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:
What are limitations or challenges with interval arithmetic?
What are some applications of interval arithmetic?
Key applications include:
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:
Related Topics
B-tree
A B-tree is a tree data structure optimized for fast indexed key lookups and writes on disk storage while keeping the tree balanced.
Skip List
A skip list is a probabilistic data structure that provides fast search and insertion over an ordered sequence using hierarchy of linked lists to skip over elements.
Probabilistic Data Structures
Probabilistic data structures are space and time efficient data structures that use randomized algorithms to provide approximate results to queries with strong guarantees.