Showing papers presented at "Symbolic Numeric Computation in 2014"

Nearly optimal computations with structured matrices

Abstract: We estimate the Boolean complexity of multiplication of structured matrices by a vector and the solution of nonsingular linear systems of equations with these matrices. We study four basic and most popular classes, that is, Toeplitz, Hankel, Cauchy and Vandermonde matrices, for which the cited computational problems are equivalent to the task of polynomial multiplication and division and polynomial and rational multipoint evaluation and interpolation. The Boolean cost estimates for the latter problems have been obtained by Kirrinnis in [10], except for rational interpolation, and we supply them now. All known Boolean cost estimates from [10] for these problems rely on using Kronecker product. This implies the d-fold precision increase for the d-th degree output, but we avoid such an increase by relying on distinct techniques based on employing FFT. Furthermore we simplify the analysis and make it more transparent by combining the representations of our tasks and algorithms both via structured matrices and via polynomials and rational functions. This also enables further extensions of our estimates to cover Trummer's important problem and computations with the popular classes of structured matrices that generalize the four cited basic matrix classes.

The euclidean distance degree

Abstract: The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for computation.

Numerical linear system solving with parametric entries by error correction

Abstract: We consider the problem of solving a full rank consistent linear system A(u)x = b(u) where the m x n matrix A and the m-dimensional vector b has entries that are polynomials in u over a field. We give an algorithm that computes the unique solution x = f(u)/g(u), which is a vector of rational functions, by evaluating the parameter u at distinct points. Those points ξλ where the matrix A evaluates to a matrix A(ξλ), with entries over the scalar field, of lower rank, or in the numeric setting to an ill-conditioned matrix, are not identified but accounted for by error-correcting code techniques. We also correct true errors where the evaluation at some u = ξλ results in an erroneous, possibly full rank consistent and well-conditioned scalar linear system. Our algorithm generalizes Welch/Berlekamp decoding of Reed/Solomon error correcting codes and their numeric floating point counterparts.We have implemented our algorithms with floating point arithmetic. For the determination of the exact numerator and denominator degrees and number of errors we use singular values based numeric rank computations. The arising linear systems for the error-corrected parametric solution are demonstrated to be well-conditioned even when the input scalars have noise. In several initial experiments we have shown that our approach is numerically stable even for larger systems m = n = 100, provided the degrees in the solution are small (≤ 2). For smaller systems m = n = 10 with higher degrees (≤ 20) the algorithm works similarly to rational function recovery. Our implementation can correct 13 true errors in both settings.

Accelerated approximation of the complex roots of a univariate polynomial

Abstract: Highly efficient and even nearly optimal algorithms have been developed for the classical problem of univariate polynomial root-finding (see, e.g., [6], [7], [4], and the bibliography therein), but this is still an area of active research. By combining some powerful techniques developed in this area we devise new algorithm for refinement of isolated complex roots that have nearly optimal Boolean complexity.

Isotopic epsilon-meshing of real algebraic space curves

Abstract: Based on an efficient generic position checking method and on a method to solve bivariate polynomial systems, we give a new algorithm to compute the topology of an algebraic space curve. Compared to the method presented by the authors, in a joint work with Lazard, the new algorithm is efficient because of two reasons. One is the bitsize of the coefficients that may appear in projections is improved. The other is that one projection is enough for most general case in the new algorithm. We also give an e-meshing of the space curve after we obtain its topology. Many nontrivial experiments show the efficiency of the algorithm.

Maximum likelihood for dual varieties

Abstract: Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. In this paper, MLE for statistical models with discrete data is studied from an algebraic statistics viewpoint. A reformulation of the MLE problem in terms of dual varieties and conormal varieties will be given. With this description, we define the dual likelihood equations and the dual MLE problem. We show that solving the dual MLE problem yields solutions to the MLE problem, and that we can solve the dual MLE problem even if we do not have the defining equations of the model itself.

Structure of symmetry of PDE: exploiting partially integrated systems

Abstract: This work is part of a sequence in which we develop and refine algorithms for computer symmetry analysis of differential equations. We show how to exploit partially integrated forms of symmetry defining systems to assist the differential elimination algorithms that uncover structure of the Lie symmetry algebras. We thus incorporate a key advantage of heuristic integration methods, that of exploiting easy integrals of simple (e.g. one term) PDE that frequently occur in such analyses. A single unified method is given that computes structure constants whether the defining system is unsolved, or has been partially or completely integrated.We also give a symbolic-numeric algorithm which for the first time can determine the structure of Lie symmetry algebras specified by defining systems that contain floating point coefficients. This algorithm incorporates a numerical version of the Cartan-Kuranishi prolongation projection algorithm from the geometry of differential equations.

Exploring univariate mixed polynomials

Abstract: We consider mixed polynomials P(z, z) of the single complex variable z with complex (or real coefficients, of degree n in z and m in z. This data is equivalent to a pair of real bivariate polynomials f(x, y) and g(x, y) obtained by separating real and imaginary parts of P. However specifying the degrees, here we focus on the case where m is small allows to investigate interesting roots structures and roots counting; intermediate between complex and real algebra. Mixed polynomials naturally appear in the study of complex polynomial matrices and complex moment problems, harmonic maps, and in recent papers dealing with Milnor fibrations.

Prony systems via decimation and homotopy continuation

Abstract: We consider polynomial systems of Prony type, appearing in many areas of mathematics Their robust numerical solution is considered to be difficult, especially in "near-colliding" situations We transform the nonlinear part of the Prony system into a Hankel-type polynomial system Combining this representation with a recently discovered "decimation" technique, we present an algorithm which applies homotopy continuation on a sequence of modified Hankel-type systems as above In this way, we are able to solve for the nonlinear variables of the original system with high accuracy when the data is perturbed

A recursive decision method for termination of linear programs

Abstract: In their CAV 2004 and 2006 papers, Tiwari and Braverman have proved that, for a class of linear programs over the reals, termination is decidable. In this paper, we propose a new algorithm to decide whether a program of the same class terminates or not. In our approach, a program with an assignment matrix having a single Jordan block or having several Jordan blocks with the same eigenvalue is treated as a basic program to which we reduce a program with arbitrary assignment matrices in a recursive process. Furthermore, if a basic program is non-terminating, our method constructs at least one point on which a given basic program does not terminate. In contrast, for a non-terminating basic program, in most cases, the methods of Tiwari and Braverman provide only a so-called N-nonterminating point. Also, different from their methods, we do not need to guess a dominant term from every loop condition in our recursive procedure.

Two variants of HJLS-PSLQ with applications

Abstract: The HJLS and PSLQ algorithms are the most popular algorithms for finding nontrivial integer relations for several real numbers. It has been already shown that PSLQ is essentially equivalent to HJLS under certain settings. We here call them HJLS-PSLQ.In the present work, we provide two variants of HJLS-PSLQ. The first one is a new modification of Bailey and Broadhurst's multi-pair version. We prove the termination of our modification, while the original multi-pair version may not terminate. The second one is an incremental version of HJLS-PSLQ. For those applications requiring to call HJLS-PSLQ many times, such as finding the minimal polynomial of an algebraic number without knowing the degree, we show the incremental version is more efficient than HJLS-PSLQ, both theoretically and practically.

Computing GCRDs of approximate differential polynomials

Abstract: Differential (Ore) type polynomials with approximate polynomial coefficients are introduced. These provide a useful representation of approximate differential operators with a strong algebraic structure, which has been used successfully in the exact, symbolic, setting. We then present an algorithm for the approximate Greatest Common Right Divisor (GCRD) of two approximate differential polynomials, which intuitively is the differential operator whose solutions are those common to the two inputs operators. More formally, given approximate differential polynomials f and g, we show how to find "nearby" polynomials f and g which have a non-trivial GCRD. Here "nearby" is under a suitably defined norm. The algorithm is a generalization of the SVD-based method of Corless et al. (1995) for the approximate GCD of regular polynomials. We work on an appropriately "linearized" differential Sylvester matrix, to which we apply a block SVD. The algorithm has been implemented in Maple and a demonstration of its robustness is presented.

The nearest polynomial to multiple given polynomials with a given zero

Abstract: The following type of problems have been well-studied in the area of symbolic-numeric computation for about twenty years: Given a polynomial f ∈ C[x] and a point z ∈ C, find the nearest polynomial f̃ ∈ C[x] to f with f̃(z) = 0. A common framework for such problems is described in [7]. In previous works, for example [3, 4, 7, 6], problems for one given polynomial were considered. Here, we consider a problem for multiple given polynomials. Through observation or by using different numerical algorithms for a given input data, we may obtain multiple polynomials being equal in theory but being slightly different each other. Thus, it is worth considering the problem for multiple polynomials. In this abstract, after the preliminaries, we define the nearest polynomial to multiple given polynomials. In the definition, we use a pair of norms to measure the nearness between polynomials. We remark the difficulty of the problem of finding the nearest polynomial depends on the norm pair. Finally, we describe an algorithm for the problem when both of the norms are the ∞-norm.

Parametrization of translational surfaces

Abstract: The algebraic translational surface is a typical modeling surface in computer aided design and architecture industry. In this paper, we give a necessary and sufficient condition so that an algebraic affine surface V is a translational surface. The proof is constructive and thus, if V is translational, we give a standard parametric representation of V.

Cleaning-up data for sparse model synthesis: when symbolic-numeric computation meets error-correcting codes

Abstract: The discipline of symbolic computation contributes to mathematical model synthesis in several ways. One is the pioneering creation of interpolation algorithms that can account for sparsity in the resulting multi-dimensional models, for example, by Zippel [12], Ben-Or and Tiwari [1], and in their recent numerical counterparts by Giesbrecht-Labahn-Lee [5] and Kaltofen-Yang-Zhi [9].

Predicting zero reductions in Gröbner basis computations

Abstract: Since Buchberger's initial algorithm for computing Grobner bases in 1965 [1] many attempts have been taken to detect zero reductions in advance. Buchberger's Product and Chain criteria may be known the most, especially in the installaton of Gebauer and Moller [9]. Signature-based criteria were first used in Faugere's F5 algorithm in 2002 [5]. Here we give a detailed discussion on zero reductions and the corresponding syzygies and explain how the different methods to predict them compare to each other. We extend the notation introduced in [4].

Mixed and componentwise condition numbers for matrix decompositions

Abstract: We present the explicit expressions of mixed and componentwise condition numbers for LU, Cholesky, and QR decompositions by using a new approach.

Automated theorem proving for special functions: the next phase

Abstract: Automated theorem proving, in a nutshell, is the combination of symbolic logic with syntactic algorithms. A formal proof calculus is chosen with two criteria in mind: expressiveness and ease of automation. These desiderata pull in opposite directions: Boolean logic and linear arithmetic are decidable, so the answers to all questions can simply be calculated, but these theories are not very expressive. At the other extreme, a dependent type theory such as the calculus of constructions used in Coq [6] is highly expressive and flexible, but complicates automation; even basic rewriting is difficult. Higher-order logic is often seen as a suitable compromise, expressive enough to reason directly about sets and functions, while still admitting substantial automation (especially in the case of Isabelle [18]).

Optimizing a linear function over a noncompact real algebraic variety

Abstract: Our aim is to compute such a polynomial Φ of the least possible degree. In [3, 4], Rostalski and Sturmfels explored dualities and their interconnections in the context of polynomial optimization (1.1). Assuming that the feasible regionX is irreducible, compact and smooth, they showed that the optimal value function Φ is represented by the defining equation of the hypersurface dual to the projective closure of X [4, Theorem 5.23]. In the present paper, we prove this conclusion is still true for a noncompact real algebraic variety X, when X is irreducible, smooth and the recession cone of the closure of the convex hull co (X) of X is pointed.

An algorithm to compute certain euler characteristics and Chern-Schwartz-MacPherson classes

Abstract: Let V be a possibly singular scheme-theoretic global complete intersection subscheme of Pn and assume that V can be written as the intersection of j hypersurfaces such that the intersection of j -- 1 of the hypersurfaces is smooth (scheme theoretically). Using a result of Fullwood [5] we develop a probabilistic algorithm to compute the Chern-Schwartz-MacPherson class (cSM) and Euler characteristic of V. This algorithm complements existing algorithms by providing performance improvements in the computation of the cSM class and Euler characteristic for schemes having the special structure described above.

Proceedings of the 2014 Symposium on Symbolic-Numeric Computation

Abstract: Algorithms that combine ideas from symbolic and numeric computation have been of increasing interest over the past decade. This has come about for several reasons: algorithms are needed for algebraic objects with imprecise or noisy data; the usual algorithms of computer algebra break down when applied to inexact values; the analytic setting itself allows many new questions to be asked. These motivations, together with the growing demand for speed, accuracy and reliability in mathematical computing, have fuelled a growing synergy between the numeric and symbolic computing fields. This fused subject has come to be known as "symbolic-numeric computation". In it, symbolic and numeric methods are combined to do more than can be done with either alone.

Numerical and geometric properties of a method for finding points on real solution components

Abstract: We consider a critical point method developed in our earlier work for finding certain solution (witness) points on real solution components of real polynomial systems of equations. The method finds points that are critical points of the distance from a plane to the component with the requirement that certain regularity conditions are satisfied. In this paper we analyze the numerical stability of the method. We aim to find at least one well conditioned witness point on each connected component by using perturbation, path tracking and projection techniques. An optimal-direction strategy and an adaptive step size control strategy for path following on high dimensional components are given.

On the interplay between asymptotic and numerical methods to solve differential equations problems

Abstract: Asymptotic and numerical methods usually represent two independent approaches to solve applied mathematical problems, in particular for ordinary or partial differential equations. Here we present some cases when both techniques are used simultaneously, in a complementary way, rather than being considered as an alternative to each other.

A note on geometric involutive bases for positive dimensional polynomial ideals and SDP methods

Abstract: This paper is motivated by [1] which gives a new symbolic-numeric approach for computing the real radical of zero dimensional polynomial systems using geometric involution and semi-definite programming (SDP) techniques. We explore the interplay between geometric involutive bases and the new SDP methods in the positive dimensional case. An important work on this topic is [5].

Safety verification of nonlinear systems based on rational invariants

Abstract: where x ∈ R is the state variable, and f(x) is a vector of rational functions in x over Q. We consider the dynamics of (1) in a bounded domain of the state space R, given by Ψ , {x ∈ R|ψ1(x) ≥ 0 ∧ · · · ∧ ψr(x) ≥ 0}, with ψi(x) ∈ Q[x] for 1 ≤ i ≤ r. We say that the system (1) is safe if all trajectories of (1) starting from any state in the initial set, can not evolve to the unsafe states. We are interested in the problem of safety verification of nonlinear system (1), described as follows.

Some ideas for the computation of matrix solvents

Abstract: We consider the matrix polynomial [EQUATION], with given coefficients [EQUATION]. A matrix [EQUATION] is called a solvent if P(S) = 0. We explore some approaches to the symbolic and numeric computation of solvents. In particular, we compute formulas for the condition number and backward error of the problem which rely on the contour integral based representation of P(S). Finally, we describe a possible approach for computing exact solvents symbolically.

A multiprecision algorithm for the solution of polynomials and polynomial eigenvalue problems

Abstract: Many applications of the real world are modelled by matrix polynomials [EQUATION] where Ai are m x m matrices, see for instance [2], [6]. A computational task encountered in this framework is computing the eigenvalues of P(x), that is, the solutions of the polynomial equation det P(x) = 0. This task is generally accomplished by reducing P(x) to a linear pencil of the kind xL -- K for suitable matrices K, L of size mn, and to solving the eigenvalue problem (λL -- K)v = 0 by means of standard numerical algorithms. A wide literature exists on this approach, we refer the reader to [7] for an example.

Finding a sparse solution of a class of linear differential equations by solving a nonlinear system

Abstract: In this paper, a new numerical method is proposed for finding a sparse solution of a class of linear differential equations with highly oscillatory coefficients. In contrast to spectral methods and finite element methods: 1) The numerical solution is represented by a linear combination of undetermined basis functions instead of a linear combination of predetermined basis functions; 2) A small nonlinear system is obtained rather than a large linear one and, the nonlinear system can be efficiently solved by Prony method; 3) The amount of computational work of our new method is independent of the parameter in the oscillatory coefficient. Some numerical examples are given to show that our new method is promising.