I have gone over these problems and I am stuck. If anyone can help me at any part of this, it will be appreciated.
4.3.1 Convergence of the bisection method
Implement the bisection method in a computer code, and compute the roots of the quadratic equation
x2 − 2 x + 0.9 = 0. Prepare and discuss a graph of the error against the iteration count, k.
4.5.2 Newton’s method
(a) Compute all zeros of the function f(x) = ln x+3−3.1 x2, accurate to the eighth decimal place.
Explain your choice of initial guess.
4.5.3 More on Newton’s method
(a) The function f(x) = x ln x has a root at x = 0. What is the rate of convergence of Newton’s
method toward this root?
4.5.4 RedlichKwong equation of state
Write a program that produces and prints a table showing the molecular volume of hydrogen for
fifteen combinations corresponding to pressure p = 1, 2, 3, 4, and 5 atm and temperature T = 200,
300, and 400 ◦K, based on the RedlichKwong equation of state (4.1.10). For the initial guess, use the
predictions of the ideal gas law. Discuss the physical significance of your results. Perry’s Chemical
Engineer’s Handbook (McGrawHill, fifth edition, pp. 3–41, 3–104) gives the following information
for hydrogen: Chemical formula: H2; Boiling Point at 1 atm: −252.7◦C; critical conditions: Tc =
−239.9◦C; Pc = 12.8 atm.
4.5.5 Viscous flow in a corner
The nonlinear equation
sin[2(x − 1)] = (1 − x) sin(2), (1)
describes viscous flow in a corner bounded by two intersecting walls with aperture angle 2; the
variable x is a measure of the strength of the flow. A trivial solution for any is x = 1. Find and
plot another solution branch, X(), in the range 0 < < .
4.6.4 A system of two equations
Compute one solution of the system
(x − 2)2 + (y − 3)3 + (x − 2.1)(y − 3.1) = 2.81, 10 e−x + 5 e1−y = 0.7468, (1)
using (a) Newton’s method, and (b) Newton’s method with the Jacobian evaluated only at the
beginning and then held constant. Compare the respective rates of convergence.
