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=== Interval Newton ===
The fixed points of the Newton operator [[Newton's method]] for a square system of functions <math>F:\mathbb{C}^n\rightarrow\mathbb{C}^n</math> correspond to the roots of <math>f</math>. In more generality, suppose that <math>G:\mathbb{C}^n\rightarrow\mathbb{C}^n</math> is a function whose fixed points correspond to the roots of <math>F</math>. Interval Newton {{citation needed|reason=don't have time to figure out how to reference a book right now|November 9, 2018}}, Krawczyk {{citation needed|reason=don't have time to figure out how to reference a book right now|November 9, 2018}}, and related methods use such functions to establish the existence and uniqueness of roots within a region <math>I</math> using topological methods:
# 1 If <math>G</math> maps a region <math>I</math> into itself, then by [[Brouwer fixed-point theorem]], <math>G</math> has at least one fixed point in <math>I</math>, and, hence <math>F</math> has at least one root in <math>I</math>.
# 2 If <math>F</math> is a [[contraction mapping]] in a region <math>I</math>, then there is at most one root of <math>F</math> in <math>I</math>.
The basic form of Newton's method replaces the input in the iterator with an interval approximation. In other words
<math display="block">N_f(I)=m(I)+J_F(I)^{-1}F(m(I))</math>
is contained within <math>I</math>, where <math>m(I)</math> is the midpoint of <math>I</math> and <math>J_f(I)</math> is an over-approximation to the set-image of the Jacobian[[Jacobian matrix and determinant]] of <math>F</math> on <math>I</math>.
=== Miranda test ===
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