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We study the link between the complexity of polynomial matrix multiplication and the complexity of solving other basic linear algebra problems on polynomial matrices. By polynomial matrices we mean <i>n</i>times <i>n</i> matrices in <b>K</b>[<i>x</i>] of degree bounded by <i>d</i>, with <b>K</b> a commutative field. Under the straight-line program model we… (More)

We present an inversion algorithm for nonsingular n × n matrices whose entries are degree d polynomials over a field. The algorithm is deterministic and, when n is a power of two, requires O˜(n 3 d) field operations for a generic input; the soft-O notation O˜indicates some missing log(nd) factors. Up to such logarithmic factors, this asymptotic complexity… (More)

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… (More)

Linear systems with structures such as Toeplitz, Vandermonde or Cauchy-likeness can be solved in O˜(α 2 n) operations, where n is the matrix size, α is its displacement rank, and O˜denotes the omission of logarithmic factors. We show that for such matrices, this cost can be reduced to O˜(α ω−1 n), where ω is a feasible exponent for matrix multiplication… (More)

Given two floating-point vectors x, y of dimension n and assuming rounding to nearest, we show that if no underflow or overflow occurs, any evaluation order for inner product returns a floating-point number r such that | r − x T y| nu|x| T |y| with u the unit roundoff. This result, which holds for any radix and with no restriction on n, can be seen as a… (More)

—In this paper we show how to reduce the computation of correctly-rounded square roots of binary floating-point data to the fixed-point evaluation of some particular integer polynomials in two variables. By designing parallel and accurate evaluation schemes for such bivariate polynomials, we show further that this approach allows for high instruction-level… (More)

Recently, some high-performance IEEE 754 single precision floating-point software has been designed, which aims at best exploiting some features (integer arithmetic, parallelism) of the STMicroelectronics ST200 Very Long Instruction Word (VLIW) processor. We review here the techniques and software tools used or developed for this design and its… (More)

— This paper presents some work in progress on fast and accurate floating-point arithmetic software for ST200-based embedded systems. We show how to use some key architectural features to design codes that achieve correct rounding-to-nearest without sacrificing for efficiency. This is illustrated with the square root function, whose implementation given… (More)

Linear systems with structures such as Toeplitz-, Vandermonde-or Cauchy-likeness can be solved in <i>O</i>~(α<sup>2</sup><i>n</i>) operations, where <i>n</i> is the matrix size, α is its displacement rank, and <i>O</i>~denotes the omission of logarithmic factors. We show that for Toeplitz-like and Vandermonde-like trices, this cost can be reduced… (More)