Numerical Methods

The quest for more constructive proofs in mathematics while retaining high levels of generality has been aided by the introduction of powerful new tools like polynomial interpolation and pinning lemmas. These approaches aim to bridge the gap between the concrete guarantees provided by constructive proofs and the wide applicability offered by probabilistic methods. Seeking Pinning…

Numerical stability refers to how sensitive an algorithm’s output is to slight changes or errors in the input data. Condition numbers quantify this sensitivity – high condition numbers imply greater numerical instability. Unstable algorithms can produce wildly inaccurate outputs even for reasonable inputs. This is problematic in computational geometry where robustness and reliability are critical….

The Real-RAM Model: Idealized Yet Limited The Real-RAM model, widely used in computational geometry algorithm analysis and design, assumes a hypothetical computing machine with infinite memory and constant-time arithmetic operations on nonnegative integers of arbitrary length. Defining properties include: Infinite memory capacity for storing integers of any size Basic arithmetic operations (+, -, *, %)…