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45-500: A finite difference is a mathematical expression of the form f  ( x + b ) − f  ( x + a ) . If a finite difference is divided by b − a , 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 Δ {\displaystyle \Delta }

90-406: A h m m ! {\displaystyle \Delta _{h}^{m-1}[T](x)=ahm\cdot h^{m-1}(m-1)!=ah^{m}m!} This completes the proof. This identity can be used to find the lowest-degree polynomial that intercepts a number of points ( x , y ) where the difference on the x -axis from one point to the next is a constant h ≠ 0 . For example, given the following points: We can use

135-449: A h m − 1 ( m − 1 ) ! {\displaystyle \Delta _{h}^{m-1}[R](x)=ah^{m-1}(m-1)!} Let S ( x ) be a polynomial of degree m . With one pairwise difference: Δ h [ S ] ( x ) = [ a ( x + h ) m + b ( x + h ) m − 1 + l.o.t. ] − [

180-447: A x m + b x m − 1 + l.o.t. ] = a h m x m − 1 + l.o.t. = T ( x ) {\displaystyle \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)} As ahm ≠ 0 , this results in a polynomial T ( x ) of degree m − 1 , with ahm as

225-486: A x n + b x n − 1 + l . o . t . {\displaystyle P(x)=ax^{n}+bx^{n-1}+l.o.t.} After n pairwise differences, the following result can be achieved, where h ≠ 0 is a real number marking the arithmetic difference: Δ h n [ P ] ( x ) = a h n n ! {\displaystyle \Delta _{h}^{n}[P](x)=ah^{n}n!} Only

270-439: A ⋅ 3 1 ⋅ 1 ! = a ⋅ 3 {\displaystyle 108=a\cdot 3^{1}\cdot 1!=a\cdot 3} It can be found that a = 36 and thus the third term of the polynomial is 36 x . Subtracting out the third term: Without any pairwise differences, it is found that the 4th and final term of the polynomial is the constant −19 . Thus, the lowest-degree polynomial intercepting all

315-409: A ⋅ 162 {\displaystyle 648=a\cdot 3^{3}\cdot 3!=a\cdot 27\cdot 6=a\cdot 162} Solving for a , it can be found to have the value 4 . Thus, the first term of the polynomial is 4 x . Then, subtracting out the first term, which lowers the polynomial's degree, and finding the finite difference again: Here, the constant is achieved after only two pairwise differences, thus

360-532: A x + b ] = a h = a h 1 1 ! {\displaystyle \Delta _{h}[Q](x)=Q(x+h)-Q(x)=[a(x+h)+b]-[ax+b]=ah=ah^{1}1!} This proves it for the base case. Let R ( x ) be a polynomial of degree m  − 1 where m ≥ 2 and the coefficient of the highest-order term be a ≠ 0 . Assuming the following holds true for all polynomials of degree m  − 1 : Δ h m − 1 [ R ] ( x ) =

405-538: 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

450-498: A desired derivative order may be constructed. 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 . Difference quotient which when taken to

495-439: A differences table, where for all cells to the right of the first y , the following relation to the cells in the column immediately to the left exists for a cell ( a + 1, b + 1) , with the top-leftmost cell being at coordinate (0, 0) : ( a + 1 , b + 1 ) = ( a , b + 1 ) − ( a , b ) {\displaystyle (a+1,b+1)=(a,b+1)-(a,b)} To find

SECTION 10

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540-435: A fixed (non-zero) value instead of approaching zero, then the right-hand side of the above equation would be written f ( x + h ) − f ( x ) h = Δ h [ f ] ( x ) h . {\displaystyle {\frac {f(x+h)-f(x)}{h}}={\frac {\Delta _{h}[f](x)}{h}}.} Hence, the forward difference divided by h approximates

585-619: 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

630-571: A more accurate approximation. If f is three times differentiable, δ h [ f ] ( x ) h − f ′ ( x ) = o ( h 2 ) . {\displaystyle {\frac {\delta _{h}[f](x)}{h}}-f'(x)=o\left(h^{2}\right).} The main problem with the central difference method, however, is that oscillating functions can yield zero derivative. If f  ( nh ) = 1 for n odd, and f  ( nh ) = 2 for n even, then f  ′( nh ) = 0 if it

675-507: A term of order h . 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. For a given polynomial of degree n ≥ 1 , expressed in the function P ( x ) , with real numbers a ≠ 0 and b and lower order terms (if any) marked as l.o.t. : P ( x ) =

720-533: Is calculated with the central difference scheme . This is particularly troublesome if the domain of f 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). In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using

765-405: Is home to sequential degrees ("higher orders") of derivation, or differentiation . This property can be generalized to all difference quotients. As this sequencing requires a corresponding boundary splintering, it is practical to break up the point range into smaller, equi-sized sections, with each section being marked by an intermediary point ( P i ), where LB = P 0 and UB = P ń ,

810-442: Is its argument , usually a point ("P") expressible on a graph. The difference between two points, themselves, is known as their Delta (Δ P ), as is the difference in their function result, the particular notation being determined by the direction of formation: The general preference is the forward orientation, as F(P) is the base, to which differences (i.e., "ΔP"s) are added to it. Furthermore, The function difference divided by

855-434: Is often used as an approximation of the derivative, typically in numerical differentiation . The derivative of a function f at a point x is defined by the limit f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h . {\displaystyle f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}.} If h has

900-427: Is the operator that maps a function f to the function Δ [ f ] {\displaystyle \Delta [f]} defined by Δ [ f ] ( x ) = f ( x + 1 ) − f ( x ) . {\displaystyle \Delta [f](x)=f(x+1)-f(x).} A difference equation is a functional equation that involves the finite difference operator in

945-532: The central difference is given by δ h [ f ] ( x ) = f ( x + h 2 ) − f ( x − h 2 ) = Δ h / 2 [ f ] ( x ) + ∇ h / 2 [ f ] ( x ) . {\displaystyle \delta _{h}[f](x)=f(x+{\tfrac {h}{2}})-f(x-{\tfrac {h}{2}})=\Delta _{h/2}[f](x)+\nabla _{h/2}[f](x).} Finite difference

SECTION 20

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990-428: The n -th order forward, backward, and central differences are given by, respectively, These equations use binomial coefficients after the summation sign shown as ( i ) . Each row of Pascal's triangle provides the coefficient for each value of i . Note that the central difference will, for odd n , have h multiplied by non-integers. This is often a problem because it amounts to changing

1035-534: The calculus of infinitesimals . Three basic types are commonly considered: forward , backward , and central finite differences. A forward difference , denoted Δ h [ f ] , {\displaystyle \Delta _{h}[f],} of a function f is a function defined as Δ h [ f ] ( x ) = f ( x + h ) − f ( x ) . {\displaystyle \Delta _{h}[f](x)=f(x+h)-f(x).} Depending on

1080-484: The limit as h approaches 0 gives the derivative of the function f . The name of the expression stems from the fact that it is the quotient of the difference of values of the function by the difference of the corresponding values of its argument (the latter is ( x + h ) - x = h in this case). The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h ). The limit of

1125-429: The n th point, equaling the degree/order: There are other derivative notations , but these are the most recognized, standard designations. The quintessential application of the divided difference is in the presentation of the definite integral, which is nothing more than a finite difference: Given that the mean value, derivative expression form provides all of the same information as the classical integral notation,

1170-410: The above central difference formula for f  ′( x + ⁠ h / 2 ⁠ ) and f  ′( x − ⁠ h / 2 ⁠ ) and applying a central difference formula for the derivative of f  ′ at x , we obtain the central difference approximation of the second derivative of f : Similarly we can apply other differencing formulas in a recursive manner. More generally,

1215-404: The application, the spacing h may be variable or constant. When omitted, h is taken to be 1; that is, Δ [ f ] ( x ) = Δ 1 [ f ] ( x ) = f ( x + 1 ) − f ( x ) . {\displaystyle \Delta [f](x)=\Delta _{1}[f](x)=f(x+1)-f(x).} A backward difference uses

1260-405: The backward difference: ∇ h [ f ] ( x ) h − f ′ ( x ) = o ( h ) → 0 as  h → 0. {\displaystyle {\frac {\nabla _{h}[f](x)}{h}}-f'(x)=o(h)\to 0\quad {\text{as }}h\to 0.} However, the central (also called centered) difference yields

1305-947: The binomial coefficients grow rapidly for large n . The relationship of these higher-order differences with the respective derivatives is straightforward, d n f d x n ( x ) = Δ h n [ f ] ( x ) h n + o ( h ) = ∇ h n [ f ] ( x ) h n + o ( h ) = δ h n [ f ] ( x ) h n + o ( h 2 ) . {\displaystyle {\frac {d^{n}f}{dx^{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).} Higher-order differences can also be used to construct better approximations. As mentioned above,

1350-410: The coefficient of the highest-order term remains. As this result is constant with respect to x , any further pairwise differences will have the value 0 . Let Q ( x ) be a polynomial of degree 1 : Δ h [ Q ] ( x ) = Q ( x + h ) − Q ( x ) = [ a ( x + h ) + b ] − [

1395-397: The coefficient of the highest-order term. Given the assumption above and m − 1 pairwise differences (resulting in a total of m pairwise differences for S ( x ) ), it can be found that: Δ h m − 1 [ T ] ( x ) = a h m ⋅ h m − 1 ( m − 1 ) ! =

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1440-502: The derivative when h is small. The error in this approximation can be derived from Taylor's theorem . Assuming that f is twice differentiable, we have Δ h [ f ] ( x ) h − f ′ ( x ) = o ( h ) → 0 as  h → 0. {\displaystyle {\frac {\Delta _{h}[f](x)}{h}}-f'(x)=o(h)\to 0\quad {\text{as }}h\to 0.} The same formula holds for

1485-460: The difference quotient (i.e., the derivative) is thus the instantaneous rate of change. By a slight change in notation (and viewpoint), for an interval [ a , b ], the difference quotient is called the mean (or average) value of the derivative of f over the interval [ a , b ]. This name is justified by the mean value theorem , which states that for a differentiable function f , its derivative f ′ reaches its mean value at some point in

1530-434: The first term, the following table can be used: This arrives at a constant 648 . The arithmetic difference is h = 3 , as established above. Given the number of pairwise differences needed to reach the constant, it can be surmised this is a polynomial of degree 3 . Thus, using the identity above: 648 = a ⋅ 3 3 ⋅ 3 ! = a ⋅ 27 ⋅ 6 =

1575-624: The first-order difference approximates the first-order derivative up to a term of order h . However, the combination Δ h [ f ] ( x ) − 1 2 Δ h 2 [ f ] ( x ) h = − f ( x + 2 h ) − 4 f ( x + h ) + 3 f ( x ) 2 h {\displaystyle {\frac {\Delta _{h}[f](x)-{\frac {1}{2}}\Delta _{h}^{2}[f](x)}{h}}=-{\frac {f(x+2h)-4f(x+h)+3f(x)}{2h}}} approximates f  ′( x ) up to

1620-440: The following result: − 306 = a ⋅ 3 2 ⋅ 2 ! = a ⋅ 18 {\displaystyle -306=a\cdot 3^{2}\cdot 2!=a\cdot 18} Solving for a , which is −17 , the polynomial's second term is −17 x . Moving on to the next term, by subtracting out the second term: Thus the constant is achieved after only one pairwise difference: 108 =

1665-410: The function values at x and x − h , instead of the values at x + h and  x : ∇ h [ f ] ( x ) = f ( x ) − f ( x − h ) = Δ h [ f ] ( x − h ) . {\displaystyle \nabla _{h}[f](x)=f(x)-f(x-h)=\Delta _{h}[f](x-h).} Finally,

1710-604: The interval of discretization. The problem may be remedied substituting the average of   δ n [ f ] (   x −   h   2   )   {\displaystyle \ \delta ^{n}[f](\ x-{\tfrac {\ h\ }{2}}\ )\ } and   δ n [ f ] (   x +   h   2   )   . {\displaystyle \ \delta ^{n}[f](\ x+{\tfrac {\ h\ }{2}}\ )~.} Forward differences applied to

1755-431: The interval. Geometrically, this difference quotient measures the slope of the secant line passing through the points with coordinates ( a , f ( a )) and ( b , f ( b )). Difference quotients are used as approximations in numerical differentiation , but they have also been subject of criticism in this application. Difference quotients may also find relevance in applications involving Time discretization , where

1800-495: The mean value form may be the preferable expression, such as in writing venues that only support/accept standard ASCII text, or in cases that only require the average derivative (such as when finding the average radius in an elliptic integral). This is especially true for definite integrals that technically have (e.g.) 0 and either π {\displaystyle \pi \,\!} or 2 π {\displaystyle 2\pi \,\!} as boundaries, with

1845-513: The point difference is known as "difference quotient": If ΔP is infinitesimal, then the difference quotient is a derivative , otherwise it is a divided difference : Regardless if ΔP is infinitesimal or finite, there is (at least—in the case of the derivative—theoretically) a point range, where the boundaries are P ± (0.5) ΔP (depending on the orientation—ΔF(P), δF(P) or ∇F(P)): Derivatives can be regarded as functions themselves, harboring their own derivatives. Thus each function

Finite difference - Misplaced Pages Continue

1890-454: The points in the first table is found: 4 x 3 − 17 x 2 + 36 x − 19 {\displaystyle 4x^{3}-17x^{2}+36x-19} 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

1935-410: 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

1980-690: 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 Károly Jordan  [ de ] (1939). Finite differences trace their origins back to one of Jost Bürgi 's algorithms ( c.  1592 ) and work by others including Isaac Newton . The formal calculus of finite differences can be viewed as an alternative to

2025-478: The width of the time step is used for the value of h. The difference quotient is sometimes also called the Newton quotient (after Isaac Newton ) or Fermat's difference quotient (after Pierre de Fermat ). The typical notion of the difference quotient discussed above is a particular case of a more general concept. The primary vehicle of calculus and other higher mathematics is the function . Its "input value"

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