AQA A-Level Mathematics I2: Simple Iterative Methods

AQA A-Level Mathematics I2: Simple Iterative Methods What is the iterative formula? How are staircase and cobweb diagrams drawn? • Draw the curve y = g(x) and the line y = x • Draw a vertical line from x₁ to a point on the curve y =g(x) • Draw a horizontal line from this point to a point on the line y = x • Repeat for n iterations • The solution α is given by α = xₙ What is the limitation of the iterative formula method? The function must converge to the solution α What is the Newton-Raphson method formula? What condition is required for the iterative formula to converge to the solution α? −1 < g'(x) < 1 for all values of x in an interval containing x₁ and α How is the Newton-Raphson method used? • Write the equation of the function • Differentiate the function • Substitute these functions into the Newton-Raphson formula • Substitute x₁ into the formula • Repeat for n iterations What are the limitations of the Newton-Raphson method? • It doesn't converge if the original approximation is close to the x-coordinate of a stationary point • It doesn't converge if f'(x) = 0 for one of the values of xₙ