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We present an optimized algorithm calculating determinant for multivariate polynomial matrix on GPU. The novel algorithm provides precise determinant for input multivariate polynomial matrix in controllable time. Our approach is based on…
Many applications in scientific computing and data science require the computation of a rank-revealing factorization of a large matrix. In many of these instances the classical algorithms for computing the singular value decomposition are…
For scientific computations on a digital computer the set of real number is usually approximated by a finite set F of "floating-point" numbers. We compare the numerical accuracy possible with difference choices of F having approximately the…
We consider the classical $k$-means clustering problem in the setting bi-criteria approximation, in which an algoithm is allowed to output $\beta k > k$ clusters, and must produce a clustering with cost at most $\alpha$ times the to the…
In practical computations, the (preconditioned) conjugate gradient (P)CG method is the iterative method of choice for solving systems of linear algebraic equations $Ax=b$ with a real symmetric positive definite matrix $A$. During the…
We develop fixed-point algorithms for the approximation of structured matrices with rank penalties. In particular we use these fixed-point algorithms for making approximations by sums of exponentials, or frequency estimation. For the basic…
Finite-precision floating point arithmetic unavoidably introduces rounding errors which are traditionally bounded using a worst-case analysis. However, worst-case analysis might be overly conservative because worst-case errors can be…
The floating-point implementation of a function on an interval often reduces to polynomial approximation, the polynomial being typically provided by Remez algorithm. However, the floating-point evaluation of a Remez polynomial sometimes…
Discrete Hahn polynomials (DHPs) and their moments are considered to be one of the efficient orthogonal moments and they are applied in various scientific areas such as image processing and feature extraction. Commonly, DHPs are used as…
A method that uses order statistics to construct multivariate distributions with fixed marginals and which utilizes a representation of the Bernstein copula in terms of a finite mixture distribution is proposed. Expectation-maximization…
In constraining iterative processes, the algorithmic operator of the iterative process is pre-multiplied by a constraining operator at each iterative step. This enables the constrained algorithm, besides solving the original problem, also…
Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into…
Certifying the positivity of trigonometric polynomials is of first importance for design problems in discrete-time signal processing. It is well known from the Riesz-Fej\'ez spectral factorization theorem that any trigonometric univariate…
In this work we present an explicit representation of the orthonormal Bernstein polynomials and demonstrate that they can be generated from a linear combination of non-orthonormal Bernstein polynomials. In addition, we report a set of $n$…
Diagonalization of a large matrix is the computational bottleneck in many applications such as electronic structure calculations. We show that a speedup of over 30% can be achieved by exploiting 32-bit floating point operations, while…
Factorization of polynomials arises in numerous areas in symbolic computation. It is an important capability in many symbolic and algebraic computation. There are two type of factorization of polynomials. One is convention polynomial…
Approximation using Fourier features is a popular technique for scaling kernel methods to large-scale problems, with myriad applications in machine learning and statistics. This method replaces the integral representation of a…
Tensors, especially higher-order tensors, are typically represented in low-rank formats to preserve the main information of the high-dimensional data while saving memory space. In practice, only a small fraction elements in high-dimensional…
Randomization has been applied to Hamiltonian simulation in a number of ways to improve the accuracy or efficiency of product formulas. Deterministic product formulas are often constructed in a symmetric way to provide accuracy of even…
We present improved algorithms for fast calculation of the inverse square root for single-precision floating-point numbers. The algorithms are much more accurate than the famous fast inverse square root algorithm and have the same or…