The pitfalls of numerical computations using floating-point numbers are well known. Existing static techniques for float- ing-point error analysis provide a single upper bound across all potential values for a floating-point variable. We present a new ...

Modular static program analyses improve over global whole-program analyses in terms of scalability at a tradeoff with analysis accuracy. This tradeoff has to-date not been explored in the context of sound floating-point roundoff error analyses; ...

Large-scale optimization problems arising from the discretization of problems involving PDEs sometimes admit solutions that can be well approximated by low-rank matrices. In this paper, we will exploit this low-rank approximation property by solving the ...

We propose a modular framework for representing the real numbers that generalizes ieee, posits, and related floating-point number systems, and which has its roots in universal codes for the positive integers such as the Elias codes. This framework ...

Roundoff errors cannot be avoided when implementing numerical programs with finite precision. The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance, for FPGAs or custom ...

The $s$-step Lanczos method can achieve an $O(s)$ reduction in data movement over the classical Lanczos method for a fixed number of iterations, allowing the potential for significant speedups on modern computers. However, although the $s$-step Lanczos ...

Writing accurate numerical software is hard because of many sources of unavoidable uncertainties, including finite numerical precision of implementations. We present a programming model where the user writes a program in a real-valued implementation and ...

In practical computation with Runge--Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can have catastrophic ...

This paper is concerned with the fast and accurate evaluation of elementary symmetric functions. We present a new compensated algorithm by applying error-free transformations to improve the accuracy of the so-called Summation Algorithm, which is used, ...

This paper proposes a counterexample-guided narrowing approach, which mutually refines analyses and testing if (possibly spurious) counterexamples are found. A prototype tool CANAT for checking roundoff errors between floating point and fixed point ...

This paper introduces an ontology-based reasoning method for requirements elicitation. We start with an ontology structure contains knowledge of functional requirements and relations among them. We then propose a framework to elicit requirements using ...

We present computationally efficient criteria that can guarantee correct ordinal ranking of Google's PageRank scores when they are computed with the power method (ordinal ranking of a list consists of assigning an ordinal number to each item in the list)...

A strict adherence to threshold pivoting in the direct solution of symmetric indefinite problems can result in substantially more work and storage than forecast by a sparse analysis of the symmetric problem. One way of avoiding this is to use static ...

It is a well-known fact that the multiplication of a number by an integer power-of-two is a very simple process in binary arithmetic. Hence, digital filters whose coefficient values are integer power-of-two are essentially multiplierless. The design of ...

In this paper we study numerical stability of the parallel Jacobi method for computing the singular values and singular subspaces of an invertible upper triangular matrix that is obtained from QR decomposition with column pivoting. We show that in this ...

We present and analyze several simple algorithms for accurately computing the sum of n floating point numbers using a wider accumulator. Let f and F be the number of significant bits in the summands and the accumulator, respectively. Then assuming gradual ...

In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed. In practice, consideration of ...