Find both Newtons and Lagranges form of the interpolating po

Find both Newton\'s and Lagrange\'s form of the interpolating polynomial of lowest degree passing through the points (-2, 19), (-1, 3), (1, 1), (2, 15). Construct the spine of degree 1 (piece-wise linear and continuous) corresponding to the above data. Write down the formulas for the Newton\'s and secant method applied for the solution of ln(x + 2)-x^2 = 0. If the starting values for the secant method are chosen to be x^(0) = 0 and x^(1) = 2, perform 2 iterations) to find the value of x.

Solution

(4).      P₃(x) = f(x₀) + f(x₀,x₁) (x-x₀) + f(x₀,x₁,x₂) (x-x₀) (x-x₁) + f(x₀,x₁,x₂,x₃) (x-x₀) (x-x₁) (x-x₂) + . . . .

                 f(x₀) = 19        f(x₀,x₁) = 3-19/-1+2 = -16

                f(x₂) = 1    f(x₀,x₁,x₂) = 1-3/1+1 = -1

                f(x₃) = 15     f(x₀,x₁,x₂,x₃) = f(x₁,x₂,x₃) - f(x₀,x₁,x₂) / (x3 - x0)

                                                        = 9-5/2+2 = 1

By substituting these values in the above equation, we get

   P₃(x) = 19 +(-16)(x+2) + 5 (x+2)(x+1) + (x+2)(x+1)(x-1)

              = x³ + 7x² - 2x - 5