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Unlike the method without pivoting, Gaussian elimination with partial pivoting consecutively applies row permutation to matrix A in order to avoid possible akk diagonal entries of matrix A being equal to zero. Gaussian elimination with partial pivoting solves the matrix equation Ax = b decomposing matrix A into a lower L and upper U triangular matrices such that PA = LU, where P is a row permut...

A matrix A of size m œ n containing items from a totally ordered universe is termed monotone if, for every i, j, 1 ‹ i < j ‹ m, the minimum value in row j lies below or to the right of the minimum in row i. Monotone matrices, and variations thereof, are known to have many important applications. In particular, the problem of computing the row minima of a monotone matrix is of import in image pr...

Let Mn,m be the set of all n × m matrices with entries in F, where F is the field of real or complex numbers. A matrix R ∈ Mn with the property Re=e, is said to be a g-row stochastic (generalized row stochastic) matrix. Let A,B∈ Mn,m, so B is said to be gw-majorized by A if there exists an n×n g-row stochastic matrix R such that B=RA. In this paper we characterize all linear operators that stro...

We focus on row sampling based approximations for matrix algorithms, in particular matrix multipication, sparse matrix reconstruction, and l2 regression. For A ∈ R (m points in d ≪ m dimensions), and appropriate row-sampling probabilities, which typically depend on the norms of the rows of the m × d left singular matrix of A (the leverage scores), we give row-sampling algorithms with linear (up...

We focus the use of row sampling for approximating matrix algorithms. We give applications to matrix multipication; sparse matrix reconstruction; and, l2 regression. For a matrix A ∈ R m×d which represents m points in d ≪ m dimensions, all of these tasks can be achieved in O(md) via the singular value decomposition (SVD). For appropriate row-sampling probabilities (which typically depend on the...

Necessary and sufficient conditions are given for a substochastic semigroup on L obtained through the Kato–Voigt perturbation theorem to be either stochastic or strongly stable. We show how such semigroups are related to piecewise deterministic Markov process, provide a probabilistic interpretation of our results, and apply them to fragmentation equations.

A binary matrix has the Consecutive Ones Property (C1P) if there exists a permutation of its columns (i.e. a sequence of column swappings) such that in the resulting matrix the 1s are consecutive in every row. A Minimal Conflicting Set (MCS) of rows is a set of rows R that does not have the C1P, but such that any proper subset of R has the C1P. In [5], Chauve et al. gave a O(∆m(n+m+ e)) time al...

We describe a new algorithm called Frequent Directions for deterministic matrix sketching in the row-updates model. The algorithm is presented an arbitrary input matrix A ∈ Rn×d one row at a time. It performed O(d`) operations per row and maintains a sketch matrix B ∈ R`×d such that for any k < ` ‖AA−BB‖2 ≤ ‖A−Ak‖F /(`− k) and ‖A− πBk(A)‖F ≤ ( 1 + k `− k ) ‖A−Ak‖F . Here, Ak stands for the mini...