Newton’s Method for linearly dependent system of equations

linear algebranewton raphsonnumerical methodssolution-verificationsystems of equations

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 !

Best Answer

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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