For some values of ri the Moran equation can be solved exactly, but often we must solve it numerically. We illustrate both methods for the equation |
0.5d + 0.5d + 0.5d + 0.25d = 1 |
Numerical solution Most graphing calculators and computer algebra systems have routines for numerical solution of equations. For example, in Mathematica the command is |
FindRoot[.5^d + .5^d + .5^d + .25^d == 1,{d,1}] |
The {d,1} specifies that d is the variable, and provides a starting guess of 1 for the solution. |
For this example, the FindRoot command gives the solution 1.72368. |
Exact solution This instance of the Moran equation can be solved
analytically when we note |
3⋅(1/2)d + (1/4)d = 1 |
can be written as |
3⋅(1/2)d + ((1/2)2)d = 1 |
Interchanging the exponents of the second term of the left side we obtain |
3⋅(1/2)d + ((1/2)d)2 = 1 |
Writing |
(1/2)d = x, |
the Moran equation becomes the quadratic equation |
3x + x2 = 1. |
The quadratic formula gives |
x = (-3 ± √(13))/2. |
Recalling |
x = (-3 + √(13))/2 |
and so |
d = Log((-3 + √(13))/2)/Log(1/2). |
This approach can be adopted to any situation in which all the scaling factors are (integer) powers of one of the scaling factors. |
Return to the Moran equation.