Industrial Mathematics - Alexiades
                      Lab 4
                  Roots and Equilibria
Find the equilibria of the single-size Ostwald ripening model we have been discussing,
i.e. find the roots of the equation

      μ x3 + c* exp( Γ/x ) = c1 ,   where  c1 := c0 + μ (x*)3,  

with parameter values:   μ=1.e-3,   c*=7.52e-7,   Γ=4.e-3,   c0= 1.05 c*.

For a specified x* (see below), your code should solve the equation for x (by calling your Newton solver), and print it out.
[ Do NOT confuse the initial size x* with initial guess(es) for Newton Method ! ]

Find the roots x1 and x2 (at full double precision: 15 decimals)
  1: when x* = 0.05 .   Verify that x1 < x2 < x* ;
  2: when x* = 0.0975 .   Verify that x1 < x* < x2;
  3: when x* = 0.08 and μ=1.e-5 .   Verify that x* < x1 < x2.

Discuss the physical meaning of these results for the single-size crystal model and what theory predicts for each case.

Submit ONLY the following:
  roots and discussion
  ========================================================== (separator line)
  your main program (that calls your Newton solver)
  ========================================================== (separator line)
  your FCN subprogram

  • The parameters pertain only to the function, so should be entered in the FCN subprogram.
  • You cannot use " * " in variable names in a code! Could use "xstar", "cstar", or some other reasonable names...