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Givens is this system of equation: $$+2x + 5y = -8$$ $$-2x - 5y = 8$$

I'm asked:

  1. Whether the system has a unique solutions
  2. Whether Newton's Method converges after only one iteration
    Well, for the first question: the system has of course infinitely many solutions because the system of equations is linearly dependent. I'm a bit unsure for the second question however. If there are infinitely many solutions, wouldn't Newton's Method in fact never converge ?


Any help/ hint would be greatly appreciated. Thanks for your help !

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    $\begingroup$ Doesn't Newton's method for solving $f(x) = 0$ simply fail if we arrive at a point where the Jacobian matrix $f'(x)$ is not invertible? In the example given in the question, Newton's method fails on the first iteration. $\endgroup$
    – littleO
    Mar 28 '20 at 11:47
  • $\begingroup$ @littleO Just that I understand it correctly: the system is linearly dependent, thus the Jacobian is already non invertible from the beginning. Thus we cannot apply Newton's Method at all, and so the question two is basically false from the beginning, since we cannot use Newton's Method at all. Is that correct ? $\endgroup$
    – Ryukyu
    Mar 28 '20 at 12:13
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    $\begingroup$ Yes, that's correct. $\endgroup$
    – littleO
    Mar 28 '20 at 12:28
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As requested, a hint: What is the Jacobi matrix of this system and what is its inverse? You need that to compute the Newton step.

One could of course also define the step using a pseudo-inverse, but that is then no longer the standard Newton method..

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