For the exercise to work correctly you’ll need to carry 16 decimal places – double
Q1) Find the value of x that satisfies
x = ln(2 + x)
Begin with x(0) = 2, and try the iteration scheme
x(k+1) = ln(2 + x(k)):
If this method converges (lets say lim x(k) as =>inf = e), determine it’s order. Repeat the
calculation, and plot log (x(k) -e) on the y axis against log (x(k-1) -e) on the x axis.
Q2) Solve (1) using the Newton-Raphson method, again beginning with x(0) = 2.
Graph log (x(k) -e) on the y axis against log (x(k-1) -e) to determine the order of convergence.
Q3) Repeat Q3 with x(0) = -1.999
Point: for a problem with multiple roots, the solution will depend on the starting
Q4) What happens with Newton-Raphson for -1:62 < x(0) < -1:01?
Point: Newton-Raphson converges rapidly when it converges, but convergence is
Q5) Use the naive method of Q1 to solve (1) beginning with x(0) = -1:5.
Point: Regions of convergence depend on the function and the method.
For#1, am I suppose to just use brute force and iterate?
Also, can someone walk me through a few steps of this method? I am not really understanding how to do this?
The Attempt at a Solution
#1) Using MATLAB, i get x= 1.14619321999999 with a difference of about
4.233406958320529e-010. Is this close enough?