85,934 research outputs found
Computing the Rank Profile Matrix
The row (resp. column) rank profile of a matrix describes the staircase shape
of its row (resp. column) echelon form. In an ISSAC'13 paper, we proposed a
recursive Gaussian elimination that can compute simultaneously the row and
column rank profiles of a matrix as well as those of all of its leading
sub-matrices, in the same time as state of the art Gaussian elimination
algorithms. Here we first study the conditions making a Gaus-sian elimination
algorithm reveal this information. Therefore, we propose the definition of a
new matrix invariant, the rank profile matrix, summarizing all information on
the row and column rank profiles of all the leading sub-matrices. We also
explore the conditions for a Gaussian elimination algorithm to compute all or
part of this invariant, through the corresponding PLUQ decomposition. As a
consequence, we show that the classical iterative CUP decomposition algorithm
can actually be adapted to compute the rank profile matrix. Used, in a Crout
variant, as a base-case to our ISSAC'13 implementation, it delivers a
significant improvement in efficiency. Second, the row (resp. column) echelon
form of a matrix are usually computed via different dedicated triangular
decompositions. We show here that, from some PLUQ decompositions, it is
possible to recover the row and column echelon forms of a matrix and of any of
its leading sub-matrices thanks to an elementary post-processing algorithm
Rank-profile revealing Gaussian elimination and the CUP matrix decomposition
Transforming a matrix over a field to echelon form, or decomposing the matrix
as a product of structured matrices that reveal the rank profile, is a
fundamental building block of computational exact linear algebra. This paper
surveys the well known variations of such decompositions and transformations
that have been proposed in the literature. We present an algorithm to compute
the CUP decomposition of a matrix, adapted from the LSP algorithm of Ibarra,
Moran and Hui (1982), and show reductions from the other most common Gaussian
elimination based matrix transformations and decompositions to the CUP
decomposition. We discuss the advantages of the CUP algorithm over other
existing algorithms by studying time and space complexities: the asymptotic
time complexity is rank sensitive, and comparing the constants of the leading
terms, the algorithms for computing matrix invariants based on the CUP
decomposition are always at least as good except in one case. We also show that
the CUP algorithm, as well as the computation of other invariants such as
transformation to reduced column echelon form using the CUP algorithm, all work
in place, allowing for example to compute the inverse of a matrix on the same
storage as the input matrix.Comment: 35 page
Symmetric indefinite triangular factorization revealing the rank profile matrix
We present a novel recursive algorithm for reducing a symmetric matrix to a
triangular factorization which reveals the rank profile matrix. That is, the
algorithm computes a factorization where is a permutation matrix,
is lower triangular with a unit diagonal and is
symmetric block diagonal with and antidiagonal
blocks. The novel algorithm requires arithmetic
operations. Furthermore, experimental results demonstrate that our algorithm
can even be slightly more than twice as fast as the state of the art
unsymmetric Gaussian elimination in most cases, that is it achieves
approximately the same computational speed. By adapting the pivoting strategy
developed in the unsymmetric case, we show how to recover the rank profile
matrix from the permutation matrix and the support of the block-diagonal
matrix. There is an obstruction in characteristic for revealing the rank
profile matrix which requires to relax the shape of the block diagonal by
allowing the 2-dimensional blocks to have a non-zero bottom-right coefficient.
This relaxed decomposition can then be transformed into a standard
decomposition at a
negligible cost
Fast Order Basis and Kernel Basis Computation and Related Problems
In this thesis, we present efficient deterministic algorithms
for polynomial matrix computation problems, including the computation
of order basis, minimal kernel basis, matrix inverse, column basis,
unimodular completion, determinant, Hermite normal form, rank and
rank profile for matrices of univariate polynomials over a field.
The algorithm for kernel basis computation also immediately provides
an efficient deterministic algorithm for solving linear systems. The
algorithm for column basis also gives efficient deterministic algorithms
for computing matrix GCDs, column reduced forms, and Popov normal
forms for matrices of any dimension and any rank.
We reduce all these problems to polynomial matrix multiplications.
The computational costs of our algorithms are then similar to the
costs of multiplying matrices, whose dimensions match the input matrix
dimensions in the original problems, and whose degrees equal the average
column degrees of the original input matrices in most cases. The use
of the average column degrees instead of the commonly used matrix
degrees, or equivalently the maximum column degrees, makes our computational
costs more precise and tighter. In addition, the shifted minimal bases
computed by our algorithms are more general than the standard minimal
bases
Settling Some Open Problems on 2-Player Symmetric Nash Equilibria
Over the years, researchers have studied the complexity of several decision
versions of Nash equilibrium in (symmetric) two-player games (bimatrix games).
To the best of our knowledge, the last remaining open problem of this sort is
the following; it was stated by Papadimitriou in 2007: find a non-symmetric
Nash equilibrium (NE) in a symmetric game. We show that this problem is
NP-complete and the problem of counting the number of non-symmetric NE in a
symmetric game is #P-complete.
In 2005, Kannan and Theobald defined the "rank of a bimatrix game"
represented by matrices (A, B) to be rank(A+B) and asked whether a NE can be
computed in rank 1 games in polynomial time. Observe that the rank 0 case is
precisely the zero sum case, for which a polynomial time algorithm follows from
von Neumann's reduction of such games to linear programming. In 2011, Adsul et.
al. obtained an algorithm for rank 1 games; however, it does not solve the case
of symmetric rank 1 games. We resolve this problem
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
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