1. Dr. Md. Mashiur Rahman Department of Physics University of Chittagong
Curve Fitting
Dr. Md. Mashiur Rahman
Department of Physics
University of Chittagong

2. Md. Mashiur Rahman Department of Physics, CU. Slide: 02
What is Curve Fitting ?
Algorithms to find out a curve which represents a given set of data, the best. x y 2 6 10 30 14 26 18 44 23 37 29 78 34 69
Md. Mashiur Rahman
Department of Physics, CU. Slide: 02

3. Md. Mashiur Rahman Department of Physics, CU. Slide: 03
What is Curve Fitting ?
Algorithms to find out a curve which represents a given set of data, the best. x y 2 6 10 30 14 26 18 44 23 37 29 78 34 69
Md. Mashiur Rahman
Department of Physics, CU. Slide: 03

4. Md. Mashiur Rahman Department of Physics, CU. Slide: 04
What is Curve Fitting ?
Algorithms to find out a curve which represents a given set of data, the best. x y 2 6 10 30 14 26 18 44 23 37 29 78 34 69
Md. Mashiur Rahman
Department of Physics, CU. Slide: 04

5. Md. Mashiur Rahman Department of Physics, CU. Slide: 05
What is Curve Fitting ?
Algorithms to find out a curve which represents a given set of data, the best. x y 2 6 10 30 14 26 18 44 23 37 29 78 34 69
Md. Mashiur Rahman
Department of Physics, CU. Slide: 05

6. Curve Fitting Algorithm
Data set:
Fitted curve:
Error or deviation: x y 2 6 10 30 14 26 18 44 23 37 29 78 34 69
Md. Mashiur Rahman
Department of Physics, CU. Slide: 06

7. Curve Fitting Algorithm
Error or deviation:
Best Fitted curve should have minimum deviation from each of the given set of data.
The most common method:
Lease Square Fitting
Md. Mashiur Rahman
Department of Physics, CU. Slide: 07

8. Md. Mashiur Rahman Department of Physics, CU. Slide: 08
Least Square Method
Developed by French Mathematician Adrien-Marie Legendre in 1805 for the study of the orbits of comets.
Theme: Summation of Square of Errors should be Least.
is minimum
Md. Mashiur Rahman
Department of Physics, CU. Slide: 08

9. Fitting a Straight Line by LSM
Fitted curve:
Straight Line
Polynomial: degree 1
is minimum
Least Square Method:
is minimum w.r.t. a0 and a1
Md. Mashiur Rahman
Department of Physics, CU. Slide: 09

10. Fitting a Straight Line by LSM
This condition satisfies the minimum of S w.r.t. a0
Md. Mashiur Rahman
Department of Physics, CU. Slide: 10

11. Fitting a Straight Line by LSM
This condition satisfies the minimum of S w.r.t. a1
Md. Mashiur Rahman
Department of Physics, CU. Slide: 11/10

12. Fitting a Straight Line by LSM
Normal
Equations
By solving the above two equations for a0 and a1 , we can get the best fitted straight line:
Md. Mashiur Rahman
Department of Physics, CU. Slide: 12

13. Fitting a Straight Line by LSM
Normal
Equations
1st normal equation can be written as:
Centroid
Least Square Fitted Line passes through the centroid of the data set.
Md. Mashiur Rahman
Department of Physics, CU. Slide: 13

14. Fitting a Straight Line by LSM
In terms of matrix, we can write:
Determinant of X
Md. Mashiur Rahman
Department of Physics, CU. Slide: 14

15. Md. Mashiur Rahman Department of Physics, CU. Slide: 15
Problem-1
Find the best fitted line by least square method for the following set of data: x 2 10 14 18 23 29 34 y 6 30 26 44 37 78 69
Least square method requires that the best fitted straight line should satisfy the following normal equations:
Md. Mashiur Rahman
Department of Physics, CU. Slide: 15

16. Md. Mashiur Rahman Department of Physics, CU. Slide: 16
Problem-1
Table: Coefficients for Normal Equations x y x2 xy 2 6 4 12 10 30
100
300 14 26
196
364 18 44
324
792 23 37
529
851 29 78
841
2262 34 69
1156
2346
130
290
3150
6927 
Md. Mashiur Rahman
Department of Physics, CU. Slide: 16

17. Md. Mashiur Rahman Department of Physics, CU. Slide: 17
Problem-1
Now the Normal Equations become,
Solutions are
So, the best fitted line becomes,
Md. Mashiur Rahman
Department of Physics, CU. Slide: 17

18. Md. Mashiur Rahman Department of Physics, CU. Slide: 18
Problem-1 x y 2 6 10 30 14 26 18 44 23 37 29 78 34 69
Md. Mashiur Rahman
Department of Physics, CU. Slide: 18

19. Md. Mashiur Rahman Department of Physics, CU. Slide: 19
Problem-1 x y
y(x) 2 6
6.712 10 30
23.472 14 26
31.852 18 44
40.231 23 37
50.706 29 78
63.276 34 69
73.751
Md. Mashiur Rahman
Department of Physics, CU. Slide: 19

20. Md. Mashiur Rahman Department of Physics, CU. Slide: 20
Problem-1 x y
y(x) 2 6
6.712 10 30
23.472 14 26
31.852 18 44
40.231 23 37
50.706 29 78
63.276 34 69
73.751
Md. Mashiur Rahman
Department of Physics, CU. Slide: 20

21. Md. Mashiur Rahman Department of Physics, CU. Slide: 21
To excel in Computational Physics
Practise!
Practise!!
Practise!!!
Md. Mashiur Rahman
Department of Physics, CU. Slide: 21