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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard {\fs32\b Interpolating 
Tabular Data with Mathcad}}
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard We are frequently 
required to interpolate tabular data; for example, the VLE data in HWK 
#4 was given in tabular form.  Another example is need to interpolate 
the steam table.  One (old fashioned) method of interpolation is to 
plot the data and draw a smooth curve through the points.  Less 
accurate is to linearly interpolate.  \par \par In this module, we 
will show how to make Mathcad do the interpolation for us.  It turns 
out that Mathcad can perform "cubic spline" interpolations to convert 
columns of x-y data into continuous functions that can be used as y = 
f(x) or x = g(y)\par \par As an example, we will take the tabular data 
from HWK #4, Prob. 4 and perform the cubic-spline interpolation.  }
.TXT 24 24 11 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard The first step is to 
enter the columns of data.  The end result is shown at left.  To get a 
column vector of placeholders to appear after the ":=", just type 
Ctrl-M or use the Math/Matrices menu.  This brings up a dialog box 
where you can enter the number of columns and rows (we used 1 column 
and 8 rows).  We then manually typed the numbers in each placeholder.}
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.EQN 15 -10 13 0 0
{0:f}NAME({0:x}NAME):{0:interp}NAME({0:pspline}NAME({0:xs}NAME,{0:ys}NAME),{0:xs}NAME,{0:ys}NAME,{0:x}NAME)
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard In this statement, we 
define a function f(x) that for a given value of x, interpolates the 
table for the corresponding value of y.  }
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}{\f1\fcharset0\fnil Times New Roman;}}\plain\cf1
\fs20 \pard This definition uses two built-in Mathcad functions: 
"interp" and "pspline".  In place of "pspline" you might substitute 
"lspline" or "cspline".  "pspline", "lspline" and "cspline" perform 
parabolic-, linear- or cubic-spline interpolations, respectively.  All 
of these functions are available (along with some documentation) using 
the {\f1\b\i f(x)} tool on the toolbar.}
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.EQN 4 0 17 0 0
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2 1 0 0 1 1 NO-TRACE-STRING                                               
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.TXT 3 47 19 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green128\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Here we have created a 
plot in which our new f(x) function is used to interpolate the tabular 
data (black squares) at 0.02 intervals in x.  The interpolated values 
are plotted as red +'s.  Note that a nice smooth curve connects the 
original tabular data.}