**Problem 23.** With \( A = \begin{bmatrix} 1 & 1 \\ 2 & 3 \\ 2 & 1 \end{bmatrix} \) and \( b = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \), use \( A = QR \) to solve the least squares problem \( Ax = b \).

Given, A=112321 and b =111T We have to solve the least squares problem Ax=b using A = QR.

Answered: Problem 23. With A Ax = b. = [11] 2 3…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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QUESTION 4 Consider a set of points in the X-Y plane: (x1.y1) (x2.2) (x3y3) (x4y4) etc. We wish to construct a line in the plane that minimizes the distance between each point and the line. There are many ways to do this, but one is the popular Least Squares Method. Let the equation of our line be y=b+mPlug in each point: y₁ =b+mx y₁=b+mx + mx ₂ y₁=b+mx E 3 =b+ FLX 21 This is a matrix equation of the form: a M= 004 X= 1x Y=XM Y= It turns out that the optimal solution can be shown to be: M=(x¹x)-1X'y Question: What is the slope of a least squares approximation to the following data points (1.1.2). (2.2.1) (3, 5.5), (4,8.75). (5,10.3)? Enter your answer in the space provided rounded to the hundreths place, You may use a software program to assist with the matrix math.
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4. Find the least-squares solution of Ax = b for A - 012 12 and b = |
4. Find the least-squares solution of Az = b for A = N and b = 211
We use the form ŷ = a + bx for the least-squares line. In some computer printouts, the least-squares equation is not given directly. Instead, the value of the constant a is given, and the coefficient b of the explanatory or predictor variable is displayed. Sometimes a is referred to as the constant, and sometimes as the intercept. Data from a report showed the following relationship between elevation (in thousands of feet) and average number of frost-free days per year in a state. A Minitab printout provides the following information. Predictor Coef SE Coef T P Constant 315.27 28.31 11.24 0.002 Elevation -31.812 3.511 -8.79 0.003 S = 11.8603 R-Sq = 96.8% Notice that "Elevation" is listed under "Predictor." This means that elevation is the explanatory variable x. Its coefficient is the slope b. "Constant" refers to a in the equation ŷ = a + bx. (a) Use the printout to write the least-squares equation. ŷ = + x (b) For each 1000-foot increase in elevation,…
We use the form ŷ = a + bx for the least-squares line. In some computer printouts, the least-squares equation is not given directly. Instead, the value of the constant a is given, and the coefficient b of the explanatory or predictor variable is displayed. Sometimes a is referred to as the constant, and sometimes as the intercept. Data from a report showed the following relationship between elevation (in thousands of feet) and average number of frost-free days per year in a state. A Minitab printout provides the following information. Predictor Coef SE Coef T P Constant 316.62 28.31 11.24 0.002 Elevation -30.516 3.511 -8.79 0.003 S = 11.8603 R-Sq = 96.2% The printout gives the value of the coefficient of determination r2. What is the value of r? Be sure to give the correct sign for r based on the sign of b. (Round your answer to four decimal places.) What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares…
Please show step-by-step solution and do not skip steps. Explain your entire process in great detail. Explain how you reached the answer you did.
Find (a) the orthogonal projection of b onto Col A and (b) a least-squares solution of Ax = b. 14 HA A = - 18 b= 1 14 7 4 a. The orthogonal projection of b onto Col A is b = (Simplify your answers.)
3. Find the the equation of the least squares regression line for the data as the linear model f(x) = ao + a₁x in the manner discussed in the textbook using the formula A = (XTX)-¹XTY. What are your values for ao and a₁?
1 2 21 Let A = -4 1 and b = 0 |. If the least-squares solution of 1 Ax = b is X1 then the value of X1 is A and the value of X2 X2 is

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Problem 23. With A Ax = b. = [11] 2 3 and b = [1 1 1]T, use A = QR to solve the least squares problem 2 1