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The linear system defining one step of Newton's method for $F(x)=0$ is the displayed first-order correction equation. A nonsingular Jacobian matrix gives a unique Newton correction. For an unconstrained stationary-point problem $F=\nabla f$, the system uses the Hessian matrix in place of $DF$.

At the current exact path point $v(\mu )$, stationarity is already zero and $Cy=\mu e$. Therefore $F_{\bar{\mu }}(v(\mu ))=(0,(\mu -\bar{\mu })e{)}^T$. Its Newton correction solvesThe matrix has no explicit dependence on the barrier parameter, so comparison with part (iv) giveswhen $J$ is nonsingular. If it is singular, the displayed vector is still a solution of the correction equations, but uniqueness of the correction cannot be inferred from differentiability of the path alone.

This step is exactly the tangent predictor in the stated first-order Taylor expansion. For nearby positive $\bar{\mu }$, it approximates the new path point with error $o(|\bar{\mu }-\mu |)$; a bounded second path derivative improves this to $O((\bar{\mu }-\mu {)}^2)$. Continuity also preserves strict positivity for sufficiently small parameter changes. Larger changes may require damping to keep both the constraint values and multiplier estimates positive.