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Termination criteria for the interval version of Newton's method for solving systems of non-linear equations
Termination criteria for the interval version of Newton's method for solving systems of non-linear equations
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机译:牛顿法求解非线性方程组的区间版本的终止准则
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摘要
One embodiment of the present invention provides a system for finding the roots of a system of nonlinear equations within an interval vector X=(X1, . . . , Xn), wherein the system of non-linear equations is specified by a vector function ƒ=(ƒ1, . . . , ƒn). The system operates by receiving a representation of the interval vector X (which is also called a box), wherein for each dimension, i, the representation of Xi includes a first floating-point number, αi, representing the left endpoint of Xi, and a second floating-point number, bi, representing the right endpoint of Xi. Next, the system performs an interval Newton step on X to produce a resulting interval vector, X′, wherein the point of expansion of the interval Newton step is a point, x, within the interval X, and wherein performing the interval Newton step involves evaluating ƒ(x) to produce an interval result ƒ1(x). The system then evaluates a first termination condition, wherein the first termination condition is TRUE if: zero is contained within ƒ1(x), J(x,X) is regular (wherein J(x,X) is the Jacobian of the function ƒevaluated with respect to x over the box X); and X is contained within X′. If the first termination condition is TRUE, the system terminates the interval Newton method and records X′ as a final bound.
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机译:本发明的一个实施例提供了一种用于在间隔向量X =(X 1 Sub>,...,X n Sub>)内找到非线性方程组的根的系统。 ,其中非线性方程组由矢量函数ƒ=(ƒ 1 Sub>,...,ƒ n Sub>)指定。该系统通过接收间隔向量X的表示形式(也称为框)进行操作,其中对于每个维度i,X i Sub>的表示形式都包括第一浮点数α< Sub> i Sub>代表X i Sub>的左端点,第二个浮点数b i Sub>代表X i的右端点我 Sub>。接下来,系统对X执行间隔牛顿步以产生结果间隔向量X',其中间隔牛顿步的扩展点是间隔X内的点x,并且其中执行间隔牛顿步涉及计算ƒ(x)以产生间隔结果ƒ 1 Sup>(x)。然后,系统评估第一终止条件,如果满足以下条件,则第一终止条件为TRUE:ƒ 1 Sup>(x)中包含零,J(x,X)是常规的(其中J(x,X )是在框X)上相对于x评估的函数ƒ的雅可比行列式; X包含在X'中。如果第一个终止条件为TRUE,则系统终止区间牛顿法并记录X'作为最终界限。
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