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Symbolic Computation

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preprint2017arXiv

Computing explicit isomorphisms with full matrix algebras over $\mathbb{F}_q(x)$

We propose a polynomial time $f$-algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over $\mathbb{F}_q$) for computing an isomorphism (if there is any) of a finite dimensional $\mathbb{F}_q(x)$-algebra $A$ given by structure constants with the algebra of $n$ by $n$ matrices with entries from $\mathbb{F}_q(x)$. The method is based on computing a finite $\mathbb{F}_q$-subalgebra of $A$ which is the intersection of a maximal $\mathbb{F}_q[x]$-order and a maximal $R$-order, where $R$ is the subring of $\mathbb{F}_q(x)$ consisting of fractions of polynomials with denominator having degree not less than that of the numerator.

preprint2016arXiv

On the computation of the HNF of a module over the ring of integers of a number field

We present a variation of the modular algorithm for computing the Hermite normal form of an $\mathcal O_K$-module presented by Cohen, where $\mathcal O_K$ is the ring of integers of a number field $K$. An approach presented in (Cohen 1996) based on reductions modulo ideals was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature. In this paper, we present a modification of the approach of Cohen to prevent the coefficient swell and we rigorously assess its complexity with respect to the size of the input and the invariants of the field $K$.

preprint2016arXiv

Exploiting chordal structure in polynomial ideals: a Gröbner bases approach

Chordal structure and bounded treewidth allow for efficient computation in numerical linear algebra, graphical models, constraint satisfaction and many other areas. In this paper, we begin the study of how to exploit chordal structure in computational algebraic geometry, and in particular, for solving polynomial systems. The structure of a system of polynomial equations can be described in terms of a graph. By carefully exploiting the properties of this graph (in particular, its chordal completions), more efficient algorithms can be developed. To this end, we develop a new technique, which we refer to as chordal elimination, that relies on elimination theory and Gröbner bases. By maintaining graph structure throughout the process, chordal elimination can outperform standard Gröbner basis algorithms in many cases. The reason is that all computations are done on "smaller" rings, of size equal to the treewidth of the graph. In particular, for a restricted class of ideals, the computational complexity is linear in the number of variables. Chordal structure arises in many relevant applications. We demonstrate the suitability of our methods in examples from graph colorings, crypt

preprint2016arXiv

Computing in quotients of rings of integers

We develop algorithms to turn quotients of rings of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and their behavior under certain quotients. We illustrate the power of our ideas in a new modular normal form algorithm for modules over rings of integers, vastly outperforming classical algorithms.

preprint2016arXiv

Congruences and Concurrent Lines in Multi-View Geometry

We present a new framework for multi-view geometry in computer vision. A camera is a mapping between $\mathbb{P}^3$ and a line congruence. This model, which ignores image planes and measurements, is a natural abstraction of traditional pinhole cameras. It includes two-slit cameras, pushbroom cameras, catadioptric cameras, and many more. We study the concurrent lines variety, which consists of $n$-tuples of lines in $\mathbb{P}^3$ that intersect at a point. Combining its equations with those of various congruences, we derive constraints for corresponding images in multiple views. We also study photographic cameras which use image measurements and are modeled as rational maps from $\mathbb{P}^3$ to $\mathbb{P}^2$ or $\mathbb{P}^1\times \mathbb{P}^1$.

preprint2016arXiv

Reverse Engineering of Irreducible Polynomials in GF(2^m) Arithmetic

Current techniques for formally verifying circuits implemented in Galois field (GF) arithmetic are limited to those with a known irreducible polynomial P(x). This paper presents a computer algebra based technique that extracts the irreducible polynomial P(x) used in the implementation of a multiplier in GF(2^m). The method is based on first extracting a unique polynomial in Galois field of each output bit independently. P(x) is then obtained by analyzing the algebraic expression in GF(2^m) of each output bit. We demonstrate that this method is able to reverse engineer the irreducible polynomial of an n-bit GF multiplier in n threads. Experiments were performed on Mastrovito and Montgomery multipliers with different P (x), including NIST-recommended polynomials and optimal polynomials for different microprocessor architectures.

preprint2016arXiv

Fast Matrix Multiplication and Symbolic Computation

The complexity of matrix multiplication (hereafter MM) has been intensively studied since 1969, when Strassen surprisingly decreased the exponent 3 in the cubic cost of the straightforward classical MM to log 2 (7) $\approx$ 2.8074. Applications to some fundamental problems of Linear Algebra and Computer Science have been immediately recognized, but the researchers in Computer Algebra keep discovering more and more applications even today, with no sign of slowdown. We survey the unfinished history of decreasing the exponent towards its information lower bound 2, recall some important techniques discovered in this process and linked to other fields of computing, reveal sample surprising applications to fast computation of the inner products of two vectors and summation of integers, and discuss the curse of recursion, which separates the progress in fast MM into its most acclaimed and purely theoretical part and into valuable acceleration of MM of feasible sizes. Then, in the second part of our paper, we cover fast MM in realistic symbolic computations and discuss applications and implementation of fast exact matrix multiplication. We first review how most of exact linear algebra can

preprint2016arXiv

Parallel Integer Polynomial Multiplication

We propose a new algorithm for multiplying dense polynomials with integer coefficients in a parallel fashion, targeting multi-core processor architectures. Complexity estimates and experimental comparisons demonstrate the advantages of this new approach.

preprint2016arXiv

Efficient sparse polynomial factoring using the Funnel heap

This work is a comprehensive extension of Abu-Salem et al. (2015) that investigates the prowess of the Funnel Heap for implementing sums of products in the polytope method for factoring polynomials, when the polynomials are in sparse distributed representation. We exploit that the work and cache complexity of an Insert operation using Funnel Heap can be refined to de- pend on the rank of the inserted monomial product, where rank corresponds to its lifetime in Funnel Heap. By optimising on the pattern by which insertions and extractions occur during the Hensel lifting phase of the polytope method, we are able to obtain an adaptive Funnel Heap that minimises all of the work, cache, and space complexity of this phase. Additionally, we conduct a detailed empirical study confirming the superiority of Funnel Heap over the generic Binary Heap once swaps to external memory begin to take place. We demonstrate that Funnel Heap is a more efficient merger than the cache oblivious k-merger, which fails to achieve its optimal (and amortised) cache complexity when used for performing sums of products. This provides an empirical proof of concept that the overlapping approach for perform- ing sums

preprint2016arXiv

Comparative study of space filling curves for cache oblivious TU Decomposition

We examine several matrix layouts based on space-filling curves that allow for a cache-oblivious adaptation of parallel TU decomposition for rectangular matrices over finite fields. The TU algorithm of \cite{Dumas} requires index conversion routines for which the cost to encode and decode the chosen curve is significant. Using a detailed analysis of the number of bit operations required for the encoding and decoding procedures, and filtering the cost of lookup tables that represent the recursive decomposition of the Hilbert curve, we show that the Morton-hybrid order incurs the least cost for index conversion routines that are required throughout the matrix decomposition as compared to the Hilbert, Peano, or Morton orders. The motivation lies in that cache efficient parallel adaptations for which the natural sequential evaluation order demonstrates lower cache miss rate result in overall faster performance on parallel machines with private or shared caches, on GPU's, or even cloud computing platforms. We report on preliminary experiments that demonstrate how the TURBO algorithm in Morton-hybrid layout attains orders of magnitude improvement in performance as the input matrices

preprint2016arXiv

Baby-Step Giant-Step Algorithms for the Symmetric Group

We study discrete logarithms in the setting of group actions. Suppose that $G$ is a group that acts on a set $S$. When $r,s \in S$, a solution $g \in G$ to $r^g = s$ can be thought of as a kind of logarithm. In this paper, we study the case where $G = S_n$, and develop analogs to the Shanks baby-step / giant-step procedure for ordinary discrete logarithms. Specifically, we compute two sets $A, B \subseteq S_n$ such that every permutation of $S_n$ can be written as a product $ab$ of elements $a \in A$ and $b \in B$. Our deterministic procedure is optimal up to constant factors, in the sense that $A$ and $B$ can be computed in optimal asymptotic complexity, and $|A|$ and $|B|$ are a small constant from $\sqrt{n!}$ in size. We also analyze randomized "collision" algorithms for the same problem.

preprint2016arXiv

OpenSBLI: A framework for the automated derivation and parallel execution of finite difference solvers on a range of computer architectures

Exascale computing will feature novel and potentially disruptive hardware architectures. Exploiting these to their full potential is non-trivial. Numerical modelling frameworks involving finite difference methods are currently limited by the 'static' nature of the hand-coded discretisation schemes and repeatedly may have to be re-written to run efficiently on new hardware. In contrast, OpenSBLI uses code generation to derive the model's code from a high-level specification. Users focus on the equations to solve, whilst not concerning themselves with the detailed implementation. Source-to-source translation is used to tailor the code and enable its execution on a variety of hardware.

preprint2016arXiv

Symbolic Representation for Analog Realization of A Family of Fractional Order Controller Structures via Continued Fraction Expansion

This paper uses the Continued Fraction Expansion (CFE) method for analog realization of fractional order differ-integrator and few special classes of fractional order (FO) controllers viz. Fractional Order Proportional-Integral-Derivative (FOPID) controller, FO[PD] controller and FO lead-lag compensator. Contemporary researchers have given several formulations for rational approximation of fractional order elements. However, approximation of the controllers studied in this paper, due to having fractional power of a rational transfer function, is not available in analog domain; although its digital realization already exists. This motivates us for applying CFE based analog realization technique for complicated FO controller structures to get equivalent rational transfer functions in terms of the controller tuning parameters. The symbolic expressions for rationalized transfer function in terms of the controller tuning parameters are especially important as ready references, without the need of running CFE algorithm every time and also helps in the synthesis of analog circuits for such FO controllers.

preprint2013arXiv

A computer algebra user interface manifesto

Many computer algebra systems have more than 1000 built-in functions, making expertise difficult. Using mock dialog boxes, this article describes a proposed interactive general-purpose wizard for organizing optional transformations and allowing easy fine grain control over the form of the result even by amateurs. This wizard integrates ideas including: * flexible subexpression selection; * complete control over the ordering of variables and commutative operands, with well-chosen defaults; * interleaving the choice of successively less main variables with applicable function choices to provide detailed control without incurring a combinatorial number of applicable alternatives at any one level; * quick applicability tests to reduce the listing of inapplicable transformations; * using an organizing principle to order the alternatives in a helpful manner; * labeling quickly-computed alternatives in dialog boxes with a preview of their results, * using ellipsis elisions if necessary or helpful; * allowing the user to retreat from a sequence of choices to explore other branches of the tree of alternatives or to return quickly to branches already visited; * allowing the user to accumulat

preprint2010arXiv

Integrating multiple sources to answer questions in Algebraic Topology

We present in this paper an evolution of a tool from a user interface for a concrete Computer Algebra system for Algebraic Topology (the Kenzo system), to a front-end allowing the interoperability among different sources for computation and deduction. The architecture allows the system not only to interface several systems, but also to make them cooperate in shared calculations.

preprint2016arXiv

Reduction-Based Creative Telescoping for Fuchsian D-finite Functions

Continuing a series of articles in the past few years on creative telescoping using reductions, we adapt Trager's Hermite reduction for algebraic functions to fuchsian D-finite functions and develop a reduction-based creative telescoping algorithm for this class of functions, thereby generalizing our recent reduction-based algorithm for algebraic functions, presented at ISSAC 2016.

preprint2016arXiv

A note about "Faster algorithms for computing Hong's bound on absolute positiveness" by K. Mehlhorn and S. Ray

We show that a linear-time algorithm for computing Hong's bound for positive roots of a univariate polynomial, described by K. Mehlhorn and S. Ray in an article "Faster algorithms for computing Hong's bound on absolute positiveness", is incorrect. We present a corrected version.

preprint2016arXiv

Solving rank-constrained semidefinite programs in exact arithmetic

We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in systems control theory and combinatorial optimization, but even in more general contexts such as polynomial optimization or real algebra. While numerical algorithms exist for solving this problem, such as interior-point or Newton-like algorithms, in this paper we propose an approach based on symbolic computation. We design an exact algorithm for solving rank-constrained semidefinite programs, whose complexity is essentially quadratic on natural degree bounds associated to the given optimization problem: for subfamilies of the problem where the size of the feasible matrix is fixed, the complexity is polynomial in the number of variables. The algorithm works under assumptions on the input data: we prove that these assumptions are generically satisfied. We also implement it in Maple and discuss practical experiments.

preprint2010arXiv

Minimal Polynomial Algorithms for Finite Sequences

We show that a straightforward rewrite of a known minimal polynomial algorithm yields a simpler version of a recent algorithm of A. Salagean.

preprint2012arXiv

Quasi-optimal multiplication of linear differential operators

We show that linear differential operators with polynomial coefficients over a field of characteristic zero can be multiplied in quasi-optimal time. This answers an open question raised by van der Hoeven.

preprint2016arXiv

Accelerated Approximation of the Complex Roots and Factors of a Univariate Polynomial

The algorithms of Pan (1995) and(2002) approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time but require precision of computing that exceeds the degree of the polynomial. This causes numerical stability problems when the degree is large. We observe, however, that such a difficulty disappears at the initial stage of the algorithms, and in our present paper we extend this stage to root-finding within a nearly optimal arithmetic and Boolean complexity bounds provided that some mild initial isolation of the roots of the input polynomial has been ensured. Furthermore our algorithm is nearly optimal for the approximation of the roots isolated in a fixed disc, square or another region on the complex plane rather than all complex roots of a polynomial. Moreover the algorithm can be applied to a polynomial given by a black box for its evaluation (even if its coefficients are not known); it promises to be of practical value for polynomial root-finding and factorization, the latter task being of interest on its own right. We also provide a new support for a winding number algorithm, which enables extension of our progress to obtaining mild init

preprint2016arXiv

Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic

Arb is a C library for arbitrary-precision interval arithmetic using the midpoint-radius representation, also known as ball arithmetic. It supports real and complex numbers, polynomials, power series, matrices, and evaluation of many special functions. The core number types are designed for versatility and speed in a range of scenarios, allowing performance that is competitive with non-interval arbitrary-precision types such as MPFR and MPC floating-point numbers. We discuss the low-level number representation, strategies for precision and error bounds, and the implementation of efficient polynomial arithmetic with interval coefficients.

preprint2016arXiv

Sparse multivariate factorization by mean of a few bivariate factorizations

We describe an algorithm to factor sparse multivariate polynomials using O(d) bivariate factorizations where d is the number of variables. This algorithm is implemented in the Giac/Xcas computer algebra system.

preprint2016arXiv

A Near-Optimal Subdivision Algorithm for Complex Root Isolation based on the Pellet Test and Newton Iteration

We describe a subdivision algorithm for isolating the complex roots of a polynomial $F\in\mathbb{C}[x]$. Given an oracle that provides approximations of each of the coefficients of $F$ to any absolute error bound and given an arbitrary square $\mathcal{B}$ in the complex plane containing only simple roots of $F$, our algorithm returns disjoint isolating disks for the roots of $F$ in $\mathcal{B}$. Our complexity analysis bounds the absolute error to which the coefficients of $F$ have to be provided, the total number of iterations, and the overall bit complexity. It further shows that the complexity of our algorithm is controlled by the geometry of the roots in a near neighborhood of the input square $\mathcal{B}$, namely, the number of roots, their absolute values and pairwise distances. The number of subdivision steps is near-optimal. For the \emph{benchmark problem}, namely, to isolate all the roots of a polynomial of degree $n$ with integer coefficients of bit size less than $τ$, our algorithm needs $\tilde O(n^3+n^2τ)$ bit operations, which is comparable to the record bound of Pan (2002). It is the first time that such a bound has been achieved using subdivision methods, and in

335 works