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										<sp count="4"/>EXAMPLES</link>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">by Jean Giraud, Richard Jackson, and Leslie Bondaryk</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">One common problem in data analysis is that of integrating over a data set which can't be accurately fit with an analytical function. Various discrete techniques can be used, or the data can be interpolated with a function that can then be used to integrate over an arbitrarily close spacing of points, yielding more accurate results.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here are two vectors that represent the same exponentially shaped data. They present several challenges for integration routines. First, they are sparse, that is, there is wide spacing between points and they cover a large range of magnitudes. X1 has approximately half the number of points as X2, spanning the same range of values. We wish to integrate over the same limits for both vectors so that we can compare five methods of numerical integration.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Generate vectors for the 51 and 25 point data sets in x and y:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Traditional numerical integration methods use quadrature rules to obtain approximate areas. The simplest of these quadrature methods are the Trapezoidal rule and Simpson's rule. The Trapezoidal rule can be applied to any vector of data, whereas Simpson's rule requires that the number of data points is odd (that is, an even number of divisions). While the Trapezoidal rule fits a straight line to each division (a straight line is drawn from data point to data point) forming trapezoidal areas, which are summed, Simpson's rule fits second degree polynomials or parabolas to three subsequent points, in many instances giving a better approximation to the integrated area.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that there is much better agreement between the two values using Simpson's rule. Sometimes, it may even be warranted to fit higher degree polynomials to the data, in which case the Newton-Coates formula may be used.</p>
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				<phrase>Newton-Coates formula</phrase>
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						<b>
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								<c val="#800000">Integration using interpolating functions</c>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">An better approach is to create an interpolating function, for example using <b>
						<u>
							<link href="./nonparam.xmcd" title="Links to section" popup="false">splines</link>
						</u>
					</b>, and then apply Mathcad's numerical integration methods to the interpolating function. The advantage of this approach is that the numerical integration methods can have an arbitrarily small step size. What is important in this approach is that the chosen interpolation function is well suited to the shape of the curve it must interpolate. We'll compare three interpolation methods: a cubic spline with parabolic endpoints (<b>pspline</b>), linear interpolation (<b>linterp</b>), and a rational function interpolation (<b>rationalint</b>).</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The <b>
						<u>
							<link href="./splines_1.xmcd" title="Links to section" popup="false">pspline</link>
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				<phrase>pspline example</phrase>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The <b>
						<u>
							<link href="./linterp.xmcd" title="Links to section" popup="false">linear interpolation</link>
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					</b>:</p>
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			<indexes>
				<phrase>linear interpolation example</phrase>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">And the <b>
						<u>
							<link href="./ratint.xmcd" title="Links to section" popup="false">rational function interpolation</link>
						</u>
					</b>:</p>
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				<phrase>rational function interpolation example</phrase>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Compare the three methods and the previous results from Simpson's rule. Since both sets of data represent the same curve, we want the values for a similar method to yield similar results.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">51 points vector </p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The integral of the actual function used to generate this data is</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Rational spline integration produces a correct value to six significant figures, and furthermore, it produces consistent values for both numbers of data points. Rational function interpolation is particularly well-suited to asymptotic data, and interpolates well between the values even in the sparse case.</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The parabolic-endpoint cubic spline does well in the case where there are more points, but fails somewhat when the data is sparse, as it cannot adequately predict the interim values for this type of curve. Linear interpolation produces much the same result as the Trapezoidal method, as we might expect, since both are drawing straight lines from one data point to the next and finding the area of the resulting trapezoid.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This comparison should convince you how important it is to choose the most appropriate interpolating method you can, as the errors propagate when you make subsequent use of the data. For example, look at the errors generated by two of the interpolation methods for the missing data points in the sparse vector.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Generate the interpolated points for the 51 point spacing using the interpolation for the 26 point spacing.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Find the associated errors with each of these points, using the second value returned by <f family="Verdana" charset="0">
						<b>
							<inlineAttr line-through="false">
								<c val="#000">rationalint</c>
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					</f>.</p>
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						<ml:real>1</ml:real>
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			<math optimize="false" disable-calc="false">
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						<ml:real>2</ml:real>
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		<region region-id="3240" left="228" top="2516.25" width="121.5" height="96" align-x="228" align-y="2526" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The error is 0 at the points which overlap in Y2, since these were used to generate the interpolating values, but at the interleaved points, the errors are very small.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If we do the same analysis using the <b>pspline</b> values for the sparse case,</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We see much larger errors, particularly at the hard-to-fit initial values on the asymptotic slope.</p>
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							<ml:real>8.3470661610450221E-05</ml:real>
							<ml:real>7.8266037601197261E-05</ml:real>
							<ml:real>2.8245036503746591E-05</ml:real>
							<ml:real>4.1522347792328804E-05</ml:real>
							<ml:real>8.0012928154570436E-06</ml:real>
							<ml:real>2.0449108433307262E-05</ml:real>
							<ml:real>1.982373710575827E-06</ml:real>
							<ml:real>9.86420134861382E-06</ml:real>
							<ml:real>2.0622351732837684E-07</ml:real>
							<ml:real>4.3736357865573383E-06</ml:real>
							<ml:real>4.3658837330555149E-08</ml:real>
							<ml:real>2.5615234715478097E-06</ml:real>
							<ml:real>-3.586903673765648E-07</ml:real>
							<ml:real>-1.0870553988292448E-06</ml:real>
							<ml:real>4.7194295859911944E-07</ml:real>
							<ml:real>7.4855307814283432E-06</ml:real>
							<ml:real>0</ml:real>
						</ml:matrix>
					</result>
				</ml:eval>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This difference accounts for the difference in the integration values.</p>
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