Related papers: Efficient implementation of elementary functions i…
Expectation Propagation is a very popular algorithm for variational inference, but comes with few theoretical guarantees. In this article, we prove that the approximation errors made by EP can be bounded. Our bounds have an asymptotic…
Recovery type a posteriori error estimators are popular, particularly in the engineering community, for their computationally inexpensive, easy to implement, and generally asymptotically exactness. Unlike the residual type error estimators,…
A method to increase the precision of feedforward networks is proposed. It requires a prior knowledge of a target function derivatives of several orders and uses this information in gradient based training. Forward pass calculates not only…
Based on two point spline approximations of arbitrary order, a series of functions that define lower bounds for sin(x) and sin(x)/x, over the interval [0,Pi/2], with increasingly low relative errors and smaller relative errors than…
We study the capability of the Fast Fourier Transform (FFT) to accelerate exact and approximate matrix multiplication without using Strassen-like divide-and-conquer. We present a simple exact algorithm running in $O(n^{2.89})$ time, which…
We show that quantum search can be used to compute the hardness to round an elementary function, that is, to determine the minimum working precision required to compute the values of an elementary function correctly rounded to a target…
New low-precision accelerators, vector instruction sets, and library functions make maximizing accuracy and performance of numerical code increasingly challenging. Two lines of work$\unicode{x2013}$traditional compilers and numerical…
We present methods for emulating the matter power spectrum by combining information from cosmological $N$-body simulations at different resolutions. An emulator allows estimation of simulation output by interpolating across the parameter…
Complexity bounds for many problems on matrices with univariate polynomial entries have been improved in the last few years. Still, for most related algorithms, efficient implementations are not available, which leaves open the question of…
This paper proposes a set of techniques to develop correctly rounded math libraries for 32-bit float and posit types. It enhances our RLibm approach that frames the problem of generating correctly rounded libraries as a linear programming…
We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, to approximate arbitrary functions. We show that the polynomial coefficients in Legendre expansion, therefore the whole series, converge to zero much more…
In recent years, a very exciting and promising method for proving lower bounds for arithmetic circuits has been proposed. This method combines the method of {\it depth reduction} developed in the works of Agrawal-Vinay [AV08], Koiran…
We consider the problem of certifying lower bounds for real-valued multivariate transcendental functions. The functions we are dealing with are nonlinear and involve semialgebraic operations as well as some transcendental functions like…
Interval arithmetic libraries provide the four elementary arithmetic operators for operand intervals bounded by floating-point numbers. Actual implementations need to make a large case analysis that considers, e.g., magnitude relations…
In modern low-power embedded platforms, floating-point (FP) operations emerge as a major contributor to the energy consumption of compute-intensive applications with large dynamic range. Experimental evidence shows that 50% of the energy…
We propose a new class of multiplier bootstraps for count functionals, ranging from a fast, approximate linear bootstrap tailored to sparse, massive graphs to a quadratic bootstrap procedure that offers refined accuracy for smaller, denser…
Fully computable a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polygonal domains. Linear finite elements are considered on anisotropic triangulations. To…
We give a deterministic algorithm for approximately counting satisfying assignments of a degree-$d$ polynomial threshold function (PTF). Given a degree-$d$ input polynomial $p(x_1,\dots,x_n)$ over $R^n$ and a parameter $\epsilon> 0$, our…
We obtain new complexity bounds for computing a triangular integral basis of a number field or a function field. We reach for function fields a softly linear cost with respect to the size of the output when the residual characteristic is…
This is a draft of a book about algorithms for performing arithmetic, and their implementation on modern computers. We are concerned with software more than hardware - we do not cover computer architecture or the design of computer…