Description
6. Consider the function f(x) = e
x − x − 2. In question 2, you showed that this function had
exactly one solution (¯x such that f(¯x) = 0) in [1, 2].
a) Apply the Bisection Methods starting at a
0 = 1 and b
0 = 2 to approximate ¯x to 4 decimal
places. (I.e., iterate until MATLAB reports the first 4 decimals remain unchanged.)
b) Apply the Method of False Position starting at a
0 = 1 and b
0 = 2 to approximate ¯x
to 4 decimal places. (I.e., iterate until MATLAB reports the first 4 decimals remain
unchanged.)
7. Consider the function
g(x) = −4 + 4x −
1
2
x
2
.
a) Starting at x
0 = 1.9, apply a fixed point iterative methods to find x
1 MATH/COSC 303
, x2
, x3
, x4
, x5
.
Computer the relative errors Rk between x
k and the closest fixed point to x
k
for each
iteration k = 0, 1, 2, …5.
b) Starting at x
0 = 2.1, apply a fixed point iterative methods to find x
1
, x2
, x3
, x4
, x5
.
Computer the relative errors Rk between x
k and the closest fixed point to x
k
for each
iteration k = 0, 1, 2, …5.
c) Starting at x
0 = 3.8, apply a fixed point iterative methods to find x
1
, x2
, x3
, x4
, x5
.
Computer the relative errors Rk between x
k and the closest fixed point to x
k
for each
iteration k = 0, 1, 2, …5
d) Starting at x
0 = 4.2, apply a fixed point iterative methods to find x
1
, x2
, x3
, x4
, x5
.
Computer the relative errors Rk between x
k and the closest fixed point to x
k
for each
iteration k = 0, 1, 2, …5
e) Compare your results to the Summarize your observations.
8. Use Netwon’s Root Finding Method (see question 5) to approximate √
27. For each iteration
compute the number of significant digits obtained.
9. Use any numerical method you desire to find x ∈ (0, 1) that solves x
cos(x) = x
x
to at least 12
significant digits.
MATH/COSC 303 Assignment 3
