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Least Squares Approximation Calculator

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Example
Created on 2024-06-20Asked by Sebastian Martin (Solvelet student)
Find the least squares approximation of the function f(x)=ex f(x) = e^x on the interval [0,1][0, 1] using the polynomial of degree 1.

Solution

To find the least squares approximation of f(x)=ex f(x) = e^x on [0,1][0, 1] using a polynomial of degree 1, we assume the approximation p(x)=a0+a1x p(x) = a_0 + a_1 x . The coefficients a0 a_0 and a1 a_1 are determined by minimizing the integral of the squared error: ∫01(ex−(a0+a1x))2 dx \int_{0}^{1} \left(e^x - (a_0 + a_1 x)\right)^2 \, dx We solve the following system of equations obtained by setting the partial derivatives of the integral with respect to a0 a_0 and a1 a_1 to zero: ∫01(ex−(a0+a1x)) dx=0 \int_{0}^{1} \left(e^x - (a_0 + a_1 x)\right) \, dx = 0 ∫01(ex−(a0+a1x))x dx=0 \int_{0}^{1} \left(e^x - (a_0 + a_1 x)\right) x \, dx = 0 Calculating these integrals, we have: ∫01ex dx−a0∫011 dx−a1∫01x dx=0 \int_{0}^{1} e^x \, dx - a_0 \int_{0}^{1} 1 \, dx - a_1 \int_{0}^{1} x \, dx = 0 [ex]01−a0[x]01−a1[x22]01=0 \left[e^x\right]_{0}^{1} - a_0 [x]_{0}^{1} - a_1 \left[\frac{x^2}{2}\right]_{0}^{1} = 0 (e−1)−a0−a12=0 (e - 1) - a_0 - \frac{a_1}{2} = 0 ∫01exx dx−a0∫01x dx−a1∫01x2 dx=0 \int_{0}^{1} e^x x \, dx - a_0 \int_{0}^{1} x \, dx - a_1 \int_{0}^{1} x^2 \, dx = 0 [exx−∫ex dx]01−a0[x22]01−a1[x33]01=0 \left[e^x x - \int e^x \, dx \right]_{0}^{1} - a_0 \left[\frac{x^2}{2}\right]_{0}^{1} - a_1 \left[\frac{x^3}{3}\right]_{0}^{1} = 0 (e−1)−1+1−a02−a13=0 (e - 1) - 1 + 1 - \frac{a_0}{2} - \frac{a_1}{3} = 0 Solving the system: (e−1)−a0−a12=0 (e - 1) - a_0 - \frac{a_1}{2} = 0 (e−1)−a02−a13=0 (e - 1) - \frac{a_0}{2} - \frac{a_1}{3} = 0 We find a0 a_0 and a1 a_1 : a0=e−1−a12 a_0 = e - 1 - \frac{a_1}{2} a1=6(e−1)−3a0 a_1 = 6 (e - 1) - 3a_0 Therefore, the least squares approximation is: p(x)=a0+a1x p(x) = a_0 + a_1 x Solved on Solvelet with Basic AI Model
Some of the related questions asked by Madison Young on Solvelet
1. Find the line of best fit for the data points (1,2)(1, 2), (2,3)(2, 3), (3,4)(3, 4), and (4,5)(4, 5) using least squares approximation,2. Conduct multiple linear regression analysis on the dataset $\{x_1 = 1, x_2 = 2, x_3 = 33. y = 5, 7, 9\}tomodeltherelationshipbetweenthepredictors to model the relationship between the predictors x_1, x_2,and, and x_3andtheresponsevariable and the response variable y$.,
DefinitionLeast square approximation is a method for finding the curve that minimizes the sum of the squares of the differences between observed values and values predicted by the model. Usage : Data Fitting, Regression Analysis, Statistical Modelling. For example, in linear regression, the least squares method finds the line \(y=mx+b\) which minimizes \(\sum(y_i - (mx_i+b))^2\) over all data points \((x_i,y_i)\).
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