Consider the following system of nonlinear equations X1 X2 x3 = cos(x1) 81 sin (23) 3...

  

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Consider the following system of nonlinear equations X1 X2 x3 = cos(x1) 81 sin (23) 3 + sin(x₁) cos(x3) 3 3 cos(x1) X2 sin (F3) + 9 3 6 + Perform three iterations by hand of successive substitution (using the obvious x = g(x) equations already given) starting from x = [1 1 1]. Compute the error relative to the exact solution x = [0 1/3 0]. • Problem 2 (2 points) Perform by hand three iterations by hand of the Newton-Raphson method to solve the same system of Problem 1, again starting from x = [1 1 1]. Compute the error relative to the exact solution. Which method performs better? Explain your answer. • Problem 3 (2 points) Solve the system of Problem 1 using a Python program: 1. Write a Python code to perform the successive iteration of Problem 1. Execute the iteration until the change in the norm of the solution relative to the previous iteration falls below = 1 x 10-4 (that is, until ||x(+¹)x(k)||/x(k)|| <e). Report the solution to the system and how many iterations it took for the method to converge. Turn in the Python code for your successive substitution implementation, and the screen output of the code. 2. Use the function scipy.optimize.fsolve to solve the system of Problem 1. Use the argument xtol to set the tolerance on the relative change in the norm of the solution to 1 x 10-4. This will allow a comparison with your successive substitution code. Report the solution and the number of function evaluations. Turn in your code. Which of the two codes converged faster? • Bonus problem (2 points): Prove that the Newton-Raphson method for solving a system of nonlinear equations converges in one iteration when the equations are linear (Hint: consider f(x) = Ax-b=0), assuming A is nonsingular. Please recall that a single example does not constitute a proof. Consider the following system of nonlinear equations X1 X2 x3 = cos(x1) 81 sin (23) 3 + sin(x₁) cos(x3) 3 3 cos(x1) X2 sin (F3) + 9 3 6 + Perform three iterations by hand of successive substitution (using the obvious x = g(x) equations already given) starting from x = [1 1 1]. Compute the error relative to the exact solution x = [0 1/3 0]. • Problem 2 (2 points) Perform by hand three iterations by hand of the Newton-Raphson method to solve the same system of Problem 1, again starting from x = [1 1 1]. Compute the error relative to the exact solution. Which method performs better? Explain your answer. • Problem 3 (2 points) Solve the system of Problem 1 using a Python program: 1. Write a Python code to perform the successive iteration of Problem 1. Execute the iteration until the change in the norm of the solution relative to the previous iteration falls below = 1 x 10-4 (that is, until ||x(+¹)x(k)||/x(k)|| <e). Report the solution to the system and how many iterations it took for the method to converge. Turn in the Python code for your successive substitution implementation, and the screen output of the code. 2. Use the function scipy.optimize.fsolve to solve the system of Problem 1. Use the argument xtol to set the tolerance on the relative change in the norm of the solution to 1 x 10-4. This will allow a comparison with your successive substitution code. Report the solution and the number of function evaluations. Turn in your code. Which of the two codes converged faster? • Bonus problem (2 points): Prove that the Newton-Raphson method for solving a system of nonlinear equations converges in one iteration when the equations are linear (Hint: consider f(x) = Ax-b=0), assuming A is nonsingular. Please recall that a single example does not constitute a proof.

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For Problem 3 you can solve the system of equations using Python and the SciPy library Heres an outl... View the full answer