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In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, ''c'' is a fixed point of the function ''f''(''x'') if and only if ''f''(''c'') = ''c''. This means ''f''(''f''(...''f''(''c'')...)) = ''fn''(''c'') = ''c'', an important terminating consideration when recursively computing ''f''. A set of fixed points is sometimes called a ''fixed set''. For example, if ''f'' is defined on the real numbers by : then 2 is a fixed point of ''f'', because ''f''(2) = 2. Not all functions have fixed points: for example, if ''f'' is a function defined on the real numbers as ''f''(''x'') = ''x'' + 1, then it has no fixed points, since ''x'' is never equal to ''x'' + 1 for any real number. In graphical terms, a fixed point means the point (''x'', ''f''(''x'')) is on the line ''y'' = ''x'', or in other words the graph of ''f'' has a point in common with that line. Points which come back to the same value after a finite number of iterations of the function are known as periodic points; a fixed point is a periodic point with period equal to one. In projective geometry, a fixed point of a projectivity has been called a double point. ==Attractive fixed points== An ''attractive fixed point'' of a function ''f'' is a fixed point ''x''0 of ''f'' such that for any value of ''x'' in the domain that is close enough to ''x''0, the iterated function sequence : converges to ''x''0. An expression of prerequisites and proof of the existence of such solution is given by the Banach fixed-point theorem. The natural cosine function ("natural" means in radians, not degrees or other units) has exactly one fixed point, which is attractive. In this case, "close enough" is not a stringent criterion at all—to demonstrate this, start with ''any'' real number and repeatedly press the ''cos'' key on a calculator (checking first that the calculator is in "radians" mode). It eventually converges to about 0.739085133, which is a fixed point. That is where the graph of the cosine function intersects the line . Not all fixed points are attractive: for example, ''x'' = 0 is a fixed point of the function ''f''(''x'') = 2''x'', but iteration of this function for any value other than zero rapidly diverges. However, if the function ''f'' is continuously differentiable in an open neighbourhood of a fixed point ''x''0, and , attraction is guaranteed. Attractive fixed points are a special case of a wider mathematical concept of attractors. An attractive fixed point is said to be a ''stable fixed point'' if it is also Lyapunov stable. A fixed point is said to be a ''neutrally stable fixed point'' if it is Lyapunov stable but not attracting. The center of a linear homogeneous differential equation of the second order is an example of a neutrally stable fixed point. 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「Fixed point (mathematics)」の詳細全文を読む スポンサード リンク
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