is decreasing on ,
and
is increasing on .
Golden Ratio:
If , then squeeze from the right and use the
new interval and the four points .
If , then squeeze from the left and use the
new interval and the four points .
Figure 1. The decision process for the golden ratio search.
(1) ,
and
(2) ,
where (to preserve the ordering ).
(3) and ,
and
(4) and .
If and only one new function evaluation is to be made in the interval , then we must have
.
Use the facts in (3) and (4) to rewrite this equation and then simplify.
,
,
,
.
Now the quadratic equation can be applied and we get
.
The value we seek is and it is often referred to as the "golden ratio." Similarly, if , then it can be shown that .
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