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## Homework Statement

For the exercise to work correctly you’ll need to carry 16 decimal places – double

precision.

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

point.

Q4) What happens with Newton-Raphson for -1:62 < x(0) < -1:01?

Point: Newton-Raphson converges rapidly when it converges, but convergence is

not guaranteed.

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.

## Homework Equations

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?