python./Implementing Newton and Broyden 14mm.

python./Implementing Newton and Broyden 14mm.

Write a program based on Newton’s method to solve the system of nonlinear equations

(x1 + 3)(x§ – 7) + 18 = 0,
sin(xze”l – 1) = 0

wuth starting ponnt xo = [-0.5 1.4]
Write a program based on Broyden’s method to solve the same system with the same starting point. Use the Jacobian at 1:0 as the initial approximate
Jacobian for Broyden’s method. Although in practice you would cheaply update the factorization of the approximate Jacobian, for this problem you
can simply solve a system at each iteration.

Compare the convergence rates of the two methods by computing the error at each iteration, given that the exact solution is x” = [0 1 ] . How
many iterations does each method require to attain full machine precision?
On the same axes. plot the error vs iteration for Newton’s method and Broyden’s method. Compute the error as ”x, – x” I I2 for each iterate xi.
Use a logarithmic scale for the y-axis.
As always, be sure your plot has correct labels, title. legend. etc.
Print how many iterations it took for each method to attain full machine precision. Be sure that you are clear about which method is which. For
example, you should have lines of output that look like “Newton: 10′ and “Broyden: 12”.

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