1. Nonlinear equations). (a) Summarise the secant method for approximating the solution of a nonlinear equation (or rootfinding problem). Hint: Explain how it works geometrically, how it compares to Newton's method and give the formula for the iterations. (b) List a few pros and cons for the secant method. (c) Consider solving the equation f(x) = 0 using the secant method, where f(x) = cos(x)-1, x [0, 5]. Use to = 0.5 and x1 = as initial guesses and apply the secant method to compute the iterates 22 and 13. Hint: these two iterates should agree with x2 and x3 in the table below. (d) Suppose all conditions for the secant method are satisfied. What will be the order of convergence? (e) Suppose you are running a Matlab implementation code to solve the above nonlinear equation using the secant method and you obtain the following table for the iterations Xk, k = 0,...,4. Compute the errors for each iteration and estimate the order of convergence numerically. Does this agree with the theory? X it 0 0.5000000000000000 1 0.7853981633974483 2 0.7363841388365822 3 0.7390581392138897 4 0.7390851493372764 Method converged in 4 iterations 1. Nonlinear equations). (a) Summarise the secant method for approximating the solution of a nonlinear equation (or rootfinding problem). Hint: Explain how it works geometrically, how it compares to Newton's method and give the formula for the iterations. (b) List a few pros and cons for the secant method. (c) Consider solving the equation f(x) = 0 using the secant method, where f(x) = cos(x)-1, x [0, 5]. Use to = 0.5 and x1 = as initial guesses and apply the secant method to compute the iterates 22 and 13. Hint: these two iterates should agree with x2 and x3 in the table below. (d) Suppose all conditions for the secant method are satisfied. What will be the order of convergence? (e) Suppose you are running a Matlab implementation code to solve the above nonlinear equation using the secant method and you obtain the following table for the iterations Xk, k = 0,...,4. Compute the errors for each iteration and estimate the order of convergence numerically. Does this agree with the theory? X it 0 0.5000000000000000 1 0.7853981633974483 2 0.7363841388365822 3 0.7390581392138897 4 0.7390851493372764 Method converged in 4 iterations