Geodesic
Newton's method and a~surface with Jacobian zero
Matematičeskie zametki, Tome 18 (1975) no. 2, pp. 253-260

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Suppose a system of nonlinear real equations $P(x)=0$, where $P$ and $x$ are n-dimensional vectors, is solved by means of the continuous analog of Newton's method. We study the behavior of the method near the surface $S$ with Jacobian zero: $S=\{x|det P'(x)=0\}$. A computational strategy is suggested in the case where the method diverges. 
@article{MZM_1975_18_2_a10,
     author = {E. I. Lin'kov},
     title = {Newton's method and a~surface with {Jacobian} zero},
     journal = {Matemati\v{c}eskie zametki},
     pages = {253--260},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_18_2_a10/}
}
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E. I. Lin'kov. Newton's method and a~surface with Jacobian zero. Matematičeskie zametki, Tome 18 (1975) no. 2, pp. 253-260. http://geodesic.mathdoc.fr/item/MZM_1975_18_2_a10/