Find local and global extrema of a function. Extreme points of a function f(x) are normally found by solving f ′ (x) = 0. A much simpler method is to evaluate f(x) for a set of discrete points in the...

Find local and global extrema of a function. Extreme points of a function f(x) are normally found by solving f ′ (x) = 0. A much simpler method is to evaluate f(x) for a set of discrete points in the interval [a, b] and look for local minima and maxima among these points. We work with n equally spaced points a = x_0
1
<>_n−1
= b, x_i
= a + ih, h = (b − a)/(n − 1).

1. First we find all local extreme points in the interior of the domain. Local minima are recognized by

f(x_i−1) > f(x_i)
_i+1), i = 1, . . . , n − 2 .

Similarly, at a local maximum point x_i
we have

f(x_i−1)
_i) > f(x_i+1), i = 1, . . . , n − 2 .

We let P_min
be the set of x values for local minima and F_min
the set of the corresponding f(x) values at these minimum points. Two sets P_max
and F_max
are defined correspondingly, containing the maximum points and their values.

2. The boundary points x = a and x = b are for algorithmic simplicity also defined as local extreme points: x = a is a local minimum if f(a)
₁), and a local maximum otherwise. Similarly, x = b is a local minimum if f(b)
_n−2), and a local maximum otherwise. The end points a and b and the corresponding function values must be added to the sets P_min, P_max, F_min, F_max.

3. The global maximum point is defined as the x value corresponding to the maximum value in F_max. The global minimum point is the x value corresponding to the minimum value in F_min.

Make a class MinMax with the following functionality:

ˆ The constructor takes f(x), a, b, and n as arguments, and calls

a method _find_extrema to compute the local and global extreme

points.

ˆ The method _find_extrema implements the algorithm above

for finding local and global extreme points, and stores the sets

Pmin, Pmax, Fmin, Fmax as list attributes in the (self) instance.

ˆ The method get_global_minimum returns the global minimum point

(x).

ˆ The method get_global_maximum returns the global maximum point

(x).

ˆ The method get_all_minima returns a list or array of all minimum

points.

ˆ The method get_all_maxima returns a list or array of all maximum

points.

ˆ The method __str__ returns a string where all the min/max points

are listed, plus the global extreme points.

Here is a sample code using class MinMax:

def f(x):

return x**2*exp(-0.2*x)*sin(2*pi*x)

m = MinMax(f, 0, 4)

print m

 The output becomes

All minima: 0.8056, 1.7736, 2.7632, 3.7584, 0

All maxima: 0.3616, 1.284, 2.2672, 3.2608, 4

Global minimum: 3.7584

Global maximum: 3.2608

Make sure that the program also works for functions without local extrema, e.g., linear functions f(x) = px + q. Name of program file: minmaxf.py.