Industrial Mathematics - Alexiades
                      Lab 4
                  Roots and Equilibria

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Find the equilibria of the single-size Ostwald ripening model we have been discussing,
i.e. find the two 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. Then find the other root.
[ Do NOT confuse the initial size x* with initial guess(es) for Newton Method ! ]

Find the roots ξ₁ and ξ₂ (at full double precision: 14 decimals)
  1: when x* = 0.05 .   Verify that ξ₁ < ξ₂ < x* ;
  2: when x* = 0.0975 .   Verify that ξ₁ < x* < ξ₂;
  3: when x* = 0.08 and μ=1.e-5 .   Verify that x* < ξ₁ < ξ₂.

In each case, discuss the physical meaning for the single-size crystal model and what theory predicts.

Submit ONLY the following   in a text file "Lab4.txt":
  Name, date, Lab4
  ======================================================= (separator line)
  roots and discussion for Case 1
  ------------------------------------------------------- (separator line)
  roots and discussion for Case 2
  ------------------------------------------------------- (separator line)
  roots and discussion for Case 3
  ======================================================= (separator line)
  your main program (that calls your Newton solver)
  ======================================================= (separator line)
  your FCN subprogram (that evaluates F and DF)

NOTE:

The parameters pertain only to the function, so should be entered in the FCN subprogram.

You cannot use " * " in variable names in a code! Can use "xstar", "cstar".