Lecture 1: Introduction; numerics; error analysis (part II)

This segment explains the fundamental properties of floating-point math, emphasizing the necessity of rounding rules due to the limitations of representing all numbers precisely. It highlights the importance of avoiding bias in rounding and the common practice of rounding to the nearest even bit to maintain symmetry and prevent skewed results in extensive calculations. This segment delves into the binary representation of floating-point numbers, explaining the concept of a leading one and how it's used to optimize storage. It also addresses exceptions to this rule, such as the number zero, and discusses the existence of positive and negative zero as distinct representations in floating-point systems. This segment focuses on the potential for bias in floating-point calculations if rounding consistently favors one direction. It illustrates how this bias can accumulate over many operations, leading to significant inaccuracies in large-scale computations. The discussion emphasizes the need for careful design of rounding rules to mitigate this issue. Floating-point math involves rounding, requiring careful rules to avoid bias. Different number systems exist: fixed-point, floating-point (singles/doubles), infinite precision (rational numbers), and bracketing (range-based). Errors stem from truncation, discretization, modeling, empirical constants, and user input. Absolute and relative errors are hard to compute directly; backward error (problem change for given approximation) is often used instead. Well-conditioned problems have small backward error implying small forward error. Condition number measures this relationship. Careful implementation is crucial to mitigate errors; even simple operations like vector norm calculation can cause overflow. This segment differentiates between absolute and relative error, explaining their limitations in practical computation due to the unknown true value. It introduces the concept of using conservative upper bounds on error as a practical alternative, providing a measure of certainty in calculations even without knowing the exact error. This segment highlights the importance of numerical stability in seemingly simple calculations. It demonstrates how calculating the norm of a vector with large values can lead to overflow errors, even though the mathematical concept is straightforward. The professor introduces a more robust method to mitigate this issue, emphasizing that mathematically equivalent methods can yield vastly different numerical results. The example of solving quadratic equations using the standard formula further illustrates how seemingly innocuous formulas can suffer from numerical instability, leading to inaccurate results. This underscores the need for careful consideration of numerical precision and the potential pitfalls of relying solely on mathematical correctness without accounting for computational limitations. This segment introduces infinite precision arithmetic using rational numbers as an example. It explains how representing numbers as fractions of integers allows for exact arithmetic without rounding errors. However, it also points out the limitations of this approach, such as the inability to represent irrational numbers and the slower computation speed compared to floating-point arithmetic.