數值微分

有限差分法

${\displaystyle {f(x+h)-f(x) \over h}.}$

${\displaystyle f'(x)=\lim _{h\to 0}{f(x+h)-f(x) \over h}.}$

${\displaystyle {f(x+h)-f(x-h) \over 2h}.}$

${\displaystyle R={{-f^{(3)}(c)} \over {6}}h^{2}}$ ,

利用浮點數的實際考量

${\displaystyle h=2{\sqrt {\varepsilon |{f(x) \over f''(x)}|}}}$  [8] （不過f"(x) = 0時不成立），而且需要有關函數的資訊。

 h:=sqrt(eps)*x;
xph:=x + h;
dx:=xph - x;
slope:=(F(xph) - F(x))/dx;


高階方法

${\displaystyle f'(x)={\frac {-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)}{12h}}+{\frac {h^{4}}{30}}f^{(5)}(c)}$

複變的方法

${\displaystyle f'(x)\approx \Im (f(x+ih))/h}$ .

${\displaystyle f^{(n)}(a)={n! \over 2\pi i}\oint _{\gamma }{f(z) \over (z-a)^{n+1}}\,\mathrm {d} z}$ ,

Lyness和Moler在1967年提出用複變數來計算數值微分[14]。Abate和Dubner提出一種用複數拉普拉斯轉換的數值反演為基礎的算法[15]

參考資料

1. ^ Richard L. Burden, J. Douglas Faires (2000), Numerical Analysis, (7th Ed), Brooks/Cole. ISBN 0-534-38216-9
2. ^ Katherine Klippert Merseth. Windows on Teaching Math: Cases of Middle and Secondary Classrooms. Teachers College Press. 2003: 34. ISBN 978-0-8077-4279-2.
3. ^ Tamara Lefcourt Ruby; James Sellers; Lisa Korf; Jeremy Van Horn; Mike Munn. Kaplan AP Calculus AB & BC 2015. Kaplan Publishing. 2014: 299. ISBN 978-1-61865-686-5.
4. ^ Andreas Griewank; Andrea Walther. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Second Edition. SIAM. 2008: 2–. ISBN 978-0-89871-659-7.
5. ^ Numerical Differentiation of Analytic Functions, B Fornberg - ACM Transactions on Mathematical Software (TOMS), 1981
6. Using Complex Variables to Estimate Derivatives of Real Functions, W Squire, G Trapp - SIAM REVIEW, 1998
7. ^ Following Numerical Recipes in C, Chapter 5.7
8. ^ p. 263 [1]
9. ^ Abramowitz & Stegun, Table 25.2
10. ^ Differential Quadrature and Its Application in Engineering: Engineering Applications, Chang Shu, Springer, 2000, ISBN 978-1-85233-209-9
11. ^ Advanced Differential Quadrature Methods, Yingyan Zhang, CRC Press, 2009, ISBN 978-1-4200-8248-7
12. ^ Martins, JRRA; Sturdza, P; Alonso, JJ. The Complex-Step Derivative Approximation. ACM Transactions on Mathematical Software. 2003, 29 (3): 245–262. doi:10.1145/838250.838251. CiteSeerX: 10.1.1.141.8002.
13. ^ 存档副本 (PDF). [2012-11-24]. （原始内容 (PDF)存档于2014-01-09）.
14. ^ Lyness, J. N.; Moler, C. B. Numerical differentiation of analytic functions. SIAM J.Numer. Anal. 1967, 4: 202–210. doi:10.1137/0704019.
15. ^ Abate, J; Dubner, H. A New Method for Generating Power Series Expansions of Functions. SIAM J. Numer. Anal. March 1968, 5 (1): 102–112. doi:10.1137/0705008.