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Jan Verschelde

Publications and source records attributed to Jan Verschelde.

At least 37 records · Page 2Linked to original sources
arXiv2017

The Method of Gauss-Newton to Compute Power Series Solutions of Polynomial Homotopies

We consider the extension of the method of Gauss-Newton from complex floating-point arithmetic to the field of truncated power series with complex floating-point coefficients. With linearization we formulate a linear system where the coefficient matrix is a series with matrix coefficients, and provide a characterization for when the matrix series is regular based on the algebraic variety of an augmented system. The structure of the linear system leads to a block triangular system. In the regular case, solving the linear system is equivalent to solving a Hermite interpolation problem. We show that this solution has cost cubic in the problem size. In general, at singular points, we rely on methods of tropical algebraic geometry to compute Puiseux series. With a few illustrative examples, we demonstrate the application to polynomial homotopy continuation.

math.NA↗
arXiv2017

Computing Tropical Prevarieties in Parallel

The computation of the tropical prevariety is the first step in the application of polyhedral methods to compute positive dimensional solution sets of polynomial systems. In particular, pretropisms are candidate leading exponents for the power series developments of the solutions. The computation of the power series may start as soon as one pretropism is available, so our parallel computation of the tropical prevariety has an application in a pipelined solver. We present a parallel implementation of dynamic enumeration. Our first distributed memory implementation with forked processes achieved good speedups, but quite often resulted in large variations in the execution times of the processes. The shared memory multithreaded version applies work stealing to reduce the variability of the run time. Our implementation applies the thread safe Parma Polyhedral Library (PPL), in exact arithmetic with the GNU Multiprecision Arithmetic Library (GMP), aided by the fast memory allocations of TCMalloc. Our parallel implementation is capable of computing the tropical prevariety of the cyclic 16-roots problem. We also report on computational experiments on the $n$-body and $n$-vortex problems; our computational results compare favorably with Gfan.

cs.MS↗
arXiv2016

Pruning Algorithms for Pretropisms of Newton Polytopes

Pretropisms are candidates for the leading exponents of Puiseux series that represent solutions of polynomial systems. To find pretropisms, we propose an exact gift wrapping algorithm to prune the tree of edges of a tuple of Newton polytopes. We prefer exact arithmetic not only because of the exact input and the degrees of the output, but because of the often unpredictable growth of the coordinates in the face normals, even for polytopes in generic position. We provide experimental results with our preliminary implementation in Sage that compare favorably with the pruning method that relies only on cone intersections.

cs.CG↗
arXiv2016

Computing all Space Curve Solutions of Polynomial Systems by Polyhedral Methods

A polyhedral method to solve a system of polynomial equations exploits its sparse structure via the Newton polytopes of the polynomials. We propose a hybrid symbolic-numeric method to compute a Puiseux series expansion for every space curve that is a solution of a polynomial system. The focus of this paper concerns the difficult case when the leading powers of the Puiseux series of the space curve are contained in the relative interior of a higher dimensional cone of the tropical prevariety. We show that this difficult case does not occur for polynomials with generic coefficients. To resolve this case, we propose to apply polyhedral end games to recover tropisms hidden in the tropical prevariety.

cs.SC↗
arXiv2015

Accelerating Polynomial Homotopy Continuation on a Graphics Processing Unit with Double Double and Quad Double Arithmetic

Numerical continuation methods track a solution path defined by a homotopy. The systems we consider are defined by polynomials in several variables with complex coefficients. For larger dimensions and degrees, the numerical conditioning worsens and hardware double precision becomes often insufficient to reach the end of the solution path. With double double and quad double arithmetic, we can solve larger problems that we could not solve with hardware double arithmetic, but at a higher computational cost. This cost overhead can be compensated by acceleration on a Graphics Processing Unit (GPU). We describe our implementation and report on computational results on benchmark polynomial systems.

cs.MS↗
arXiv2015

Solving Polynomial Systems in the Cloud with Polynomial Homotopy Continuation

Polynomial systems occur in many fields of science and engineering. Polynomial homotopy continuation methods apply symbolic-numeric algorithms to solve polynomial systems. We describe the design and implementation of our web interface and reflect on the application of polynomial homotopy continuation methods to solve polynomial systems in the cloud. Via the graph isomorphism problem we organize and classify the polynomial systems we solved. The classification with the canonical form of a graph identifies newly submitted systems with systems that have already been solved.

cs.MS↗
arXiv2015

Tracking Many Solution Paths of a Polynomial Homotopy on a Graphics Processing Unit

Polynomial systems occur in many areas of science and engineering. Unlike general nonlinear systems, the algebraic structure enables to compute all solutions of a polynomial system. We describe our massive parallel predictor-corrector algorithms to track many solution paths of a polynomial homotopy. The data parallelism that provides the speedups stems from the evaluation and differentiation of the monomials in the same polynomial system at different data points, which are the points on the solution paths. Polynomial homotopies that have tens of thousands of solution paths can keep a sufficiently large amount of threads occupied. Our accelerated code combines the reverse mode of algorithmic differentiation with double double and quad double precision to compute more accurate results faster.

cs.MS↗
arXiv2014

GPU acceleration of Newton's method for large systems of polynomial equations in double double and quad double arithmetic

In order to compensate for the higher cost of double double and quad double arithmetic when solving large polynomial systems, we investigate the application of NVIDIA Tesla K20C general purpose graphics processing unit. The focus on this paper is on Newton's method, which requires the evaluation of the polynomials, their derivatives, and the solution of a linear system to compute the update to the current approximation for the solution. The reverse mode of algorithmic differentiation for a product of variables is rewritten in a binary tree fashion so all threads in a block can collaborate in the computation. For double arithmetic, the evaluation and differentiation problem is memory bound, whereas for complex quad double arithmetic the problem is compute bound. With acceleration we can double the dimension and get results that are twice as accurate in about the same time.

cs.DC↗
arXiv2014

Computing all Affine Solution Sets of Binomial Systems

To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible decomposition of a variety is typically understood in affine space, including also those components with zero coordinates. For the problem of computing solution sets in the intersection of some coordinate planes, the direct application of a polyhedral method fails, because the original facial structure of the Newton polytopes may alter completely when selected variables become zero. Our new proposed method enumerates all factors contributing to a generalized permanent and toric solutions as a special case of this enumeration. For benchmark problems such as the adjacent 2-by-2 minors of a general matrix, our methods scale much better than the witness set representations of numerical algebraic geometry.

cs.SC↗
arXiv2014

Modernizing PHCpack through phcpy

PHCpack is a large software package for solving systems of polynomial equations. The executable phc is menu driven and file oriented. This paper describes the development of phcpy, a Python interface to PHCpack. Instead of navigating through menus, users of phcpy solve systems in the Python shell or via scripts. Persistent objects replace intermediate files.

cs.MS↗
arXiv2013

A Polyhedral Method to Compute All Affine Solution Sets of Sparse Polynomial Systems

To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible decomposition of a variety is typically understood in affine space, including also those components with zero coordinates. We present a polyhedral method to compute all affine solution sets of a polynomial system. The method enumerates all factors contributing to a generalized permanent. Toric solution sets are recovered as a special case of this enumeration. For sparse systems as adjacent 2-by-2 minors our methods scale much better than the techniques from numerical algebraic geometry.

cs.SC↗
arXiv2013

Polyhedral Methods for Space Curves Exploiting Symmetry Applied to the Cyclic n-roots Problem

We present a polyhedral algorithm to manipulate positive dimensional solution sets. Using facet normals to Newton polytopes as pretropisms, we focus on the first two terms of a Puiseux series expansion. The leading powers of the series are computed via the tropical prevariety. This polyhedral algorithm is well suited for exploitation of symmetry, when it arises in systems of polynomials. Initial form systems with pretropisms in the same group orbit are solved only once, allowing for a systematic filtration of redundant data. Computations with cddlib, Gfan, PHCpack, and Sage are illustrated on cyclic $n$-roots polynomial systems.

math.NA↗
arXiv2013

Orthogononalization on a general purpose graphics processing unit with double double and quad double arithmetic

Our problem is to accurately solve linear systems on a general purpose graphics processing unit with double double and quad double arithmetic. The linear systems originate from the application of Newton's method on polynomial systems. Newton's method is applied as a corrector in a path following method, so the linear systems are solved in sequence and not simultaneously. One solution path may require the solution of thousands of linear systems. In previous work we reported good speedups with our implementation to evaluate and differentiate polynomial systems on the NVIDIA Tesla C2050. Although the cost of evaluation and differentiation often dominates the cost of linear system solving in Newton's method, because of the limited bandwidth of the communication between CPU and GPU, we cannot afford to send the linear system to the CPU for solving during path tracking. Because of large degrees, the Jacobian matrix may contain extreme values, requiring extended precision, leading to a significant overhead. This overhead of multiprecision arithmetic is our main motivation to develop a massively parallel algorithm. To allow overdetermined linear systems we solve linear systems in the least squares sense, computing the QR decomposition of the matrix by the modified Gram-Schmidt algorithm. We describe our implementation of the modified Gram-Schmidt orthogonalization method for the NVIDIA Tesla C2050, using double double and quad double arithmetic. Our experimental results show that the achieved speedups are sufficiently high to compensate for the overhead of one extra level of precision.

cs.MS↗
arXiv2012

PHCpack in Macaulay2

The Macaulay2 package PHCpack.m2 provides an interface to PHCpack, a general-purpose polynomial system solver that uses homotopy continuation. The main method is a numerical blackbox solver which is implemented for all Laurent systems. The package also provides a fast mixed volume computation, the ability to filter solutions, homotopy path tracking, and a numerical irreducible decomposition method. As the size of many problems in applied algebraic geometry often surpasses the capabilities of symbolic software, this package will be of interest to those working on problems involving large polynomial systems.

math.AG↗
arXiv2012

Computing Puiseux Series for Algebraic Surfaces

In this paper we outline an algorithmic approach to compute Puiseux series expansions for algebraic surfaces. The series expansions originate at the intersection of the surface with as many coordinate planes as the dimension of the surface. Our approach starts with a polyhedral method to compute cones of normal vectors to the Newton polytopes of the given polynomial system that defines the surface. If as many vectors in the cone as the dimension of the surface define an initial form system that has isolated solutions, then those vectors are potential tropisms for the initial term of the Puiseux series expansion. Our preliminary methods produce exact representations for solution sets of the cyclic $n$-roots problem, for $n = m^2$, corresponding to a result of Backelin.

cs.SC↗
arXiv2012

Evaluating polynomials in several variables and their derivatives on a GPU computing processor

In order to obtain more accurate solutions of polynomial systems with numerical continuation methods we use multiprecision arithmetic. Our goal is to offset the overhead of double double arithmetic accelerating the path trackers and in particular Newton's method with a general purpose graphics processing unit. In this paper we describe algorithms for the massively parallel evaluation and differentiation of sparse polynomials in several variables. We report on our implementation of the algorithmic differentiation of products of variables on the NVIDIA Tesla C2050 Computing Processor using the NVIDIA CUDA compiler tools.

cs.MS↗
arXiv2011

Quality Up in Polynomial Homotopy Continuation by Multithreaded Path Tracking

Speedup measures how much faster we can solve the same problem using many cores. If we can afford to keep the execution time fixed, then quality up measures how much better the solution will be computed using many cores. In this paper we describe our multithreaded implementation to track one solution path defined by a polynomial homotopy. Limiting quality to accuracy and confusing accuracy with precision, we strive to offset the cost of multiprecision arithmetic running multithreaded code on many cores.

cs.DC↗