Optimized look-ahead recurrences for adjacent rows in the Padé table

A new look-ahead algorithm for recursively computing Padé approximants is introduced. It generates a subsequence of the Padé approximants on two adjacent rows (defined by fixed numerator degree) of the Padé table. Its two basic versions reduce to the classical Levinson and Schur algorithms if no look-ahead is required. The new algorithm can be viewed as a combination of the look-ahead sawtooth and the look-ahead Levinson and Schur algorithms that we proposed before, but now the look-ahead step size is minimal (as in the sawtooth version) and the computational costs are as low as in the least expensive competing algorithms (including our look-ahead Levinson and Schur algorithms). The underlying recurrences link well-conditioned basic pairs,i.e., pairs of sufficiently different neighboring Padé forms.The algorithm can be used to solve Toeplitz systems of equationsTx = b. In this application it comes in several versions: anO(N ²) Levinson-type form, anO(N ²) Schur-type form, and a superfastO(N log² N) Schur-type version. As an option of the first two versions, the corresponding block LDU decompositions ofT ^–1 orT, respectively, can be found.     

Calculating a function of a matrix with a real spectrum

Let T be a square matrix with a real spectrum, and let f be an analytic function. The problem of the approximate calculation of f(T) is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that T is triangular and its diagonal entries t_ii are arranged in increasing order. To avoid calculations using the differences t_ii ? t_jj with close (including equal) t_ii and t_jj, it is proposed to represent T in a block form and calculate the two main block diagonals using interpolating polynomials. The rest of the f(T) entries can be calculated using the Parlett recurrence algorithm. It is also proposed to perform some scalar operations (such as the building of interpolating polynomials) with an enlarged number of significant decimal digits.

     

A Recursive Test for P-Matrices

The time complexity for testing whether an n-by-n real matrix is a P-matrix is reduced from O(2ⁿ n ³) to O(2 ⁿ ) by applying recursively a criterion for P-matrices based on Schur complementation. A Matlab program implementing the associated algorithm is provided.     

Numerical computation of an analytic singular value decomposition of a matrix valued function

Summary This paper extends the singular value decomposition to a path of matricesE(t). An analytic singular value decomposition of a path of matricesE(t) is an analytic path of factorizationsE(t)=X(t)S(t)Y(t) ^T whereX(t) andY(t) are orthogonal andS(t) is diagonal. To maintain differentiability the diagonal entries ofS(t) are allowed to be either positive or negative and to appear in any order. This paper investigates existence and uniqueness of analytic SVD's and develops an algorithm for computing them. We show that a real analytic pathE(t) always admits a real analytic SVD, a full-rank, smooth pathE(t) with distinct singular values admits a smooth SVD. We derive a differential equation for the left factor, develop Euler-like and extrapolated Euler-like numerical methods for approximating an analytic SVD and prove that the Euler-like method converges.Partial support received from SFB 343, Diskrete Strukturen in der Mathematik, Universität BielefeldPartial support received from FSP Mathematisierung, Universität BielefeldPartial support received from FSP Mathematisierung, Universität BielefeldPartial support received from National Science Foundation grant CCR-8820882. Some support was also received from the University of Kansas through International Travel Fund 560478 and General Research Allocations # 3758-20-0038 and #3692-20-0038.     

Jottings

Abstract

The Public Commission of the University: The Role of Scholars in an Industrial, Urban and Corporate Society by John F. A. Taylor. New York: New York University Press, 1981, 220 pages, $17.50.     

On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices

A stronger result on the limiting distribution of the eigenvalues of random Hermitian matrices of the form A + XTX*, originally studied in Mar enko and Pastur, is presented. Here, X(N × n), T(n × n), and A(N × N) are independent, with X containing i.i.d. entries having finite second moments, T is diagonal with real (diagonal) entries, A is Hermitian, and n/N → c > 0 as N → ∞. Under additional assumptions on the eigenvalues of A and T, almost sure convergence of the empirical distribution function of the eigenvalues of A + XTX* is proven with the aid of Stieltjes transforms, taking a more direct approach than previous methods.     

A divide and conquer approach to computing the mean first passage matrix for Markov chains via Perron complement reductions

Let M_T be the mean first passage matrix for an n‐state ergodic Markov chain with a transition matrix T. We partition T as a 2×2 block matrix and show how to reconstruct M_T efficiently by using the blocks of T and the mean first passage matrices associated with the non‐overlapping Perron complements of T. We present a schematic diagram showing how this method for computing M_T can be implemented in parallel. We analyse the asymptotic number of multiplication operations necessary to compute M_T by our method and show that, for large size problems, the number of multiplications is reduced by about 1/8, even if the algorithm is implemented in serial. We present five examples of moderate sizes (of orders 20–200) and give the reduction in the total number of flops (as opposed to multiplications) in the computation of M_T. The examples show that when the diagonal blocks in the partitioning of T are of equal size, the reduction in the number of flops can be much better than 1/8. Copyright © 2001 John Wiley & Sons, Ltd.     

A fast method to block-diagonalize a Hankel matrix

In this paper, we consider an approximate block diagonalization algorithm of an n×n real Hankel matrix in which the successive transformation matrices are upper triangular Toeplitz matrices, and propose a new fast approach to compute the factorization in O(n ²) operations. This method consists on using the revised Bini method (Lin et al., Theor Comp Sci 315: 511–523, 2004). To motivate our approach, we also propose an approximate factorization variant of the customary fast method based on Schur complementation adapted to the n×n real Hankel matrix. All algorithms have been implemented in Matlab and numerical results are included to illustrate the effectiveness of our approach.     

Perturbation analysis for the generalized Schur complement of a positive semi‐definite matrix

Let and S=C?B^HA^?B be the generalized Schur complement of A?0 in P. In this paper, some perturbation bounds of S are presented which generalize the result of Stewart (Technical Report TR‐95‐38, University of Maryland, 1995) and enrich the perturbation theory for the Schur complement. Also, an error estimate for the smallest perturbation of C, which lowers the rank of P, is discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

Skew quasisymmetric Schur functions and noncommutative Schur functions

Recently a new basis for the Hopf algebra of quasisymmetric functions QSym, called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions as examples, and give rise to a new poset LC that is analogous to Young's lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions NSym. This basis of NSym is dual to the basis of quasisymmetric Schur functions and its elements are the pre-image of the Schur functions under the forgetful map χ:NSym→Sym. We prove that the multiplicative structure constants of the noncommutative Schur functions, equivalently the coefficients of the skew quasisymmetric Schur functions when expanded in the quasisymmetric Schur basis, are nonnegative integers, satisfying a Littlewood–Richardson rule analogue that reduces to the classical Littlewood–Richardson rule under χ.As an application we show that the morphism of algebras from the algebra of Poirier–Reutenauer to Sym factors through NSym. We also extend the definition of Schur functions in noncommuting variables of Rosas–Sagan in the algebra NCSym to define quasisymmetric Schur functions in the algebra NCQSym. We prove these latter functions refine the former and their properties, and project onto quasisymmetric Schur functions under the forgetful map. Lastly, we show that by suitably labeling LC, skew quasisymmetric Schur functions arise in the theory of Pieri operators on posets.     

On the positivity of high-degree Schur classes of an ample vector bundle

Let X be a smooth projective variety of dimension n,and let E be an ample vector bundle over X.We show that any Schur class of E,lying in the cohomology group of bidegree(n-1,n-1),has a representative which is strictly positive in the sense of smooth forms.This conforms the prediction of Griffiths conjecture on the positive polynomials of Chern classes/forms of an ample vector bundle on the form level,and thus strengthens the celebrated positivity results of Fulton and Lazarsfeld(1983)for certain degrees.     

Decomposition of polynomial matrices into factors with prescribed canonical diagonal forms

We give a test for decomposability of a polynomial matrix over an arbitrary infinite field into factors with prescribed canonical diagonal forms whose product is the canonical diagonal form of the given matrix. We exhibit a method of actually constructing such decompositions of polynomial matrices.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 17–19.     

Metric characterizations of proper interval graphs and tree-clique graphs

A connected graph G is a tree-clique graph if there exists a spanning tree T (a compatible tree) such that every clique of G is a subtree of T. When T is a path the connected graph G is a proper interval graph which is usually defined as intersection graph of a family of closed intervals of the real line such that no interval contains another. We present here metric characterizations of proper interval graphs and extend them to tree-clique graphs. This is done by demonstrating “local” properties of tree-clique graphs with respect to the subgraphs induced by paths of a compatible tree. © 1996 John Wiley & Sons, Inc.     

Second Order Cones for Maximal Monotone Operators via Representative Functions

It is shown that various first and second order derivatives of the Fitzpatrick and Penot representative functions for a maximal monotone operator T, in a reflexive Banach space, can be used to represent differential information associated with the tangent and normal cones to the Graph T. In particular we obtain formula for the proto-derivative, as well as its polar, the normal cone to the graph of T. First order derivatives are shown to be useful in recognising points of single-valuedness of T. We show that a strong form of proto-differentiability to the graph of T, is often associated with single valuedness of T. The second author’s research was funded by NSERC and the Canada Research Chair programme, and the first author’s by ARC grant number DP0664423. This study was commenced between August and December 2005 while the first author was visiting Dalhousie University.     

The shifted plactic monoid

We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman’s mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood–Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux–Schützenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.     

Geometry of the Normal Bundle of a Submanifold*

 We study the geometric behavior of the normal bundle T ^⊥ M of a submanifold M of a Riemannian manifold . We compute explicitely the second fundamental form of T ^⊥ M and look at the relation between the minimality of T ^⊥ M and M. Finally we show that the Maslov forms with respect to a suitable connection of the pair (T ^⊥ M, are null. Received March 14, 2001; in revised form February 11, 2002     

The domination and competition graphs of a tournament

Vertices x and y dominate a tournament T if for all vertices z ≠ x, y, either x beats z or y beats z. Let dom(T) be the graph on the vertices of T with edges between pairs of vertices that dominate T. We show that dom(T) is either an odd cycle with possible pendant vertices or a forest of caterpillars. While this is not a characterization, it does lead to considerable information about dom(T). Since dom(T) is the complement of the competition graph of the tournament formed by reversing the arcs of T, complementary results are obtained for the competition graph of a tournament. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 103–110, 1998