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A finite difference is a mathematical expression of the form . If a finite difference is divided by , 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. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences. Today, the term "finite difference" is often taken as synonymous with finite difference approximations of derivatives, especially in the context of numerical methods. Finite difference approximations are finite difference quotients in the terminology employed above. Finite differences have also been the topic of study as abstract self-standing mathematical objects, e.g. in works by George Boole (1860), L. M. Milne-Thomson (1933), and Károly Jordan (1939), tracing its origins back to Isaac Newton. In this viewpoint, the formal calculus of finite differences is an alternative to the calculus of infinitesimals.〔Jordán, op. cit., p. 1 and Milne-Thomson, p. xxi. Milne-Thomson, Louis Melville (2000): ''The Calculus of Finite Differences'' (Chelsea Pub Co, 2000) ISBN 978-0821821077〕 ==Forward, backward, and central differences== Three forms are commonly considered: forward, backward, and central differences.〔〔〔 A forward difference is an expression of the form : Depending on the application, the spacing ''h'' may be variable or constant. When omitted, ''h'' is taken to be 1: . A backward difference uses the function values at ''x'' and ''x'' − ''h'', instead of the values at ''x'' + ''h'' and ''x'': : Finally, the central difference is given by : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Finite difference」の詳細全文を読む スポンサード リンク
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