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14.5 Divide and Conquer for Finding One Dimensional Extrema

The divide and conquer algorithm for solving equations discussed in the last chapter can be applied to finding extrema of functions of one variable, and therefore of functions defined parametrically on a curve.

Suppose for example you seek a local maximum of F on a curve.

To get started you need to find three values of the parameter used to define the curve, such that the value of F is greater at the point on the curve defined by the middle parameter value than it is at the other two points.

Suppose these parameter values are a , b and c . You can then examine the function F for parameter values a + b 2 and b + c 2 . The maximum of the value of F at these two points and at b will be the center of three parameter values with the same property as a , b and c , namely the value of F at the middle one greater than its value at the others, but the size of the larger parameter interval between the center one and another will go down by a factor of two after this step.

Iterating will close in on a solution exactly as in the zero finding algorithm.

Exercises:

14.9 Explain why the maximum of the value of F at a + b 2 , b + c 2 and b is larger than the two values at point closest around it among these and a and c .

14.10 You can divide the domain of the parameter into intervals of size d and make a list of values of F at the points on the curve between these intervals. From this you can locate local maxima on the list of values, and start this procedure at each. Set up a spreadsheet to do this for the curve x = 5 sin ⁡ t , y = 3 cos ⁡ 3 t , z = sin ⁡ 2 t , for t = 0 to 2 π , and F = x 2 + y 2 + z 2 . Find the local maxima to 5 decimal places.

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