values from the iteration x cosx arex0 08 x1

Question

Values from the iteration x = cos(x) are:

x₀ = 0.8, x₁ = 0.696707, x₂ = 0.766959, x₃ = 0.720024, x₄ = 0.751790, x₅ = 0.730468.

a) Calculate the sequence {y_n} from Aitken's ?² method. Given that the solution of x = cos x is 0.739085133 . . . , ?nd the rations between consecutive errors in {xn} (which approach a constant A) and also in {y_n} (which approach another constant, say B). Con?rm that {y_n} is converging twice as fast as {x_n} (i.e. that B = A²).

b) Use Steffensen's method: from x₀, x₁, x₂ calculate y₀. Then start the iteration again from x₃ = y₀, and from x₃, x₄, x₅ calculate y₁.