線性化

函數的線性化

f(x)=x^2在(x, f(x))的近似值

${\displaystyle y=(f(a)+f'(a)(x-a))}$

多變數函數的線性化

${\displaystyle f(x,y)\approx f(a,b)+\left.{\frac {\partial f(x,y)}{\partial x}}\right|_{a,b}(x-a)+\left.{\frac {\partial f(x,y)}{\partial y}}\right|_{a,b}(y-b)}$

${\displaystyle f({\mathbf {x} })\approx f({\mathbf {p} })+\left.{\nabla f}\right|_{\mathbf {p} }\cdot ({\mathbf {x} }-{\mathbf {p} })}$

線性化的應用

${\displaystyle {\frac {d\mathbf {x} }{dt}}=\mathbf {F} (\mathbf {x} ,t)}$,

${\displaystyle {\frac {d\mathbf {x} }{dt}}\approx \mathbf {F} (\mathbf {x_{0}} ,t)+D\mathbf {F} (\mathbf {x_{0}} ,t)\cdot (\mathbf {x} -\mathbf {x_{0}} )}$

參考資料

1. ^ The linearization problem in complex dimension one dynamical systems at Scholarpedia. [2020-04-10]. （原始内容存档于2018-07-04）.
2. ^
3. ^ Leonov, G. A.; Kuznetsov, N. V. Time-Varying Linearization and the Perron effects. International Journal of Bifurcation and Chaos. 2007, 17 (4): 1079–1107. doi:10.1142/S0218127407017732.
4. Moffatt, Mike. (2008) Dotdash State-Space Approach页面存档备份，存于互联网档案馆 Economics Glossary; Terms Beginning with S. Accessed June 19, 2008.
5. ^ Bagwell, S.; Ledger, P. D.; Gil, A. J.; Mallett, M.; Kruip, M. A linearised hp–finite element framework for acousto-magneto-mechanical coupling in axisymmetric MRI scanners. International Journal for Numerical Methods in Engineering. 2017, 112 (10): 1323–1352. doi:10.1002/nme.5559.