What is interval arithmetic

Interval arithmetic is a system for computing with intervals representing uncertain numbers rather than fixed real numbers. It gives results guaranteed to contain the actual value within machine precision.

Intervals bound numerical error and uncertainty arising from rounding, measurement limitations, and incomplete data. Standard arithmetic operators like addition, subtraction, multiplication are extended to mathematical intervals to yield interval results.

For example, [1.5, 2.0] + [2.5, 3.0] = [4.0, 5.0]. The interval result captures all possible values from adding any numbers in the input ranges.

Interval arithmetic provides a robust computational foundation for building reliable systems like databases, constraint solvers, control systems and graphics. Numerical instability is avoided by propagating error bounds.

Sophisticated data structures like B-trees, Bloom filters, distributed hash tables and skip lists employ interval techniques for key operations like range queries, hashes, and sorting. Interval arithmetic enables mathematical algorithms to run reliably and efficiently on real-world data.

How does it work?

In interval arithmetic, numbers are represented by an upper and lower bound e.g. [2.5, 3.0] rather than a single value.

Operators like +, -, x are redefined on intervals to produce guaranteed enveloping supersets of actual point-value arithmetic. This captures the impact of uncertainty and rounding errors.

Why is it important? Where is it used?

Interval arithmetic yields rigorous numerical analysis. It has applications in control theory, signal processing, computational geometry, global optimization, and computer graphics where accounting for rounding errors and uncertainty is critical.

It prevents overconfidence in faulty precision and allows guaranteed interval bounds on computations. Numeric subtleties are handled rigorously.

FAQ

How are traditional arithmetic operations modified for intervals?

Basic interval arithmetic operations:

• x + y = [x1 + y1, x2 + y2]
• x - y = [x1 - y2, x2 - y1]
• x * y = [min(pairs), max(pairs)]
• x / y = x * (1/y)

What are limitations or challenges with interval arithmetic?

• Overestimation due to dependency problem.
• Computational complexity compared to floating point.
• Lack of native hardware support.
• Difficulty representing special values like NaN or Inf.

What are some applications of interval arithmetic?

Key applications include:

• Bounding rounding errors and uncertainty
• Error analysis and numerical stability
• Optimization and constraint solving
• Safety critical computations like flight control
• Graphics and geometry problems

How does interval arithmetic contrast with probabilistic approaches?

Interval arithmetic gives guaranteed bounds vs probabilistic confidence levels. It represents lack of knowledge rather than frequency-based probabilities.

References:

• [Book] Advances in Numerical Analysis Emphasizing Interval Data by CRC Press
• [Article] Interval Arithmetic: from Principles to Implementation
• [Post] Running Windowing Queries in Stream Processing
• [Post] Probabilistic Data Structures in Streaming: Count-Min Sketch