a closed-form expression to describe a data set, rather than simply draw a line through it, but there are circumstances where the spline technique is perferable, or, in fact, the only one feasible.<sp count="2" />See, for example, the <b><u><link href="./splinesvspoly.xmcd" title="Links to section" popup="false">comparison of B-splines with a polynomial fit</link></u></b>.</p></text></region><region region-id="1052" left="36" top="374.25" width="327" height="84" align-x="36" align-y="384" 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">Different kinds of splines depend on different basis functions, that is, the type of function which will be used to interpolate between one data point and the next.<sp count="2" />Depending on the form of your data, you may wish to use interpolation methods with special properties, like the <b><u><link href="./ratint.xmcd" title="Links to section" popup="false">rational spline</link></u></b>, which is appropriate for asymptotic data, or data with a <b><u><link href="./refrindex.xmcd" title="Links to Application" popup="false">known rational function</link></u></b> for a physical desciption.</p></text></region><region region-id="1352" left="36" top="470.25" width="324.75" height="180" 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="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Note that polynomials splines, such as the built-in <b><u><link href="./splines_1.xmcd" title="Links to section" popup="false">lspline, pspline, and cspline</link></u></b>, or the new <b><u><link href="./polint.xmcd" title="Links to section" popup="false">polyint</link></u></b>, are a very poor fit to functions such as ln(x) and sqrt(x), because these functions have poles in the value and first derivative. A <b><link href="./ratint.xmcd" title="Links to section" popup="false"><u>rational function</u></link></b> is a much better choice here because it has the pole to match the original function. </p><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit" /><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Additionally, polynomial splines return very suspect values near the end points, hence the choice of boundary conditions <f family="Verdana" charset="0"><b><c val="#000"><inlineAttr line-through="false">lspline</inlineAttr></c></b></f>, <b><inlineAttr line-through="false"><f family="Verdana" charset="0"><c val="#000">pspline</c></f></inlineAttr></b> and <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">cspline</c></inlineAttr></b></f>. The so called "natural boundary conditions" used in <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">cspline</c></inlineAttr></b></f> are probably the statistically worst option for the general problem. These end point issues mean that extrapolation beyond the data fitting region should not be done, and, in the case of <link href="./statspline.xmcd" title="Links to section" popup="false"><b><u>B-Splines</u></b></link>, this is generally enforced by making the function zero outside the data range. </p></text></region><region region-id="1354" left="0" top="660" width="6000" height="6" align-x="0" align-y="660" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1355" left="36" top="674.25" width="321" height="108" align-x="36" align-y="684" 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">On the other hand, a rational function may be a good approximation to the true function which generated the data. If care is taken, it can be extrapolated beyond the endpoints. As a general rule of thumb, a data set with several turning points is well fit by polynomials (gaussians and lorentzians are exceptions) while data which is very smooth (for example, monotonic in derivative) is better fit by rational functions. This smoothness is what makes rational functions better candidates for data extrapolation than polynomial functions.</p></text></region><region region-id="1356" left="36" top="787.5" width="60.75" height="12.75" align-x="36" align-y="798" 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">Cubic splines</p></text><indexes><phrase>cubic splines</phrase><phrase>splines, cubic</phrase></indexes></region><region region-id="1003" left="36" top="812.25" width="297.75" height="96" 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="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"><c val="#000">The first step in cubic spline interpolation is to compute a vector of second derivatives for the spline.<sp count="2" />For the built-in mathcad cubic spline functions, you have a choice of "linear" endpoints </c><c val="#000">(</c><b><c val="#000">lspline</c></b><c val="#000">), "parabolic" endpoints (</c><b><c val="#000">pspline</c></b><c val="#000">) or "cubic" endpoints (</c><b><c val="#000">cspline</c></b><c val="#000">).<sp count="2" />In this simulation, the choice of splines is done by setting the global variable </c><f family="Times New Roman" charset="0"><i><inlineAttr font-weight="normal" line-through="false"><c val="#000">type</c></inlineAttr></i></f><c val="#000"> below using to the values </c><f family="Times New Roman" charset="0"><c val="#000"><inlineAttr font-weight="normal" line-through="false">0 = </inlineAttr></c></f><c val="#000"><b>lspline</b></c><c val="#000">, </c><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">1 = </c></inlineAttr></f><b><c val="#000">pspline</c></b><c val="#000">,</c><c val="#000"><b><sp /></b></c><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">2 = </c></inlineAttr></f><b><c val="#000">cspline</c></b><c val="#000">.</c></p></text></region><region region-id="1016" left="36" top="932.25" width="114" height="12" align-x="36" align-y="942" 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"><c val="#000">We define the function</c></p></text></region><region region-id="909" left="66" top="957" width="302.25" height="12.75" align-x="126.75" 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:function><ml:id>spline</ml:id><ml:boundVars><ml:id>n</ml:id><ml:id>x</ml:id><ml:id>y</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>if</ml:id><ml:sequence><ml:apply><ml:equal /><ml:id>n</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:id>pspline</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id></ml:sequence></ml:apply><ml:apply><ml:id>if</ml:id><ml:sequence><ml:apply><ml:equal /><ml:id>n</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:id>cspline</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id></ml:sequence></ml:apply><ml:apply><ml:id>lspline</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id></ml:sequence></ml:apply></ml:sequence></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="4" /></region><region region-id="1017" left="36" top="980.25" width="124.5" height="12" align-x="36" align-y="990" 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">for comparison purposes.</p></text></region><region region-id="1018" left="36" top="998.25" width="107.25" height="12" align-x="36" align-y="1008" 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">First specify the data</p></text></region><region region-id="912" left="66" top="1017" width="25.5" height="12.75" align-x="79.5" align-y="1026" 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>6</ml:real></ml:define></math><rendering item-idref="5" /></region><region region-id="1007" left="144" top="1017" width="35.25" height="12.75" align-x="152.25" align-y="1026" 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>N</ml:id></ml:range></ml:define></math><rendering item-idref="6" /></region><region region-id="1010" left="228" top="1017" width="21.75" height="18" align-x="240" align-y="1026" 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:id>i</ml:id></ml:define></math><rendering item-idref="7" /></region><region region-id="1014" left="66" top="1041" width="44.25" height="12.75" align-x="75.75" align-y="1050" 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:id>sort</ml:id><ml:id>x</ml:id></ml:apply></ml:define></math><rendering item-idref="8" /></region><region region-id="1015" left="144" top="1041" width="62.25" height="18" align-x="157.5" align-y="1050" 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>i</ml:id></ml:apply><ml:apply><ml:minus /><ml:apply><ml:id>rnd</ml:id><ml:real>2</ml:real></ml:apply><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="9" /></region><region region-id="1019" left="36" top="1064.25" width="288" height="60" 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">The vector of second derivatives returned by the spline routines is found by solving a linear system of equations. The arrays describing this system are set up below.<sp count="2" />Note that changing the type of endpoint derivatives only affects a few elements of the coefficient matrix. </p></text></region><region region-id="919" left="66" top="1131" width="51.75" height="12.75" align-x="75" align-y="1140" 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>r</ml:id><ml:range><ml:real>1</ml:real><ml:apply><ml:minus /><ml:id>N</ml:id><ml:real>1</ml:real></ml:apply></ml:range></ml:define></math><rendering item-idref="10" /></region><region region-id="1020" left="36" top="1154.25" width="107.25" height="12" align-x="36" align-y="1164" 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 constant vector:</p></text></region><region region-id="921" left="66" top="1179" width="114.75" height="38.25" align-x="80.25" align-y="1200" 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>b</ml:id><ml:id>r</ml:id></ml:apply><ml:apply><ml:minus /><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>y</ml:id><ml:apply><ml:plus /><ml:id>r</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>y</ml:id><ml:id>r</ml:id></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:plus /><ml:id>r</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>r</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>y</ml:id><ml:id>r</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>y</ml:id><ml:apply><ml:minus /><ml:id>r</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>r</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:minus /><ml:id>r</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="11" /></region><region region-id="1023" left="222" top="1191" width="27.75" height="18" align-x="237.75" align-y="1200" 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>b</ml:id><ml:real>0</ml:real></ml:apply><ml:real>0</ml:real></ml:define></math><rendering item-idref="12" /></region><region region-id="1024" left="276" top="1191" width="30.75" height="18" align-x="294.75" align-y="1200" 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>b</ml:id><ml:id>N</ml:id></ml:apply><ml:real>0</ml:real></ml:define></math><rendering item-idref="13" /></region><region region-id="1021" left="36" top="1226.25" width="113.25" height="12" align-x="36" align-y="1236" 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 coefficient matrix:</p></text></region><region region-id="925" left="66" top="1251" width="40.5" height="18" align-x="94.5" 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:define><ml:apply><ml:indexer /><ml:id>M</ml:id><ml:sequence><ml:real>0</ml:real><ml:real>0</ml:real></ml:sequence></ml:apply><ml:real>1</ml:real></ml:define></math><rendering item-idref="14" /></region><region region-id="1026" left="144" top="1251" width="46.5" height="18" align-x="178.5" 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:define><ml:apply><ml:indexer /><ml:id>M</ml:id><ml:sequence><ml:id>N</ml:id><ml:id>N</ml:id></ml:sequence></ml:apply><ml:real>1</ml:real></ml:define></math><rendering item-idref="15" /></region><region region-id="927" left="66" top="1281" width="80.25" height="33" align-x="101.25" align-y="1302" 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>r</ml:id><ml:apply><ml:minus /><ml:id>r</ml:id><ml:real>1</ml:real></ml:apply></ml:sequence></ml:apply><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>r</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:minus /><ml:id>r</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:apply><ml:real>6</ml:real></ml:apply></ml:define></math><rendering item-idref="16" /></region><region region-id="928" left="174" top="1281" width="81" height="33" align-x="199.5" align-y="1302" 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>r</ml:id><ml:id>r</ml:id></ml:sequence></ml:apply><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:plus /><ml:id>r</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:minus /><ml:id>r</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:apply><ml:real>3</ml:real></ml:apply></ml:define></math><rendering item-idref="17" /></region><region region-id="1025" left="276" top="1281" width="81.75" height="33" align-x="312" align-y="1302" 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>r</ml:id><ml:apply><ml:plus /><ml:id>r</ml:id><ml:real>1</ml:real></ml:apply></ml:sequence></ml:apply><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:plus /><ml:id>r</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>r</ml:id></ml:apply></ml:apply><ml:real>6</ml:real></ml:apply></ml:define></math><rendering item-idref="18" /></region><region region-id="1357" left="0" top="1320" width="6000" height="6" align-x="0" align-y="1320" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1022" left="36" top="1328.25" width="155.25" height="12" 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 type specific assignments:</p></text></region><region region-id="931" left="66" top="1353" width="202.5" height="38.25" align-x="94.5" align-y="1374" 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:real>0</ml:real><ml:real>1</ml:real></ml:sequence></ml:apply><ml:apply><ml:id>if</ml:id><ml:sequence><ml:apply><ml:equal /><ml:id>type</ml:id><ml:real>1</ml:real></ml:apply><ml:real>-1</ml:real><ml:apply><ml:id>if</ml:id><ml:sequence><ml:apply><ml:equal /><ml:id>type</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:apply><ml:real>0</ml:real></ml:sequence></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="19" /></region><region region-id="932" left="66" top="1407" width="132" height="38.25" align-x="94.5" align-y="1428" 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:real>0</ml:real><ml:real>2</ml:real></ml:sequence></ml:apply><ml:apply><ml:id>if</ml:id><ml:sequence><ml:apply><ml:equal /><ml:id>type</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>0</ml:real></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:apply><ml:real>0</ml:real></ml:sequence></ml:apply></ml:define></math><rendering item-idref="20" /></region><region region-id="933" left="66" top="1461" width="243.75" height="38.25" align-x="110.25" align-y="1482" 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>N</ml:id><ml:apply><ml:minus /><ml:id>N</ml:id><ml:real>1</ml:real></ml:apply></ml:sequence></ml:apply><ml:apply><ml:id>if</ml:id><ml:sequence><ml:apply><ml:equal /><ml:id>type</ml:id><ml:real>1</ml:real></ml:apply><ml:real>-1</ml:real><ml:apply><ml:id>if</ml:id><ml:sequence><ml:apply><ml:equal /><ml:id>type</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>N</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:minus /><ml:id>N</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:minus /><ml:id>N</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:minus /><ml:id>N</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:apply></ml:apply><ml:real>0</ml:real></ml:sequence></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="21" /></region><region region-id="934" left="66" top="1515" width="173.25" height="38.25" align-x="110.25" align-y="1536" 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>N</ml:id><ml:apply><ml:minus /><ml:id>N</ml:id><ml:real>2</ml:real></ml:apply></ml:sequence></ml:apply><ml:apply><ml:id>if</ml:id><ml:sequence><ml:apply><ml:equal /><ml:id>type</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:div /><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:minus /><ml:id>N</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>N</ml:id></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:minus /><ml:id>N</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:minus /><ml:id>N</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:apply></ml:apply><ml:real>0</ml:real></ml:sequence></ml:apply></ml:define></math><rendering item-idref="22" /></region><region region-id="1029" left="36" top="1568.25" width="93.75" height="12" align-x="36" align-y="1578" 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">Set the type here:</p></text></region><region region-id="937" left="66" top="1593" width="34.5" height="12.75" align-x="89.25" align-y="1602" 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>type</ml:id><ml:real>1</ml:real></ml:globalDefine></math><rendering item-idref="23" /></region><region region-id="938" left="126" top="1590.75" width="54" height="14.25" align-x="126" align-y="1602" 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"><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false"><c val="#000">0 = </c></inlineAttr></f><b><c val="#000">lspline</c></b></p></text></region><region region-id="939" left="198" top="1592.25" width="53.25" height="12" align-x="198" align-y="1602" 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"><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">1 = </c></inlineAttr></f><b><c val="#000">pspline</c></b></p></text></region><region region-id="940" left="270" top="1592.25" width="52.5" height="12" align-x="270" align-y="1602" 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"><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">2 = </c></inlineAttr></f><b><c val="#000">cspline</c></b></p></text></region><region region-id="1030" left="36" top="1616.25" width="288" height="24" align-x="36" align-y="1626" 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"><c val="#000">The value returned by the corresponding spline function is given by solving the system </c><f family="Times New Roman" charset="0"><i><inlineAttr font-weight="normal" line-through="false"><c val="#000">M</c></inlineAttr></i></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000"><sp /></c></inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">* </c></inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><i><c val="#000">vs</c></i></inlineAttr></f><inlineAttr font-weight="normal" line-through="false"><c val="#000"><f family="Times New Roman" charset="0"> = </f></c></inlineAttr><f family="Times New Roman" charset="0"><i><inlineAttr font-weight="normal" line-through="false"><c val="#000">b</c></inlineAttr></i></f><c val="#000"> for </c><f family="Times New Roman" charset="0"><i><inlineAttr font-weight="normal" line-through="false"><c val="#000">vs</c></inlineAttr></i></f><c val="#000">.</c></p></text></region><region region-id="942" left="66" top="1695" width="52.5" height="18.75" align-x="81.75" align-y="1710" 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>vs</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:pow /><ml:id>M</ml:id><ml:real>-1</ml:real></ml:apply><ml:id>b</ml:id></ml:apply></ml:define></math><rendering item-idref="24" /></region><region region-id="1028" left="144" top="1651.5" width="81" height="111.75" align-x="159" align-y="1710" 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>vs</ml:id></ml:eval></math><rendering item-idref="25" /></region><region region-id="944" left="66" top="1795.5" width="149.25" height="166.5" align-x="138.75" align-y="1878" 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>spline</ml:id><ml:sequence><ml:id>type</ml:id><ml:id>x</ml:id><ml:id>y</ml:id></ml:sequence></ml:apply></ml:eval><resultFormat><general radix="dec" complex-threshold="10" precision="3" show-trailing-zeros="false" exponential-threshold="3" /><matrix display-style="auto" expand-nested-arrays="false" /><unit format-units="false" simplify-units="false" /><table item-idref="26" /></resultFormat></math><rendering item-idref="27" /></region><region region-id="945" left="36" top="1986" width="6000" height="6" align-x="36" align-y="1986" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1033" left="36" top="1993.5" width="89.25" height="12.75" align-x="36" align-y="2004" 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">The interp function</p></text><indexes><phrase>interp</phrase><phrase>functions, interp</phrase></indexes></region><region region-id="1031" left="36" top="2018.25" width="287.25" height="24" align-x="36" align-y="2028" 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 point at which we want to evaluate the spline is denoted by t and is defined as a global variable below. </p></text></region><region region-id="1032" left="36" top="2054.25" width="282" height="36" align-x="36" align-y="2064" 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">Determine which two x values t lies between, using the first two if t is less than all of them.<sp count="2" />Use the last two x values if t is greater than all of them. </p></text></region><region region-id="948" left="66" top="2109" width="53.25" height="12.75" align-x="76.5" align-y="2118" 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:apply><ml:minus /><ml:id>N</ml:id><ml:real>1</ml:real></ml:apply></ml:range></ml:define></math><rendering item-idref="28" /></region><region region-id="1035" left="144" top="2109" width="27.75" height="18" align-x="159.75" align-y="2118" 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:real>1</ml:real></ml:apply><ml:real>1</ml:real></ml:define></math><rendering item-idref="29" /></region><region region-id="1036" left="36" top="2132.25" width="274.5" height="48" align-x="36" align-y="2142" 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 statement determines the indices of the two x values that bound t.<sp count="2" />The last two values are used if t lies above the range, and the first two values are used if t lies below the range.</p></text></region><region region-id="993" left="66" top="2190.75" width="162.75" height="166.5" align-x="96" 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:apply><ml:matcol /><ml:id>u</ml:id><ml:apply><ml:mult style="default" /><ml:real>0</ml:real><ml:id>k</ml:id></ml:apply></ml:apply><ml:program><ml:ifThen><ml:apply><ml:lessOrEqual /><ml:id>t</ml:id><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:real>0</ml:real></ml:apply></ml:apply><ml:localDefine><ml:id>index</ml:id><ml:matrix rows="2" cols="1"><ml:real>0</ml:real><ml:real>1</ml:real></ml:matrix></ml:localDefine></ml:ifThen><ml:ifThen><ml:apply><ml:greaterOrEqual /><ml:id>t</ml:id><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>N</ml:id></ml:apply></ml:apply><ml:localDefine><ml:id>index</ml:id><ml:matrix rows="2" cols="1"><ml:apply><ml:minus /><ml:id>N</ml:id><ml:real>1</ml:real></ml:apply><ml:id>N</ml:id></ml:matrix></ml:localDefine></ml:ifThen><ml:otherwise><ml:program><ml:localDefine><ml:id>i</ml:id><ml:real>0</ml:real></ml:localDefine><ml:while><ml:apply><ml:lessThan /><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>i</ml:id></ml:apply><ml:id>t</ml:id></ml:apply><ml:localDefine><ml:id>i</ml:id><ml:apply><ml:plus /><ml:id>i</ml:id><ml:real>1</ml:real></ml:apply></ml:localDefine></ml:while><ml:localDefine><ml:id>index</ml:id><ml:matrix rows="2" cols="1"><ml:apply><ml:minus /><ml:id>i</ml:id><ml:real>1</ml:real></ml:apply><ml:id>i</ml:id></ml:matrix></ml:localDefine></ml:program></ml:otherwise></ml:program></ml:define></math><rendering item-idref="30" /></region><region region-id="1037" left="36" top="2366.25" width="35.25" height="12" align-x="36" align-y="2376" 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">Define:</p></text></region><region region-id="953" left="66" top="2391" width="33.75" height="19.5" align-x="80.25" align-y="2400" 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:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="31" /></region><region region-id="1038" left="144" top="2391" width="33.75" height="19.5" align-x="158.25" align-y="2400" 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>x2</ml:id><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="32" /></region><region region-id="1039" left="228" top="2391" width="36.75" height="19.5" align-x="243.75" align-y="2400" 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:apply><ml:indexer /><ml:id>y</ml:id><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="33" /></region><region region-id="1040" left="312" top="2391" width="36.75" height="19.5" align-x="327.75" align-y="2400" 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>y2</ml:id><ml:apply><ml:indexer /><ml:id>y</ml:id><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="34" /></region><region region-id="957" left="66" top="2421" width="41.25" height="19.5" align-x="81.75" align-y="2430" 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>v1</ml:id><ml:apply><ml:indexer /><ml:id>vs</ml:id><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="35" /></region><region region-id="1041" left="144" top="2421" width="41.25" height="19.5" align-x="159.75" align-y="2430" 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>v2</ml:id><ml:apply><ml:indexer /><ml:id>vs</ml:id><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="36" /></region><region region-id="959" left="66" top="2451" width="50.25" height="12.75" align-x="81" align-y="2460" 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>hx</ml:id><ml:apply><ml:minus /><ml:id>x2</ml:id><ml:id>x1</ml:id></ml:apply></ml:define></math><rendering item-idref="37" /></region><region region-id="1042" left="144" top="2451" width="54.75" height="12.75" align-x="160.5" align-y="2460" 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>hy</ml:id><ml:apply><ml:minus /><ml:id>y2</ml:id><ml:id>y1</ml:id></ml:apply></ml:define></math><rendering item-idref="38" /></region><region region-id="1043" left="228" top="2451" width="54.75" height="12.75" align-x="244.5" align-y="2460" 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>hv</ml:id><ml:apply><ml:minus /><ml:id>v2</ml:id><ml:id>v1</ml:id></ml:apply></ml:define></math><rendering item-idref="39" /></region><region region-id="1044" left="36" top="2474.25" width="280.5" height="24" align-x="36" align-y="2484" 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 portion of the cubic spline defined on the interval from x1 to x2 is given by</p></text></region><region region-id="963" left="66" top="2510.25" width="309" height="33.75" align-x="88.5" align-y="2532" 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>p</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:id>hv</ml:id><ml:id>hx</ml:id></ml:apply><ml:apply><ml:div /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:id>x</ml:id><ml:id>x1</ml:id></ml:apply></ml:parens><ml:real>3</ml:real></ml:apply><ml:real>6</ml:real></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:id>v1</ml:id><ml:apply><ml:div /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:id>x</ml:id><ml:id>x1</ml:id></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:div /><ml:id>hy</ml:id><ml:id>hx</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:id>hx</ml:id><ml:real>6</ml:real></ml:apply><ml:parens><ml:apply><ml:plus /><ml:id>v2</ml:id><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:id>v1</ml:id></ml:apply></ml:apply></ml:parens></ml:apply></ml:apply></ml:parens><ml:parens><ml:apply><ml:minus /><ml:id>x</ml:id><ml:id>x1</ml:id></ml:apply></ml:parens></ml:apply></ml:apply><ml:id>y1</ml:id></ml:apply></ml:define></math><rendering item-idref="40" /></region><region region-id="964" left="36" top="2562" width="6000" height="6" align-x="36" align-y="2562" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1045" left="36" top="2582.25" width="111.75" height="12" align-x="36" align-y="2592" 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"><c val="#000">Specify the value of </c><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" font-style="normal" line-through="false"><c val="#000">t</c></inlineAttr></f><c val="#000">:</c></p></text></region><region region-id="966" left="66" top="2607" width="30.75" height="12.75" align-x="74.25" align-y="2616" 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>t</ml:id><ml:real>4.03</ml:real></ml:globalDefine></math><rendering item-idref="41" /></region><region region-id="1046" left="36" top="2630.25" width="260.25" height="24" align-x="36" align-y="2640" 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"><c val="#000">The value returned by </c><b><c val="#000">interp</c></b><c val="#000"> is shown below along with that given by the construction of this section. </c></p></text></region><region region-id="968" left="66" top="2673" width="67.5" height="12.75" align-x="87" align-y="2682" 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>p</ml:id><ml:id>t</ml:id></ml:apply></ml:eval></math><rendering item-idref="42" /></region><region region-id="969" left="66" top="2703" width="278.25" height="12.75" align-x="77.25" align-y="2712" 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>v</ml:id><ml:apply><ml:id>if</ml:id><ml:sequence><ml:apply><ml:equal /><ml:id>type</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:id>lspline</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id></ml:sequence></ml:apply><ml:apply><ml:id>if</ml:id><ml:sequence><ml:apply><ml:equal /><ml:id>type</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:id>pspline</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id></ml:sequence></ml:apply><ml:apply><ml:id>cspline</ml:id><ml:sequence><ml:id>x</ml:id><ml:id>y</ml:id></ml:sequence></ml:apply></ml:sequence></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="43" /></region><region region-id="970" left="66" top="2733" width="113.25" height="12.75" align-x="132.75" 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:eval><ml:apply><ml:id>interp</ml:id><ml:sequence><ml:id>v</ml:id><ml:id>x</ml:id><ml:id>y</ml:id><ml:id>t</ml:id></ml:sequence></ml:apply></ml:eval></math><rendering item-idref="44" /></region><region region-id="1358" left="36" top="2762.25" width="51" height="12" align-x="36" align-y="2772" 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"><b><c val="#800000">B-splines</c></b></p></text><indexes><phrase>B-splines</phrase><phrase>splines, B-splines</phrase></indexes></region><region region-id="1145" left="36" top="2798.25" width="328.5" height="60" align-x="36" align-y="2808" 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"><b><u><link href="./statspline.xmcd" title="Links to section" popup="false">B-splines</link></u></b> are based on polynomial splines, but they use a set of points on the x-axis, called knots, to determine the piecewise<sp count="2" />polynomials.<sp count="2" />The polynomials, or basis functions, are used to fit the data by finding a sum of these functions which passes through the data.</p></text></region><region region-id="1158" left="228" top="2874" width="48.75" height="78.75" align-x="239.25" align-y="2916" 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>u</ml:id><ml:matrix rows="5" cols="1"><ml:real>0</ml:real><ml:real>0.75</ml:real><ml:real>1.25</ml:real><ml:real>1.75</ml:real><ml:real>2.75</ml:real></ml:matrix></ml:define></math><rendering item-idref="45" /></region><region region-id="1152" left="66" top="2877.75" width="46.5" height="95.25" align-x="79.5" align-y="2928" 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="6" cols="1"><ml:real>0</ml:real><ml:real>0.5</ml:real><ml:real>1</ml:real><ml:real>1.5</ml:real><ml:real>2</ml:real><ml:real>2.5</ml:real></ml:matrix></ml:define></math><rendering item-idref="46" /></region><region region-id="1153" left="144" top="2877.75" width="45" height="95.25" align-x="156" align-y="2928" 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="6" cols="1"><ml:real>1</ml:real><ml:real>2</ml:real><ml:real>3.1</ml:real><ml:real>2.9</ml:real><ml:real>1.7</ml:real><ml:real>2.8</ml:real></ml:matrix></ml:define></math><rendering item-idref="47" /></region><region region-id="1159" left="228" top="2967" width="23.25" height="12.75" align-x="239.25" align-y="2976" 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>d</ml:id><ml:real>2</ml:real></ml:define></math><rendering item-idref="48" /></region><region region-id="1165" left="36" top="2990.25" width="304.5" height="39.75" align-x="36" align-y="3000" 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 any set of four consecutive knots k we define a piecewise quadratic function (because <region region-id="702" left="234" top="3003" width="427.5" height="12.75" align-x="244.5" align-y="3012" 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>d</ml:id></ml:eval></math><rendering item-idref="49" /></region><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">in our example</inlineAttr></f>) as follows.</p></text></region><region region-id="1169" left="36" top="3038.25" width="30.75" height="12" align-x="36" align-y="3048" 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">Define</p></text></region><region region-id="1170" left="66" top="3057" width="123.75" height="44.25" align-x="102.75" 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:define><ml:function><ml:id>B1</ml:id><ml:boundVars><ml:id>k</ml:id><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:div /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>3</ml:real></ml:apply><ml:id>x</ml:id></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>3</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:parens><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>3</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:parens></ml:apply></ml:apply></ml:define></math><rendering item-idref="50" /></region><region region-id="1171" left="66" top="3111" width="163.5" height="44.25" align-x="102.75" 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:function><ml:id>B2</ml:id><ml:boundVars><ml:id>k</ml:id><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:div /><ml:apply><ml:mult style="default" /><ml:apply><ml:neg /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>3</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:parens></ml:apply><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>2</ml:real></ml:apply><ml:id>x</ml:id></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>3</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:parens><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:parens></ml:apply><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:parens></ml:apply></ml:apply></ml:define></math><rendering item-idref="51" /></region><region region-id="1172" left="66" top="3171" width="163.5" height="44.25" align-x="102.75" 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:function><ml:id>B3</ml:id><ml:boundVars><ml:id>k</ml:id><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:div /><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>3</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:parens><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>1</ml:real></ml:apply><ml:id>x</ml:id></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>3</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:parens><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>2</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:parens></ml:apply><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:parens></ml:apply></ml:apply></ml:define></math><rendering item-idref="52" /></region><region region-id="1173" left="0" top="3228" width="6000" height="6" align-x="0" align-y="3228" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1174" left="36" top="3248.25" width="288.75" height="12" align-x="36" align-y="3258" 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 we define our B-spline basis functions of degree d as</p></text></region><region region-id="1175" left="66" top="3279" width="225.75" height="78" align-x="98.25" align-y="3288" 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>B</ml:id><ml:boundVars><ml:id>k</ml:id><ml:id>x</ml:id></ml:boundVars></ml:function><ml:program><ml:ifThen><ml:apply><ml:lessThan /><ml:apply><ml:lessThan /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>0</ml:real></ml:apply><ml:id>x</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:id>B1</ml:id><ml:sequence><ml:id>k</ml:id><ml:id>x</ml:id></ml:sequence></ml:apply><ml:apply><ml:id>B2</ml:id><ml:sequence><ml:id>k</ml:id><ml:id>x</ml:id></ml:sequence></ml:apply></ml:apply><ml:apply><ml:id>B3</ml:id><ml:sequence><ml:id>k</ml:id><ml:id>x</ml:id></ml:sequence></ml:apply></ml:apply></ml:ifThen><ml:ifThen><ml:apply><ml:lessThan /><ml:apply><ml:lessOrEqual /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>1</ml:real></ml:apply><ml:id>x</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:plus /><ml:apply><ml:id>B1</ml:id><ml:sequence><ml:id>k</ml:id><ml:id>x</ml:id></ml:sequence></ml:apply><ml:apply><ml:id>B2</ml:id><ml:sequence><ml:id>k</ml:id><ml:id>x</ml:id></ml:sequence></ml:apply></ml:apply></ml:ifThen><ml:ifThen><ml:apply><ml:lessThan /><ml:apply><ml:lessOrEqual /><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>2</ml:real></ml:apply><ml:id>x</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>k</ml:id><ml:real>3</ml:real></ml:apply></ml:apply><ml:apply><ml:id>B1</ml:id><ml:sequence><ml:id>k</ml:id><ml:id>x</ml:id></ml:sequence></ml:apply></ml:ifThen><ml:otherwise><ml:real>0</ml:real></ml:otherwise></ml:program></ml:define></math><rendering item-idref="53" /></region><region region-id="1178" left="36" top="3374.25" width="316.5" height="36" align-x="36" align-y="3384" 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 above are general formulas for the B-spline basis of degree 2 for any set of knots.<sp count="2" />Similar definitions exist for degrees 1 and 3.<sp count="2" />Now we continue our construction for our data.<sp count="2" /></p></text></region><region region-id="1179" left="36" top="3434.25" width="298.5" height="24" align-x="36" align-y="3444" 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 next step is to extend the given set of knots by adding a few knots to either end of the original set</p></text></region><region region-id="1186" left="66" top="3471" width="87.75" height="18" align-x="79.5" align-y="3480" 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>t</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:parens></ml:apply></ml:apply></ml:define></math><rendering item-idref="54" /></region><region region-id="1210" left="222" top="3471" width="80.25" height="18" align-x="235.5" align-y="3480" 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>t</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:real>0</ml:real></ml:apply><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:parens></ml:apply></ml:define></math><rendering item-idref="55" /></region><region region-id="1188" left="66" top="3501" width="57.75" height="12.75" align-x="75" align-y="3510" 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>t</ml:id><ml:apply><ml:id>stack</ml:id><ml:sequence><ml:id>t</ml:id><ml:id>u</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="56" /></region><region region-id="1212" left="144" top="3501" width="45.75" height="12.75" align-x="155.25" align-y="3510" 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:apply><ml:id>last</ml:id><ml:id>u</ml:id></ml:apply></ml:define></math><rendering item-idref="57" /></region><region region-id="1190" left="66" top="3525" width="101.25" height="18" align-x="90" align-y="3534" 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>t</ml:id><ml:apply><ml:plus /><ml:id>n</ml:id><ml:real>3</ml:real></ml:apply></ml:apply><ml:apply><ml:plus /><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:id>n</ml:id></ml:apply><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:id>n</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:apply><ml:minus /><ml:id>n</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:apply></ml:parens></ml:apply></ml:define></math><rendering item-idref="58" /></region><region region-id="1211" left="222" top="3525" width="108.75" height="18" align-x="246" align-y="3534" 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>t</ml:id><ml:apply><ml:plus /><ml:id>n</ml:id><ml:real>4</ml:real></ml:apply></ml:apply><ml:apply><ml:plus /><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:id>n</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:id>n</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>u</ml:id><ml:apply><ml:minus /><ml:id>n</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:apply></ml:parens></ml:apply></ml:apply></ml:define></math><rendering item-idref="59" /></region><region region-id="1215" left="66" top="3555" width="56.25" height="144.75" align-x="74.25" align-y="3630" 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>t</ml:id></ml:eval></math><rendering item-idref="60" /></region><region region-id="1217" left="144" top="3588" width="53.25" height="78.75" align-x="154.5" align-y="3630" 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>u</ml:id></ml:eval></math><rendering item-idref="61" /></region><region region-id="1218" left="36" top="3710.25" width="292.5" height="48" align-x="36" align-y="3720" 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">With these knots we construct our B-spline basis functions by taking four consecutive knots (there are 6 such the same number as our data points)) and using our function B above.<sp count="2" />Define</p></text></region><region region-id="1219" left="66" top="3771.75" width="99.75" height="83.25" align-x="104.25" align-y="3816" 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>Bp</ml:id><ml:boundVars><ml:id>n</ml:id><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:id>B</ml:id><ml:sequence><ml:matrix rows="4" cols="1"><ml:apply><ml:indexer /><ml:id>t</ml:id><ml:id>n</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>t</ml:id><ml:apply><ml:plus /><ml:id>n</ml:id><ml:real>1</ml:real></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>t</ml:id><ml:apply><ml:plus /><ml:id>n</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:indexer /><ml:id>t</ml:id><ml:apply><ml:plus /><ml:id>n</ml:id><ml:real>3</ml:real></ml:apply></ml:apply></ml:matrix><ml:id>x</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="62" /></region><region region-id="1220" left="36" top="3866.25" width="105" height="12" align-x="36" align-y="3876" 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">our basis functions.<sp count="2" /></p></text></region><region region-id="1221" left="0" top="3888" width="6000" height="6" align-x="0" align-y="3888" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1222" left="36" top="3896.25" width="220.5" height="12" align-x="36" align-y="3906" 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">For this particular set of knots they look like</p></text></region><region region-id="1224" left="66" top="3921" width="83.25" height="12.75" align-x="76.5" align-y="3930" 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>s</ml:id><ml:range><ml:sequence><ml:real>-1.5</ml:real><ml:real>-1.49</ml:real></ml:sequence><ml:real>4.75</ml:real></ml:range></ml:define></math><rendering item-idref="63" /></region><region region-id="1088" left="12" top="3942" width="294.75" height="197.25" align-x="12" align-y="3942" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="64" /><rendering item-idref="65" /></region><region region-id="1227" left="36" top="4148.25" width="271.5" height="24" align-x="36" align-y="4158" 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 may wish to change the knots u at the outset and see how the basis functions change.</p></text></region><region region-id="1228" left="36" top="4184.25" width="291.75" height="15" align-x="36" align-y="4194" 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 perform our interpolation, we find numbers A<sub>j</sub> such that</p></text></region><region region-id="1229" left="66" top="4213.5" width="99.75" height="43.5" align-x="85.5" align-y="4236" 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>f</ml:id><ml:id>x</ml:id></ml:apply><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id>j</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>A</ml:id><ml:id>j</ml:id></ml:apply><ml:apply><ml:id>Bp</ml:id><ml:sequence><ml:id>j</ml:id><ml:id>x</ml:id></ml:sequence></ml:apply></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:real>5</ml:real></ml:bounds></ml:apply></ml:apply></math><rendering item-idref="66" /></region><region region-id="1230" left="36" top="4268.25" width="201" height="12" align-x="36" align-y="4278" 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">interpolates the data.<sp count="2" />That is, we want</p></text></region><region region-id="1231" left="66" top="4297.5" width="99.75" height="43.5" align-x="79.5" align-y="4320" 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:indexer /><ml:id>Y</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id>j</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>A</ml:id><ml:id>j</ml:id></ml:apply><ml:apply><ml:id>Bp</ml:id><ml:sequence><ml:id>j</ml:id><ml:apply><ml:indexer /><ml:id>X</ml:id><ml:id>i</ml:id></ml:apply></ml:sequence></ml:apply></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:real>5</ml:real></ml:bounds></ml:apply></ml:apply></math><rendering item-idref="67" /></region><region region-id="1232" left="36" top="4352.25" width="199.5" height="12" align-x="36" align-y="4362" 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 is just a linear system solvable as</p></text></region><region region-id="1235" left="66" top="4383" width="32.25" height="12.75" align-x="74.25" align-y="4392" 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:real>5</ml:real></ml:range></ml:define></math><rendering item-idref="68" /></region><region region-id="1251" left="144" top="4383" width="33" height="12.75" align-x="153" align-y="4392" 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:real>5</ml:real></ml:range></ml:define></math><rendering item-idref="69" /></region><region region-id="1359" left="30" top="4455" width="72.75" height="18" align-x="59.25" align-y="4464" 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>M1</ml:id><ml:sequence><ml:id>i</ml:id><ml:id>j</ml:id></ml:sequence></ml:apply><ml:apply><ml:id>Bp</ml:id><ml:sequence><ml:id>j</ml:id><ml:apply><ml:indexer /><ml:id>X</ml:id><ml:id>i</ml:id></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="70" /></region><region region-id="1360" left="108" top="4413.75" width="277.5" height="95.25" align-x="126.75" align-y="4464" 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>M1</ml:id></ml:eval></math><rendering item-idref="71" /></region><region region-id="1249" left="0" top="4536" width="6000" height="6" align-x="0" align-y="4536" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1265" left="66" top="4587" width="76.5" height="12.75" align-x="79.5" align-y="4596" 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:apply><ml:id>lsolve</ml:id><ml:sequence><ml:id>M1</ml:id><ml:id>Y</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="72" /></region><region region-id="1266" left="222" top="4545.75" width="73.5" height="95.25" align-x="234.75" align-y="4596" 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>A</ml:id></ml:eval></math><rendering item-idref="73" /></region><region region-id="1270" left="36" top="4628.25" width="30.75" height="12" align-x="36" align-y="4638" 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">Define</p></text></region><region region-id="1268" left="66" top="4645.5" width="101.25" height="43.5" align-x="86.25" align-y="4668" 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:summation /><ml:lambda><ml:boundVars><ml:id>j</ml:id></ml:boundVars><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:indexer /><ml:id>A</ml:id><ml:id>j</ml:id></ml:apply><ml:apply><ml:id>Bp</ml:id><ml:sequence><ml:id>j</ml:id><ml:id>x</ml:id></ml:sequence></ml:apply></ml:apply></ml:parens></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:real>5</ml:real></ml:bounds></ml:apply></ml:define></math><rendering item-idref="74" /></region><region region-id="1271" left="36" top="4712.25" width="132" height="12" align-x="36" align-y="4722" 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 we have our function:</p></text></region><region region-id="1272" left="54" top="4740" width="311.25" height="143.25" align-x="54" align-y="4740" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="75" /><rendering item-idref="76" /></region><region region-id="1289" left="36" top="4910.25" width="302.25" height="12" align-x="36" align-y="4920" 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">So what does bspline return exactly?<sp count="2" />The output of bspline </p></text></region><region region-id="1108" left="96" top="4935" width="94.5" height="12.75" align-x="108.75" align-y="4944" 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>Q</ml:id><ml:apply><ml:id>bspline</ml:id><ml:sequence><ml:id>X</ml:id><ml:id>Y</ml:id><ml:id>u</ml:id><ml:id>d</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="77" /></region><region region-id="1290" left="36" top="4958.25" width="115.5" height="12" align-x="36" align-y="4968" 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">has three components:</p></text></region><region region-id="1110" left="96" top="4983" width="33.75" height="18" align-x="112.5" align-y="4992" 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:indexer /><ml:id>Q</ml:id><ml:real>0</ml:real></ml:apply></ml:eval></math><rendering item-idref="78" /></region><region region-id="1291" left="36" top="5006.25" width="268.5" height="36" align-x="36" align-y="5016" 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">is just a flag to interp that the output came from bspline as opposed to say pspline.<sp count="2" />The third component</p></text></region><region region-id="1112" left="96" top="5046.75" width="126.75" height="29.25" align-x="112.5" align-y="5064" 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:indexer /><ml:id>Q</ml:id><ml:real>2</ml:real></ml:apply></ml:eval></math><rendering item-idref="79" /></region><region region-id="1292" left="36" top="5084.25" width="113.25" height="12" align-x="36" align-y="5094" 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">is just the node values</p></text></region><region region-id="1114" left="108" top="5112" width="53.25" height="78.75" align-x="118.5" align-y="5154" 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>u</ml:id></ml:eval></math><rendering item-idref="80" /></region><region region-id="1115" left="0" top="5202" width="6000" height="6" align-x="0" align-y="5202" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1301" left="36" top="5210.25" width="283.5" height="12" align-x="36" align-y="5220" 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">wrapped into the columns of an array.<sp count="2" />The other output</p></text></region><region region-id="1304" left="66" top="5242.5" width="219" height="45.75" align-x="82.5" align-y="5268" 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:indexer /><ml:id>Q</ml:id><ml:real>1</ml:real></ml:apply></ml:eval></math><rendering item-idref="81" /></region><region region-id="1302" left="36" top="5312.25" width="297.75" height="36" align-x="36" align-y="5322" 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">contains the coefficients of the piecewise polynomials determined by the fit on each interval specified by the knots.<sp count="2" />For example</p></text></region><region region-id="1305" left="66" top="5356.5" width="96" height="45.75" align-x="101.25" align-y="5382" 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:matcol /><ml:parens><ml:apply><ml:indexer /><ml:id>Q</ml:id><ml:real>1</ml:real></ml:apply></ml:parens><ml:real>0</ml:real></ml:apply></ml:eval></math><rendering item-idref="82" /></region><region region-id="1309" left="36" top="5420.25" width="272.25" height="12" align-x="36" align-y="5430" 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">are the coefficients of the quadratic fit on the interval</p></text></region><region region-id="1121" left="72" top="5448.75" width="78" height="29.25" align-x="107.25" align-y="5466" 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:matcol /><ml:parens><ml:apply><ml:indexer /><ml:id>Q</ml:id><ml:real>2</ml:real></ml:apply></ml:parens><ml:real>0</ml:real></ml:apply></ml:eval></math><rendering item-idref="83" /></region><region region-id="1310" left="36" top="5498.25" width="93" height="12" align-x="36" align-y="5508" 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">so we could define</p></text></region><region region-id="1312" left="66" top="5527.5" width="90" height="51.75" align-x="93" align-y="5556" 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>p0</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:mult style="default" /><ml:apply><ml:matcol /><ml:parens><ml:apply><ml:indexer /><ml:id>Q</ml:id><ml:real>1</ml:real></ml:apply></ml:parens><ml:real>0</ml:real></ml:apply><ml:matrix rows="3" cols="1"><ml:real>1</ml:real><ml:id>x</ml:id><ml:apply><ml:pow /><ml:id>x</ml:id><ml:real>2</ml:real></ml:apply></ml:matrix></ml:apply></ml:define></math><rendering item-idref="84" /></region><region region-id="1124" left="198" top="5527.5" width="90" height="51.75" align-x="225" align-y="5556" 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>p1</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:mult style="default" /><ml:apply><ml:matcol /><ml:parens><ml:apply><ml:indexer /><ml:id>Q</ml:id><ml:real>1</ml:real></ml:apply></ml:parens><ml:real>1</ml:real></ml:apply><ml:matrix rows="3" cols="1"><ml:real>1</ml:real><ml:id>x</ml:id><ml:apply><ml:pow /><ml:id>x</ml:id><ml:real>2</ml:real></ml:apply></ml:matrix></ml:apply></ml:define></math><rendering item-idref="85" /></region><region region-id="1311" left="36" top="5606.25" width="300" height="12" align-x="36" align-y="5616" 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">These functions interpolate on the appropriate subintervals:</p></text></region><region region-id="1319" left="66" top="5637" width="34.5" height="18" align-x="83.25" align-y="5646" 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:indexer /><ml:id>X</ml:id><ml:real>0</ml:real></ml:apply></ml:eval></math><rendering item-idref="86" /></region><region region-id="1331" left="144" top="5637" width="33" height="18" align-x="159.75" align-y="5646" 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:indexer /><ml:id>Y</ml:id><ml:real>0</ml:real></ml:apply></ml:eval></math><rendering item-idref="87" /></region><region region-id="1346" left="222" top="5637" width="51.75" height="18" align-x="256.5" align-y="5646" 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>p0</ml:id><ml:apply><ml:indexer /><ml:id>X</ml:id><ml:real>0</ml:real></ml:apply></ml:apply></ml:eval></math><rendering item-idref="88" /></region><region region-id="1321" left="66" top="5667" width="41.25" height="18" align-x="83.25" align-y="5676" 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:indexer /><ml:id>X</ml:id><ml:real>1</ml:real></ml:apply></ml:eval></math><rendering item-idref="89" /></region><region region-id="1333" left="144" top="5667" width="33" height="18" align-x="159.75" align-y="5676" 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:indexer /><ml:id>Y</ml:id><ml:real>1</ml:real></ml:apply></ml:eval></math><rendering item-idref="90" /></region><region region-id="1347" left="222" top="5667" width="51.75" height="18" align-x="256.5" align-y="5676" 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>p0</ml:id><ml:apply><ml:indexer /><ml:id>X</ml:id><ml:real>1</ml:real></ml:apply></ml:apply></ml:eval></math><rendering item-idref="91" /></region><region region-id="1323" left="66" top="5697" width="34.5" height="18" align-x="83.25" align-y="5706" 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:indexer /><ml:id>X</ml:id><ml:real>2</ml:real></ml:apply></ml:eval></math><rendering item-idref="92" /></region><region region-id="1335" left="144" top="5697" width="39.75" height="18" align-x="159.75" align-y="5706" 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:indexer /><ml:id>Y</ml:id><ml:real>2</ml:real></ml:apply></ml:eval></math><rendering item-idref="93" /></region><region region-id="1348" left="222" top="5697" width="58.5" height="18" align-x="256.5" align-y="5706" 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>p1</ml:id><ml:apply><ml:indexer /><ml:id>X</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:eval></math><rendering item-idref="94" /></region><region region-id="1349" left="36" top="5738.25" width="135" height="12" align-x="36" align-y="5748" 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 those are B-spline fits.</p></text></region><region region-id="1351" left="6" top="5772" width="427.5" height="5.25" align-x="6" align-y="5772" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" 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