李 喆,周 蕊

(长春理工大学 理学院,长春 130022)

定理1[10]设A∈n×n.若对某个b∈n,c∈n,(n+1)×(n+1)矩阵非奇异,则rankA=n-1当且仅当线性方程组的解满足条件f=0.

定理2[11]输入一区间矩阵A∈In×n及区间右端列向量b∈In,若Verifylss函数[12]成功输出区间向量X⊂In,则X满足条件对某个⊆X.

利用区间牛顿迭代法可以验证非线性方程的解.

定理3[11]令f:为可微函数,X=(x1,x2)∈I且给定假设0∉f′(X),利用区间运算,定义若⊂X,则X内包含f的唯一解.若Ø,则对所有的x∈X,f(x)≠0.

定义边界矩阵

(1)

其中b,c∈n.则当时,矩阵非奇异,且在向量附近,线性方程组

(2)

本文利用隐行列式方法[13]计算梯度f(ε),其理论基于数值二分法[8,10].对线性方程组(2)关于每个变量εi求导,得

(3)

因此可以通过求解k个具有相同系数矩阵,但不同右端列向量的线性方程组得到f(ε).

下面基于文献[14]的结果通过将非线性方程f(ε)=0的某些变元做特定化方法,将验证f(ε)=0的解转化为验证具有一个变元非线性方程的解.设

(4)

定义

(5)

1) 选择i0满足式(4).

3) 若步骤2)不收敛,则输出算法失败.

引入参数向量ε=(ε1,ε2,ε3,ε4,ε5,ε6),定义参数区间对称矩阵

1) 确定i0=1;

2) 令E1=(-0.088,-0.086);

3) 计算参数向量U=(0.077 4,0.077 5);

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