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Finite difference
A finite difference is a mathematical expression of the form f(x + b) − f(x + a). Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation.
The difference operator, commonly denoted , is the operator that maps a function f to the function defined by 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. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences.
![{\displaystyle \Delta [f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/978a8399050eeaf584e6c0f7a57196f57d9bdb71)
=f(x+1)-f(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15da4d892ccab64158a6e556b809b3f365da7057)
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 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 (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 the calculus of infinitesimals.
Three basic types are commonly considered: forward, backward, and central finite differences.
A forward difference, denoted of a function f is a function defined as
![{\displaystyle \Delta _{h}[f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f27d5cf82d8c60c1fc01eeaef360596b74082d75)
=f(x+h)-f(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b4b1b463ac6a532d0089dad4ef7f7ec1a91acb9)
Depending on the application, the spacing h may be variable or constant. When not specified, the default value for h is 1; that is,
=\Delta _{1}[f](x)=f(x+1)-f(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c34fb96bcb57be4378a378d93aa636c18f9bda94)
A backward difference uses the function values at x and x − h, instead of the values at x + h and x:
=f(x)-f(x-h)=\Delta _{h}[f](x-h).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a100757e5e1977e4c05a91e35bacf18bc054954c)
Hub AI
Finite difference AI simulator
(@Finite difference_simulator)
Finite difference
A finite difference is a mathematical expression of the form f(x + b) − f(x + a). Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation.
The difference operator, commonly denoted , is the operator that maps a function f to the function defined by 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. 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 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 (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 the calculus of infinitesimals.
Three basic types are commonly considered: forward, backward, and central finite differences.
A forward difference, denoted of a function f is a function defined as
Depending on the application, the spacing h may be variable or constant. When not specified, the default value for h is 1; that is,
A backward difference uses the function values at x and x − h, instead of the values at x + h and x: