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• #### 8. Publishing with MATLAB

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Textbook by: Serhat Beyenir. E-mail the author

# Interpolation

Module by: Serhat Beyenir. E-mail the author

Summary: Interpolation with MATLAB.

Linear interpolation is one of the most common techniques for estimating values between two given data points. For example, when using steam tables, we often have to carry out interpolations. With this technique, we assume that the function between the two points is linear. MATLAB has a built-in interpolation function.

## The interp1 Function

Give an x-y table, y_new = interp1(x,y,x_new) interpolates to find y_new. Consider the following examples:

### Example 1

To demonstrate how the interp1 function works, let us create an x-y table with the following commands;


x = 0:5;
y = [0,20,60,68,77,110];

To tabulate the output, we can use

[x',y']

The result is

ans =

0     0
1    20
2    60
3    68
4    77
5   110

Suppose we want to find the corresponding value for 1.5 or interpolate for 1.5. Using y_new = interp1(x,Y,x_new) syntax, let us type in:

y_new=interp1(x,y,1.5)

y_new =

40


### Example 2

The table we created above has only 6 elements in it and suppose we need a more detailed table. In order to do that, instead of a single new x value, we can define an array of new x values, the interp1 function returns an array of new y values:


new_x = 0:0.2:5;
new_y = interp1(x,y,new_x);

Let's display this table

[new_x',new_y']

The result is

ans =

0         0
0.2000    4.0000
0.4000    8.0000
0.6000   12.0000
0.8000   16.0000
1.0000   20.0000
1.2000   28.0000
1.4000   36.0000
1.6000   44.0000
1.8000   52.0000
2.0000   60.0000
2.2000   61.6000
2.4000   63.2000
2.6000   64.8000
2.8000   66.4000
3.0000   68.0000
3.2000   69.8000
3.4000   71.6000
3.6000   73.4000
3.8000   75.2000
4.0000   77.0000
4.2000   83.6000
4.4000   90.2000
4.6000   96.8000
4.8000  103.4000
5.0000  110.0000


### Example 3

Using the table below, find the internal energy of steam at 215 ˚C and the temperature if the internal energy is 2600 kJ/kg (use linear interpolation).

Table 1: An extract from Steam Tables
Temperature [C] Internal Energy [kJ/kg]
100 2506.7
150 2582.8
200 2658.1
250 2733.7
300 2810.4
400 2967.9
500 3131.6

First let us enter the temperature and energy values

temperature = [100, 150, 200, 250, 300, 400, 500];
energy = [2506.7, 2582.8, 2658.1, 2733.7, 2810.4, 2967.9, 3131.6];
[temperature',energy']

returns

ans =

1.0e+003 *

0.1000    2.5067
0.1500    2.5828
0.2000    2.6581
0.2500    2.7337
0.3000    2.8104
0.4000    2.9679
0.5000    3.1316

issue the following for the first question:

new_energy = interp1(temperature,energy,215)

returns

new_energy =

2.6808e+003

Now, type in the following for the second question:

new_temperature = interp1(energy,temperature,2600)

returns

new_temperature =

161.4210


## Summary of Key Points

1. Linear interpolation is a technique for estimating values between two given data points,
2. Problems involving steam tables often require interpolated data,
3. MATLAB has a built-in interpolation function.

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