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We present a class of Newton-like methods to enclose solutions of systems of nonlinear equations. Theorems are derived concerning the feasibility of the method, its global convergence, its speed and the quality of enclosure.

For contractive interval functions $\left[g\right]$ we show that $\left[g\right]\left({\left[x\right]}_{ϵ}^{k_0}\right)\subseteq \int \left({\left[x\right]}_{ϵ}^{k_0}\right)$ results from the iterative process ${\left[x\right]}^{k+1}:=\left[g\right]\left({\left[x\right]}_{ϵ}^k\right)$ after finitely many iterations if one uses the epsilon-inflated vector ${\left[x\right]}_{ϵ}^k$ as input for $\left[g\right]$ instead of the original output vector ${\left[x\right]}^k$. Applying Brouwer’s fixed point theorem, zeros of various mathematical problems can be verified in this way.

A necessary and sufficient to guarantee feasibility of the interval Gaussian algorithms for a class of matrices. We apply the interval Gaussian algorithm to an $n\times n$ interval matrix $\left[A\right]$ the comparison matrix $〈\left[A\right]〉$ of which is irreducible and diagonally dominant. We derive a new necessary and sufficient criterion for the feasibility of this method extending a recently given sufficient criterion.