Exercise "RootFinding and Optimization"

1. Implement the Modified Newton's method and the downhill simplex method.
2. Find the minima of the Rosenbrock's banana function, f(x,y)=(1-x)²+100(y-x²)².
3. Find the minima of the Himmelblau's function, f(x,y)=(x²+y-11)²+(x+y²-7)².
4. Solve the system of equations (1-x)=0, 10(y-x²)=0.

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5. Implement the Broyden's quasi-Newton's method.
6. Compare the three methods with respect to the number of calls of the function under optimization.
7. Compare the following two strategies of for expansion (after successful reflection) in the downhill simplex method:
   1. perform "expansion" against the old centroid
   2. perform "expansion" against the new centroid
8. Make some other interesting examples.