Note: you are not permitted to use syms in MATLAB
1. (15 pts.) The following cubic polynomial has three real roots
p(x) = 256×3
+ 1279x 510
(a) Plot p(x) for :9 x 2:1 and indicate the locations of the three roots. What are the
exact roots of p(x)?
(b) Write a MATLAB code to run Newton’s method with x0 = 1:6 and discuss the conver-
(c) Write a MATLAB code to run the secant method starting with x0 = 1:6 and x1 = 1:4
and discuss the convergence?
(d) Write a MATLAB code to run the bisection method with the initial bracket [:9; 2:1] and
discuss the convergence.
2. (10 pts.) Plot the two curves in the uv plane implied by the following systems. Then nd
all solutions exactly using simple algebra.
3. (10 pts.) Write a MATLAB code to perform Newton’s method for the three systems from
problem #2 above. For each system, demonstrate the correctness of your implementation by
showing quadratic convergence.
4. (10 pts.) Using your Newton’s method codes from #3 above, determine an initial guesses
for which the algorithm will converge to each of roots of the system. That is to say that if
the system has p roots, you need to determine initial guesses for which the algorithm will
converge to each of the p roots.