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881.
K.V. Fernando 《Linear algebra and its applications》2007,422(1):77-99
This report may be considered as a non-trivial extension of an unpublished report by William Kahan (Accurate Eigenvalues of a symmetric tri-diagonal matrix, Technical Report CS 41, Computer Science Department, Stanford University, 1966). His interplay between matrix theory and computer arithmetic led to the development of algorithms for computing accurate eigenvalues and singular values. His report is generally considered as the precursor for the development of IEEE standard 754 for binary arithmetic. This standard has been universally adopted by virtually all PC, workstation and midrange hardware manufactures and tens of billions of such machines have been produced. Now we use the features in this standard to improve the original algorithm.In this paper, we describe an algorithm in floating-point arithmetic to compute the exact inertia of a real symmetric (shifted) tridiagonal matrix. The inertia, denoted by the integer triplet (π, ν, ζ), is defined as the number of positive, negative and zero eigenvalues of a real symmetric (or complex Hermitian) matrix and the adjective exact refers to the eigenvalues computed in exact arithmetic. This requires the floating-point computation of the diagonal matrix D of the LDLt factorization of the shifted tridiagonal matrix T − τI with +∞ and −∞ rounding modes defined in IEEE 754 standard. We are not aware of any other algorithm which gives the exact answer to a numerical problem when implemented in floating-point arithmetic in standard working precisions. The guaranteed intervals for eigenvalues are obtained by bisection or multisection with this exact inertia information. Similarly, using the Golub-Kahan form, guaranteed intervals for singular values of bidiagonal matrices can be computed. The diameter of the eigenvalue (singular value) intervals depends on the number of shifts with inconsistent inertia in two rounding modes. Our algorithm not only guarantees the accuracy of the solutions but is also consistent across different IEEE 754 standard compliant architectures. The unprecedented accuracy provided by our algorithms could be also used to debug and validate standard floating-point algorithms for computation of eigenvalues (singular values). Accurate eigenvalues (singular values) are also required by certain algorithms to compute accurate eigenvectors (singular vectors).We demonstrate the accuracy of our algorithms by using standard matrix examples. For the Wilkinson matrix, the eigenvalues (in IEEE double precision) are very accurate with an (open) interval diameter of 6 ulps (units of the last place held of the mantissa) for one of the eigenvalues and lesser (down to 2 ulps) for others. These results are consistent across many architectures including Intel, AMD, SGI and DEC Alpha. However, by enabling IEEE double extended precision arithmetic in Intel/AMD 32-bit architectures at no extra computational cost, the (open) interval diameters were reduced to one ulp, which is the best possible solution for this problem. We have also computed the eigenvalues of a tridiagonal matrix which manifests in Gauss-Laguerre quadrature and the results are extremely good in double extended precision but less so in double precision. To demonstrate the accuracy of computed singular values, we have also computed the eigenvalues of the Kac30 matrix, which is the Golub-Kahan form of a bidiagonal matrix. The tridiagonal matrix has known integer eigenvalues. The bidiagonal Cholesky factor of the Gauss-Laguerre tridiagonal is also included in the singular value study. 相似文献
882.
A. Yu. Trynin 《Siberian Mathematical Journal》2007,48(5):929-938
We obtain some upper and lower estimates for the sequences of the Lebesgue functions and constants of the Whittaker operators for continuous functions. We give an analog of Nevai’s formula for the Lagrange-Chebyshev and Lagrange-Laguerre interpolation polynomials for the operators under consideration. Its “local” version is established.
相似文献
$L_n (f,x) = \sum\limits_{k = 0}^n {\frac{{\sin (nx - k\pi )}}{{nx - k\pi }}} f\left( {\frac{{k\pi }}{n}} \right)$
883.
We establish sufficient conditions for a cohomology class of a discrete subgroup Γ of a connected semisimple Lie group with finite center to be representable by a bounded differential form on the quotient by Γ of the associated symmetric space; furthermore if \(\rho : \Gamma\to\mathrm{PU}(1,q)\) is any representation of any discrete subgroup Γ of SU (1, p), we give an explicit closed bounded differential form on the quotient by Γ of complex hyperbolic space which is a representative for the pullback via ρ of the Kähler class of PU(1,q). If G,G′ are Lie groups of Hermitian type, we generalize to representations \(\rho : \Gamma\to G'\) of lattices Γ < G the invariant defined in [Burger, M., Iozzi, A.: Bounded cohomology and representation variates in PU (1,n). Preprint announcement, April 2000] for which we establish a Milnor–Wood type inequality. As an application we study maximal representations into PU(1, q) of lattices in SU(1,1). 相似文献
884.
关于正整数的六边形数部分 总被引:2,自引:0,他引:2
吕忠田 《纯粹数学与应用数学》2007,23(3):377-380
对任意正整数n,设b(n)表示n的六边形数部分.即就是b(n)=m(2m-1),如果m(2m-1)≤n<(m 1)(2m 1),n∈N.本文主要目的是研究一个Dirichlet 级数的收敛性,并给出f(2)的一个精确的计算公式. 相似文献
885.
J. Cichon J. D. Mitchell M. Morayne 《Transactions of the American Mathematical Society》2007,359(5):2059-2074
If is a metric space, then and denote the semigroups of continuous and Lipschitz mappings, respectively, from to itself. The relative rank of modulo is the least cardinality of any set where generates . For a large class of separable metric spaces we prove that the relative rank of modulo is uncountable. When is the Baire space , this rank is . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.
886.
Hacene Serrai Arijitt Borthakur Lotfi Senhadji Ravinder Reddy Navin Bansal 《Journal of magnetic resonance (San Diego, Calif. : 1997)》2000,142(2):341
The application of continuous wavelet transform (CWT) analysis technique is presented to analyze multiple-quantum-filtered (MQF) 23Na magnetic resonance spectroscopy (MRS) data. CWT acts on the free-induction-decay (FID) signal as a time-frequency variable filter. The signal-to-noise ratio (SNR) and frequency resolution of the output filter are locally increased. As a result, MQF equilibrium longitudinal magnetization and the apparent fast and slow transverse relaxation times are accurately estimated. A developed iterative algorithm based on frequency signal detection and components extraction, already proposed, was used to estimate the values of the signal parameters by analyzing simulated time-domain MQF signals and data from an agarose gel. The results obtained were compared to those obtained by measurement of signal height in frequency domain as a function of MQF preparation time and those obtained by a simple time-domain curve fitting. The comparison indicates that the CWT approach provides better results than the other tested methods that are generally used for MQF 23Na MRS data analysis, especially when the SNR is low. The mean error on the estimated values of the amplitude signal and the apparent fast and slow transverse relaxation times for the simulated data were 2.19, 6.63, and 16.17% for CWT, signal height in frequency domain, and time-domain curve fitting methods, respectively. Another major advantage of the proposed technique is that it allows quantification of MQF 23Na signal from a single FID and, thus, reduces the experiment time dramatically. 相似文献
887.
本文考虑具有连续时滞的Lotka-Volterra竞争系统。给出了该系统的稳定性分析。利用Hopf分支定理,我们证明了由时滞引起的不稳定性可以导致系统产生稳定的极限环。利用Birkhoff定理证明了系统可能出现循环解。最后,我们给出数值实例,验证了上述结果。 相似文献
888.
丁协平 《应用数学和力学(英文版)》1991,12(1):85-92
In this paper,we prove some intersection theorems concerning noncompact sets withH-convex sections which generalize the corresponding results of Ma.Fan,Tarafdar,Lassonde and Shin-Tan to H-spaces without the linear structure and to noncompact setting.An application to von Neumann type minimax theorems is given. 相似文献
889.
动态力学温度谱的分析及应用 总被引:1,自引:0,他引:1
臧昆 《高等学校化学学报》1990,11(2):197-201
本文由动态力学温度谱求得松弛谱,并从松弛谱的变化分析了热处理条件对试样超分子结构的影响,进而讨论了WLF方程中常数之比值C1/C2与超分子结构的关系。 相似文献
890.
第三相界面对高分子共混物粗化过程的影响研究 总被引:8,自引:3,他引:5
共混物相分离的机理已有研究[1].但对共混物分散相的粗化过程的研究则不多见.1977年,Cahn等[2]预言第三相界面与低分子共混物之间存在的浸润作用对相分离过程应有较大影响.近年Tanaka等[3]的研究结果表明,在几何空间受限的条件下,含有小分子... 相似文献