Editing Newton's method (section)

===={{mvar|k}} variables, {{mvar|k}} functions{{anchor|multidimensional}}====
One may also use Newton's method to solve systems of {{mvar|k}} equations, which amounts to finding the (simultaneous) zeroes of {{mvar|k}} continuously differentiable functions <math>f:\R^k\to \R.</math> This is equivalent to finding the zeroes of a single vector-valued function <math>F:\R^k\to \R^k.</math> In the formulation given above, the scalars {{mvar|x<sub>n</sub>}} are replaced by vectors {{math|'''x'''{{sub|{{var|n}}}}}} and instead of dividing the function {{math|{{var|f}}({{var|x}}{{sub|{{var|n}}}})}} by its derivative {{math|{{var|{{prime|f}}}}({{var|x}}{{sub|{{var|n}}}})}} one instead has to left multiply the function {{math|{{var|F}}('''x'''{{sub|{{var|n}}}})}} by the inverse of its {{math|{{var|k}} × {{var|k}}}} [[Jacobian matrix]] {{math|{{var|J}}{{sub|{{var|F}}}}('''x'''{{sub|{{var|n}}}})}}.<ref name=":3">{{Cite book |last1=Burden |first1=Burton |url=https://archive.org/details/numericalanaly00burd/ |title=Numerical Analysis |last2=Fairs |first2=J. Douglas |last3=Reunolds |first3=Albert C |date=July 1981 |publisher=Prindle, Weber & Schmidt |isbn=0-87150-314-X |edition=2nd |location=Boston, MA, United States |oclc=1036752194 |pages=448–452 |language=en}}</ref><ref>{{Cite book |last= Evans |first=Gwynne A. |url-access= registration|url=https://archive.org/details/practicalnumeric0000evan/ |title=Practical Numerical Analysis |date=1995 |publisher=John Wiley & Sons|isbn=0471955353 |location= Chichester |publication-date=1995 |pages=30–33 |language=en | oclc=1319419671 }}</ref><ref>{{Cite book |last1=Demidovich |first1=Boris Pavlovich |url=https://archive.org/details/computational-mathematics/mode/2up |title=Computational Mathematics |last2=Maron |first2=Isaak Abramovich |date=1981 |publisher=MIR Publishers |isbn=9780828507042 |edition=Third |location=Moscow  |pages=460–478 |language=en}}</ref> This results in the expression

<math display="block">\mathbf{x}_{n+1} = \mathbf{x}_{n} - J_F(\mathbf{x}_n)^{-1} F(\mathbf{x}_n) .</math>

or, by solving the [[system of linear equations]]

<math display="block">J_F(\mathbf{x}_n) (\mathbf{x}_{n+1} - \mathbf{x}_n) = -F(\mathbf{x}_n)</math>

for the unknown {{math|'''x'''{{sub|{{var|n}} + 1}} − '''x'''{{sub|{{var|n}}}}}}.<ref>{{cite book |last1=Kiusalaas |first1=Jaan |title=Numerical Methods in Engineering with Python 3 |date=March 2013 |publisher=Cambridge University Press |location=New York |isbn=978-1-107-03385-6 |pages=175–176 |edition=3rd |url=https://www.cambridge.org/9781107033856}}</ref>