Improving the numerical stability of fast matrix multiplication

Fast algorithms for matrix multiplication, namely those that perform asymptotically fewer scalar operations than the classical algorithm, have been considered primarily of theoretical interest. Apart from Strassen's original algorithm, few fast algorithms have been efficiently implemented or used in practical applications. However, there exist many practical alternatives to Strassen's algorithm with varying performance and numerical properties. Fast algorithms are known to be numerically stable, but because their error bounds are slightly weaker than the classical algorithm, they are not used even in cases where they provide a performance benefit. We argue in this paper that the numerical sacrifice of fast algorithms, particularly for the typical use cases of practical algorithms, is not prohibitive, and we explore ways to improve the accuracy both theoretically and empirically. The numerical accuracy of fast matrix multiplication depends on properties of the algorithm and of the input matrices, and we consider both contributions independently. We generalize and tighten previous error analyses of fast algorithms and compare their properties. We discuss algorithmic techniques for improving the error guarantees from two perspectives: manipulating the algorithms, and reducing input anomalies by various forms of diagonal scaling. Finally, we benchmark performance and demonstrate our improved numerical accuracy

MEMS 411: The Pill Cutter

The goal of this design project was to improve the current method of pill cutting for STL Hills Pharmacy. Pharmacists and other medical professionals cut hundreds of pills weekly. Their current methods can only cut one pill at a time with an error rate of 50%. Pills are often split without enough precision and need to be thrown out. Our project was designed to have a 10% maximum error rate, require 1/4 of the force, and split pills 4 times as fast

Magic: The Gathering Is Turing Complete

Magic: The Gathering is a popular and famously complicated trading card game about magical combat. In this paper we show that optimal play in real-world Magic is at least as hard as the Halting Problem. This provides a positive answer to the question "is there a real-world game where perfect play is undecidable under the rules in which it is typically played?", a question that has been open for a decade [David Auger and Oliver Teytaud, 2012; Erik D. Demaine and Robert A. Hearn, 2009]. To do this, we present a methodology for embedding an arbitrary Turing machine into a game of Magic such that the first player is guaranteed to win the game if and only if the Turing machine halts. Our result applies to how real Magic is played, can be achieved using standard-size tournament-legal decks, and does not rely on stochasticity or hidden information. Our result is also highly unusual in that all moves of both players are forced in the construction. This shows that even recognising who will win a game in which neither player has a non-trivial decision to make for the rest of the game is undecidable. We conclude with a discussion of the implications for a unified computational theory of games and remarks about the playability of such a board in a tournament setting

Systems biology analysis of drivers underlying hallmarks of cancer cell metabolism.

Malignant transformation is often accompanied by significant metabolic changes. To identify drivers underlying these changes, we calculated metabolic flux states for the NCI60 cell line collection and correlated the variance between metabolic states of these lines with their other properties. The analysis revealed a remarkably consistent structure underlying high flux metabolism. The three primary uptake pathways, glucose, glutamine and serine, are each characterized by three features: (1) metabolite uptake sufficient for the stoichiometric requirement to sustain observed growth, (2) overflow metabolism, which scales with excess nutrient uptake over the basal growth requirement, and (3) redox production, which also scales with nutrient uptake but greatly exceeds the requirement for growth. We discovered that resistance to chemotherapeutic drugs in these lines broadly correlates with the amount of glucose uptake. These results support an interpretation of the Warburg effect and glutamine addiction as features of a growth state that provides resistance to metabolic stress through excess redox and energy production. Furthermore, overflow metabolism observed may indicate that mitochondrial catabolic capacity is a key constraint setting an upper limit on the rate of cofactor production possible. These results provide a greater context within which the metabolic alterations in cancer can be understood

MEMS 411: In-Beach Pressure Sampler Design Report

Claudio would like a device to put into the beach sand, around nest-depth, tomeasure and record data that can determine pressure as waves crash and recede