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<li class="navelem"><a class="el" href="class_quant_lib_1_1_cubic_interpolation.html">CubicInterpolation</a> </li>
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<!-- doxytag: class="QuantLib::CubicInterpolation" --><!-- doxytag: inherits="QuantLib::Interpolation" -->
<p>Cubic interpolation between discrete points.
<a href="class_quant_lib_1_1_cubic_interpolation.html#details">More...</a></p>
<p><code>#include <ql/math/interpolations/cubicinterpolation.hpp></code></p>
<div class="dynheader">
Inheritance diagram for CubicInterpolation:</div>
<div class="dyncontent">
<div class="center"><img src="class_quant_lib_1_1_cubic_interpolation__inherit__graph.png" border="0" usemap="#_cubic_interpolation_inherit__map" alt="Inheritance graph"/></div>
<map name="_cubic_interpolation_inherit__map" id="_cubic_interpolation_inherit__map">
<area shape="rect" id="node2" href="class_quant_lib_1_1_interpolation.html" title="base class for 1-D interpolations." alt="" coords="21,6,115,37"/></map>
<center><span class="legend">[<a href="graph_legend.html">legend</a>]</span></center></div>
<p><a href="class_quant_lib_1_1_cubic_interpolation-members.html">List of all members.</a></p>
<table class="memberdecls">
<tr><td colspan="2"><h2><a name="pub-types"></a>
Public Types</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top">enum  </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#aeccb633d9226f37b800589483ea2f0f6">DerivativeApprox</a> { <br/>
  <a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#aeccb633d9226f37b800589483ea2f0f6a73a1396ca8adb89d7ff481ff55974c1e">Spline</a>,
<a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#aeccb633d9226f37b800589483ea2f0f6aace9569c3a723c414afe7e6332958312">SplineOM1</a>,
<a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#aeccb633d9226f37b800589483ea2f0f6a5230bf2465bab97aecd7087ee1a8ce1e">SplineOM2</a>,
<a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#aeccb633d9226f37b800589483ea2f0f6a8908df67d5fb494d1799a270564789f8">FourthOrder</a>,
<br/>
  <a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#aeccb633d9226f37b800589483ea2f0f6a6c56130d695e3aea964232dacfa2c298">Parabolic</a>,
<a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#aeccb633d9226f37b800589483ea2f0f6ac194b4cb0838c7c484b13eb6d6dc0010">FritschButland</a>,
<a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#aeccb633d9226f37b800589483ea2f0f6ab1ddc152184a0f96b8f6e9be8168e708">Akima</a>,
<a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#aeccb633d9226f37b800589483ea2f0f6a065fa14c1b83af108fdae0c713695951">Kruger</a>
<br/>
}</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">enum  </td><td class="memItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#af3393571fa8a8daa4ee5c06613b26555">BoundaryCondition</a> { <br/>
  <a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#af3393571fa8a8daa4ee5c06613b26555a7922e3c8bf97083083c9261300dcfedf">NotAKnot</a>,
<a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#af3393571fa8a8daa4ee5c06613b26555aaf8392865aea8f24355012391183d9e9">FirstDerivative</a>,
<a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#af3393571fa8a8daa4ee5c06613b26555a23de8c854eb282410ba10963837c94b5">SecondDerivative</a>,
<a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#af3393571fa8a8daa4ee5c06613b26555a6db09e3bc054df6f996a45de2ccfeacd">Periodic</a>,
<br/>
  <a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#af3393571fa8a8daa4ee5c06613b26555a1e1c27164b52b5cdf7df7f110dbb0fd3">Lagrange</a>
<br/>
}</td></tr>
<tr><td colspan="2"><h2><a name="pub-methods"></a>
Public Member Functions</h2></td></tr>
<tr><td class="memTemplParams" colspan="2">template<class I1 , class I2 > </td></tr>
<tr><td class="memTemplItemLeft" align="right" valign="top"> </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#a4660490e94d17a41507f676a5eb41a15">CubicInterpolation</a> (const I1 &xBegin, const I1 &xEnd, const I2 &yBegin, <a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#aeccb633d9226f37b800589483ea2f0f6">CubicInterpolation::DerivativeApprox</a> da, bool monotonic, <a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#af3393571fa8a8daa4ee5c06613b26555">CubicInterpolation::BoundaryCondition</a> leftCond, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> leftConditionValue, <a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#af3393571fa8a8daa4ee5c06613b26555">CubicInterpolation::BoundaryCondition</a> rightCond, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> rightConditionValue)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a294c76c9e77213bbb3da550152033826"></a><!-- doxytag: member="QuantLib::CubicInterpolation::primitiveConstants" ref="a294c76c9e77213bbb3da550152033826" args="() const " -->
const std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > & </td><td class="memItemRight" valign="bottom"><b>primitiveConstants</b> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a4119d6ba4d3f11be628a129a7a2bfe6f"></a><!-- doxytag: member="QuantLib::CubicInterpolation::aCoefficients" ref="a4119d6ba4d3f11be628a129a7a2bfe6f" args="() const " -->
const std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > & </td><td class="memItemRight" valign="bottom"><b>aCoefficients</b> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="aaca6dd10570ae2dbc6e5ea6ab8a7e9bb"></a><!-- doxytag: member="QuantLib::CubicInterpolation::bCoefficients" ref="aaca6dd10570ae2dbc6e5ea6ab8a7e9bb" args="() const " -->
const std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > & </td><td class="memItemRight" valign="bottom"><b>bCoefficients</b> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a7746999062e0736bd067da1e60548069"></a><!-- doxytag: member="QuantLib::CubicInterpolation::cCoefficients" ref="a7746999062e0736bd067da1e60548069" args="() const " -->
const std::vector< <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> > & </td><td class="memItemRight" valign="bottom"><b>cCoefficients</b> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a7f8698aec8c4a956e68c3a9d6b19f23a"></a><!-- doxytag: member="QuantLib::CubicInterpolation::monotonicityAdjustments" ref="a7f8698aec8c4a956e68c3a9d6b19f23a" args="() const " -->
const std::vector< bool > & </td><td class="memItemRight" valign="bottom"><b>monotonicityAdjustments</b> () const </td></tr>
</table>
<hr/><a name="details" id="details"></a><h2>Detailed Description</h2>
<div class="textblock"><p>Cubic interpolation between discrete points. </p>
<p><a class="el" href="class_quant_lib_1_1_cubic.html" title="Cubic interpolation factory and traits">Cubic</a> interpolation is fully defined when the ${f_i}$ function values at points ${x_i}$ are supplemented with ${f^'_i}$ function derivative values.</p>
<p>Different type of first derivative approximations are implemented, both local and non-local. Local schemes (Fourth-order, Parabolic, Modified Parabolic, Fritsch-Butland, Akima, Kruger) use only $f$ values near $x_i$ to calculate each $f^'_i$. Non-local schemes (Spline with different boundary conditions) use all ${f_i}$ values and obtain ${f^'_i}$ by solving a linear system of equations. Local schemes produce $C^1$ interpolants, while the spline schemes generate $C^2$ interpolants.</p>
<p>Hyman's monotonicity constraint filter is also implemented: it can be applied to all schemes to ensure that in the regions of local monotoniticity of the input (three successive increasing or decreasing values) the interpolating cubic remains monotonic. If the interpolating cubic is already monotonic, the Hyman filter leaves it unchanged preserving all its original features.</p>
<p>In the case of $C^2$ interpolants the Hyman filter ensures local monotonicity at the expense of the second derivative of the interpolant which will no longer be continuous in the points where the filter has been applied.</p>
<p>While some non-linear schemes (Modified Parabolic, Fritsch-Butland, Kruger) are guaranteed to be locally monotonic in their original approximation, all other schemes must be filtered according to the Hyman criteria at the expense of their linearity.</p>
<p>See R. L. Dougherty, A. Edelman, and J. M. Hyman, "Nonnegativity-, Monotonicity-, or Convexity-Preserving CubicSpline and
Quintic Hermite Interpolation" Mathematics Of Computation, v. 52, n. 186, April 1989, pp. 471-494.</p>
<dl class="todo"><dt><b><a class="el" href="todo.html#_todo000033">Possible enhancements:</a></b></dt><dd>implement missing schemes (FourthOrder and ModifiedParabolic) and missing boundary conditions (Periodic and Lagrange).</dd></dl>
<dl class="test"><dt><b><a class="el" href="test.html#_test000042">Tests:</a></b></dt><dd>to be adapted from old ones.</dd></dl>
</div><hr/><h2>Member Enumeration Documentation</h2>
<a class="anchor" id="aeccb633d9226f37b800589483ea2f0f6"></a><!-- doxytag: member="QuantLib::CubicInterpolation::DerivativeApprox" ref="aeccb633d9226f37b800589483ea2f0f6" args="" -->
<div class="memitem">
<div class="memproto">
<table class="memname">
<tr>
<td class="memname">enum <a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#aeccb633d9226f37b800589483ea2f0f6">DerivativeApprox</a></td>
</tr>
</table>
</div>
<div class="memdoc">
<dl><dt><b>Enumerator: </b></dt><dd><table border="0" cellspacing="2" cellpadding="0">
<tr><td valign="top"><em><a class="anchor" id="aeccb633d9226f37b800589483ea2f0f6a73a1396ca8adb89d7ff481ff55974c1e"></a><!-- doxytag: member="Spline" ref="aeccb633d9226f37b800589483ea2f0f6a73a1396ca8adb89d7ff481ff55974c1e" args="" -->Spline</em> </td><td>
<p>Spline approximation (non-local, non-monotonic, linear[?]). Different boundary conditions can be used on the left and right boundaries: see <a class="el" href="class_quant_lib_1_1_boundary_condition.html" title="Abstract boundary condition class for finite difference problems.">BoundaryCondition</a>. </p>
</td></tr>
<tr><td valign="top"><em><a class="anchor" id="aeccb633d9226f37b800589483ea2f0f6aace9569c3a723c414afe7e6332958312"></a><!-- doxytag: member="SplineOM1" ref="aeccb633d9226f37b800589483ea2f0f6aace9569c3a723c414afe7e6332958312" args="" -->SplineOM1</em> </td><td>
<p>Overshooting minimization 1st derivative. </p>
</td></tr>
<tr><td valign="top"><em><a class="anchor" id="aeccb633d9226f37b800589483ea2f0f6a5230bf2465bab97aecd7087ee1a8ce1e"></a><!-- doxytag: member="SplineOM2" ref="aeccb633d9226f37b800589483ea2f0f6a5230bf2465bab97aecd7087ee1a8ce1e" args="" -->SplineOM2</em> </td><td>
<p>Overshooting minimization 2nd derivative. </p>
</td></tr>
<tr><td valign="top"><em><a class="anchor" id="aeccb633d9226f37b800589483ea2f0f6a8908df67d5fb494d1799a270564789f8"></a><!-- doxytag: member="FourthOrder" ref="aeccb633d9226f37b800589483ea2f0f6a8908df67d5fb494d1799a270564789f8" args="" -->FourthOrder</em> </td><td>
<p>Fourth-order approximation (local, non-monotonic, linear) </p>
</td></tr>
<tr><td valign="top"><em><a class="anchor" id="aeccb633d9226f37b800589483ea2f0f6a6c56130d695e3aea964232dacfa2c298"></a><!-- doxytag: member="Parabolic" ref="aeccb633d9226f37b800589483ea2f0f6a6c56130d695e3aea964232dacfa2c298" args="" -->Parabolic</em> </td><td>
<p>Parabolic approximation (local, non-monotonic, linear) </p>
</td></tr>
<tr><td valign="top"><em><a class="anchor" id="aeccb633d9226f37b800589483ea2f0f6ac194b4cb0838c7c484b13eb6d6dc0010"></a><!-- doxytag: member="FritschButland" ref="aeccb633d9226f37b800589483ea2f0f6ac194b4cb0838c7c484b13eb6d6dc0010" args="" -->FritschButland</em> </td><td>
<p>Fritsch-Butland approximation (local, monotonic, non-linear) </p>
</td></tr>
<tr><td valign="top"><em><a class="anchor" id="aeccb633d9226f37b800589483ea2f0f6ab1ddc152184a0f96b8f6e9be8168e708"></a><!-- doxytag: member="Akima" ref="aeccb633d9226f37b800589483ea2f0f6ab1ddc152184a0f96b8f6e9be8168e708" args="" -->Akima</em> </td><td>
<p>Akima approximation (local, non-monotonic, non-linear) </p>
</td></tr>
<tr><td valign="top"><em><a class="anchor" id="aeccb633d9226f37b800589483ea2f0f6a065fa14c1b83af108fdae0c713695951"></a><!-- doxytag: member="Kruger" ref="aeccb633d9226f37b800589483ea2f0f6a065fa14c1b83af108fdae0c713695951" args="" -->Kruger</em> </td><td>
<p>Kruger approximation (local, monotonic, non-linear) </p>
</td></tr>
</table>
</dd>
</dl>
</div>
</div>
<a class="anchor" id="af3393571fa8a8daa4ee5c06613b26555"></a><!-- doxytag: member="QuantLib::CubicInterpolation::BoundaryCondition" ref="af3393571fa8a8daa4ee5c06613b26555" args="" -->
<div class="memitem">
<div class="memproto">
<table class="memname">
<tr>
<td class="memname">enum <a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#af3393571fa8a8daa4ee5c06613b26555">BoundaryCondition</a></td>
</tr>
</table>
</div>
<div class="memdoc">
<dl><dt><b>Enumerator: </b></dt><dd><table border="0" cellspacing="2" cellpadding="0">
<tr><td valign="top"><em><a class="anchor" id="af3393571fa8a8daa4ee5c06613b26555a7922e3c8bf97083083c9261300dcfedf"></a><!-- doxytag: member="NotAKnot" ref="af3393571fa8a8daa4ee5c06613b26555a7922e3c8bf97083083c9261300dcfedf" args="" -->NotAKnot</em> </td><td>
<p>Make second(-last) point an inactive knot. </p>
</td></tr>
<tr><td valign="top"><em><a class="anchor" id="af3393571fa8a8daa4ee5c06613b26555aaf8392865aea8f24355012391183d9e9"></a><!-- doxytag: member="FirstDerivative" ref="af3393571fa8a8daa4ee5c06613b26555aaf8392865aea8f24355012391183d9e9" args="" -->FirstDerivative</em> </td><td>
<p>Match value of end-slope. </p>
</td></tr>
<tr><td valign="top"><em><a class="anchor" id="af3393571fa8a8daa4ee5c06613b26555a23de8c854eb282410ba10963837c94b5"></a><!-- doxytag: member="SecondDerivative" ref="af3393571fa8a8daa4ee5c06613b26555a23de8c854eb282410ba10963837c94b5" args="" -->SecondDerivative</em> </td><td>
<p>Match value of second derivative at end. </p>
</td></tr>
<tr><td valign="top"><em><a class="anchor" id="af3393571fa8a8daa4ee5c06613b26555a6db09e3bc054df6f996a45de2ccfeacd"></a><!-- doxytag: member="Periodic" ref="af3393571fa8a8daa4ee5c06613b26555a6db09e3bc054df6f996a45de2ccfeacd" args="" -->Periodic</em> </td><td>
<p>Match first and second derivative at either end. </p>
</td></tr>
<tr><td valign="top"><em><a class="anchor" id="af3393571fa8a8daa4ee5c06613b26555a1e1c27164b52b5cdf7df7f110dbb0fd3"></a><!-- doxytag: member="Lagrange" ref="af3393571fa8a8daa4ee5c06613b26555a1e1c27164b52b5cdf7df7f110dbb0fd3" args="" -->Lagrange</em> </td><td>
<p>Match end-slope to the slope of the cubic that matches the first four data at the respective end </p>
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<hr/><h2>Constructor & Destructor Documentation</h2>
<a class="anchor" id="a4660490e94d17a41507f676a5eb41a15"></a><!-- doxytag: member="QuantLib::CubicInterpolation::CubicInterpolation" ref="a4660490e94d17a41507f676a5eb41a15" args="(const I1 &xBegin, const I1 &xEnd, const I2 &yBegin, CubicInterpolation::DerivativeApprox da, bool monotonic, CubicInterpolation::BoundaryCondition leftCond, Real leftConditionValue, CubicInterpolation::BoundaryCondition rightCond, Real rightConditionValue)" -->
<div class="memitem">
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<table class="memname">
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<td class="memname"><a class="el" href="class_quant_lib_1_1_cubic_interpolation.html">CubicInterpolation</a> </td>
<td>(</td>
<td class="paramtype">const I1 & </td>
<td class="paramname"><em>xBegin</em>, </td>
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<tr>
<td class="paramkey"></td>
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<td class="paramtype">const I1 & </td>
<td class="paramname"><em>xEnd</em>, </td>
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<td class="paramkey"></td>
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<td class="paramtype">const I2 & </td>
<td class="paramname"><em>yBegin</em>, </td>
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<td class="paramkey"></td>
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<td class="paramtype"><a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#aeccb633d9226f37b800589483ea2f0f6">CubicInterpolation::DerivativeApprox</a> </td>
<td class="paramname"><em>da</em>, </td>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">bool </td>
<td class="paramname"><em>monotonic</em>, </td>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#af3393571fa8a8daa4ee5c06613b26555">CubicInterpolation::BoundaryCondition</a> </td>
<td class="paramname"><em>leftCond</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>leftConditionValue</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="el" href="class_quant_lib_1_1_cubic_interpolation.html#af3393571fa8a8daa4ee5c06613b26555">CubicInterpolation::BoundaryCondition</a> </td>
<td class="paramname"><em>rightCond</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td>
<td class="paramname"><em>rightConditionValue</em> </td>
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<td>)</td>
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<dl class="pre"><dt><b>Precondition:</b></dt><dd>the <img class="formulaInl" alt="$ x $" src="form_134.png"/> values must be sorted. </dd></dl>
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