Jean-Guillaume Dumas

LINBOX: A GENERIC LIBRARY FOR EXACT LINEAR ALGEBRA

Black box techniques [12] are enabling exact linear algebra computations of a scale well beyond anything previously possible. The development of new and interesting algorithms has proceeded apace for the past two decades. It is time for the dissemination of these algorithms in an easily used software library so that the mathematical community may readily take advantage of their power. LinBox is that library. In this paper, we describe the design of this generic library, sketch its current range of capabilities, and give several examples of its use. The examples include a solution of Trefethen’s “Hundred Digit Challenge” problem #7 [14] and the computation of all the homology groups of simplicial complexes using the Smith normal form [8]. Exact black box methods are currently successful on sparse matrices with hundreds of thousands of rows and columns and having several million nonzero entries. The main reason large problems can be solved by black box methods is that they require much less memory in general than traditional eliminationbased methods do. This fact is widely used in the numerical computation area. We refer for instance to the templates for linear system solution and eigenvalue problems [2,1]. This has also led the computer algebra community to a considerable interest in black box methods. Since Wiedemann’s seminal paper [16], many developments have been proposed especially to adapt Krylov or Lanczos methods to fast exact algorithms. We refer to [5] and references therein for a review of problems and solutions. LinBox supplies efficient black box solutions for a variety of problems including linear equations and matrix normal forms with the guiding design principle of re-usability. The most essential and driving design criterion for LinBox is that it is generic with respect to the domain of computation. This is because there are many and various representations of finite fields each of which is advantageous to use for some algorithm under some circumstance. The integral and rational number capabilities depend heavily on modular

Memory efficient scheduling of Strassen-Winograd's matrix multiplication algorithm

We propose several new schedules for Strassen-Winograds matrix multiplication algorithm, they reduce the extra memory allocation requirements by three different means: by introducing a few pre-additions, by overwriting the input matrices, or by using a first recursive level of classical multiplication. In particular, we show two fully in-place schedules: one having the same number of operations, if the input matrices can be overwritten; the other one, slightly increasing the constant of the leading term of the complexity, if the input matrices are read-only. Many of these schedules have been found by an implementation of an exhaustive search algorithm based on a pebble game.

Exact sparse matrix-vector multiplication on GPU's and multicore architectures

We propose different implementations of the sparse matrix-dense vector multiplication (SpMV) for finite fields and rings Z /m Z. We take advantage of graphic card processors (GPU) and multi-core architectures. Our aim is to improve the speed of SpMV in the LinBox library, and henceforth the speed of its black-box algorithms. Besides, we use this library and a new parallelisation of the sigma-basis algorithm in a parallel block Wiedemann rank implementation over finite fields.

Cartesian effect categories are Freyd-categories

Most often, in a categorical semantics for a programming language, the substitution of terms is expressed by composition and finite products. However this does not deal with the order of evaluation of arguments, which may have major consequences when there are side-effects. In this paper Cartesian effect categories are introduced for solving this issue, and they are compared with strong monads, Freyd-categories and Haskells Arrows. It is proved that a Cartesian effect category is a Freyd-category where the premonoidal structure is provided by a kind of binary product, called the sequential product. The universal property of the sequential product provides Cartesian effect categories with a powerful tool for constructions and proofs. To our knowledge, both effect categories and sequential products are new notions.

Essentially optimal interactive certificates in linear algebra

Certificates to a linear algebra computation are additional data structures for each output, which can be used by a---possibly randomized---verification algorithm that proves the correctness of each output. The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size <i>N</i>, meaning <i>N</i> times a factor <i>N</i>^<i>o</i>(1), i.e., a factor <i>N</i>^{<i>η</i>(<i>N</i>)} with lim_{<i>N</i> → ∞} η(<i>N</i>) = 0. We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an <i>n × n</i> dense integer matrix <i>A</i>. Our certificates can be verified in Monte-Carlo bit complexity (<i>n</i>² log ||<i>A</i>||)^1+o(1), where log ||A|| is the bit size of the integer entries, solving an open problem in [Kaltofen, Nehring, Saunders, Proc. ISSAC 2011] subject to computational hardness assumptions. Second, we give algorithms that compute certificates for the rank of sparse or structured <i>n × n</i> matrices over an abstract field, whose Monte Carlo verification complexity is 2 matrix-times-vector products + <i>n</i>^1+o(1) arithmetic operations in the field. For example, if the <i>n × n</i> input matrix is sparse with <i>n</i>^1+o(1) non-zero entries, our rank certificate can be verified in <i>n</i>^1+o(1) field operations. This extends also to integer matrices with only an extra log ||<i>A</i>||^1+o(1) factor. All our certificates are based on interactive verification protocols with the interaction removed by a Fiat-Shamir identification heuristic. The validity of our verification procedure is subject to standard computational hardness assumptions from cryptography.

A duality between exceptions and states

Computational effects may often be interpreted in the Kleisli category of a monad or in the coKleisli category of a comonad. The duality between monads and comonads corresponds, in general, to a symmetry between construction and observation, for instance between raising an exception and looking up a state. Thanks to the properties of adjunction one may go one step further: the coKleisli-on-Kleisli category of a monad provides a kind of observation with respect to a given construction, while dually the Kleisli-on-coKleisli category of a comonad provides a kind of construction with respect to a given observation. In the previous examples this gives rise to catching an exception and updating a state. However, the interpretation of computational effects is usually based on a category which is not self-dual, like the category of sets. This leads to a breaking of the monad-comonad duality. For instance, in a distributive category the state effect has much better properties than the exception effect. This remark provides a novel point of view on the usual mechanism for handling exceptions. The aim of this paper is to build an equational semantics for handling exceptions based on the coKleisli-on-Kleisli category of the monad of exceptions. We focus on n-ary functions and conditionals. We propose a programmer’s language for exceptions and we prove that it has the required behaviour with respect to n-ary functions and conditionals.In this short note we study the semantics of two basic computational effects, exceptions and states, from a new point of view. In the handling of exceptions we dissociate the control from the elementary operation that recovers from the exception. In this way it becomes apparent that there is a duality, in the categorical sense, between exceptions and states.

Decorated proofs for computational effects: States

The syntax of an imperative language does not mention explicitly the state, while its denotational semantics has to mention it. In this paper we show that the equational proofs about an imperative language may hide the state, in the same way as the syntax does.

LINBOX founding scope allocation, parallel building blocks, and separate compilation

As a building block for a wide range of applications, computational exact linear algebra has to conciliate efficiency and genericity. The goal of the LinBox project is to address this problem in the design of an efficient general-purpose C++ opensource library for exact linear algebra over the integers, the rationals, and finite fields. Matrices can be either dense, sparse or black box (i.e. viewed as a linear operator, acting on vectors only). The library proposes a set of high level linear algebra solutions, such as the rank, the determinant, the solution of a linear system, the Smith normal form, the echelon form, the characteristic polynomial, etc.

Q-adic transform revisited

We present an algorithm to perform a simultaneous modular reduction of several residues. This enables to compress polynomials into integers and perform several modular operations with machine integer arithmetic. The idea is to convert the X-adic representation of modular polynomials, with X an indeterminate, to a q-adic representation where q is an integer larger than the field characteristic. With some control on the different involved sizes it is then possible to perform some of the q-adic arithmetic directly with machine integers or floating points. Depending also on the number of performed numerical operations one can then convert back to the q-adic or X-adic representation and eventually mod out high residues. In this note we present a new version of both conversions: more tabulations and a way to reduce the number of divisions involved in the process are presented. The polynomial multiplication is then applied to arithmetic and linear algebra in small finite field extensions.

On Newton–Raphson Iteration for Multiplicative Inverses Modulo Prime Powers

We study algorithms for the fast computation of modular inverses. Newton-Raphson iteration over p-adic numbers gives a recurrence relation computing modular inverse modulo pm, that is logarithmic in m. We solve the recurrence to obtain an explicit formula for the inverse. Then, we study different implementation variants of this iteration and show that our explicit formula is interesting for small exponent values but slower or large exponent, say of more than 700 bits. Overall, we thus propose a hybrid combination of our explicit formula and the best asymptotic variants. This hybrid combination yields then a constant factor improvement, also for large exponents.