Why Floating Point Accuracy Isn't Enough for Stable Algorithms скачать в хорошем качестве

Why Floating Point Accuracy Isn't Enough for Stable Algorithms

Did you know that having accurate floating point operations at every step doesn't guarantee your algorithm is stable? 🤯 When you're running computations for millions or billions of steps, even tiny errors can compound in dangerous ways. The key insight: errors need to average out to zero over time. If your errors consistently bias in one direction (always too high or always too low), you'll get exponential instability—even with high per-step accuracy! This is why we need to prove stability, not just accuracy. Stability means your errors are sometimes high, sometimes low, balancing out over the long run. Classic example: Euler's method applied to the exponential function shows small per-step errors that grow exponentially because each over-approximation creates more over-approximation. Small bias + many steps = big problems! This is a fundamental concept in numerical analysis that trips up a lot of people when they're implementing numerical algorithms. #NumericalAnalysis #ScientificComputing #computationalmathematics #NumericalStability #Julialang #julia #SciML #Mathematics #Algorithm #modelingandsimulation