New condition numbers for matrices and linear systems

New condition numbers for matrices and linear systems are proposed, based on the dependence of the relative errors in the result upon the relative errors of the data.     

Mixed and componentwise condition numbers for rectangular structured matrices

Abstract  We provide explicit expressions for both mixed and componentwise structured condition numbers for several classes of structured rectangular matrices: upper triangular, Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Such expressions for many other classes of matrices can be similarly derived. Mathematics Subject Classification (2000): 15A12, 65F35     

Monoton finite element matrices and their computed condition numbers

Certain finite element stiffness matrices appearing in the discretization of second order problems possess positive inverses (are monoton). Some classical theorems concerning such positive matrices are expediently evoked to compute numerical lower and upper bounds of any desired closeness on the maximum and spectral condition numbers of these matrices.     

Eigenvalue assignment for real matrices of specified form

The purpose of this note is to report on various existence conditions found in connection with the problem of assigning the eigenvalues of a realnth-order matrix constrained by the specification of certain zero entries. This problem finds applications in many areas, but it was originally motivated by attempts to obtain desired responses from annth-order dynamic system governed by a set of linear homogeneous differential equations of the formdot{x} = AxwhereAis the constant real system matrix to be found.     

Eigenvalue location of certain matrices arising in convergence analysis problems

We give simple formulas for the eigenvalues of a certain type of matrices occurring in some convergence analysis problems. Application of the result to a pseudo-linear regression algorithm gives explicit necessary and sufficient conditions for local convergence.     

On the spectral condition of rectangular vandermonde matrices

Let V_n,m be a rectangular n×m Vandermonde matrix with real nodes and let the number m of nodes be greater than the number n of powers; in this paper we find, under mild restrictions on the nodes, that the behavior of the spectral conditioning of V_n,m for m→∞ is essentially the square root of the conditioning of the Hilbert matrix of order n.     

Managing uncertainty and discontinuous condition numbers in geometric computation

We consider operations on subdivision surfaces under the strict robustness requirement that these floating-point computations return an object with the same topological form as the true solution. The problems involved may however be ill-conditioned, and defined in terms of uncertain data, and even supplementary interval arithmetic may not ensure robustness. Trapping mechanisms are therefore proposed to resolve this difficulty.     

区间矩阵二次稳定的充分必要条件

提出了区间矩阵二次稳定的充分必要条件,以及相应的稳定裕度的计算方法.结论以线性矩阵不等式(LMI)的形式给出.利用功能强大的LMI工具,求解非常方便.所给实例表明,该方法用于确定区间矩阵的鲁棒稳定性及其稳定裕度,非常有效.     

Perturbation analysis and condition numbers of symmetric algebraic Riccati equations

This paper is devoted to the perturbation analysis of symmetric algebraic Riccati equations. Based on our perturbation analysis, the upper bounds for the normwise, mixed and componentwise condition numbers are presented. The results are demonstrated by our preliminary numerical experiments.     

Stability analysis of interval matrices: another sufficient condition

A sufficient condition for the stability of interval matrices is presented, based on a Lyapunov approach. The condition, while requiring the solution of a Lyapunov matrix equation, removes the restrictions imposed by Heinen ∥1984). Examples given illustrate the improvement of the proposed condition over the ones given by Heinen and Bialas (1983) and Daoyi (1985).     

Experiments with data perturbations to study condition numbers and numerical stability

Ill-conditioning as well as roundoff errors lead to inaccuracies in scientific applications. Random perturbations in the initial data allow to derive an error estimation. By varying the size of these perturbations and under some reasonable assumption on the rounding errors, we can find an interval of perturbations where a log-linear least-squares fit gives an estimation of the regularity and the condition number of the problem.     

Sufficient condition for the positive definiteness of symmetric interval matrices

A sufficient condition for the positive definiteness of symmetric interval matrices is obtained.     

Symmetric triangular approximations of fuzzy numbers under a general condition and properties

We consider the set \(\mathcal {P}\) of real parameters associated to a fuzzy number, in a general form which includes the most important characteristics already introduced for fuzzy numbers. We find the set \(\mathcal {P}_{\mathrm{s}}\subset \mathcal {P}\) with the property that for any given fuzzy number there exists at least a symmetric triangular fuzzy number which preserves a fixed parameter \(p\in \mathcal {P}\). We compute the symmetric triangular approximation of a fuzzy number which preserves the parameter \(p\in \mathcal {P }_{\mathrm{s}}\). The uniqueness is an immediate consequence; therefore, an approximation operator is obtained. The properties of scale and translation invariance, additivity and continuity of this operator are studied. Some applications related with value and expected value, as important parameters, are given too.     

A necessary and sufficient condition for the positive-definiteness of interval symmetric matrices

A necessary and sufficient condition for the positive-definiteness of interval symmetric matrices is obtained, and a useful application is shown.     

On condition numbers in the cyclic reduction processes of a tridiagonal matrix

The cyclic reduction method is a direct method for solving tridiagonal linear systems. At the first step of this method, a tridiagonal coefficient matrix is transformed into a pentadiagonal form. In this article, we prove that the condition number for eigenvalues of some classes of coefficient matrices always decreases after the first step of the cyclic reduction method.     

Level-2 condition numbers for least-squares solution of Kronecker product linear systems

Kronecker product linear systems arise more and more often in the real world. In this paper, we consider the minimum norm least-squares solution of Kronecker product linear systems (A? B)x=b by extending some known results from nonsingular cases to singular or rectangular cases. We investigate the level-2 condition numbers, and the corresponding bounds of these condition numbers.     

A strict bound to the condition number of bordered positive definite matrices

An easily computable strict bound is derived for the condition number in theL ₂ norm of bordered positive definite matrices.