[Math] Matlab: least square method

Question

Specify a linear function in terms of the least squares method approximates the set point table. Then calculate the sum of squares deviations of this linear function in given points.

xi=[1.25 2.34 4.15 5.12 6.05 6.45];
yi=[-1.3 -1.6 -2.3 -3.1 -3.8 -4.1];

I assume that the required polynomial is second-degree, and the answer is:
P = -0.5467x – 0.3894

How to format following equation in Matlab?

sum = $\sum_{i=0}^{n}[p(x_{i})-y_{i}]^2$

Answer 1

symbolic toolbox is not the usual way to do least square method in MATLAB, the most used function is polyfit and polyval, in which polyfit will return the coefficients set $\{a_k\}$ in the following fitting polynomial: $$ p_n(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 $$ simply type in

a = polyfit(x,y,1); % fitting polynomial has degree 1

and you will find that a = [-0.54668 -0.38939] which coincides with what you give.

If you use second degree a = polyfit(x,y,2);, a will be [-0.074948 0.033439 -1.234].

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For the second question, to evaluate $\displaystyle\sum\limits_{i=0}^{n}[p(x_{i})-y_{i}]^2$, say you have two $(n+1)$-array xi and yi, then the most vectorized command to compute this explicitly is, supposedly you have your p give as above:

p = @(x)-0.5467*x-0.3894
S = sum((p(xi)-yi).^2,2)

noted the dot before the exponential hat, it is for the vectorized arithmetic operation in MATLAB. Or simply use the built-in Euclidean norm function norm which returns the $l^2$-norm of a sequence:

S = norm(p(xi)-yi); 
S = S^2;

will give you the same result.