Strategies for spectrum slicing based on restarted Lanczos methods

In the context of symmetric-definite generalized eigenvalue problems, it is often required to compute all eigenvalues contained in a prescribed interval. For large-scale problems, the method of choice is the so-called spectrum slicing technique: a shift-and-invert Lanczos method combined with a dynamic shift selection that sweeps the interval in a smart way. This kind of strategies were proposed initially in the context of unrestarted Lanczos methods, back in the 1990’s. We propose variations that try to incorporate recent developments in the field of Krylov methods, including thick restarting in the Lanczos solver and a rational Krylov update when moving from one shift to the next. We discuss a parallel implementation in the SLEPc library and provide performance results.     

一种求二元有理插值函数的方法

给出一种方法可直接计算基于矩形节点的二元有理插值函数的分母在节点处的值 ,进而判断相应的二元有理插值函数是否存在 .此方法运用灵活 ,适用范围广 ,在相应的有理插值函数存在时 ,能给出它的具体表达式 .此外 ,我们还针对文中两个主要逆矩阵 ,给出了相应的递推公式 ,避免了求逆计算 .     

Error analysis of symplectic Lanczos method for Hamiltonian eigenvalue problem

A rounding error analysis for the symplectic Lanczos method is given to solve the large-scale sparse Hamiltonian eigenvalue problem. If no breakdown occurs in the method, then it can be shown that the Hamiltonian structure preserving requirement does not destroy the essential feature of the nonsymmetric Lanczos algorithm. The relationship between the loss of J-orthogonality among the symplectic Lanczos vectors and the convergence of the Ritz values in the symmetric Lanczos algorithm is discussed. It is demonstrated that under certain assumptions the computed J-orthogonal Lanczos vectors lose the J-orthogonality when some Ritz values begin to converge. Our analysis closely follows the recent works of Bai and Fabbender. Selected from Journal of Mathematical Research and Exposition, 2004, 24(1): 91–106     

On the denominator values and barycentric weights of rational interpolants

We improve upon the method of Zhu and Zhu [A method for directly finding the denominator values of rational interpolants, J. Comput. Appl. Math. 148 (2002) 341–348] for finding the denominator values of rational interpolants, reducing considerably the number of arithmetical operations required for their computation. In a second stage, we determine the points (if existent) which can be discarded from the rational interpolation problem. Furthermore, when the interpolant has a linear denominator, we obtain a formula for the barycentric weights which is simpler than the one found by Berrut and Mittelmann [Matrices for the direct determination of the barycentric weights of rational interpolation, J. Comput. Appl. Math. 78 (1997) 355–370]. Subsequently, we give a necessary and sufficient condition for the rational interpolant to have a pole.     

Barycentric rational interpolation with asymptotically monitored poles

We present a method for asymptotically monitoring poles to a rational interpolant written in barycentric form. Theoretical and numerical results are given to show the potential of the proposed interpolant.     

C2-rational cubic spline involving tension parameters

In the present paper, C¹-piecewise rational cubic spline function involving tension parameters is considered which produces a monotonie interpolant to a given monotonie data set. It is observed that under certain conditions the interpolant preserves the convexity property of the data set. The existence and uniqueness of a C²-rational cubic spline interpolant are established. The error analysis of the spline interpolant is also given.     

The Linear Rational Pseudospectral Method with Preassigned Poles

We present a linear rational pseudospectral (collocation) method with preassigned poles for solving boundary value problems. It consists in attaching poles to the trial polynomial so as to make it a rational interpolant. Its convergence is proved by transforming the problem into an associated boundary value problem. Numerical examples demonstrate that the rational pseudospectral method is often more efficient than the polynomial method.     

Rational Krylov sequence methods for eigenvalue computation

Algorithms to solve large sparse eigenvalue problems are considered. A new class of algorithms which is based on rational functions of the matrix is described. The Lanczos method, the Arnoldi method, the spectral transformation Lanczos method, and Rayleigh quotient iteration all are special cases, but there are also new algorithms which correspond to rational functions with several poles. In the simplest case a basis of a rational Krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of Hessenberg matrices.     

C1 monotone cubic Hermite interpolant

Constraining an interpolation to be shape preserving is a well established technique for modelling scientific data. Many techniques express the constraint variables in terms of abstract quantities that are difficult to relate to either physical values or the geometric properties of the interpolant. In this paper, we construct a piecewise monotonic interpolant where the degrees of freedom are expressed in terms of the weights of the rational Bézier cubic interpolant.     

Shape Preserving Interpolation to Convex Data

1.IntroductionA fundamental problem in computer graphics is the drawing of a smooth curve through aset of data points(xi,fi) (i=0 ,1 ,… ,n) .In many applications,particularly in scientificvisualisation,the y- values are depenenton the x- values and it is…     

A method for directly finding the denominator values of rational interpolants

In this paper, we give an algorithm for directly finding the denominator values of rational interpolants at the nodes, and present an expression for the corresponding rational interpolant when the latter exists. With these denominator values, our method also provides information concerning the existence of the interpolant and the presence of unattainable points and poles.     

关于有理插值函数存在性的研究

在本文中 ,我们利用 Newton插值多项式 ,改进了 [1 ]中的方法 ,使其能更简便 ,快速 ,严谨地判别有理插值函数的存在性 ,并在其存在时给出相应的插值有理函数的具体表达式 .     

Rational interpolation with a single variable pole

In this paper the necessary and sufficient conditions for given data to admit a rational interpolant in _k,1 with no poles in the convex hull of the interpolation points is studied. A method for computing the interpolant is also provided.Partially supported by DGICYT-0121.     

A matrix rational Lanczos method for model reduction in large‐scale first‐ and second‐order dynamical systems

In the present paper, we describe an adaptive modified rational global Lanczos algorithm for model‐order reduction problems using multipoint moment matching‐based methods. The major problem of these methods is the selection of some interpolation points. We first propose a modified rational global Lanczos process and then we derive Lanczos‐like equations for the global case. Next, we propose adaptive techniques for choosing the interpolation points. Second‐order dynamical systems are also considered in this paper, and the adaptive modified rational global Lanczos algorithm is applied to an equivalent state space model. Finally, some numerical examples will be given.     

二元向量值有理插值存在性的一种判别方法

文章给出了对于矩形网格上基于二元Newton插值公式的二元向量值有理插值存在性的充要条件.在存在的情况下,建立了具有显式表达式的不同于向量连分式的二元向量值有理插值函数,并且这种方法具有承袭性.最后给出的实例说明了这种算法的有效性.     

Chebyshev rational interpolation

The Lanczos method and its variants can be used to solve efficiently the rational interpolation problem. In this paper we present a suitable fast modification of a general look-ahead version of the Lanczos process in order to deal with polynomials expressed in the Chebyshev orthogonal basis. The proposed approach is particularly suited for approximating analytic functions by means of rational interpolation at certain nodes located on the boundary of an elliptical region of the complex plane. In fact, in this case it overcomes some of the numerical difficulties which limited the applicability of the look-ahead Lanczos process for determining the coefficients both of the numerators and of the denominators with respect to the standard power basis. This revised version was published online in August 2006 with corrections to the Cover Date.     

An algorithm for generalized rational interpolation

A recursive algorithm for the construction of the generalized form of the interpolating rational function is derived. This generalization of the Neville-Aitken algorithm constructs a table of all possible rational interpolants in implicit form. The algorithm may be simply modified so that it does not break down when a singularity occasionally appears. The coefficients of the interpolant and the evaluation of the interpolant at an arbitrary point may be easily calculated.     

Note on Rational Interpolants

<正> In this note we present a constructive proof of symmetrical determinantal formulas forthe numerator and denominator of an ordinary rational interpolant,consider the confluencecase and give new determinantal formulas of the rational interpolant by means of Lagrange'sbasis functions.     

Model reduction of state space systems via an implicitly restarted Lanczos method

The nonsymmetric Lanczos method has recently received significant attention as a model reduction technique for large-scale systems. Unfortunately, the Lanczos method may produce an unstable partial realization for a given, stable system. To remedy this situation, unexpensive implicit restarts are developed which can be employed to stabilize the Lanczos generated model.This work was supported in part by ARPA (US Army ORA4466.01), by ARPA (Grant 60NANB2D1272), by the Department of Energy (Contract DE-FG0f-91ER25103) and by the National Science Foundation (Grants CCR-9209349 and CCR-9120008).     

A multivariate QD-like algorithm

The problem of constructing a univariate rational interpolant or Padé approximant for given data can be solved in various equivalent ways: one can compute the explicit solution of the system of interpolation or approximation conditions, or one can start a recursive algorithm, or one can obtain the rational function as the convergent of an interpolating or corresponding continued fraction.In case of multivariate functions general order systems of interpolation conditions for a multivariate rational interpolant and general order systems of approximation conditions for a multivariate Padé approximant were respectively solved in [6] and [9]. Equivalent recursive computation schemes were given in [3] for the rational interpolation case and in [5] for the Padé approximation case. At that moment we stated that the next step was to write the general order rational interpolants and Padé approximants as the convergent of a multivariate continued fraction so that the univariate equivalence of the three main defining techniques was also established for the multivariate case: algebraic relations, recurrence relations, continued fractions. In this paper a multivariate qd-like algorithm is developed that serves this purpose.