is decreasing on ,

and

is increasing on .

**Golden Ratio:**

*and are less than .*

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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