黑龙江大学自然科学学报
黑龍江大學自然科學學報
흑룡강대학자연과학학보
JOURNAL OF NATURAL SCIENCE OF HEILONGJIANG UNIVERSITY
2004年
4期
1-3
,共3页
矩阵特征值问题%对称三对角矩阵%QR(QL)方法%Wilkinson位移%总体收敛性
矩陣特徵值問題%對稱三對角矩陣%QR(QL)方法%Wilkinson位移%總體收斂性
구진특정치문제%대칭삼대각구진%QR(QL)방법%Wilkinson위이%총체수렴성
matrix eigenvalue problem%symmetric tridiagonal matrix%QL(QR) Algorithm%Wilkinson Shift%global convergence
很多实际问题,如求结构振动的固有频率,动力系统稳定性的临界值等常常归结为计算对称矩阵的特征值,而首选的计算方法是先把该矩阵正交相似变换成一个对称三对角矩阵,再对这个对称三对角矩阵用带位移的QR(QL)方法.1968年J.H.Wilkinson给出对称三对角矩阵带位移的QR方法的第一个总体收敛定理,他证明了带Wilkinson位移的QR方法的总体收敛性,这是QR(QL)方法的理论基础,但他的证明太复杂.1978年W.Hoffman和B.N.Parlett又给出一个新证明,这是一个很精彩的证明,但也不是很简单.在此给出一简单而初等的证明,很适宜放在教材中.
很多實際問題,如求結構振動的固有頻率,動力繫統穩定性的臨界值等常常歸結為計算對稱矩陣的特徵值,而首選的計算方法是先把該矩陣正交相似變換成一箇對稱三對角矩陣,再對這箇對稱三對角矩陣用帶位移的QR(QL)方法.1968年J.H.Wilkinson給齣對稱三對角矩陣帶位移的QR方法的第一箇總體收斂定理,他證明瞭帶Wilkinson位移的QR方法的總體收斂性,這是QR(QL)方法的理論基礎,但他的證明太複雜.1978年W.Hoffman和B.N.Parlett又給齣一箇新證明,這是一箇很精綵的證明,但也不是很簡單.在此給齣一簡單而初等的證明,很適宜放在教材中.
흔다실제문제,여구결구진동적고유빈솔,동력계통은정성적림계치등상상귀결위계산대칭구진적특정치,이수선적계산방법시선파해구진정교상사변환성일개대칭삼대각구진,재대저개대칭삼대각구진용대위이적QR(QL)방법.1968년J.H.Wilkinson급출대칭삼대각구진대위이적QR방법적제일개총체수렴정리,타증명료대Wilkinson위이적QR방법적총체수렴성,저시QR(QL)방법적이론기출,단타적증명태복잡.1978년W.Hoffman화B.N.Parlett우급출일개신증명,저시일개흔정채적증명,단야불시흔간단.재차급출일간단이초등적증명,흔괄의방재교재중.
In many practice problems such as to know the resonnance frequency of a structure and to know the critical value for the stability of a dynamical system,often need to compute the eigenvalues of a symmetric matrix. The chief method to compute the eigenvalues of a symmetric matrix is first to transformate the matrix to a symmetric tridiagonal matrix similarly and orthogonally, then to use the QR(QL) method with shift to the symmetric tridiagonal matrix. For any unreducible symmetric tridiagonal, it is always convergent when use QR(QL) Algorithm with Wilkinson shift. This is a basic theorem of QR(QL) Algorithm theory. The first proof of this theorem was given by J.H. Wilkinson at 1968. The proof is very complicated [7]. 1978, W.Hoffman and B.N.Parlett gave an other proof. It is a very nice proof [1]. But it is not so simple too. A new proof of this theorem is given, simply and elementarily. The new proof satisfies to write in textbooks.
