摘要
首先构造了一个数列,找出数列满足的递推关系,将递推关系采用矩阵的形式表示,计算出矩阵的n个特征值,对特征矩阵进行初等变换,求出特征向量,得到可逆矩阵,根据特征值理论,求出相似对角阵,确定矩阵与一对角阵的相似关系,由此推出矩阵的n次幂与对角矩阵的n次幂是相似的。然后,利用特征值和特征向量,导出数列的通项,通项中含有复数的n次方,当n较大时计算通项比较麻烦,为此引入虚数表示方法,将通项表达式中有关的系数采用三角式表示。进而,由数列的各项均为正整数,当n较小时,通项与真值偏差微小,断定出通项的真值,当n较大时,由于舍入误差的积累,通项与其真值的偏差大些,必须减小舍入误差。最后,对所得的通项给予验证得出结果是正确的,方法是可行的。
This article constructs a sequence,finds out the recurrence relations,expresses the recurrence relation on the motrix form,counts out n eigenvalues of the matrix and elementary transformation of the feature matrix,finds the feature vector,and gets invertible matrix.According to eigenvalue theory,the author obtains similar matrix of a diagonal matrix,determines the similarity relation between the matrix and its diagonal matrix,and deduces that the n-th power matrix and its n-th power diagonal matrix are similar.Then by eigenvalues and eigenvectors,the author exports the general term of sequence.This general term contains plural n-th power,so when n is large enough,to compute the general term can lead to a mass.For this reason,the imaginary representation is introduced,and the general term is expressed by the triangular-type.As each term of sequence is positive integer,when n is smaller,the deviation between the general term and the true value is very small.Therefore,the true value of the general term can be determined.When n is larger,because the rounding errors are accumulated,general term and its deviation from the true value may be larger.In order to ensure the accuracy of calculation,rounding errors must be reduced.Finally,the general term is proved,and the method is feasible.
出处
《沈阳师范大学学报（自然科学版）》
CAS
2011年第3期347-351,共5页
Journal of Shenyang Normal University:Natural Science Edition
基金
辽宁省教育厅高等学校科学研究项目(20060842)
关键词
特征值
相似矩阵
递推关系
通项
eigenvalue
similar matrices
recursion relation
general term
