Difference between revisions of "Interpolation"

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=== Example ===
 
=== Example ===
  
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Using a third degree polynomial, approximate the density of air at the following temperatures: 50, 300, 500 °F.
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Using a third degree polynomial and the data in the table above, approximate the density of air at 50 °F and 300 °F.
  
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We require 4 points to fit a third degree polynomial, <math>a_0 \, a_1 \, a_2 \, a_3</math>.  We will choose the 4 points closest to the temperature that we wish to interpolate to.
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We require 4 points to fit a third degree polynomial, <math>a_0, \, a_1, \, a_2, \, a_3</math>.  We will choose the 4 points closest to the temperature that we wish to interpolate to.
  
 
==== Implementation in Matlab ====
 
==== Implementation in Matlab ====
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# Set up and solve the linear system to obtain the polynomial coefficients <math>a_0 \ldots a_3</math>.
 
# Set up and solve the linear system to obtain the polynomial coefficients <math>a_0 \ldots a_3</math>.
 
# Evaluate the polynomial to obtain the temperature.
 
# Evaluate the polynomial to obtain the temperature.
−
<ul><ul>
+
 
 +
The MATLAB code to obtain the density at 50 &deg;F is
 +
<ul>
 
<source lang="matlab">
 
<source lang="matlab">
−
Ti = [ -9.7  80  170  260  350  440  530 620  710  800  890  980 1070 ]';
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% Set the data points from the table.
−
rhoi=[ 1.39 1.16 0.99 0.87 0.78 0.69 0.63 0.58 0.54 0.50 0.46 0.44 0.41 ]';
+
Ti = [ -9.7  80  170  260  350  440  530 ...
 +
        620  710  800  890  980 1070 ]';
 +
rhoi=[ 1.39 1.16 0.99 0.87 0.78 0.69 0.63 ...
 +
      0.58 0.54 0.50 0.46 0.44 0.41 ]';
  
 +
% Set the temperature we want to evaluate the density at.
 
T = 50;
 
T = 50;
 +
 +
% determine the set of points that we will use to construct the interpolant.
 
points = 1:4;
 
points = 1:4;
 +
 +
% set up the linear system
 
A = [ ones(4,1), ones(4,1).*Ti(points), ones(4,1)*Ti(points).^2, ones(4,1)*Ti(points).^3 ];
 
A = [ ones(4,1), ones(4,1).*Ti(points), ones(4,1)*Ti(points).^2, ones(4,1)*Ti(points).^3 ];
 
b = rhoi(points);
 
b = rhoi(points);
 +
 +
% solve for the polynomial coefficients
 
a = A\b;
 
a = A\b;
  
 +
% evaluate the density.
 
rho = a(0) + a(1)*T + a(2)*T^2 + a(3)*T^3;
 
rho = a(0) + a(1)*T + a(2)*T^2 + a(3)*T^3;
 
</source>
 
</source>
 
</ul>
 
</ul>
 +
 
For <var>T=300</var> we would repeat the above with the only change being:
 
For <var>T=300</var> we would repeat the above with the only change being:
 
<ul><source lang="matlab">
 
<ul><source lang="matlab">
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points=4:8;
 
points=4:8;
 
</source></ul>
 
</source></ul>
−
For <var>T=500</var> we would change
 
−
<ul><source lang="matlab">
 
−
T=500;
 
−
points=5:9;
 
−
</source></ul>
 
−
</ul>
 
  
 
== Cubic Spline Interpolation ==
 
== Cubic Spline Interpolation ==

Revision as of 16:48, 1 September 2008

Linear Interpolation

A linear function may be written as f(x) = a_0 + a_1 x. Given any two data points, ( x_1, y_1 ), and (x_2,y_2), we can determine a linear function that exactly passes through these points. We can do this by solving for a_0 and a_1. Since there are two unknowns we require two equations. They are:

\begin{align} y_1 &= a_0 + a_1 x_1 \\ y_2 &= a_0 + a_1 x_2 \end{align}

These may be written in matrix form as

\left[ \begin{array}{cc} 1 & x_1 \\ 1 & x_2 \end{array} \right] \left( \begin{array}{c} a_0 \\ a_1 \end{array} \right) = \left( \begin{array}{c} y_1 \\ y_2 \end{array} \right)

Given values for ( x_1, y_1 ), and (x_2,y_2), these equations may be easily solved in MATLAB. But since they are so simple, we can easily by hand to obtain a general solution,

\begin{align} a_0 &= \frac{x_1 y_2 -x_2 y_1}{x_1-x_2} \\ a_1 &= \frac{y_1-y_2}{x_1-x_2} \end{align}

This gives the coefficients, which may then be substituted into the original equation f(x) = a_0 + a_1 x and simplified to obtain

f(x) = \left( \frac{y_2-y_1}{x_2-x_1} \right) \left( x-x_1 \right) + y_1.

This is a very convenient equation for performing linear interpolation.

Example

The density of air at various temperatures and atmospheric pressure is given in the following table

Temperature (°F) 1.39 1.16 0.99 0.87 0.78 0.69 0.63 0.58 0.54 0.50 0.46 0.44 0.41
Density (kg/m3) -10 80 170 260 350 440 530 620 710 800 890 980 1070

Estimate the density of air at 32 °F. Using linear interpolation, we have \rho(T)=a_0 + a_1 T. We define (T_1,\rho_1)=(-9.7,1.39) and (T_2,\rho_2)=(80,1.161). Now we can calculate

\begin{align} \rho(32) &\approx \frac{\rho_2 -\rho_1}{T_2-T_1}\left( T - T_1 \right) + \rho_1 \\ &= \frac{1.16-1.39}{-9.7-80} \left( 32 + 9.7 \right) + 1.39 \\ &= 1.28. \end{align}

Implementation in Matlab

There are three ways to do interpolation in MATLAB.

  1. Solve the linear system. 
     T1=-9.7;  T2=80;
 rho1=1.39; rho2=1.16;
 T = 32;
 A = [ 1 T1; 1 T2 ];
 b = [ rho1; rho2 ];
 a = A\b;
 rho = a(1) + a(2)*T;
  2. Use the general equation we derived above: 
     T1=-9.7; T2-80;
 rho1=1.39; rho2=1.16;
 T = 32;
 rho = (rho2-rho1)/(T2-T1) * (T-T1) + rho1;
  3. Use MATLAB's built-in interpolation tool 
     Ti = [-9.7 80];
 rhoi = [1.39 1.16];
 rho = interp1( Ti, rhoi, 32 );

All three of these methods provide exactly the same answer. However, using MATLAB's interp1 function allows you to use a vector for the points that you want to interpolate to. For example, if we wanted the density at 30, 50, 70, 10, and 110 °F, we could accomplish this by: 

  Ti = [-9.7 80 170 ];
  rhoi = [1.39 1.16 0.995];
  T = 30:20:110;
  rho = interp1( Ti, rhoi, T );

Implementation in Excel

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Polynomial Interpolation

Polynomial interpolation is a simple extension of linear interpolation. In general, an nth degree polynomial is given as
f(x)=\sum_{i=0}^n a_i x^i.

If n=1 then we recover a first-degree polynomial, which is linear. The formula gives f(x)=\sum_{i=0}^1 a_i x^i = a_0 + a_1 x.

In general, if we want to interpolate a set of data using an nth degree polynomial, then we must determine n+1 coefficients, a_0 \ldots a_n. Therefore, we require n+1 points to interpolate using an nth degree polynomial.

The equations that must be solved are given as

\left[\begin{array}{ccccc} 1 & x_{1} & x_{1}^2 & \cdots & x_{1}^{n}\\ 1 & x_{2} & x_{2}^2 & \cdots & x_{2}^{n}\\ \vdots & \vdots & \vdots & \cdots & \vdots \\ 1 & x_{n+1} & x_{n+1}^2 & \cdots & x_{n+1}^{n}\end{array}\right]\left(\begin{array}{c} a_{0}\\ a_{1}\\ \vdots\\ a_{n}\end{array}\right) = \left(\begin{array}{c} y_{1}\\ y_{2}\\ \vdots\\ y_{n+1}\end{array}\right)

Solving these equations provides the values for the coefficients, a_0 \ldots a_n. Once we know these coefficients, we can evaluate the polynomial at any point.

Example

Using a third degree polynomial and the data in the table above, approximate the density of air at 50 °F and 300 °F.

We require 4 points to fit a third degree polynomial, a_0, \, a_1, \, a_2, \, a_3. We will choose the 4 points closest to the temperature that we wish to interpolate to.

Implementation in Matlab

Our procedure is:

  1. Determine which points we need to use.
  2. Set up and solve the linear system to obtain the polynomial coefficients a_0 \ldots a_3.
  3. Evaluate the polynomial to obtain the temperature.

The MATLAB code to obtain the density at 50 °F is 

    % Set the data points from the table.
Ti = [ -9.7   80  170  260  350  440  530 ...
        620  710  800  890  980 1070 ]';
rhoi=[ 1.39 1.16 0.99 0.87 0.78 0.69 0.63 ...
       0.58 0.54 0.50 0.46 0.44 0.41 ]';

% Set the temperature we want to evaluate the density at.
T = 50;

% determine the set of points that we will use to construct the interpolant.
points = 1:4;

% set up the linear system
A = [ ones(4,1), ones(4,1).*Ti(points), ones(4,1)*Ti(points).^2, ones(4,1)*Ti(points).^3 ];
b = rhoi(points);

% solve for the polynomial coefficients
a = A\b;

% evaluate the density.
rho = a(0) + a(1)*T + a(2)*T^2 + a(3)*T^3;

For T=300 we would repeat the above with the only change being: 

    T=300;
points=4:8;

Cubic Spline Interpolation

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Example

Implementation in Matlab

Lagrange Polynomial Interpolation

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Given n_p points (x_k,y_k), the n^\mathrm{th} order Lagrange polynomial that interpolates these function values, f(x) are expressed as

f(x)=\sum_{k=0}^{n} y_{k} L_{k}(x),

where L_{k}(x) is the Lagrange polynomial given by

L_{k}(x)=\prod_{\stackrel{i=0}{i\ne k}}^{n}\frac{x-x_{i}}{x_{k}-x_{i}},

and n_p = n+1. In other words, for an n^\mathrm{th} order interpolation, we require n_p=n+1 points.

The \prod operator represents the continued product, and is analogous to the \sum operator for summations. For example, \prod_{i=1}^{3} (x+i) = (x+1)(x+2)(x+3).

Example

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