Page 156 - Compact Numerical Methods For Computers

Optimisation and nonlinear equations 145
TABLE 12.1. Nonlinear least-
squares data of example 12.2.

i Y i l
1 5·308
2 7·24
3 9·638
4 12·866
5 17·069
6 23·192
7 31·443
8 38·558
9 50·156
10 62·948
11 75·995
12 91·972

where Z and b are constants. In this simple example, the equations reduce to

or

so that

However, in general, the system will involve more than one commodity and will
not offer a simple analytic solution.
Example 12.4. Root-finding
In the economic analysis of capital projects, a measure of return on investment
that is commonly used is the internal rate of return r. This is the rate of interest
applicable over the life of the project which causes the net present value of the
project at the time of the first investment to be zero. Let y li be the net revenue of
the project, that is, revenue or income minus loss or investment, in the ith time
period. This has a present value at the first time period of
i - 1
y /(1 + 0·01r)
li
where r is the interest rate in per cent per period. Thus the total present value at
the beginning of the first time period is

where K is the number of time periods in the life of the project. By setting
b = 1/(1 + 0·01r)
this problem is identified as a polynomial root-finding problem (12.8).