Jacobi矩阵的逆特征问题

本文研究了两个Jacobi矩阵的逆特征问题：I给定实数λ,μ(λ＞μ）和n维非零实向量x，y，求n阶Jacobi矩阵J,使Jx=λx，Jy=μy,且λ＞λ2（J）＞…＞λi-1（J）＞μ＞λi＋1（J）…＞λn（J），或λi（J）＞λ2（J）＞…＞λi－1（J）＞λ＞λi＋1（J）＞…＞λn-1（J）＞μ·II给定实数λ，μ（λ＞μ）和n维非零实向量x，y，求n阶Jacobi矩阵J，使Jx＝λx，Jy＝μy，且λ1（J）＞λ2（J）＞…＞λi－1（J）＞λ＞μ＞λi＋2（J）＞…＞λn（J）．文中给出了问题I；II有唯一解的充要条件，并给出了解的表达式．     

一类具有转向点问题的n阶近似解

用Langer变换和Olver变换求得一类具有转向点问题的n阶近似解:y(x)=v(x)ψ(x),其中ψ=λ12-14×(x2-1)14,2332=-λx∫11-τ2dτ,v(z)=A(z,λ)ξ(λ23z)+B(z,λ)'ζ(λ23z).并探讨了其特征值问题,得到λn=4n+1112,n=0,1,2….由此给出了该类问题的解的一般性结论.     

一个不等式的初等证明

文 [1]提出如下猜想 :设λ≥ 1,x,y,z >0 ,则xλx +y+yλy +z+zλz +x　≤ 3λ+1(1)文 [2 ]用导数证明了 (1)式 ,本文给出简明的初等证明 .证明　由已知得 xλx +y,yλy +z,zλz +x三式中必有两个同时不大于 (或不小于 ) 1λ +1,不妨设为 xλx +y 和yλy +z.于是有(xλx +y - 1λ +1) (yλy +z -1λ+1)≥ 0即 xλx +y+yλy +z≤(1+λ) xy(λx +y) (λy +z) +1λ +1(2 )由柯西不等式有(λx +y) (λy +z)≥ (λ xy +yz) 2 .代入 (2 )得　 xλx +y +yλy +z ≤(λ+1) xλ x +z +1λ+1(3)又　 (λz +x) (λ+1)≥ (λ z +x ) 2(4)于是 ,由 (3)、(…     

一个分式不等式的探讨

文 [1]提出如下有趣问题 :设λ、μ、ν为不全为零的非负实数 ,求使不等式xλx+ μy +νz + yλy+ μz +νx +zλz+ μx+νy ≥ 3λ+ μ+ν (1)对任意正实数x ,y ,z都成立的充要条件 .经探讨 ,我们得到了下面的定理 1　当λ、μ、ν≥ 0且 μ ,ν不全为零时 (若 μ =ν =0 ,λ ≠ 0 ,则 (1)为恒等式 ) ,(1)对任意x ,y,z>0成立的充要条件是2λ≤ μ +ν .证明　用 ∑f(x ,y ,z)表示 f(x ,y ,z)+ f(y ,z ,x) + f(z ,x ,y) ,经演算有∑x(λy + μz+νx) (λz+ μx +νz)=λμν∑x3 + (λ3 + μ3 +ν3 + 3λμν)xyz +(λ2 μ+ μ2 ν+ν2 λ) …     

一个含双参数的分式不等式及其应用

本文给出了一个新颖的涉及正实数a,b,c,x,y,z含双参数λ,μ的分式不等式,并举例说明其应用,同时在文章最后还给出了一个猜想.定理设x,y,z∈R ,a,b,c∈R ,λμ≥-1,若λ>0,则有(x μy)a2(λ 1)x (λμ 1)y z (y μz)b2(λ 1)y (λμ 1)z x (z μx)c2(λ 1)z (λμ 1)x y≤1λ[(a     

简单自反DTS(υ,λ)的存在谱

一个Directed三元系DTS(υ，λ)=(X，B)是自反的，如果它与它的逆(x，B^-1)同构，其中B^-1={(z，y，x)；(x，y，z)∈B}．继已给出SCDTS(υ，λ)的存在谱之后，又给出简单SCDTS(υ，λ)的存在谱。     

实对称五对角矩阵的两类广义特征值反问题

讨论了如下两类广义特征值反问题:(i)由给定的三个互异的特征对和给定的实对称正定五对角矩阵构造一个实对称五对角矩阵;(ii)由给定的三个互异特征对和给定的全对称正定五对角矩阵构造一个全对称五对角矩阵.利用线性方程组理论、对称向量和反对称向量的性质,分别得到了两类反问题存在唯一解的充要条件,并给出了解的表达式和数值算法;最后通过数值例子说明了算法的有效性.     

关于Jacobi矩阵逆特征值问题的扰动分析

１预备 若不特别说明,本文沿用［６］中记号． Ｈｏｃｈｓｔａｄｔ于１９６７年提出如下问题［１］： 问题Ⅰ　给定两组实数｛λ｝ｎｊ＝１＝１和｛μ｝ｎ＝１ｉ＝１,满足构造一个ｎ阶实对称三对角矩阵Ｊｎ,使得λ１,…λｎ为人的特征值,而Ｊｎ－１阶顺序主子阵的特征值为μ１,…,μｎ－１． 问题Ⅱ　给定一组实数｛λｊ｝ｎｊ＝１,满足构造一个ｎ阶全对称三对角矩阵Ｊｎ（ｓ）,使得Ｊｎ（ｓ）的特征值为λ１,λ２,…λｎ． ｄｅ Ｂｏｏｒ和Ｇｏｌｕｂ［４］提出如下问题： 问题Ⅲ　给定两组实数满足构造ｎ阶实对称三对角矩阵Ｊ…     

Bargmann空间中无界Gribov-Intissar算子的谱逼近（英文）

在[Adv.Math.(China),2015,44(3):335-353]中,我们研究了经典Bargmann空间Bo中的非自伴算子H_μ:H_μ=S_μ+H_λ,其中S_μ=μz d/(dz),H_λ=iλ(z(d~2)/(dz~2)+z~2 d/(dz)),i~2=-1,参数μ,λ都是实数.我们给出了H_μ的谱分析和H_μ的广义特征向量的渐近分析.设ek(z)=(z~k)/((k!)~(1/2)),k=1,2,…是B0的正交基.算子H_μ可以被一列三对角矩阵逼近,此三对角矩阵的主对角线元素为β_k=μk,次对角线元素α_k=iλk(k+1)~(1/2),1≤k≤n,n∈N.对于μ∈C和λ∈C,本文主要研究上述矩阵的特征值z_(k,n)(μ,λ)的局部化,它是多项式P_(n+1)~(μ,λ)(z)的零点,P_(n+1)~(μ,λ)(z)满足三项递推关系:若"∈R和λ∈R,则上述矩阵是复对称的.在这种情况下,我们证明了R上有界变分复值函数∈(z)的存在性,它使得权重为∈(z)的多项式P_n~(μ,λ)(z)是正交的.我们也考虑了H_μ的扰动H_λ'=S_λ'+H_λ,其中S_λ'=λ'z~2(d~2)/(dz~2)+S_μ,λ'∈R,H_λ可以被矩阵(h_(jk)~λ)_(j,k=1)~∞表示.证明了可以通过S_λ'的特征值和有限矩阵(h_(jk)~λ)_(j,k=1)~n的特征值的组合来逼近H_λ'的特征值.     

由主特征对构造对称三对角矩阵

本文讨论了由给定的两个特征对(λ1，x^(1))和(λ2，x^(2))来重构一个对称三对角矩阵的问题．给出了保证λ1为Tn的最大特征值的条件．     

由谱数据数值稳定地构造实对称带状矩阵

§1.引言 设r,n是正整数并且0r有a_(ij)=0.     

实对称五对角矩阵逆特征值问题

1 引 言 对于n阶实对称矩阵A=(aij),r是一个正整数,且1≤r≤n-1,当|i-j|>r时,aij=0(i,j=1,2,…,n),至少有一个i使得ai,i+r≠0,则称矩阵A是带宽为2r+1的实对称带状矩阵.特别地,当r=1时,称A为实对称三对角矩阵;当r=2时,称A为实对称五对角矩阵. 实对称带状矩阵逆特征值问题应用十分广泛,这类问题不仅来自微分方程逆特征值问     

关于代数特征值反问题对称情况可解的充分条件

§1.引言 本文讨论下述特征值反问题的可解性: 问题 G.设A_0=(a_(ij)~((0)))和A_k=(a_(ij)~((k)))(k=1,…,n)是一组n+1个n×n实对称矩阵,λ_1,…,λ_n是n个不同的实数.求实数c_1,…,c_n使得矩阵A_0+sum from k-1 to n C_k·A_k的特征值为λ_1,…,λ_n. [1]和[2]曾给出此问题可解的充分条件.本文应用Rothe不动点定理[3]给出问题G可解的另外两个充分条件.本文的结果可判定[1]和[2]中定理所不能判定的某些问题     

用3个向量对构造梁振动系统的刚度矩阵

针对梁的离散化模型的刚度矩阵是五对角矩阵,梁振动反问题的实质是实对称五对角矩阵的特征值反问题.该文利用向量对、Moore-Penrose广义逆给出了实对称五对角矩阵向量对反问题存在唯一解的条件,并结合矩阵分块讨论了双对称五对角矩阵向量对反问题解存在唯一的条件,进而计算了次对角线位置元素为负,其它位置元素均为正的实对称五对角矩阵特征值反问题.由于构造梁的离散模型需要的数据可由测试得到,故而其结果适合于模态分析、系统结构的分析与设计等方面应用.最后给出了数值算例,通过数值讨论说明方法的有效性.     

On the numerical solution of ill‐conditioned linear systems by regularization and iteration

We propose to reduce the (spectral) condition number of a given linear system by adding a suitable diagonal matrix to the system matrix, in particular by shifting its spectrum. Iterative procedures are then adopted to recover the solution of the original system. The case of real symmetric positive definite matrices is considered in particular, and several numerical examples are given. This approach has some close relations with Riley's method and with Tikhonov regularization. Moreover, we identify approximately the aforementioned procedure with a true action of preconditioning.     

Numerical methods for solving some matrix feasibility problems

In this paper, we design two numerical methods for solving some matrix feasibility problems, which arise in the quantum information science. By making use of the structured properties of linear constraints and the minimization theorem of symmetric matrix on manifold, the projection formulas of a matrix onto the feasible sets are given, and then the relaxed alternating projection algorithm and alternating projection algorithm on manifolds are designed to solve these problems. Numerical examples show that the new methods are feasible and effective.     

Robustness and applicability of Markov chain Monte Carlo algorithms for eigenvalue problems

In this paper we analyse applicability and robustness of Markov chain Monte Carlo algorithms for eigenvalue problems. We restrict our consideration to real symmetric matrices.

Almost Optimal Monte Carlo (MAO) algorithms for solving eigenvalue problems are formulated. Results for the structure of both – systematic and probability error are presented. It is shown that the values of both errors can be controlled independently by different algorithmic parameters. The results present how the systematic error depends on the matrix spectrum. The analysis of the probability error is presented. It shows that the close (in some sense) the matrix under consideration is to the stochastic matrix the smaller is this error. Sufficient conditions for constructing robust and interpolation Monte Carlo algorithms are obtained. For stochastic matrices an interpolation Monte Carlo algorithm is constructed.

A number of numerical tests for large symmetric dense matrices are performed in order to study experimentally the dependence of the systematic error from the structure of matrix spectrum. We also study how the probability error depends on the balancing of the matrix.     

Symmetric-triangular decomposition and its applications part II: Preconditioners for indefinite systems

As an application of the symmetric-triangular (ST) decomposition given by Golub and Yuan (2001) and Strang (2003), three block ST preconditioners are discussed here for saddle point problems. All three preconditioners transform saddle point problems into a symmetric and positive definite system. The condition number of the three symmetric and positive definite systems are estimated. Therefore, numerical methods for symmetric and positive definite systems can be applied to solve saddle point problems indirectly. A numerical example for the symmetric indefinite system from the finite element approximation to the Stokes equation is given. Finally, some comments are given as well. AMS subject classification (2000) 65F10     

Residual Selection in A Projection Method for Convex Minimization Problems

We study a certain class of min-max optimization problems and we show that the solutions of such problems are, in some sense, degenerate, Two examples are given:minimization of the spectral radius of a symmetric matrix, minimization of the largest eigenvalue of a symmetric matrix     

一类特殊微分代数方程组的性质及应用

一类特殊微分代数方程组的性质及应用顾金生，胡显承（清华大学应用数学系）ＴＨＥＰＲＯＰＥＲＴＩＥＳＯＦＡＫＩＮＤＯＦＤＩＦＦＥＲＥＮＴＩＡＬ／ＡＬＧＥＢＲＡＩＣＥＱＵＡＴＩＯＮＳＡＮＤＴＨＥＩＲＡＰＰＬＩＣＡＴＩＯＮＳ￥ＧｕＪｉｎ－ｓｈｅｎｇ；ＨｕＸｉ...