Английская Википедия:Finite difference

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Шаблон:Use American English Шаблон:Short description Шаблон:Use mdy dates A finite difference is a mathematical expression of the form Шаблон:Math. If a finite difference is divided by Шаблон:Math, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.

The difference operator, commonly denoted <math>\Delta</math> is the operator that maps a function Шаблон:Mvar to the function <math>\Delta[f]</math> defined by

<math>\Delta[f](x)= f(x+1)-f(x).</math>

A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations, specially in the solving methods. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences.

In numerical analysis, finite differences are widely used for approximating derivatives, and the term "finite difference" is often used as an abbreviation of "finite difference approximation of derivatives". Finite difference approximations are finite difference quotients in the terminology employed above.

Finite differences were introduced by Brook Taylor in 1715 and have also been studied as abstract self-standing mathematical objects in works by George Boole (1860), L. M. Milne-Thomson (1933), and Шаблон:Interlanguage link (1939). Finite differences trace their origins back to one of Jost Bürgi's algorithms (Шаблон:Circa) and work by others including Isaac Newton. The formal calculus of finite differences can be viewed as an alternative to the calculus of infinitesimals.

Basic types

Файл:Finite difference method.svg

The three types of the finite differences. The central difference about x gives the best approximation of the derivative of the function at x.

Three basic types are commonly considered: forward, backward, and central finite differences.

Шаблон:Anchor A forward difference, denoted <math>\Delta_h[f],</math> of a function Шаблон:Mvar is a function defined as

<math> \Delta_h[f](x) = f(x + h) - f(x). </math>

Depending on the application, the spacing Шаблон:Mvar may be variable or constant. When omitted, Шаблон:Mvar is taken to be 1; that is,

<math> \Delta[f](x) = \Delta_1[f](x) =f(x+1)-f(x) .</math>

Шаблон:Anchor A backward difference uses the function values at Шаблон:Mvar and Шаблон:Math, instead of the values at Шаблон:Math and Шаблон:Mvar:

<math> \nabla_h[f](x) = f(x) - f(x-h)=\Delta_h[f](x-h). </math>

Шаблон:Anchor Finally, the central difference is given by

<math> \delta_h[f](x) = f(x+\tfrac{h}2)-f(x-\tfrac{h}2)=\Delta_{h/2}[f](x)+\nabla_{h/2}[f](x).</math>

Relation with derivatives

Шаблон:Anchor Finite difference is often used as an approximation of the derivative, typically in numerical differentiation.

The derivative of a function Шаблон:Mvar at a point Шаблон:Mvar is defined by the limit

<math> f'(x) = \lim_{h\to0} \frac{f(x+h) - f(x)}{h}. </math>

If Шаблон:Mvar has a fixed (non-zero) value instead of approaching zero, then the right-hand side of the above equation would be written

<math> \frac{f(x + h) - f(x)}{h} = \frac{\Delta_h[f](x)}{h}. </math>

Hence, the forward difference divided by Шаблон:Mvar approximates the derivative when Шаблон:Mvar is small. The error in this approximation can be derived from Taylor's theorem. Assuming that Шаблон:Mvar is twice differentiable, we have

<math> \frac{\Delta_h[f](x)}{h} - f'(x) = o(h)\to 0 \quad \text{as }h \to 0. </math>

The same formula holds for the backward difference:

<math> \frac{\nabla_h[f](x)}{h} - f'(x) = o(h)\to 0 \quad \text{as }h \to 0. </math>

However, the central (also called centered) difference yields a more accurate approximation. If Шаблон:Mvar is three times differentiable,

<math> \frac{\delta_h[f](x)}{h} - f'(x) = o\left(h^2\right) . </math>

The main problemШаблон:Citation needed with the central difference method, however, is that oscillating functions can yield zero derivative. If Шаблон:Math for Шаблон:Mvar odd, and Шаблон:Math for Шаблон:Mvar even, then Шаблон:Math if it is calculated with the central difference scheme. This is particularly troublesome if the domain of Шаблон:Mvar is discrete. See also Symmetric derivative.

Authors for whom finite differences mean finite difference approximations define the forward/backward/central differences as the quotients given in this section (instead of employing the definitions given in the previous section).

Higher-order differences

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 Шаблон:Math and Шаблон:Math and applying a central difference formula for the derivative of Шаблон:Math at Шаблон:Mvar, we obtain the central difference approximation of the second derivative of Шаблон:Mvar:

Second-order central

<math> f(x) \approx \frac{\delta_h^2[f](x)}{h^2} = \frac{ \frac{f(x+h) - f(x)}{h} - \frac{f(x) - f(x-h)}{h} }{h} = \frac{f(x+h) - 2 f(x) + f(x-h)}{h^{2}} . </math>

Similarly we can apply other differencing formulas in a recursive manner.

Second order forward

<math> f(x) \approx \frac{\Delta_h^2[f](x)}{h^2} = \frac{ \frac{f(x+2h) - f(x+h)}{h} - \frac{f(x+h) - f(x)}{h} }{h} = \frac{f(x+2h) - 2 f(x+h) + f(x)}{h^{2}} . </math>

Second order backward

<math> f(x) \approx \frac{\nabla_h^2[f](x)}{h^2} = \frac{ \frac{f(x) - f(x-h)}{h} - \frac{f(x-h) - f(x-2h)}{h} }{h} = \frac{f(x) - 2 f(x-h) + f(x - 2h)}{h^{2}} . </math>

More generally, the Шаблон:Mvarth order forward, backward, and central differences are given by, respectively,

Forward

<math>\Delta^n_h[f](x) = \sum_{i = 0}^{n} (-1)^{n-i} \binom{n}{i} f\bigl(x + i h\bigr),</math>

Backward

<math>\nabla^n_h[f](x) = \sum_{i = 0}^{n} (-1)^i \binom{n}{i} f(x - ih),</math>

Central

<math>\delta^n_h[f](x) = \sum_{i = 0}^{n} (-1)^i \binom{n}{i} f\left(x + \left(\frac{n}{2} - i\right) h\right).</math>

These equations use binomial coefficients after the summation sign shown as Шаблон:Math. Each row of Pascal's triangle provides the coefficient for each value of Шаблон:Mvar.

Note that the central difference will, for odd Шаблон:Mvar, have Шаблон:Mvar 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 Шаблон:Math and Шаблон:Math.

Forward differences applied to a sequence are sometimes called the binomial transform of the sequence, and have a number of interesting combinatorial properties. 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 Шаблон:Mvar.

The relationship of these higher-order differences with the respective derivatives is straightforward,

<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\left(h^2\right).</math>

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 Шаблон:Mvar. However, the combination

<math> \frac{\Delta_h[f](x) - \frac12 \Delta_h^2[f](x)}{h} = - \frac{f(x+2h)-4f(x+h)+3f(x)}{2h} </math>

approximates Шаблон:Math up to a term of order Шаблон:Math. This can be proven by expanding the above expression in Taylor series, or by using the calculus of finite differences, explained below.

If necessary, the finite difference can be centered about any point by mixing forward, backward, and central differences.

Polynomials

For a given polynomial of degree Шаблон:Math, expressed in the function Шаблон:Math, with real numbers Шаблон:Math and Шаблон:Math and lower order terms (if any) marked as Шаблон:Math:

<math>P(x) = ax^n + bx^{n-1} + l.o.t.</math>

After Шаблон:Math pairwise differences, the following result can be achieved, where Шаблон:Math is a real number marking the arithmetic difference:

<math>\Delta_h^n [P](x) = ah^nn!</math>

Only the coefficient of the highest-order term remains. As this result is constant with respect to Шаблон:Math, any further pairwise differences will have the value Шаблон:Math.

Inductive proof

Base case

Let Шаблон:Math be a polynomial of degree Шаблон:Math:

<math display=block>\Delta_h [Q](x) = Q(x + h) - Q(x) = [a(x + h) + b] - [ax + b] = ah = ah^11!</math>

This proves it for the base case.

Inductive step

Let Шаблон:Math be a polynomial of degree Шаблон:Math where Шаблон:Math and the coefficient of the highest-order term be Шаблон:Math. Assuming the following holds true for all polynomials of degree Шаблон:Math:

<math>\Delta_h^{m-1} [R](x) = ah^{m-1}(m-1)!</math>

Let Шаблон:Math be a polynomial of degree Шаблон:Math. With one pairwise difference:

<math display=block>\Delta_h [S](x) = [a(x+h)^{m} + b(x+h)^{m-1} + \text{l.o.t.}] - [ax^m + bx^{m-1} + \text{l.o.t.}] = ahmx^{m-1} + \text{l.o.t.} = T(x)</math>

As Шаблон:Math, this results in a polynomial Шаблон:Math of degree Шаблон:Math, with Шаблон:Math as the coefficient of the highest-order term. Given the assumption above and Шаблон:Math pairwise differences (resulting in a total of Шаблон:Math pairwise differences for Шаблон:Math), it can be found that:

<math>\Delta_h^{m-1} [T](x) = ahm \cdot h^{m-1}(m-1)! = ah^mm!</math>

This completes the proof.

Application

This identity can be used to find the lowest-degree polynomial that intercepts a number of points Шаблон:Math where the difference on the x-axis from one point to the next is a constant Шаблон:Math. For example, given the following points:

x	y
1	4
4	109
7	772
10	2641
13	6364

We can use a differences table, where for all cells to the right of the first Шаблон:Math, the following relation to the cells in the column immediately to the left exists for a cell Шаблон:Math, with the top-leftmost cell being at coordinate Шаблон:Math:

<math>(a+1, b+1) = (a, b+1) - (a, b)</math>

To find the first term, the following table can be used:

Шаблон:Math	Шаблон:Math	Шаблон:Math	Шаблон:Math	Шаблон:Math
1	4
4	109	105
7	772	663	558
10	2641	1869	1206	648
13	6364	3723	1854	648

This arrives at a constant Шаблон:Math. The arithmetic difference is Шаблон:Math, as established above. Given the number of pairwise differences needed to reach the constant, it can be surmised this is a polynomial of degree Шаблон:Math. Thus, using the identity above:

<math>648 = a \cdot 3^3 \cdot 3! = a \cdot 27 \cdot 6 = a \cdot 162</math>

Solving for Шаблон:Math, it can be found to have the value Шаблон:Math. Thus, the first term of the polynomial is Шаблон:Math.

Then, subtracting out the first term, which lowers the polynomial's degree, and finding the finite difference again:

Шаблон:Math	Шаблон:Math	Шаблон:Math	Шаблон:Math
1	Шаблон:Math
4	Шаблон:Math	−147
7	Шаблон:Math	−453	−306
10	Шаблон:Math	−759	−306
13	Шаблон:Math	−1065	−306

Here, the constant is achieved after only two pairwise differences, thus the following result:

<math>-306 = a \cdot 3^2 \cdot 2! = a \cdot 18</math>

Solving for Шаблон:Math, which is Шаблон:Math, the polynomial's second term is Шаблон:Math.

Moving on to the next term, by subtracting out the second term:

Шаблон:Math	Шаблон:Math	Шаблон:Math
1	Шаблон:Math
4	Шаблон:Math	108
7	Шаблон:Math	108
10	Шаблон:Math	108
13	Шаблон:Math	108

Thus the constant is achieved after only one pairwise difference:

<math>108 = a \cdot 3^1 \cdot 1! = a \cdot 3</math>

It can be found that Шаблон:Math and thus the third term of the polynomial is Шаблон:Math. Subtracting out the third term:

Шаблон:Math	Шаблон:Math
1	Шаблон:Math
4	Шаблон:Math
7	Шаблон:Math
10	Шаблон:Math
13	Шаблон:Math

Without any pairwise differences, it is found that the 4th and final term of the polynomial is the constant Шаблон:Math. Thus, the lowest-degree polynomial intercepting all the points in the first table is found:

<math>4x^3 - 17x^2 + 36x - 19</math>

Arbitrarily sized kernels

Using linear algebra one can construct finite difference approximations which utilize an arbitrary number of points to the left and a (possibly different) number of points to the right of the evaluation point, for any order derivative. This involves solving a linear system such that the Taylor expansion of the sum of those points around the evaluation point best approximates the Taylor expansion of the desired derivative. Such formulas can be represented graphically on a hexagonal or diamond-shaped grid. 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. Finite difference approximations for non-standard (and even non-integer) stencils given an arbitrary stencil and a desired derivative order may be constructed.

Properties

• For all positive Шаблон:Mvar and Шаблон:Mvar <math display="block">\Delta^n_{kh} (f, x) = \sum\limits_{i_1=0}^{k-1} \sum\limits_{i_2=0}^{k-1} \cdots \sum\limits_{i_n=0}^{k-1} \Delta^n_h \left(f, x+i_1h+i_2h+\cdots+i_nh\right).</math>
• Leibniz rule: <math display="block">\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>

In differential equations

Шаблон:Main article An important application of finite differences is in numerical analysis, especially in numerical differential equations, which aim at the numerical solution of ordinary and partial differential equations. 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 methods.

Common applications of the finite difference method are in computational science and engineering disciplines, such as thermal engineering, fluid mechanics, etc.

Newton's series

The Newton series consists of the terms of the Newton forward difference equation, named after Isaac Newton; in essence, it is the Gregory–Newton interpolation formula (named after Isaac Newton and James Gregory), first published in his Principia Mathematica in 1687, namely the discrete analog of the continuous Taylor expansion,

Шаблон:Equation box 1 which holds for any polynomial function Шаблон:Mvar and for many (but not all) analytic functions. (It does not hold when Шаблон:Mvar is exponential type <math>\pi</math>. This is easily seen, as the sine function vanishes at integer multiples of <math>\pi</math>; the corresponding Newton series is identically zero, as all finite differences are zero in this case. Yet clearly, the sine function is not zero.) Here, the expression

<math>\binom{x}{k} = \frac{(x)_k}{k!}</math>

is the binomial coefficient, and

<math>(x)_k=x(x-1)(x-2)\cdots(x-k+1)</math>

is the "falling factorial" or "lower factorial", while the empty product Шаблон:Math is defined to be 1. In this particular case, there is an assumption of unit steps for the changes in the values of Шаблон:Math of the generalization below.

Note the formal correspondence of this result to Taylor's theorem. Historically, this, as well as the Chu–Vandermonde identity,

<math>(x+y)_n=\sum_{k=0}^n \binom{n}{k} (x)_{n-k} \,(y)_k ,</math>

(following from it, and corresponding to the binomial theorem), are included in the observations that matured to the system of umbral calculus.

Newton series expansions can be superior to Taylor series expansions when applied to discrete quantities like quantum spins (see Holstein–Primakoff transformation), bosonic operator functions or discrete counting statistics.

To illustrate how one may use Newton's formula in actual practice, consider the first few terms of doubling the Fibonacci sequence Шаблон:Math One can find a polynomial that reproduces these values, by first computing a difference table, and then substituting the differences that correspond to Шаблон:Math (underlined) into the formula as follows, <math display=block> \begin{matrix}

\begin{array}{|c||c|c|c|} \hline

x & f=\Delta^0 & \Delta^1 & \Delta^2 \\

\hline 1&\underline{2}& & \\

& &\underline{0}& \\

2&2& &\underline{2} \\

& &2& \\

3&4& & \\ \hline \end{array}

&

\quad \begin{align} 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)\\

\\

& =2 \cdot 1 + 0 \cdot \dfrac{x-1}{1} + 2 \cdot \dfrac{(x-1)(x-2)}{2} \\

\\

& =2 + (x-1)(x-2) \\ \end{align} \end{matrix} </math>

For the case of nonuniform steps in the values of Шаблон:Mvar, Newton computes the divided differences, <math display=block>\Delta _{j,0}=y_j,\qquad \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\},\qquad \Delta 0_k=\Delta _{0,k}</math> the series of products,

<math>{P_0}=1,\quad \quad P_{k+1}=P_k\cdot \left( \xi -x_k \right) ,</math>

and the resulting polynomial is the scalar product,

<math>f(\xi ) = \Delta 0 \cdot P\left( \xi \right)</math> .

In analysis with [[p-adic number|Шаблон:Mvar-adic numbers]], Mahler's theorem states that the assumption that Шаблон:Mvar is a polynomial function can be weakened all the way to the assumption that Шаблон:Mvar is merely continuous.

Carlson's theorem provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series does not, in general, exist.

The Newton series, together with the Stirling series and the Selberg series, is a special case of the general difference series, all of which are defined in terms of suitably scaled forward differences.

In a compressed and slightly more general form and equidistant nodes the formula reads

<math>f(x)=\sum_{k=0}\binom{\frac{x-a}h}{k} \sum_{j=0}^k (-1)^{k-j}\binom{k}{j}f(a+j h).</math>

Calculus of finite differences

The forward difference can be considered as an operator, called the difference operator, which maps the function Шаблон:Mvar to Шаблон:Math. This operator amounts to

<math>\Delta_h = T_h-I, </math>

where Шаблон:Math is the shift operator with step h, defined by Шаблон:Math, and Шаблон:Mvar is the identity operator.

The finite difference of higher orders can be defined in recursive manner as Шаблон:Math. Another equivalent definition is Шаблон:Math.

The difference operator Шаблон:Math is a linear operator, as such it satisfies Шаблон:Math.

It also satisfies a special Leibniz rule indicated above, Шаблон:Math. Similar statements hold for the backward and central differences.

Formally applying the Taylor series with respect to Шаблон:Mvar, yields the formula

<math> \Delta_h = hD + \frac{1}{2!} h^2D^2 + \frac{1}{3!} h^3D^3 + \cdots = \mathrm{e}^{hD} - I , </math>

where Шаблон:Mvar denotes the continuum derivative operator, mapping Шаблон:Mvar to its derivative Шаблон:Math. The expansion is valid when both sides act on analytic functions, for sufficiently small Шаблон:Mvar. Thus, Шаблон:Math, and formally inverting the exponential yields

<math> hD = \ln(1+\Delta_h) = \Delta_h - \tfrac{1}{2} \, \Delta_h^2 + \tfrac{1}{3} \, \Delta_h^3 - \cdots. </math>

This formula holds in the sense that both operators give the same result when applied to a polynomial.

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 Шаблон:Math mentioned at the end of the section Higher-order differences.

The analogous formulas for the backward and central difference operators are

<math> hD = -\ln(1-\nabla_h) \quad\text{and}\quad hD = 2 \operatorname{arsinh}\left(\tfrac12 \, \delta_h\right). </math>

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 (Шаблон:Math limits),

A large number of formal differential relations of standard calculus involving functions Шаблон:Math thus map systematically to umbral finite-difference analogs involving Шаблон:Math.

For instance, the umbral analog of a monomial Шаблон:Mvar is a generalization of the above falling factorial (Pochhammer k-symbol),

<math>~(x)_n\equiv \left(xT_h^{-1}\right)^n=x (x-h) (x-2h) \cdots \bigl(x-(n-1)h\bigr),</math>

so that

<math>\frac{\Delta_h}{h} (x)_n=n (x)_{n-1} ,</math>

hence the above Newton interpolation formula (by matching coefficients in the expansion of an arbitrary function Шаблон:Math in such symbols), and so on.

For example, the umbral sine is

<math>\sin \left(x\,T_h^{-1}\right) = x -\frac{(x)_3}{3!} + \frac{(x)_5}{5!} - \frac{(x)_7}{7!} + \cdots</math>

As in the continuum limit, the eigenfunction of Шаблон:Math also happens to be an exponential,

<math>\frac{\Delta_h}{h}(1+\lambda h)^\frac{x}{h} =\frac{\Delta_h}{h} e^{\ln (1+\lambda h) \frac{x}{h}}= \lambda e^{\ln (1+\lambda h) \frac{x}{h}} ,</math>

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. This umbral exponential thus amounts to the exponential generating function of the Pochhammer symbols.

Thus, for instance, the Dirac delta function maps to its umbral correspondent, the cardinal sine function,

<math>\delta (x) \mapsto \frac{\sin \left[ \frac{\pi}{2}\left(1+\frac{x}{h}\right) \right]}{ \pi (x+h) },</math>

and so forth. Difference equations can often be solved with techniques very similar to those for solving differential equations.

The inverse operator of the forward difference operator, so then the umbral integral, is the indefinite sum or antidifference operator.

Rules for calculus of finite difference operators

Analogous to rules for finding the derivative, we have:

• Constant rule: If Шаблон:Mvar is a constant, then

<math>\Delta c = 0</math>

• Linearity: if Шаблон:Mvar and Шаблон:Mvar are constants,

<math>\Delta (a\ f + b\ g) = a \ \Delta f + b \ \Delta g</math>

All of the above rules apply equally well to any difference operator as to Шаблон:Math, including Шаблон:Math and Шаблон:Math.

• Product rule:

<math> \begin{align} \Delta (f g) &= f \,\Delta g + g \,\Delta f + \Delta f \,\Delta g \\[4pt]

\nabla (f g) &= f \,\nabla g + g \,\nabla f - \nabla f \,\nabla g \end{align}</math>

• Quotient rule:

<math>\nabla \left( \frac{f}{g} \right) = \left. \left( \det \begin{bmatrix} \nabla f & \nabla g \\ f & g \end{bmatrix} \right) \right/ \left( g \cdot \det {\begin{bmatrix} g & \nabla g \\ 1 & 1 \end{bmatrix}}\right) </math>

or

<math>\nabla\left( \frac{f}{g} \right)= \frac {g \,\nabla f - f \,\nabla g}{g \cdot (g - \nabla g)}</math>

• Summation rules:

<math>\begin{align} \sum_{n=a}^b \Delta f(n) &= f(b+1)-f(a) \\ \sum_{n=a}^{b} \nabla f(n) &= f(b)-f(a-1) \end{align}</math>

See references.

Generalizations

• A generalized finite difference is usually defined as <math display="block">\Delta_h^\mu[f](x) = \sum_{k=0}^N \mu_k f(x+kh),</math> where Шаблон:Math is its coefficient vector. An infinite difference is a further generalization, where the finite sum above is replaced by an infinite series. Another way of generalization is making coefficients Шаблон:Math depend on point Шаблон:Mvar: Шаблон:Math, thus considering weighted finite difference. Also one may make the step Шаблон:Mvar depend on point Шаблон:Mvar: Шаблон:Math. Such generalizations are useful for constructing different modulus of continuity.
• The generalized difference can be seen as the polynomial rings Шаблон:Math. It leads to difference algebras.
• Difference operator generalizes to Möbius inversion over a partially ordered set.
• As a convolution operator: Via the formalism of incidence algebras, difference operators and other Möbius inversion can be represented by convolution with a function on the poset, called the Möbius function Шаблон:Mvar; for the difference operator, Шаблон:Mvar is the sequence Шаблон:Nowrap.

Multivariate finite differences

Finite differences can be considered in more than one variable. They are analogous to partial derivatives in several variables.

Some partial derivative approximations are:

<math display=block>\begin{align} f_{x}(x,y) &\approx \frac{f(x+h ,y) - f(x-h,y)}{2h} \\ f_{y}(x,y) &\approx \frac{f(x,y+k ) - f(x,y-k)}{2k} \\ f_{xx}(x,y) &\approx \frac{f(x+h ,y) - 2 f(x,y) + f(x-h,y)}{h^2} \\ f_{yy}(x,y) &\approx \frac{f(x,y+k) - 2 f(x,y) + f(x,y-k)}{k^2} \\ 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} . \end{align}</math>

Alternatively, for applications in which the computation of Шаблон:Mvar is the most costly step, and both first and second derivatives must be computed, a more efficient formula for the last case is

<math display=block> 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>

since the only values to compute that are not already needed for the previous four equations are Шаблон:Math and Шаблон:Math.

References

• Richardson, C. H. (1954): An Introduction to the Calculus of Finite Differences (Van Nostrand (1954) online copy
• Mickens, R. E. (1991): Difference Equations: Theory and Applications (Chapman and Hall/CRC) Шаблон:ISBN

External links

• Шаблон:Springer
• Table of useful finite difference formula generated using Mathematica
• D. Gleich (2005), Finite Calculus: A Tutorial for Solving Nasty Sums
• Discrete Second Derivative from Unevenly Spaced Points