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| {{Use mdy dates|date=September 2011}}
| | So many people take their wellness as a given. Your diet program ought to give attention to your overall health. With great diet and a wellness diet, attaining and looking after an effective weight will come in a natural way. Keep reading for more information.<br><br> |
| A '''finite difference''' is a mathematical expression of the form {{math|''f''(''x'' + ''b'') − ''f''(''x'' + ''a'')}}. If a finite difference is divided by {{math|''b'' − ''a''}}, one gets a [[difference quotient]]. The approximation of [[derivative]]s by finite differences plays a central role in [[finite difference method]]s for the [[numerical analysis|numerical]] solution of [[differential equation]]s, especially [[boundary value problem]]s.
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| [[Recurrence_relation#Relationship_to_difference_equations_narrowly_defined|Recurrence relation]]s can be written as difference equations by replacing iteration notation with finite differences.
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| ==Forward, backward, and central differences==
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| Three forms are commonly considered: forward, backward, and central differences.
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| A '''forward difference''' is an expression of the form
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| :<math> \Delta_h[f](x) = f(x + h) - f(x). \ </math>
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| Depending on the application, the spacing ''h'' may be variable or constant.
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| A '''backward difference''' uses the function values at ''x'' and ''x'' − ''h'', instead of the values at ''x'' + ''h'' and ''x'':
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| :<math> \nabla_h[f](x) = f(x) - f(x-h). \ </math>
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| {{anchor|central difference}}
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| Finally, the '''central difference''' is given by
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| :<math> \delta_h[f](x) = f(x+\tfrac12h)-f(x-\tfrac12h). \ </math>
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| == Relation with derivatives ==
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| The [[derivative]] of a function {{mvar|f}} at a point {{mvar| x}} is defined by the [[limit of a function|limit]]
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| :<math> f'(x) = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}. </math>
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| If {{mvar|h}} has a fixed (non-zero) value instead of approaching zero, then the right-hand side of the above equation would be written
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| :<math> \frac{f(x + h) - f(x)}{h} = \frac{\Delta_h[f](x)}{h}. </math>
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| Hence, the forward difference divided by {{mvar|h}} approximates the derivative when {{mvar|h}} is small. The error in this approximation can be derived from [[Taylor's theorem]]. Assuming that {{mvar|f}} is continuously differentiable, the error is
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| :<math> \frac{\Delta_h[f](x)}{h} - f'(x) = O(h) \quad (h \to 0). </math>
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| The same formula holds for the backward difference:
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| :<math> \frac{\nabla_h[f](x)}{h} - f'(x) = O(h). </math>
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| However, the central difference yields a more accurate approximation. Its error is proportional to square of the spacing (if {{mvar|f}} is twice continuously differentiable; that is, the second derivative of the function, {{mvar|f"}}, is continuous for all {{mvar|x}}):
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| :<math> \frac{\delta_h[f](x)}{h} - f'(x) = O(h^{2}) . \!</math>
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| The main problem with the central difference method, however, is that oscillating functions can yield zero derivative. If {{math|''f''(''nh''){{=}}1}} for {{mvar|n}} odd, and {{math|''f''(''nh''){{=}}2}} for {{mvar|n}} even, then {{math|''f ' '' (''nh''){{=}}0}} if it is calculated with the central difference scheme. This is particularly troublesome if the domain of {{mvar|f}} is discrete.
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| ==Higher-order differences== <!-- this section is linked to further down in the article -->
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| In an analogous way one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for {{math|''f ' ''(''x''+''h''/2)}} and {{math|''f ' ''(''x''−''h''/2)}} and applying a central difference formula for the derivative of {{mvar| f '}} at {{mvar|x}}, we obtain the central difference approximation of the second derivative of {{mvar|f}}:
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| '''2nd order central'''
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| :<math> f''(x) \approx \frac{\delta_h^2[f](x)}{h^2} = \frac{f(x+h) - 2 f(x) + f(x-h)}{h^{2}} . </math>
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| Similarly we can apply other differencing formulas in a recursive manner.
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| '''2nd order forward'''
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| :<math> f''(x) \approx \frac{\Delta_h^2[f](x)}{h^2} = \frac{f(x+2h) - 2 f(x+h) + f(x)}{h^{2}} . </math>
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| More generally, the '''{{mvar|n}}-th order forward, backward, and central''' differences are given by, respectively,
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| '''Forward'''
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| :<math>\Delta^n_h[f](x) =
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| \sum_{i = 0}^{n} (-1)^i \binom{n}{i} f(x + (n - i) h),
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| </math> | |
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| '''Backward'''
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| :<math>\nabla^n_h[f](x) =
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| \sum_{i = 0}^{n} (-1)^i \binom{n}{i} f(x - ih),
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| </math>
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| '''Central'''
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| :<math>\delta^n_h[f](x) =
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| \sum_{i = 0}^{n} (-1)^i \binom{n}{i} f\left(x + \left(\frac{n}{2} - i\right) h\right).
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| </math>
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| Note that the central difference will, for odd {{mvar|n}}, have {{mvar|h}} multiplied by non-integers. This is often a problem because it amounts to changing the interval of discretization. The problem may be remedied taking the average of <math>\delta^n[f](x - h/2)</math> and <math>\delta^n[f](x + h/2)</math>.
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| Forward differences applied to a [[sequence]] are sometimes called the [[binomial transform]] of the sequence, and have a number of interesting combinatorial properties.
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| Forward differences may be evaluated using the [[Nörlund–Rice integral]]. The integral representation for these types of series is interesting, because the integral can often be evaluated using [[asymptotic expansion]] or [[saddle-point]] techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large {{mvar|n}}.
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| The relationship of these higher-order differences with the respective derivatives is straightforward,
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| :<math>\frac{d^n f}{d x^n}(x) = \frac{\Delta_h^n[f](x)}{h^n}+O(h) = \frac{\nabla_h^n[f](x)}{h^n}+O(h) = \frac{\delta_h^n[f](x)}{h^n} + O(h^2).</math>
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| Higher-order differences can also be used to construct better approximations. As mentioned above, the first-order difference approximates the first-order derivative up to a term of order {{mvar|h}}. However, the combination
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| :<math> \frac{\Delta_h[f](x) - \frac12 \Delta_h^2[f](x)}{h} = - \frac{f(x+2h)-4f(x+h)+3f(x)}{2h} </math>
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| approximates ''f'''(''x'') up to a term of order {{math|''h''<sup>2</sup>}}. This can be proven by expanding the above expression in [[Taylor series]], or by using the calculus of finite differences, explained below.
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| If necessary, the finite difference can be centered about any point by mixing forward, backward, and central differences.
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| ===Arbitrarily sized kernels===
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| Using a little linear algebra, one can fairly easily construct approximations, which sample an arbitrary number of points to the left and a (possibly different) number of points to the right of the center point, for any order of derivative. This involves solving a linear system such that the Taylor expansion of the sum of those points, around the center point, well approximates the Taylor expansion of the desired derivative.
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| This is useful for differentiating a function on a grid, where, as one approaches the edge of the grid, one must sample fewer and fewer points on one side.
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| The details are outlined in these [http://commons.wikimedia.org/wiki/File:FDnotes.djvu notes].
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| ===Properties===
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| * For all positive ''k'' and ''n''
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| :<math>\Delta^n_{kh} (f, x) = \sum\limits_{i_1=0}^{k-1} \sum\limits_{i_2=0}^{k-1} ... \sum\limits_{i_n=0}^{k-1} \Delta^n_h (f, x+i_1h+i_2h+...+i_nh).</math>
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| * [[Leibniz rule (generalized product rule)|Leibniz rule]]:
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| :<math>\Delta^n_h (fg, x) = \sum\limits_{k=0}^n \binom{n}{k} \Delta^k_h (f, x) \Delta^{n-k}_h(g, x+kh).</math>
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| ==Finite difference methods==
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| An important application of finite differences is in [[numerical analysis]], especially in [[numerical partial differential equations|numerical differential equations]], which aim at the numerical solution of [[ordinary differential equation|ordinary]] and [[partial differential equation]]s respectively. The idea is to replace the derivatives appearing in the differential equation by finite differences that approximate them. The resulting methods are called [[finite difference method]]s.
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| Common applications of the finite difference method are in computational science and engineering disciplines, such as [[thermal engineering]], [[fluid mechanics]], etc.
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| ==Newton's series==
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| The '''[[Newton polynomial|Newton series]]''' consists of the terms of the '''Newton forward difference equation''', named after [[Isaac Newton]]; in essence, it is the '''Newton interpolation formula''', first published in his ''[[Philosophiæ Naturalis Principia Mathematica|Principia Mathematica]]'' in 1687,<ref>Newton, Isaac, (1687). [http://books.google.com/books?id=KaAIAAAAIAAJ&dq=sir%20isaac%20newton%20principia%20mathematica&as_brr=1&pg=PA466#v=onepage&q&f=false ''Principia'', Book III, Lemma V, Case 1]</ref> namely the discrete analog of the continuum Taylor expansion,
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| {{Equation box 1
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| |indent =::
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| |equation = <math>f(x)=\sum_{k=0}^\infty\frac{\Delta^k [f](a)}{k!} ~(x-a)_k
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| = \sum_{k=0}^\infty {x-a \choose k}~ \Delta^k [f](a) ~,
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| </math>
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| |cellpadding= 6
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| |border
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| |border colour = #0073CF
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| |background colour=#F9FFF7}}
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| which holds for any [[polynomial]] function ''f'' and for most (but not all) [[analytic function]]s. Here, the expression
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| :<math>{x \choose k} = \frac{(x)_k}{k!}</math>
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| is the [[binomial coefficient]], and
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| :<math>(x)_k=x(x-1)(x-2)\cdots(x-k+1)</math>
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| is the "[[falling factorial]]" or "lower factorial", while the [[empty product]] (''x'')<sub>0</sub> is defined to be 1. In this particular case, there is an assumption of unit steps for the changes in the values of ''x'', ''h''=1 of the generalization below.
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| Note also the formal correspondence of this result to [[Taylor's theorem]]. Historically, this, as well as the [[Chu-Vandermonde identity]],
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| :<math>(x+y)_n=\sum_{k=0}^n {n \choose k} (x)_{n-k} ~(y)_k ~,</math>
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| (following from it, and corresponding to the [[binomial theorem]]), are included in the observations which matured to the system of the [[umbral calculus]].
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| To illustrate how one may use Newton's formula in actual practice, consider the first few terms of the [[Fibonacci sequence]] f = 2, 2, 4... One can find a [[polynomial]] that reproduces these values, by first computing a difference table, and then substituting the differences which correspond to x<sub>0</sub> (underlined) into the formula as follows,
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| :<math>
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| \begin{matrix}
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| \begin{array}{|c||c|c|c|}
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| \hline
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| x & f=\Delta^0 & \Delta^1 & \Delta^2 \\
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| \hline
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| 1&\underline{2}& & \\
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| & &\underline{0}& \\
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| 2&2& &\underline{2} \\
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| & &2& \\
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| 3&4& & \\
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| \hline
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| \end{array}
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| &
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| \quad \begin{matrix}
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| f(x)=\Delta^0 \cdot 1 +\Delta^1 \cdot \dfrac{(x-x_0)_1}{1!} + \Delta^2 \cdot \dfrac{(x-x_0)_2}{2!} \quad (x_0=1)\\
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| \\
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| =2 \cdot 1 + 0 \cdot \dfrac{x-1}{1} + 2 \cdot \dfrac{(x-1)(x-2)}{2} \\
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| \\
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| =2 + (x-1)(x-2) \\
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| \end{matrix}
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| \end{matrix}
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| </math>
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| For the case of nonuniform steps in the values of ''x'', Newton computes the [[divided differences]],
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| :<math>\Delta _{j,0}=y_j,\quad \quad \Delta _{j,k}=\frac{\Delta _{j+1,k-1}-\Delta _{j,k-1}}{x_{j+k}-x_j}\quad \ni \quad \left\{ k>0,\ \ j\le \max \left( j \right)-k \right\},\quad \quad \Delta 0_k=\Delta _{0,k}</math>
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| the series of products,
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| :<math>{P_0}=1,\quad \quad P_{k+1}=P_k\cdot \left( \xi -x_k \right) ~,</math>
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| and the resulting polynomial is the [[scalar product]], <math>f(\xi ) = \Delta 0 \cdot P\left( \xi \right)</math> .<ref>[[Robert D. Richtmyer|Richtmeyer, D.]] and Morton, K.W., (1967). ''Difference Methods for Initial Value Problems'', 2nd ed., Wiley, New York.</ref>
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| In analysis with [[p-adic number]]s, [[Mahler's theorem]] states that the assumption that ''f'' is a polynomial function can be weakened all the way to the assumption that ''f'' is merely continuous.
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| [[Carlson's theorem]] provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series will not, in general, exist.
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| The Newton series, together with the [[Stirling series]] and the [[Selberg class|Selberg series]], is a special case of the general [[difference series]], all of which are defined in terms of suitably scaled forward differences.
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| In a compressed and slightly more general form and equidistant nodes the formula reads
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| :<math>f(x)=\sum_{k=0}{\frac{x-a}h \choose k} \sum_{j=0}^k (-1)^{k-j}{k\choose j}f(a+j h).</math>
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| ==Calculus of finite differences==
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| The forward difference can be considered as a difference [[Operator (mathematics)|operator]],<ref>[[George Boole|Boole, George]], (1872). ''A Treatise On The Calculus of Finite Differences'', 2nd ed., Macmillan and Company. [http://archive.org/details/cu31924031240934 On line]. Also, [Dover edition 1960]</ref><ref>Jordan, Charles, (1939/1965). "Calculus of Finite Differences", Chelsea Publishing. On-line: [http://books.google.com/books?hl=en&lr=&id=3RfZOsDAyQsC&oi=fnd&pg=PA1&ots=AqSuAgOKs3&sig=fzPpAdvnzp7sG6PorqIe5qFjD2Q#v=onepage]</ref> which maps the function {{mvar|''f''}} to {{math|Δ<sub>''h''</sub>[''f'' ]}}. This operator amounts to
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| ::<math>\Delta_h = T_h-I, \,</math>
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| where {{math|''T''<sub>''h''</sub>}} is the [[shift operator]] with step ''h'', defined by {{math|''T''<sub>''h''</sub>[''f'' ](''x'') {{=}} ''f''(''x''+''h'')}}, and {{mvar|''I''}} is the [[identity operator]].
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| The finite difference of higher orders can be defined in recursive manner as {{math| Δ<sub>''h''</sub><sup>''n''</sup> ≡ Δ<sub>''h''</sub> (Δ<sub>''h''</sub><sup>''n''-1</sup>}}). Another equivalent definition is {{math| Δ<sub>''h''</sub><sup>''n''</sup> {{=}} [''T''<sub>''h''</sub> −''I'']<sup>''n''</sup>}}.
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| The difference operator {{math|Δ<sub>''h''</sub>}} is a [[linear operator]] and it satisfies a special [[Leibniz rule (generalized product rule)|Leibniz rule]] indicated above,
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| {{math|Δ<sub>''h''</sub>(''f''(''x'')''g''(''x'')) {{=}} (Δ<sub>''h''</sub>''f''(''x'')) ''g''(''x''+''h'') + ''f''(''x'') (Δ<sub>''h''</sub>''g''(''x''))}}. Similar statements hold for the backward and central differences.
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| Formally applying the [[Taylor series]] with respect to ''h'', yields the formula
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| :<math> \Delta_h = hD + \frac{1}{2} h^2D^2 + \frac{1}{3!} h^3D^3 + \cdots = \mathrm{e}^{hD} - I ~, </math>
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| where ''D'' denotes the continuum derivative operator, mapping ''f'' to its derivative ''f'''. The expansion is valid when both sides act on [[analytic function]]s, for sufficiently small ''h''. Thus, {{math|''T''<sub>''h''</sub>{{=}}e<sup>''hD''</sup>}}, and formally inverting the exponential yields
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| :<math> hD = \log(1+\Delta_h) = \Delta_h - \tfrac{1}{2} \Delta_h^2 + \tfrac{1}{3} \Delta_h^3 + \cdots. \, </math>
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| This formula holds in the sense that both operators give the same result when applied to a polynomial.
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| Even for analytic functions, the series on the right is not guaranteed to converge; it may be an [[asymptotic series]]. However, it can be used to obtain more accurate approximations for the derivative. For instance, retaining the first two terms of the series yields the second-order approximation to {{math|''f’''(''x'')}} mentioned at the end of the [[#Higher-order differences|section ''Higher-order differences'']].
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| The analogous formulas for the backward and central difference operators are
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| :<math> hD = -\log(1-\nabla_h) \quad\text{and}\quad hD = 2 \, \operatorname{arsinh}(\tfrac12\delta_h). </math>
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| The calculus of finite differences is related to the [[umbral calculus]] of combinatorics. This remarkably systematic correspondence is due to the identity of the [[commutators]] of the umbral quantities to their continuum analogs ({{math|''h''→0}} limits),
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| {{Equation box 1
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| |indent =::
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| |equation =
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| <math> \Bigl[ \frac{\Delta_h}{h} ~,~ x\, T^{-1}_h \Bigr] = [ D ~,~ x ] = I ~.</math>
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| |cellpadding= 6
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| |border
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| |border colour = #0073CF
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| |background colour=#F9FFF7}}
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| A large number of formal differential relations of standard calculus involving
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| functions {{math|''f''(''x'')}} thus ''map systematically to umbral finite-difference analogs'' involving {{math|''f''(''xT''<sub>h</sub><sup>−1</sup>)}}.
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| For instance, the umbral analog of a monomial ''x''<sup>n</sup> is a generalization of the above falling factorial ([[Pochhammer k-symbol]]),
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| :<math>~(x)_n\equiv (xT_h^{-1})^n=x (x-h) (x-2h) \cdots (x-(n-1)h)</math> ,
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| so that
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| ::<math>\frac{\Delta_h}{h} ~(x)_n=n ~(x)_{n-1} ~,</math>
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| hence the above Newton interpolation formula (by matching coefficients in the expansion of an arbitrary function ''f''(''x'') in such symbols), and so on.
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| For example, the umbral sine is
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| :<math>\sin (x\,T_h^{-1}) = x -\frac{(x)_3}{3!} + \frac{(x)_5}{5!} - \frac{(x)_7}{7!} + ... ~.</math>
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| As in the continuum limit, the eigenfunction of {{math| Δ<sub>''h''</sub> /''h''}} also happens ''to be an exponential'',
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| ::<math>\frac{\Delta_h}{h}~(1+\lambda h)^{x/h} =\frac{\Delta_h}{h} ~e^{\ln (1+\lambda h) ~x/h}= \lambda ~e^{\ln (1+\lambda h) ~x/h} ~,</math>
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| and hence ''Fourier sums of continuum functions are readily mapped to umbral Fourier sums faithfully'', i.e., involving the same Fourier coefficients multiplying these umbral basis exponentials.<ref>{{cite journal |last =Zachos|first =C.| authorlink =Cosmas Zachos| year =2008| title =Umbral Deformations on Discrete Space-Time | journal =International Journal of Modern Physics A| volume =23 | issue=13| pages =2005–2014 | doi = 10.1142/S0217751X08040548 }}</ref> This umbral exponential thus amounts to the exponential [[generating function]] of the [[Pochhammer symbol]]s.
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| Thus, for instance, the [[Dirac delta function]] maps to its umbral correspondent, the [[Sinc function|cardinal sine function]],
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| :<math>\delta (x) \mapsto \frac{\sin \bigl[ \frac{\pi}{2}(1+x/h) \bigr]}{ \pi (x+h) }~,</math>
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| and so forth.<ref>{{cite doi|10.3389/fphy.2013.00015|noedit}}</ref> [[Difference equation]]s can often be solved with techniques very similar to those for solving [[differential equation]]s.
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| The inverse operator of the forward difference operator, so then the umbral integral, is the [[indefinite sum]] or antidifference operator.
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| ==Rules for calculus of finite difference operators==
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| Analogous to [[Derivative#Rules for finding the derivative|rules for finding the derivative]], we have:
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| * '''Constant rule''': If ''c'' is a [[Constant (mathematics)|constant]], then
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| :<math>\Delta c = 0{\,}</math>
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| * '''[[Linearity of differentiation|Linearity]]''': if ''a'' and ''b'' are [[Constant (mathematics)|constants]],
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| :<math>\Delta (a f + b g) = a \,\Delta f + b \,\Delta g</math>
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| All of the above rules apply equally well to any difference operator, including <math>\nabla</math> as to <math>\Delta</math>.
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| * '''[[Product rule]]''':
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| :<math> \Delta (f g) = f \,\Delta g + g \,\Delta f + \Delta f \,\Delta g </math>
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| :<math> \nabla (f g) = f \,\nabla g + g \,\nabla f - \nabla f \,\nabla g </math>
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| * '''[[Quotient rule]]''':
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| :<math>\nabla \left( \frac{f}{g} \right) = \frac{1}{g} \det \begin{bmatrix} \nabla f & \nabla g \\ f & g \end{bmatrix}
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| \left( \det {\begin{bmatrix} g & \nabla g \\ 1 & 1 \end{bmatrix}}\right)^{-1} </math>
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| ::or
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| :<math>\nabla\left( \frac{f}{g} \right)= \frac {g \,\nabla f - f \,\nabla g}{g \cdot (g - \nabla g)}</math>
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| :<math>\Delta\left( \frac{f}{g} \right)= \frac {g \,\Delta f - f \,\Delta g}{g \cdot (g + \Delta g)}</math>
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| * '''Summation rules''':
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| :<math>\sum_{n=a}^{b} \Delta f(n) = f(b+1)-f(a)</math>
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| :<math>\sum_{n=a}^{b} \nabla f(n) = f(b)-f(a-1)</math>
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| <br /><ref>{{cite book|last=Levy|first=H.|coauthor=Lessman, F.|title=Finite Difference Equations|year=1992|publisher=Dover|isbn=0-486-67260-3}}</ref><ref>Ames, W. F., (1977). ''Numerical Methods for Partial Differential Equations'', Section 1.6. Academic Press, New York. ISBN 0-12-056760-1.</ref><ref>[[Francis B. Hildebrand|Hildebrand, F. B.]], (1968). ''Finite-Difference Equations and Simulations'', Section 2.2, Prentice-Hall, Englewood Cliffs, New Jersey.</ref><ref>{{Cite journal
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| | first1 = Philippe | last1 = Flajolet
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| | authorlink2 = Robert Sedgewick (computer scientist) | first2 = Robert | last2 = Sedgewick
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| | url = http://www-rocq.inria.fr/algo/flajolet/Publications/mellin-rice.ps.gz
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| | title = Mellin transforms and asymptotics: Finite differences and Rice's integrals
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| | journal=Theoretical Computer Science
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| | volume = 144 | issue = 1–2 | year = 1995 | pages = 101–124
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| | doi = 10.1016/0304-3975(94)00281-M
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| | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.</ref>
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| ==Generalizations==
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| *A '''generalized finite difference''' is usually defined as
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| :<math>\Delta_h^\mu[f](x) = \sum_{k=0}^N \mu_k f(x+kh),</math>
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| where <math>\mu = (\mu_0,\ldots,\mu_N)</math> is its coefficients vector. An '''infinite difference''' is a further generalization, where the finite sum above is replaced by an [[Series (mathematics)|infinite series]]. Another way of generalization is making coefficients <math>\mu_k</math> depend on point <math>x</math> : <math>\mu_k=\mu_k(x)</math>, thus considering '''weighted finite difference'''. Also one may make step <math>h</math> depend on point <math>x</math> : <math>h=h(x)</math>. Such generalizations are useful for constructing different [[modulus of continuity]].
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| *The generalized difference can be seen as the polynomial rings <math>R[T_h]</math> . It leads to difference algebras.
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| *Difference operator generalizes to [[Möbius inversion]] over a [[partially ordered set]].
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| *As a convolution operator: Via the formalism of [[incidence algebra]]s, difference operators and other Möbius inversion can be represented by [[convolution]] with a function on the poset, called the [[Möbius function]] μ; for the difference operator, μ is the sequence (1, −1, 0, 0, 0, ...).
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| ==Finite difference in several variables==
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| Finite differences can be considered in more than one variable. They are analogous to [[partial derivative]]s in several variables.
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| Some partial derivative approximations are (using central step method):
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| :<math> f_{x}(x,y) \approx \frac{f(x+h ,y) - f(x-h,y)}{2h} \ </math>
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| :<math> f_{y}(x,y) \approx \frac{f(x,y+k ) - f(x,y-k)}{2k} \ </math>
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| :<math> f_{xx}(x,y) \approx \frac{f(x+h ,y) - 2 f(x,y) + f(x-h,y)}{h^2} \ </math>
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| :<math> f_{yy}(x,y) \approx \frac{f(x,y+k) - 2 f(x,y) + f(x,y-k)}{k^2} \ </math>
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| :<math> f_{xy}(x,y) \approx \frac{f(x+h,y+k) - f(x+h,y-k) - f(x-h,y+k) + f(x-h,y-k)}{4hk} ~. </math>
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| Alternatively, for applications in which the computation of {{mvar|f}} is the most costly step, and both first and second derivatives must be computed, a more efficient formula for the last case is
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| :<math> f_{xy}(x,y) \approx \frac{f(x+h, y+k) - f(x+h, y) - f(x, y+k) + 2 f(x,y) - f(x-h, y) - f(x, y-k) + f(x-h, y-k)}{2hk} ~,</math>
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| since the only values to be computed which are not already needed for the previous four equations are {{math|''f''(''x''+''h'', ''y''+''k'')}} and {{math|''f''(''x''−''h'', ''y''−''k'')}}.
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| ==See also==
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| {{columns-list|3|
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| * [[Carlson's theorem]]
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| * [[Divided differences]]
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| * [[Finite difference coefficients]]
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| * [[Finite difference method]]
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| * [[Five-point stencil]]
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| * [[Gilbreath's conjecture]]
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| * [[Lagrange polynomial]]
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| *[[Central differencing scheme]]
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| * [[Upwind differencing scheme for convection]]
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| * [[Modulus of continuity]]
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| * [[Newton polynomial]]
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| * [[Nörlund–Rice integral]]
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| * [[Numerical differentiation]]
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| * [[Sheffer sequence]]
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| * [[Summation by parts]]
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| * [[Table of Newtonian series]]
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| * [[Taylor series]]
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| * [[Time scale calculus]]
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| * [[Umbral calculus]]
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| }}
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| == References ==
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| <references/>
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| == External links ==
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| * {{springer|title=Finite-difference calculus|id=p/f040230}}
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| * [http://reference.wolfram.com/mathematica/tutorial/NDSolvePDE.html#c:4 Table of useful finite difference formula generated using [[Mathematica]] ]
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| * [http://www.stanford.edu/~dgleich/publications/finite-calculus.pdf Finite Calculus: A Tutorial for Solving Nasty Sums]
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| * [http://mathformeremortals.wordpress.com/2013/01/12/a-numerical-second-derivative-from-three-points/ Discrete Second Derivative from Unevenly Spaced Points]
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| {{DEFAULTSORT:Finite Difference}}
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| [[Category:Finite differences| ]]
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| [[Category:Numerical differential equations]]
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| [[Category:Mathematical analysis]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Linear operators in calculus]]
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