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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 10.4

# Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 10.4

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594 CHAPTER 10 NUMERICAL METHODS 10.4 APPLICATIONS OF NUMERICAL METHODS Applications of Gaussian Elimination with Pivoting In Section 2.5 you used least squares regression analysis to find linear mathematical models that best fit a set of n points in the plane. This procedure can be extended to cover polyno- mial models of any degree as follows. Note that if this system of equations reduces to x i a 0 x i 2 a 1 x i y i , na 0 2 x i a 1 y i m 1 Regression Analysis for Polynomials The least squares regression polynomial of degree m for the points is given by where the coefficients are determined by the following system of linear equa- tions. x i m a 0 x i m 1 a 1 x i m 2 a 2 . . . x i 2 m a m x i m y i . . . x i 2 a 0 x i 3 a 1 x i 4 a 2 . . . x i m 2 a m x i 2 y i x i a 0 x i 2 a 1 x i 3 a 2 . . . x i m 1 a m x i y i na 0 x i a 1 x i 2 a 2 . . . x i m a m y i m 1 y a m x m a m 1 x m 1 . . . a 2 x 2 a 1 x a 0 , . . . , x n , y n x 1 , y 1 , x 2 , y 2 ,

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which has a solution of and Exercise 16 asks you to show that this formula is equivalent to the matrix formula for linear regression that was presented in Section 2.5. Example 1 illustrates the use of regression analysis to find a second-degree polynomial model. E X A M P L E 1 Least Squares Regression Analysis The world population in billions for the years between 1965 and 2000, is shown in Table 10.9. (Source: U.S. Census Bureau) TABLE 10.9 Year 1965 1970 1975 1980 1985 1990 1995 2000 Population 3.36 3.72 4.10 4.46 4.86 5.28 5.69 6.08 Find the second-degree least squares regression polynomial for these data and use the resulting model to predict the world population for 2005 and 2010. Solution Begin by letting represent 1965, represent 1970, and so on. So the collection of points is given by which yields So the system of linear equations giving the coefficients of the quadratic model is Gaussian elimination with pivoting on the matrix 8 4 44 4 44 64 44 64 452 37.55 2.36 190.86 44 a 0 64 a 1 452 a 2 190.86. 4 a 0 44 a 1 64 a 2 2.36 8 a 0 4 a 1 44 a 2 37.55 y a 2 x 2 a 1 x a 0 8 i 1 x i 2 y i 190.86. 8 i 1 x i y i 2.36, 8 i 1 y i 37.55, 8 i 1 x i 4 452, 8 i 1 x i 3 64, 8 i 1 x i 2 44, 8 i 1 x i 4, n 8, 1, 5.28 , 2, 5.69 , 3, 6.08 , 4, 3.36 , 3, 3.72 , 2, 4.10 , 1, 4.46 , 0, 4.86 , x 3 x 4 a 0 y i n a 1 x i n . a 1 n x i y i x i y i n x i 2 x i 2 SECTION 10.4 APPLICATIONS OF NUMERICAL METHODS 595
produces So by back substitution you find the solution to be and the regression quadratic is Figure 10.1 compares this model with the given points. To predict the world population for 2005, let obtaining Similarly, the prediction for 2010 is E X A M P L E 2 Least Squares Regression Analysis Find the third-degree least squares regression polynomial for the points Solution For this set of points the linear system becomes 441 a 0 2275 a 1 12,201 a 2 67,171 a 3 1258. 91 a 0 441 a 1 2275 a 2 12,201 a 3 242 21 a 0 91 a 1 441 a 1 2275 a 3 52 7 a 0 21 a 1 91 a 2 441 a 3 14 x i 3 a 0 x i 4 a 1 x i 5 a 2 x i 6 a 3 x i 3 y i x i 2 a 0 x i 3 a 1 x i 4 a 2 x i 5 a 3 x i 2 y i x i a 0 x i 2 a 1 x i 3 a 2 x i 4 a 3 x i y i na 0 x i a 1 x i 2 a 2 x i 3 a 3 y i 0, 0 , 1, 2 , 2, 3 , 3, 2 , 4, 1 , 5, 2 , 6, 4 .

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