** 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 x**^{3} + c* exp( Γ/x ) = c_{1} , where **c**_{1} := c_{0} + μ (x*)^{3},

with parameter values:
**μ=1.e-3, c*=7.52e-7, Γ=4.e-3, c**_{0}= 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

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your main program (that calls your Newton solver)

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your FCN subprogram

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!
Could use "xstar", "cstar", or some other reasonable names...