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suppose to have a function $F(x,y,z) = [ f_1(x,y,z),f_2(x,y,z),f_3(x,y,z)]$ and that $f_1$ depend only by x, $f_2$ depends only by y and $f_3$ depends only by z. Now if I apply newton method I can write $[d_x,d_y,d_z] = -J^{-1}(x,y,z)*F(x,y,z)$.

The question is if I exchange the rows of $F$ for example in this way $F_{new}(x,y,z) = [ f_2(x,y,z),f_1(x,y,z),f_3(x,y,z)]$ it seems I can write in the same way this $[d_x,d_y,d_z] = -J^{-1}(x,y,z)*F_{new}(x,y,z)$ , but intuitively I expect something like this $[d_y,d_x,d_z] = -J^{-1}(x,y,z)*F_{new}(x,y,z)$.

Can you please explain me why the inversion of rows doesn't affect the steps retrieved?

Carl
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1 Answers1

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This is because the permutation that swaps these rows is also applied to the Jacobian. Let this permutation be $P$, then we have $$ F_{new} = PF\implies J_{new} = PJ. $$ The Newton step $s_{new}$ is then $$ s_{new} = -J_{new}^{-1}F_{new}(x_k) = -J^{-1}P^{-1}PF = -J^{-1}F(x_k) = s $$

whpowell96
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