AdvancedTopic 90 of 104Engineering Mathematics
Numerical error and conditioning
Round-off and truncation errors can grow; ill-conditioned problems amplify them.
In depth
Floating-point numbers have finite precision (about 16 digits for doubles), producing round-off error. Truncation error arises from approximating processes. The condition number measures how input errors are amplified; ill-conditioned problems lose digits even with stable algorithms.
Key points
- Subtracting nearly equal numbers loses precision
- Machine epsilon ≈ 2.2×10⁻¹⁶
- Stable vs unstable algorithms
Example0.1+0.2 is not exactly 0.3 in binary floating point.
What is Numerical error and conditioning?
Round-off and truncation errors can grow; ill-conditioned problems amplify them.
Why is it important to understand numerical error and conditioning?
Numerical methods always introduce error, so you must know how rounding and approximation errors grow and when a problem is sensitive to them.
What is an ill-conditioned problem?
A problem whose output changes a lot for a small change in input. The condition number measures this sensitivity, and a large value warns of trouble.
Why can subtracting two nearly equal numbers be dangerous?
Round-off errors become large relative to the small difference, which is catastrophic cancellation. Reformulate the calculation to avoid it.
How is Numerical error and conditioning used in practice?
Typical applications include finding why a matrix solve gives wrong answers with nearly singular data, checking a finite difference result for truncation error and choosing a step size for an ODE solver that stays accurate.
What should I learn before Numerical error and conditioning?
Study "Monte Carlo simulation" first (topic 89 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Numerical error and conditioning?
Subtracting nearly equal numbers loses precision. Machine epsilon ≈ 2.2×10⁻¹⁶. Stable vs unstable algorithms.
Can you give an example of Numerical error and conditioning?
0.1+0.2 is not exactly 0.3 in binary floating point.
What comes after Numerical error and conditioning?
The next topic is "Root-finding methods", which builds directly on this one.
How difficult is Numerical error and conditioning?
It is classed as Advanced in Engineering Mathematics (topic 90 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.