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| In mathematics, to approximate a derivative to an arbitrary order of accuracy, it is possible to use the [[finite difference]]. A finite difference can be '''central''', '''forward''' or '''backward'''.
| | I'm Eldon and I live in Preston South. <br>Ι'm intеrested іn Occupational Therapy, Crocheting аnd Danish art. I lіke travelling ɑnd reading fantasy.<br><br>Heгe is my web рage; asvab for marines; [http://www.hm38hm.com/asvab-practice-test/ www.hm38hm.com], |
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| ==Central finite difference==
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| This table contains the coefficients of the central differences, for several order of accuracy:<ref name=fornberg>{{Citation | last1=Fornberg | first1=Bengt | title=Generation of Finite Difference Formulas on Arbitrarily Spaced Grids | doi=10.1090/S0025-5718-1988-0935077-0 | year=1988 | journal=[[Mathematics of Computation]] | issn=0025-5718 | volume=51 | issue=184 | pages=699–706}}.</ref>
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| <!-- replaces Image:Coeff_der_cent_eng.jpg -->
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| {| class="wikitable" style="text-align:center"
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| |-
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| ! Derivative
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| ! Accuracy
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| ! −4
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| ! −3
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| ! −2
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| ! −1
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| ! 0
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| ! 1
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| ! 2
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| ! 3
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| ! 4
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| |-
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| | rowspan="4" | 1
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| || 2 || || || || −1/2 || 0|| 1/2|| || ||
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| |-
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| || 4 || || || 1/12 || −2/3 || 0|| 2/3|| −1/12 || ||
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| |-
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| || 6 || || −1/60 || 3/20 || −3/4 || 0 || 3/4 || −3/20 || 1/60 ||
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| |- style="border-bottom: 2px solid #aaa;"
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| || 8 ||1/280 || −4/105 || 1/5 || −4/5 || 0 || 4/5 || −1/5 || 4/105 || −1/280
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| |-
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| | rowspan="4" | 2
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| || 2 || || || || 1 || −2|| 1|| || ||
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| |-
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| || 4 || || || −1/12 || 4/3 || −5/2|| 4/3|| −1/12 || ||
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| |-
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| || 6 || || 1/90 || −3/20 || 3/2 || −49/18 || 3/2 || −3/20 || 1/90 ||
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| |- style="border-bottom: 2px solid #aaa;"
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| || 8 ||−1/560 || 8/315 || −1/5 || 8/5 || −205/72 || 8/5 || −1/5 || 8/315 || −1/560
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| |-
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| | rowspan="3" | 3
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| || 2 || || || −1/2 || 1 || 0|| −1|| 1/2 || ||
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| |-
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| || 4 || || 1/8 || −1 || 13/8 || 0|| −13/8|| 1 || −1/8 ||
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| |- style="border-bottom: 2px solid #aaa;"
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| || 6 || −7/240 || 3/10 || −169/120 || 61/30 ||0 || −61/30|| 169/120 || −3/10 || 7/240
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| |-
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| | rowspan="3" | 4
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| || 2 || || || 1 || −4 || 6|| −4|| 1 || ||
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| |-
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| || 4 || || −1/6 || 2 || −13/2 || 28/3|| −13/2|| 2 || −1/6 ||
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| |-
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| || 6 || 7/240 || −2/5 || 169/60 || −122/15 ||91/8 || −122/15|| 169/60 || −2/5 || 7/240
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| |}
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| For example, the third derivative with a second-order accuracy is
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| : <math>\displaystyle f'''(x_{0}) \approx \displaystyle \frac{-\frac{1}{2}f(x_{-2}) + f(x_{-1}) -f(x_{+1}) +\frac{1}{2}f(x_{+2})}{h^3_x} + O\left(h_x^2 \right) </math>
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| where <math> h_x </math> represents a uniform grid spacing between each finite difference interval.
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| ==Forward and backward finite difference==
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| This table contains the coefficients of the forward differences, for several order of accuracy:<ref name=fornberg/>
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| {| class="wikitable" style="text-align:center"
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| |-
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| ! Derivative
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| ! Accuracy
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| ! 0
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| ! 1
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| ! 2
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| ! 3
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| ! 4
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| ! 5
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| ! 6
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| ! 7
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| ! 8
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| |-
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| | rowspan="6" | 1
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| || 1 || −1 || 1 || || || || || || ||
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| |-
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| || 2 || −3/2 || 2 || −1/2 || || || || || ||
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| |-
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| || 3 || −11/6 || 3 || −3/2|| 1/3 || || || || ||
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| |-
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| || 4 || −25/12 || 4 || −3 || 4/3 || −1/4|| || || ||
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| |-
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| || 5 || −137/60 || 5 || −5 || 10/3 || −5/4 || 1/5 || || ||
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| |- style="border-bottom: 2px solid #aaa;"
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| || 6 || −49/20 || 6 || −15/2 || 20/3 || −15/4 || 6/5 || −1/6 || ||
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| |-
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| | rowspan="6" | 2
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| || 1 || 1 || −2 || 1 || || || || || ||
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| |-
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| || 2 || 2 || −5 || 4 || −1 || || || || ||
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| |-
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| || 3 || 35/12 || −26/3 || 19/2 || −14/3 || 11/12 || || || ||
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| |-
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| || 4 || 15/4 || −77/6 || 107/6 || −13 || 61/12 || −5/6|| || ||
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| |-
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| || 5 || 203/45 || −87/5 || 117/4 || −254/9 || 33/2 || −27/5 || 137/180 || ||
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| |- style="border-bottom: 2px solid #aaa;"
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| || 6 || 469/90 || −223/10 || 879/20 || −949/18 || 41 || −201/10 || 1019/180 || −7/10 ||
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| |-
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| | rowspan="6" | 3
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| || 1 || −1 || 3 || −3 || 1 || || || || ||
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| |-
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| || 2 || −5/2 || 9 || −12 || 7 || −3/2|| || || ||
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| |-
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| || 3 || −17/4 || 71/4 || −59/2 || 49/2 || −41/4 || 7/4 || || ||
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| |-
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| || 4 || −49/8 || 29 || −461/8 || 62 || −307/8 || 13 || −15/8 || ||
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| |-
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| || 5 || −967/120 || 638/15 || −3929/40 || 389/3 || −2545/24 || 268/5 || −1849/120 || 29/15 ||
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| |- style="border-bottom: 2px solid #aaa;"
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| || 6 || −801/80 || 349/6 || −18353/120 || 2391/10 || −1457/6 || 4891/30 || −561/8 || 527/30 || −469/240
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| |-
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| | rowspan="5" | 4
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| || 1 || 1 || −4 || 6 || −4 || 1|| || || ||
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| |-
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| || 2 || 3 || −14 || 26 || −24 || 11 || −2 || || ||
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| |-
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| || 3 || 35/6 || −31 || 137/2 || −242/3 || 107/2 || −19 || 17/6 || ||
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| |-
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| || 4 || 28/3 || −111/2 || 142 || −1219/6 || 176 || −185/2 || 82/3 || −7/2 ||
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| |- style="border-bottom: 2px solid #aaa;"
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| || 5 || 1069/80 || −1316/15 || 15289/60 || −2144/5 || 10993/24 || −4772/15 || 2803/20 || −536/15 || 967/240
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| |}
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| For example, the first derivative with a third-order accuracy and the second derivative with a second-order accuracy are
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| : <math>\displaystyle f'(x_{0}) \approx \displaystyle \frac{-\frac{11}{6}f(x_{0}) + 3f(x_{+1}) -\frac{3}{2}f(x_{+2}) +\frac{1}{3}f(x_{+3}) }{h_{x}} + O\left(h_{x}^3 \right), </math>
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| : <math>\displaystyle f''(x_{0}) \approx \displaystyle \frac{2f(x_{0}) - 5f(x_{+1}) + 4f(x_{+2}) - f(x_{+3}) }{h_{x}^2} + O\left(h_{x}^2 \right), </math>
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| while the corresponding backward approximations are given by
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| : <math>\displaystyle f'(x_{0}) \approx \displaystyle \frac{\frac{11}{6}f(x_{0}) - 3f(x_{-1}) +\frac{3}{2}f(x_{-2}) -\frac{1}{3}f(x_{-3}) }{h_{x}} + O\left(h_{x}^3 \right), </math>
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| : <math>\displaystyle f''(x_{0}) \approx \displaystyle \frac{2f(x_{0}) - 5f(x_{-1}) + 4f(x_{-2}) - f(x_{-3}) }{h_{x}^2} + O\left(h_{x}^2 \right). </math>
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| In general, to get the coefficients of the backward approximations, give all odd derivatives listed in the table the opposite sign, whereas for even derivatives the signs stay the same.
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| ==See also==
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| * [[Finite difference method]]
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| * [[Finite difference]]
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| * [[Five-point stencil]]
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| == References ==
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| {{reflist}}
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| {{Numerical PDE}}
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| {{DEFAULTSORT:Finite Difference Coefficient}}
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| [[Category:Finite differences]]
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| [[Category:Numerical differential equations]]
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I'm Eldon and I live in Preston South.
Ι'm intеrested іn Occupational Therapy, Crocheting аnd Danish art. I lіke travelling ɑnd reading fantasy.
Heгe is my web рage; asvab for marines; www.hm38hm.com,