It include the basic definition of curve fitting and it's applications in mathematical and non-mathematically with the help of linear algebra and matlab.

1. Submitting to -
Naveen Kumar S B
Professor
Science & Humanities
Presented by -
Gowtham CR – PES1201801581
Nithin Chandra R – PES1201801340
Branch & Section – EEE 4’A

2. Curve Fitting is the process of constructing a curve, or
mathematical function, that has the best fit to a series of
data points.

3.  Matrix is one of the most thought and
used form or method to simplify such
problems.
 It’s the same case here, we used
matrix method which includes matrix
inversion.
 Matrix method is also helpful in storing
the inputs(co-ordinate points) in the
form of matrix.

4.  To find the curve equation using the inputs(co-ordinate points), it
need’s users effort to solve for it each and every time.
 So, coding it wouldn't just nullify the users efforts but also he/she
can visualize the curve by plotting it .
 The code is user friendly which even permits the user to choose the
degree of polynomial which should less them number of inputs.
 More the no.of inputs more close it is to the expected equation.

5. Any polynomial of order ‘n’ can be expressed as:
Y = a*xn+ b*x(n-1) + c*x(n-2) + ….. + k*x(n-(n-1) + C
In here, by substituting the value of ‘x’ and ‘Y’ accordingly, We find a
set of linear equations with constants a, b, c,….., k, C to be unknowns.
Solving for those and
substituting them back to the
general equation, we could find
the expected equation.

6.  We’ve used matrices and inverse method to find the best fitting curve –
 Let’s consider the degree of polynomial m=3. And the number of co-
ordinate points, which should be greater than or equal to m+1, to be
n=4.
 We know that any cubic equation should be of the form
Y = Ax3+Bx2+Cx+D.
 Substituting ‘x’ and ‘Y’ values of each of the co-ordinate points. We get
5 equations each with 4 common unknowns A, B, C and D.

7.  Now we use linear algebra to find these unknowns by inserting the
values of ‘Y’ in a matrix and the co-efficient of A, B, C and D in another
matrix.
 Let's, name these matrixes Q and P respectively.
 P*A = Q, wherein A is the matrix of Constants.
A = inverse (P)*Q || if inverse of P exists or else
A = (inverse (transpose (P)*(P)))*(transpose (P)*Q) || in which
left inverse for the matrix ‘P’ is given by (inverse (transpose
(P)*(P)))*(transpose (P))
 We got all the constants which are placed back in the general
equation.
 The required polynomial if found out and it has been plotted in the
graph.

8. Advantages -
 As mentioned before reduces human efforts and user can view the
equation’s graphical representation.
 Could be used any number of times and for any degree of
polynomial.
 Could be used to convert any non-polynomial( exponential,
trigonometric, etc..,) into a approximate polynomial function.
Disadvantages –
 Non-polynomial output can’t be obtained.

11. For the input
a. Degree of the Polynomial (m) = 3
b. Number of co-ordinate points (n) = 4
c. (0, -12), (1, -12), (-1, -6), (2, 0) are the co-ordinate points which are
in (x, y) format.