least squares method<\/a> makes it possible to find a relationship that best describes the available data as a whole, while practical examples will show how this approach is used to construct approximating polynomials and find approximate function values.<\/p>\nApproximation of Experimental Data: Problem Statement<\/h2>\n
Suppose that a scientific or engineering experiment produces a set of points<\/p>\n
\\[
\n(x_0,y_0),\\quad(x_1,y_1),\\quad\\dots,\\quad(x_n,y_n).
\n\\]<\/p>\n
Each value \\( x_i \\) corresponds to an experimental value \\( y_i \\), where<\/p>\n
\\[
\ni=0,1,\\dots,n.
\n\\]<\/p>\n
We need to find an approximating function<\/p>\n
\\[
\ny=f(x),
\n\\]<\/p>\n
whose values at the given points differ as little as possible from the experimental values \\( y_i \\).<\/p>\n
At first glance, this problem may seem similar to interpolation. However, there is an important difference between interpolation and approximation. An interpolation polynomial must pass exactly through all the given points:<\/p>\n
\\[
\nP(x_i)=y_i.
\n\\]<\/p>\n
For an approximating function, this condition is not required. In the general case, its values at the points \\( x_i \\) may differ from the given values:<\/p>\n
\\[
\nP(x_i)\\ne y_i.
\n\\]<\/p>\n
This is especially important when working with experimental data. Because of measurement errors, individual points may deviate from the overall pattern, so a function that passes exactly through every point does not always describe the relationship being studied in the best way.<\/p>\n
Thus, the goal of approximation is to construct a function that does not necessarily reproduce every value exactly but describes the entire data set sufficiently well.<\/p>\n
Approximating Polynomial: Choosing an Approximate Relationship<\/h2>\n
Different mathematical relationships can be used as approximating functions. In this article, we will consider polynomial approximation, which means that we will look for an approximating function in the form of a polynomial of degree \\( m \\), where \\( m<n \\):<\/p>\n
\\[
\nP_m(x)=a_0+a_1\\cdot x+a_2\\cdot x^2+\\dots+a_m\\cdot x^m.
\n\\]<\/p>\n
Here, \\( a_0,a_1,\\dots,a_m \\) are unknown coefficients that must be determined from the given experimental data.<\/p>\n
At the point \\( x_i \\), the value of the approximating polynomial is<\/p>\n
\\[
\nP_m(x_i)=a_0+a_1\\cdot x_i+a_2\\cdot x_i^2+\\dots+a_m\\cdot x_i^m.
\n\\]<\/p>\n
Now compare this value with the corresponding experimental value \\( y_i \\). The difference<\/p>\n
\\[
\nP_m(x_i)-y_i
\n\\]<\/p>\n
can be treated as the deviation of the approximating function from the given data at the point \\( x_i \\).<\/p>\n
If there are many data points, a natural question arises: how can we evaluate the quality of the approximation for the entire data set at once? To do this, we need a criterion that takes all these deviations into account at the same time.<\/p>\n
Least Squares Method: Criterion for Approximation Quality<\/h2>\n
Individual deviations can be either positive or negative. If we simply add them together, values with opposite signs may partially or completely cancel each other out. In that case, a small sum would not necessarily mean that the approximating function describes the data well.<\/p>\n
To avoid this cancellation, the Least Squares Method uses the squares of the deviations. For all the given points, we form the sum<\/p>\n
\\[
\nS=\\sum_{i=0}^{n}\\left(P_m(x_i)-y_i\\right)^2.
\n\\]<\/p>\n
Substituting the expression for the approximating polynomial gives<\/p>\n
\\[
\nS=\\sum_{i=0}^{n}\\left(a_0+a_1\\cdot x_i+a_2\\cdot x_i^2+\\dots+a_m\\cdot x_i^m-y_i\\right)^2.
\n\\]<\/p>\n
The quantity \\( S \\) represents the overall deviation of the approximating polynomial from the experimental data. The smaller its value, the better the polynomial fits the given points.<\/p>\n
Therefore, we need to determine the coefficients \\( a_0,a_1,\\dots,a_m \\) for which the sum of squared deviations has the smallest possible value. To do this, we now move on to the conditions for minimizing the function \\( S \\).<\/p>\n
Minimizing the Sum of Deviations: Constructing the System of Normal Equations<\/h2>\n
The function \\( S \\) depends on the unknown coefficients \\( a_0,a_1,\\dots,a_m \\). To find its minimum, we calculate the partial derivative with respect to each coefficient and set it equal to zero:<\/p>\n
\\[
\n\\left\\{
\n\\begin{array}{l}
\n\\dfrac{\\partial S}{\\partial a_0}=0,\\\\[6pt]
\n\\dfrac{\\partial S}{\\partial a_1}=0,\\\\[6pt]
\n\\vdots\\\\[6pt]
\n\\dfrac{\\partial S}{\\partial a_m}=0.
\n\\end{array}
\n\\right.
\n\\]<\/p>\n
Let us write these conditions in more detail:<\/p>\n
\\[
\n\\left\\{
\n\\begin{array}{l}
\n\\dfrac{\\partial S}{\\partial a_0}
\n=2\\cdot\\displaystyle\\sum_{i=0}^{n}
\n\\left(a_0+a_1\\cdot x_i+a_2\\cdot x_i^2+\\cdots+a_m\\cdot x_i^m-y_i\\right)=0,\\\\[8pt]
\n\\dfrac{\\partial S}{\\partial a_1}
\n=2\\cdot\\displaystyle\\sum_{i=0}^{n}
\n\\left(a_0+a_1\\cdot x_i+a_2\\cdot x_i^2+\\cdots+a_m\\cdot x_i^m-y_i\\right)\\cdot x_i=0,\\\\[8pt]
\n\\vdots\\\\[8pt]
\n\\dfrac{\\partial S}{\\partial a_m}
\n=2\\cdot\\displaystyle\\sum_{i=0}^{n}
\n\\left(a_0+a_1\\cdot x_i+a_2\\cdot x_i^2+\\cdots+a_m\\cdot x_i^m-y_i\\right)\\cdot x_i^m=0.
\n\\end{array}
\n\\right.
\n\\]<\/p>\n
Under appropriate conditions on the given data points, the solution of the resulting system determines the coefficients for which the function \\( S \\) reaches its minimum value.<\/p>\n
The common factor \\( 2 \\) can be removed because it does not affect the solution of the system. We then expand the sums and group the terms according to the unknown coefficients \\( a_0,a_1,\\dots,a_m \\).<\/p>\n
As a result, we obtain the system of normal equations:<\/p>\n
\\[
\n\\left\\{
\n\\begin{array}{l}
\n(n+1)\\cdot a_0+a_1\\cdot\\displaystyle\\sum_{i=0}^{n}x_i+a_2\\cdot\\displaystyle\\sum_{i=0}^{n}x_i^2+\\cdots+a_m\\cdot\\displaystyle\\sum_{i=0}^{n}x_i^m=\\displaystyle\\sum_{i=0}^{n}y_i,\\\\[8pt]
\na_0\\cdot\\displaystyle\\sum_{i=0}^{n}x_i+a_1\\cdot\\displaystyle\\sum_{i=0}^{n}x_i^2+a_2\\cdot\\displaystyle\\sum_{i=0}^{n}x_i^3+\\cdots+a_m\\cdot\\displaystyle\\sum_{i=0}^{n}x_i^{m+1}=\\displaystyle\\sum_{i=0}^{n}y_i\\cdot x_i,\\\\[8pt]
\n\\vdots\\\\[8pt]
\na_0\\cdot\\displaystyle\\sum_{i=0}^{n}x_i^m+a_1\\cdot\\displaystyle\\sum_{i=0}^{n}x_i^{m+1}+a_2\\cdot\\displaystyle\\sum_{i=0}^{n}x_i^{m+2}+\\cdots+a_m\\cdot\\displaystyle\\sum_{i=0}^{n}x_i^{2\\cdot m}=\\displaystyle\\sum_{i=0}^{n}y_i\\cdot x_i^m.
\n\\end{array}
\n\\right.
\n\\]<\/p>\n
All sums containing \\( x_i \\) and \\( y_i \\) are calculated from the given experimental data. Once these sums have been evaluated, we obtain a system of linear equations for the unknown coefficients \\( a_0,a_1,\\dots,a_m \\).<\/p>\n
By solving this system, we determine the coefficients of the approximating polynomial<\/p>\n
\\[
\nP_m(x)=a_0+a_1\\cdot x+a_2\\cdot x^2+\\dots+a_m\\cdot x^m.
\n\\]<\/p>\n
The solution of the system of normal equations determines the polynomial that best fits the given data according to the least-squares criterion. Now let us see how this procedure is applied in practice.<\/p>\n
Least Squares Method: Practical Examples of Function Approximation<\/h2>\n
Now let us apply the system of normal equations derived above to specific data sets. First, we will construct a first-degree polynomial, then consider quadratic approximation, and finally use the resulting relationship to calculate an approximate function value.<\/p>\n
Example 1. Use the least squares method to find a first-degree polynomial that approximates the function given in the table<\/h3>\n
![\"Flowchart](\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2026\/08\/least-squares-method1.jpg\")
<\/p>\n","protected":false},"excerpt":{"rendered":"