Polynomial interpolation: Objective: given n distinct points ( x1 , y1 ), ( x2 , y 2 ), , ( xn , y n ) , find a polynomial a 0 a1 x a 2 x 2 a n 1 x n 1 of degree n-1 or less, 10 interpolates the data. 8 (x4,y4) y 6 (x3,y3) 2 4 (x1,y1) 0 (x2,y2) -3 -2 -1 0 1 2 3 x Special case: y a0 a1 x . Given ( x1 , y1 ) and ( x2 , y2 ) , find the line 4 5 (x2,y2) (x1,y1) 3 y 6 7 [solution:] 0.0 0.2 0.4 0.6 x 1 0.8 1.0 , that The unknown coefficients a0 and a1 can be found by solving the system of two linear equations a 0 a1 x1 y1 a 0 a1 x 2 y 2 . General case: a0 , a1 ,, an1 can be found by solving for the following system of linear equations: a 0 a1 x1 a 2 x12 a n 1 x1n 1 y1 a 0 a1 x 2 a 2 x 22 a n 1 x 2n 1 y 2 a 0 a1 x n a 2 x n2 a n 1 x nn 1 y n 1 1 1 x1 x2 xn x1n 1 a 0 y1 x 2n 1 a1 y 2 x nn 1 a n 1 y n Thus, the augmented matrix used in Gauss-Jordan reduction is 1 1 1 x1 x2 xn x12 x1n 1 x 22 x 2n 1 x n2 x nn 1 y1 y2 . yn Example: Determine the equation of the polynomial of degree 2 whose graph passes through the points (1,6), (2,3) and (3,2). [solution:] 2 Suppose the polynomial of degree 2 is y a 0 a1 x a 2 x 2 . Then, the corresponding system of linear equations is a 0 a1 a 2 6 a 0 2a1 4a 2 3 . a 0 3a1 9a 2 2 Thus, the augmented matrix is 1 1 1 6 1 2 2 2 3 inreduced row echelon form 1 3 3 2 2 1 0 0 11 0 1 0 6 0 0 1 1 a 0 11, a1 6, a 2 1 y x 2 6 x 11 3

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# 3.1 Polynomial interpolation