Exercise "Root Finding"

1. (6 points) Modified Newton's method with back-tracking linesearch

   • Implement the Modified Newton's method (with the simple backtracking linesearch algorithm) with numerical evaluation of the derivatives in the Jacobian matrix.

   • Solve the system of equations

     
     A x y = 1 ,
     
     exp(-x) + exp(-y) = 1 + 1/A, 
     
     

     where A=10000.

   • Find the minimum of the Rosenbrock's valley function,

     
     f(x,y) = (1-x)2+100(y-x2)2
     

     by searching for the roots of its gradient.

   • Find the minimum of the Himmelblau's function

     
     f(x,y) = (x2+y-11)2+(x+y2-7)2
     

     by searching for the roots of its gradient.

   • Make some other interesting examples.

2. (3 points)

   Implement the version of the modified Newton's method where the derivatives are supplied by the user.

   Find out the effectiveness of this implementation (as compared with the previous implementation) on some interesting examples by counting the number of function calls.

3. (1 point)

   Implement a more refined linear search with, say, quadratic interpolation.

   Compare the number of function evaluations using the refined linear search and the simple backtracking.

4. (0 points)

   Implement a linear search with qubic interpolation.