Definition
Error analysis is the systematic study of the uncertainty, approximations, and mistakes inherent in mathematical modeling, numerical computation, and physical measurement. It quantifies the difference between an estimated or measured value and the true, exact value.
How to read: Absolute error E equals the absolute value of x hat minus x. Relative error E sub rel equals absolute error divided by the absolute value of the true x. Meaning / when to use: Absolute error measures the raw magnitude of the mistake. Relative error normalizes the error against the true value, providing context (e.g., an error of 1 meter is massive when measuring a room, but negligible when measuring the Earth).
Why It Matters
In computer science, numbers are stored with finite precision (floating-point arithmetic). Every calculation introduces tiny rounding errors. In complex algorithms like solving differential equations, these tiny errors can compound exponentially, turning a slightly inaccurate estimate into catastrophic divergence. Error analysis guarantees that the output of a program or measurement is reliable and bounds the “worst-case scenario.”
Core Concepts
- Truncation Error: The error introduced by cutting off an infinite mathematical process (like using only the first 5 terms of a Taylor series instead of all infinite terms).
- Round-off Error: The error introduced by a computer’s inability to store numbers with infinite precision.
- Propagation of Error: How errors in initial inputs amplify or dampen as they pass through subsequent mathematical operations.
- Tolerance: A pre-defined acceptable threshold for error used to determine when an iterative algorithm should stop running.