Editing Bisection method (section)

=== Methods based on degree computation ===
Some of these methods are based on computing the [[topological degree]], which for a bounded region <math>\Omega \subseteq \mathbb{R}^n</math> and a differentiable function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^n</math> is defined as a sum over its roots:
:<math>\deg(f, \Omega) := \sum_{y\in f^{-1}(\mathbf{0})} \sgn \det(Df(y))</math>,
where <math>Df(y)</math> is the [[Jacobian matrix]], <math>\mathbf{0} = (0,0,...,0)^T</math>, and
:<math>\sgn(x) = \begin{cases}
1, & x>0 \\
0, & x=0 \\
-1, & x<0 \\
\end{cases}</math>
is the [[sign function]].<ref>{{cite journal |last1=Polymilis |first1=C. |last2=Servizi |first2=G. |last3=Turchetti |first3=G. |last4=Skokos |first4=Ch. |last5=Vrahatis |first5=M. N. |journal=Libration Point Orbits and Applications |title=Locating Periodic Orbits by Topological Degree Theory |date=May 2003 |pages=665–676 |doi=10.1142/9789812704849_0031 |arxiv=nlin/0211044 |isbn=978-981-238-363-1 }}</ref> In order for a root to exist, it is sufficient that <math>\deg(f, \Omega) \neq 0</math>, and this can be verified using a [[surface integral]] over the boundary of <math>\Omega</math>.<ref>{{Cite journal |last=Kearfott |first=Baker |date=1979-06-01 |title=An efficient degree-computation method for a generalized method of bisection |url=https://doi.org/10.1007/BF01404868 |journal=Numerische Mathematik |language=en |volume=32 |issue=2 |pages=109–127 |doi=10.1007/BF01404868 |s2cid=122058552 |issn=0945-3245|url-access=subscription }}</ref>