結(jié)構(gòu)方陣秩虧為1的可信性驗證

李 喆,周 蕊

(長春理工大學 理學院,長春 130022)

定理1[10]設(shè)A∈n×n.若對某個b∈n,c∈n,(n+1)×(n+1)矩陣非奇異,則rankA=n-1當且僅當線性方程組的解滿足條件f=0.

定理2[11]輸入一區(qū)間矩陣A∈In×n及區(qū)間右端列向量b∈In,若Verifylss函數(shù)[12]成功輸出區(qū)間向量X?In,則X滿足條件對某個?X.

利用區(qū)間牛頓迭代法可以驗證非線性方程的解.

定理3[11]令f:為可微函數(shù),X=(x1,x2)∈I且給定假設(shè)0?f′(X),利用區(qū)間運算,定義若?X,則X內(nèi)包含f的唯一解.若?,則對所有的x∈X,f(x)≠0.

定義邊界矩陣

(1)

其中b,c∈n.則當時,矩陣非奇異,且在向量附近,線性方程組

(2)

本文利用隱行列式方法[13]計算梯度f(ε),其理論基于數(shù)值二分法[8,10].對線性方程組(2)關(guān)于每個變量εi求導,得

(3)

因此可以通過求解k個具有相同系數(shù)矩陣,但不同右端列向量的線性方程組得到f(ε).

下面基于文獻[14]的結(jié)果通過將非線性方程f(ε)=0的某些變元做特定化方法,將驗證f(ε)=0的解轉(zhuǎn)化為驗證具有一個變元非線性方程的解.設(shè)

(4)

定義

(5)

1) 選擇i0滿足式(4).

3) 若步驟2)不收斂,則輸出算法失敗.

引入?yún)?shù)向量ε=(ε1,ε2,ε3,ε4,ε5,ε6),定義參數(shù)區(qū)間對稱矩陣

1) 確定i0=1;

2) 令E1=(-0.088,-0.086);

3) 計算參數(shù)向量U=(0.077 4,0.077 5);

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