不同方法的微分公式

数值微分方法	${{{\text{d}}f} \mathord{\left/ {\vphantom {{{\text{d}}f} {{\text{d}}x}}} \right. } {{\text{d}}x}} = \left( \cdot \right)$	截断误差	${{{{\text{d}}^2}f} \mathord{\left/ {\vphantom {{{{\text{d}}^2}f} {{\text{d}}{x^2}}}} \right. } {{\text{d}}{x^2}}} = \left( \cdot \right)$	截断误差
FDM	$\frac{{f(x + h) - f(x)}}{h}$	$O(h)$	$\frac{{f(x + 2h) - 2f(x + h) + f(x)}}{{{h^2}}}$	$O(h)$
CDM	$\frac{{f(x + h) - f(x - h)}}{{2h}}$	$O({h^2})$	$\frac{{f(x + h) - 2f(x) + f(x - h)}}{{{h^2}}}$	$O({h^2})$
CSDA	$\frac{{{\Re _i}\left[ {f\left( {x + ih} \right)} \right]}}{h}$	$O({h^2})$	$\frac{2}{{{h^2}}}\left[ {f\left( x \right) - {\Re _i}\left[ {f\left( {x + ih} \right)} \right]} \right]$	$O({h^2})$
HDSDA	$\frac{{{\Im _{{\varepsilon _1}}}\left[ {f\left( {x + {h_{\text{1}}}{\varepsilon _1} + {h_{\text{2}}}{\varepsilon _2}} \right)} \right]}}{{{h_{\text{1}}}}}$	无	$\frac{{{\Im _{{\varepsilon _1}{\varepsilon _2}}}\left[ {f\left( {x + {h_{\text{1}}}{\varepsilon _1} + {h_{\text{2}}}{\varepsilon _2}} \right)} \right]}}{{{h_{\text{1}}}{h_{\text{2}}}}}$	无

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数值微分方法	${{{\text{d}}f} \mathord{\left/ {\vphantom {{{\text{d}}f} {{\text{d}}x}}} \right. } {{\text{d}}x}} = \left( \cdot \right)$	截断误差	${{{{\text{d}}^2}f} \mathord{\left/ {\vphantom {{{{\text{d}}^2}f} {{\text{d}}{x^2}}}} \right. } {{\text{d}}{x^2}}} = \left( \cdot \right)$	截断误差
FDM	$\frac{{f(x + h) - f(x)}}{h}$	$O(h)$	$\frac{{f(x + 2h) - 2f(x + h) + f(x)}}{{{h^2}}}$	$O(h)$
CDM	$\frac{{f(x + h) - f(x - h)}}{{2h}}$	$O({h^2})$	$\frac{{f(x + h) - 2f(x) + f(x - h)}}{{{h^2}}}$	$O({h^2})$
CSDA	$\frac{{{\Re _i}\left[ {f\left( {x + ih} \right)} \right]}}{h}$	$O({h^2})$	$\frac{2}{{{h^2}}}\left[ {f\left( x \right) - {\Re _i}\left[ {f\left( {x + ih} \right)} \right]} \right]$	$O({h^2})$
HDSDA	$\frac{{{\Im _{{\varepsilon _1}}}\left[ {f\left( {x + {h_{\text{1}}}{\varepsilon _1} + {h_{\text{2}}}{\varepsilon _2}} \right)} \right]}}{{{h_{\text{1}}}}}$	无	$\frac{{{\Im _{{\varepsilon _1}{\varepsilon _2}}}\left[ {f\left( {x + {h_{\text{1}}}{\varepsilon _1} + {h_{\text{2}}}{\varepsilon _2}} \right)} \right]}}{{{h_{\text{1}}}{h_{\text{2}}}}}$	无