Applied Numerical Methods
Homework 1
Due by 11:59 pm Monday Oct 9th . Submit your work via Gradescope. For theoretical problems,
you have to show complete work/calculations to demonstrate how you get the answers. For computational problems, you have to submit MATLAB code, with figures (if the problem required you
to).
Problem 1 (Computational)
Read about the vectorization techniques in MATLAB: https://www.mathworks.com/help/matlab/
matlab_prog/vectorization.html, do the following tasks without the use of for-loop. For each
part, your code must be no more than 2 lines.
(a) Construct the following 100 × 1 column vector
1,
1
1
1
1
,
, ··· , 2, ··· ,
22 32
k
1002
T
(Hint: Check the uses of colon (:) for vector creation and period (.) for array operation in MATLAB)
(a) Compute
100
P
sin(2k)
2
k=1 k + 1
(Hint: sum)
Problem 2 (Theoretical)
Use the Bisection method to find a solution accurate to within 10−1 for f (x) = x3 − 7x2 + 14x − 6
on the interval [0, 1].
Problem 3 (Computational)
2
(a) Use MATLAB to plot the graphs of f (x) = x2 − 1 and g(x) = e1−x on the interval [−2, 0].
(b) For the equation f (x) = g(x), use the Bisection method to approximate a root accurate to
within 10−3 on the interval [−2, 0].
1
Problem 4 (Theoretical)
(a) Let f (x) = −x3 − cos x and x0 = −1. Use Newton’s method to find x2 .
(b) Could x0 = 0 be used?
2