| 单词 | saddle point |
| 释义 | saddle point A point at which a function of two variables takes a maximum for movement in some directions and a minimum for movement in other directions. The term is borrowed from geography, where a saddle is a low point in a range of hills. For example, if y = x2 - z2, ∂y/∂x = 2x and ∂2 y/∂x2 = 2; thus if x is varied holding z constant, y has a minimum at x = 0. But ∂y/∂z = -2z and ∂2 y/∂z2 = -2; thus if z is varied holding x constant, y takes a maximum at z = 0. The origin is therefore a saddle point for this function. The directions in which movement is considered need not correspond to the axes. Consider for example y = x2 + z2 + 3xz. Movement along the axes gives minima at the origin, as does movement in the direction where dz = dx; but movement in the direction where dz = -dx gives dy/dx = -x and d2 y/dx2 = -2; thus if x and z are varied along one diagonal, y has a minimum, but for movement along the other diagonal y has a maximum at the origin. |
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