Difference between revisions of "Interpolation"

Revision as of 16:09, 1 September 2008

1 Linear Interpolation
- 1.1 Example
  - 1.1.1 Implementation in Matlab
  - 1.1.2 Implementation in Excel
2 Polynomial Interpolation
- 2.1 Example
  - 2.1.1 Implementation in Matlab
3 Cubic Spline Interpolation
- 3.1 Example
  - 3.1.1 Implementation in Matlab
4 Lagrange Polynomial Interpolation
- 4.1 Example

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.

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;

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;

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

{{Stub}|section}}

Polynomial Interpolation

Polynomial interpolation is a simple extension of linear interpolation. In general, an n^th 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 n^th degree polynomial, then we must determine n+1 coefficients, $a_0 \ldots a_n$ . Therefore, we require n+1 points to interpolate using an n^th 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, approximate the density of air at the following temperatures: 50, 300, 500 °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:

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

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 ]';

T = 50;
points = 1:4;
A = [ ones(4,1), ones(4,1).*Ti(points), ones(4,1)*Ti(points).^2, ones(4,1)*Ti(points).^3 ];
b = rhoi(points);
a = A\b;

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;

For T=500 we would change

T=500;
points=5:9;

Cubic Spline Interpolation

This section is a stub and needs to be expanded.

If you can provide information or finish this section you're welcome to do so and then remove this message afterwards.

Example

Implementation in Matlab

Lagrange Polynomial Interpolation

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, $\prod_{i=1}^{3} (x+i) = (x+1)(x+2)(x+3)$ .

@@ Line 1: / Line 1: @@
-{{Stub|page}}
 == Linear Interpolation ==
+A linear function may be written as
+<math>f(x) = a_0 + a_1 x.</math>
+Given any two data points, <math>( x_1, y_1 )</math>, and <math>(x_2,y_2)</math>, we can determine a linear function that exactly passes through these points.  We can do this by solving for <math>a_0</math> and <math>a_1</math>.  Since there are two unknowns we require two equations.  They are:
+<center><math>\begin{align}
+ y_1 &= a_0 + a_1 x_1 \\
+ y_2 &= a_0 + a_1 x_2
+\end{align}</math></center>
+These may be written in [[LinearAlgebra#Linear_Systems_of_Equations|matrix form]] as
+<center><math>
+\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)
+</math></center>
+Given values for <math>( x_1, y_1 )</math>, and <math>(x_2,y_2)</math>, these equations may be easily [[Linear_Systems_in_Matlab|solved in MATLAB]].  But since they are so simple, we can easily by hand to obtain a general solution,
+<center><math>\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}</math></center>
+This gives the coefficients, which may then be substituted into the original equation <math>f(x) = a_0 + a_1 x</math> and simplified to obtain
+<center><math>
+  f(x) = \left( \frac{y_2-y_1}{x_2-x_1} \right) \left( x-x_1 \right) + y_1.
+</math></center>
+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
+{| border="1" cellpadding="3" cellspacing="0" align="center" style="text-align:center"
+|-
+! Temperature (&deg;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/m<sup>3</sup>)
+| -10 || 80 || 170 || 260 || 350 || 440 || 530 || 620 || 710 || 800 || 890 || 980 || 1070
+|}
+Estimate the density of air at 32 &deg;F.  Using linear interpolation, we have <math>\rho(T)=a_0 + a_1 T</math>.
+We define <math>(T_1,\rho_1)=(-9.7,1.39)</math> and <math>(T_2,\rho_2)=(80,1.161)</math>.  Now we can calculate
+:<math>\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}</math>
 ==== Implementation in Matlab ====
+There are three ways to do interpolation in MATLAB.
+<ol>
+<li> Solve the linear system.
+<source lang="matlab">
+ 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;
+</source>
+<li> Use the general equation we derived above:
+<source lang="matlab">
+ T1=-9.7; T2-80;
+ rho1=1.39; rho2=1.16;
+ T = 32;
+ rho = (rho2-rho1)/(T2-T1) * (T-T1) + rho1;
+</source>
+<li> Use MATLAB's built-in interpolation tool
+<source lang="matlab">
+ Ti = [-9.7 80];
+ rhoi = [1.39 1.16];
+ rho = interp1( Ti, rhoi, 32 );
+</source>
+</ol>
+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 &deg;F, we could accomplish this by:
+<source lang="matlab">
+  Ti = [-9.7 80 170 ];
+  rhoi = [1.39 1.16 0.995];
+  T = 30:20:110;
+  rho = interp1( Ti, rhoi, T );
+</source>
 ==== Implementation in Excel ====
+{{Stub}|section}}
@@ Line 13: / Line 92: @@
 == Polynomial Interpolation ==
+Polynomial interpolation is a simple extension of linear interpolation.  In general, an <var>n<sup>th</sup></var> degree polynomial is given as <center><math>f(x)=\sum_{i=0}^n a_i x^i.</math></center>
+If <var>n=1</var> then we recover a first-degree polynomial, which is linear.  The formula gives <math>f(x)=\sum_{i=0}^1 a_i x^i = a_0 + a_1 x</math>.
+In general, if we want to interpolate a set of data using an <var>n<sup>th</sup></var> degree polynomial, then we must determine <var>n+1</var> coefficients, <math>a_0 \ldots a_n</math>.  Therefore, we require <var>n+1</var> points to interpolate using an <var>n<sup>th</sup></var> degree polynomial.
+The equations that must be solved are given as
+<center><math>
+\left[\begin{array}{ccccc}
+& x_{1} & x_{1}^2 & \cdots & x_{1}^{n}\\
+& x_{2} & x_{2}^2 & \cdots & x_{2}^{n}\\
+\vdots & \vdots & \vdots & \cdots & \vdots \\
+& 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)
+</math></center>
+Solving these equations provides the values for the coefficients, <math>a_0 \ldots a_n</math>.  Once we know these coefficients, we can evaluate the polynomial at any point.
 === Example ===
+Using a third degree polynomial, approximate the density of air at the following temperatures: 50, 300, 500 &deg;F.
+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 ====
+Our procedure is:
+# Determine which points we need to use.
+# 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.
+<ul><ul>
+<source lang="matlab">
+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 ]';
+T = 50;
+points = 1:4;
+A = [ ones(4,1), ones(4,1).*Ti(points), ones(4,1)*Ti(points).^2, ones(4,1)*Ti(points).^3 ];
+b = rhoi(points);
+a = A\b;
+rho = a(0) + a(1)*T + a(2)*T^2 + a(3)*T^3;
+</source>
+</ul>
+For <var>T=300</var> we would repeat the above with the only change being:
+<ul><source lang="matlab">
+T=300;
+points=4:8;
+</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 ==
+{{Stub|section}}
 === Example ===

Difference between revisions of "Interpolation"

Revision as of 16:09, 1 September 2008

Contents

Linear Interpolation

Example

Implementation in Matlab

Implementation in Excel

Polynomial Interpolation

Example

Implementation in Matlab

Cubic Spline Interpolation

Example

Implementation in Matlab

Lagrange Polynomial Interpolation

Example

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