The Role Of Functionals in Fitting A Curve To A Polynomial

 In numerical analysis the functional - in the form:

F(a,b,c) =  å ⁴ _n =1  (ax_i  ² +  bx_i   +  c   -  y_i)  ² 

Often plays a key role in fitting a curve to a polynomial of the form:

f(x) = ax ² +  bx  +  c 

Generally starting with being given a table, e.g.

We are then tasked with fitting a polynomial of form a x ² +  bx  + c  to the set of points,   (x₁ , y_1 ,  x₂ , y_{2  ,} x₃ , y_{3   ,}  x₄ , y₄ )say as displayed in the  table:

To proceed we  must choose a, b, c to   minimize the functional, such that:

F(a,b,c) =  å ⁴ _n =1  (ax_i  ² +  bx_i   +  c   -  y_i)  ² 

This can be done by solving the linear system of equations:

  ¶f/¶a = 0,        ¶f/¶b = 0,  ¶f/¶c = 0                                                                                                          

 Computation based on the respective values for  (x_i,  y_i) are shown below:

[1] ¶f/¶a =  2 (ax_i  ² +  bx_i   +  c   -y_i)x_i ²  = 0

Then:

(-2,1): 32a -  16b + 8c - 8 = 0

(-1, 1): 2a    - 2b   +2c + 2 = 0

(0, 2):   0

(1, 1): 2a  + 2b + 2c - 2  = 0

            ------------------------

Summing: 36a - 16b +12c - 8 = 0

[2] ¶f/¶b =  2 (ax_i  ² +  bx_i   +  c   -y_i)x_i   = 0

(-2,1 ): - 16a + 8b - 4c + 4 = 0

(-1, -1):  -2a  - 2b - 2c -2  = 0

(0, 2):      0

(1, 1):     2a  + 2b  + 2c  - 2 = 0

---------------------------------

     Sum:   16a + 12b  - 4c = 0

[3] ¶f/¶c =  2 (ax_i  ² +  bx_i   +  c   -y_i)   = 0

(-2, 1):  8a - 4b + 2c - 2 = 0

(-1, -1): 2a - 2b + 2c + 2 = 0

(0, 2): 2c - 4 = 0

(1, 1): 2a -4b + 8c - 6 = 0

 -----------------------------

Sum:  12a - 4b + 8c - 6 = 0

From the 3 sums we arrive at a 3 x 3 system given by:

36a - 16b +12c - 8 = 0

16a + 12b  - 4c = 0

12a - 4b + 8c - 6 = 0

Written in matrix form:

The values for a, b, c can then either be found using Gaussian elimination with pivotal condensation or matrix inversion. We choose the latter as it is more straightforward and amenable to a good computing program.  Using Mathcad, for example, we obtain:

Thus: a = 0.25, b = 0.55, c = 0.65

The unknowns a, b and c can also be computed using the Gaussian elimination calculator below. 

Gaussian elimination calculator (onlinemschool.com)

Simply insert the original array values for M and the values for v - and see all the steps that would normally be needed and the solutions for a, b and c.

The calculator will also perform a check of the Mathcad result.

Suggested Problem:

 Fit a second degree polynomial  to the ordered pairs in the table below:

Hence, find a, b and c for the polynomial: ax_i ² +  bx_i   +  c