Solutions To Numerical Analysis Problems (5) Least Squares Approximations

 1)  Obtain the line: y = mx + b which best fits the following data points:

(0.10, 0.10),  (0.20, 0.20), (0.30, 0.30), (0.40, 0.40), (0.50, 0.50)  

Solution:

It is evident on inspection: y_obs =  x_obs    and m =1 (constant slope)

Further b = 0, so the least squares line has D ² = 0:

This yields:  y = x 

Graphing:

2) Apply the method of least squares to obtain the line y = mx + b which best fits the points:  (0,1), (1,2), (2,3)

Solution:

Using the data we prepare the table:

For which: å (D ² )=  14- 12b + 3b ² - 16m +  5m ² 

¶f/¶m = 10m + 6m – 16 = 0

Or: 16m = 16  so m = 1

¶f/¶b = 6b -12  + 6b = 0

So:  12b = 12  and b = 1.0

Hence:  y  = x + 1 is best fit line.

Graph is below:

3) In examining the frequency F of subflares within regions of sunspot area (A)* the following table of data is obtained:

Apply the method of least squares to obtain the line F = mA + b which best fits the points

Solution:

Use same technique as shown above, and we obtain:

m = 2.693  and b =   0.613

So best fit line to data:  F = 2.693A +  0.613

Graph is shown below: