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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 Redlich-Kwong 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 Redlich-Kwong 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 (McGraw-Hill, 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.
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