Error Analysis Summary

Numerical Analysis · Axiom Academy

Let's review the key concepts of numerical errors and how to control them. Absolute Error: The raw difference between true and approximate values Relative Error: Normalized by true value - better for comparing accuracies Significant Digits: Approximately - counts correct decimal places Why It Matters: Relative error is usually more meaningful because 0.001 error means different things for values of 1 vs 1,000,000 Round-off Error: From finite precision representation (e.g., 0.1 can't be exact in binary) Truncation Error: From approximating infinite processes (e.g., stopping a Taylor series early) Propagation: How errors grow or shrink through calculations Key Insight: These two error types often trade off - smaller steps reduce truncation error but may increase round-off Machine Epsilon: for double precision (~15-16 decimal digits max) Binary Fractions: Only powers of 2 (0.5, 0.25, 0.125...) are exact 0.1 + 0.2 ≠ 0.3: The classic floating-point surprise - always use tolerance comparisons Safe Comparison: Use instead of direct equality Condition Number: measures problem sensitivity Well-conditioned: Small input changes cause small output changes ( ) Ill-conditioned: Small input changes cause large output changes ( ) Numerical Stability: A stable algorithm doesn't make errors worse than necessary Example Recap: Computing Error Given: True value , approximation Interpret: About 3.3 significant digits - acceptable for rough estimates but not precision work

This is the written version of the interactive lesson above. See the full Numerical Analysis course.