I have a quick question. I have a system of residuals in the form of

$$R_i= (\sum_{j=1}^n a_{ij}x_j)-b_i$$

where $a_{ij}$ and $b_{i}$ are constants. I am trying to show that applying Newton-Raphson to this system of equations will converge to a solution in one iteration, independent of the initial guess. Does anyone have any ideas on demonstrating this?

  • $\begingroup$ The linear approximation to a linear function is the same function... $\endgroup$ – Ian Sep 19 '18 at 6:16
  • $\begingroup$ Right, I just have a hard time proving this mathematically with a system of equations. $\endgroup$ – Patrick Sep 19 '18 at 7:18
  • $\begingroup$ Well, show the Jacobian of $f(x)=Ax-b$ is $A$, then check $A(x_0-A^{-1}(Ax_0-b))=b$. In the end this is exactly what I said, the linear approximation of $f(x)$ at any $x_0$ is just $f(x)$ itself. So because Newton's method sets the linear approximation equal to zero in each step, $f$ winds up equal to zero in the first step too. $\endgroup$ – Ian Sep 19 '18 at 14:41

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