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A notation system was previously presented which can notate any rational frequency in free Just Intonation. Transposition of music is carried out by multiplying each member of a set of frequencies by a single frequency. Transposition of JI…
Recent renewed interest in optimizing and analyzing floating-point programs has lead to a diverse array of new tools for numerical programs. These tools are often complementary, each focusing on a distinct aspect of numerical programming.…
Training with larger number of parameters while keeping fast iterations is an increasingly adopted strategy and trend for developing better performing Deep Neural Network (DNN) models. This necessitates increased memory footprint and…
Floating-point arithmetic is error-prone and unintuitive. Floating-point debuggers instrument programs to monitor floating-point arithmetic at run time and flag numerical issues. They estimate residues, i.e., the difference between actual…
Today, almost all computer systems use IEEE-754 floating point to represent real numbers. Recently, posit was proposed as an alternative to IEEE-754 floating point as it has better accuracy and a larger dynamic range. The configurable…
These notes introduce quantum computation and quantum error correction, emphasising the importance of stabilisers and the mathematical foundations in basic Lie theory. We begin by using the double cover map $\mathrm{SU}_2 \rightarrow…
We describe an algorithm for converting one bipartite quantum state into another using only local operations and classical communication, which is much simpler than the original algorithm given by Nielsen [Phys. Rev. Lett. 83, 436 (1999)].…
A class of two-bit bit flipping algorithms for decoding low-density parity-check codes over the binary symmetric channel was proposed in [1]. Initial results showed that decoders which employ a group of these algorithms operating in…
The typical model for measurement noise in quantum error correction is to randomly flip the binary measurement outcome. In experiments, measurements yield much richer information - e.g., continuous current values, discrete photon counts -…
There are now several comprehensive web applications, stand-alone computer programs and computer algebra functions that, given a floating point number such as 6.518670730718491, can return concise nonfloat constants such as 3 arctan 2 + ln…
This note shows how to compute, to high relative accuracy under mild assumptions, complex Jacobi rotations for diagonalization of Hermitian matrices of order two, using the correctly rounded functions $\mathtt{cr\_hypot}$ and…
Calculations of highly excited and delocalized molecular vibrational states are computationally challenging tasks, which strongly depends on the choice of coordinates for describing vibrational motions. We introduce a new method that…
TwoPunctures is perhaps the most widely-adopted code for generating binary black hole "puncture" initial data and interpolating these (spectral) data onto evolution grids. In typical usage, the bulk of this code's run time is spent in its…
In this work a rationalized algorithm for calculating the quotient of two complex numbers is presented which reduces the number of underlying real multiplications. The performing of a complex number division using the naive method takes 4…
Modern AI hardware, such as Nvidia's Blackwell architecture, is increasingly embracing low-precision floating-point (FP) formats to handle the pervasive activation outliers in Large Language Models (LLMs). Despite this industry trend, a…
We discuss efficient algorithms for the accurate forward and reverse evaluation of the discrete Fourier-Bessel transform (dFBT) as numerical tools to assist in the 2D polar convolution of two radially symmetric functions, relevant, e.g., to…
Fixed-point number representation is commonly employed in digital VLSI designs that have stringent hardware efficiency constraints. However, fixed-point numbers cover a relatively small dynamic range for a given bitwidth. In contrast,…
We introduce a novel method for encoding integers using smooth real-valued functions whose integral properties implicitly reflect discrete quantities. In contrast to classical representations, where the integer appears as an explicit…
We generalize the binary quantum counting algorithm of Lesovik, Suslov, and Blatter [Phys. Rev. A 82, 012316 (2010)] to higher counting bases. The algorithm makes use of qubits, qutrits, and qudits to count numbers in a base 2, base 3, or…
This paper provides a bound on the number of numeric operations (fixed or floating point) that can safely be performed before accuracy is lost. This work has important implications for control systems with safety-critical software, as these…