Underflow is a form of rounding error that occurs when a number near zero is rounded to zero.

Underflow

Overflow is a form of rounding error that occurs when a large number is rounded to ∞ or -∞.

Overflow

This function is used to predict the probabilities of values in a multinoulli distribution. This function must be stabilized against underflow and overflow.

Softmax Function

Conditioning refers to how rapidly the output of a function changes in response to small changes in its input. This sensitivity is quantified by the $$\textbf{condition number}$$, which measures the rate of change of the function as the input varies.

Function Conditioning

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- Computation of transforms
- Computations in finite fields
- Computations on matrices
- Computations on polynomials
- Number-theoretic computations
- Interpolation
- Numerical differentiation
- Interval arithmetic
- Arbitrary-precision arithmetic
- Automatic differentiation
- Mesh generation
- Discretization

Numerical Analysis

Constrained optimization is the mathematical process of finding the minimum or maximum value of an objective function, $$f(x)$$, over a restricted set of inputs, $$\mathbb{S}$$. The points in $$\mathbb{S}$$ that satisfy the problem's constraints are called feasible points.

Constrained Optimization

For a positive integer n, an n-point Gauss–Legendre rule uses the n roots of the degree-n Legendre polynomial and their corresponding weights. For unweighted integration on [-1,1], the ideal mathematical rule integrates every polynomial of degree at most 2n−1 exactly. This guarantee does not extend to arbitrary functions or all higher-degree polynomials. Numerical approximations to nodes, weights, and arithmetic can introduce error; this caveat is a scoped inference, not a universal library error bound.

Source: https://dlmf.nist.gov/3.5

Polynomial exactness of Gauss–Legendre quadrature

For a real function f defined throughout a closed interval X, an interval evaluation F(X) satisfies inclusion when f(x) belongs to F(X) for every x in X.

For finite real intervals [a,b] and [c,d], outward-rounded addition gives [down(a+c),up(b+d)]. Here down returns a representable lower bound on the exact sum, and up returns a representable upper bound. Guaranteed containment requires exact arithmetic or suitable directed rounding, together with supported operations and correct implementation of their bounds; merely storing two floating-point endpoints is insufficient.

Conditions and limits:
- An enclosure need not have the smallest possible width and does not represent statistical confidence.
- Using a no-rounding policy with inexact floating-point arithmetic removes the containment guarantee.
- Input conversions, rounding-state management, exceptional values, exponent limits, and support for particular functions require appropriate handling.
- Correct scalar directed rounding supports endpoint bounds but does not by itself establish a complete interval algorithm.

Sources:
- [Boost Interval Arithmetic Library — Introduction and Overview](https://www.boost.org/doc/libs/latest/libs/numeric/interval/doc/interval.htm)
- [Boost Interval Arithmetic Library — Rounding Policies](https://www.boost.org/doc/libs/latest/libs/numeric/interval/doc/rounding.htm)
- [GNU MPFR — Directed rounding](https://www.mpfr.org/mpfr-current/mpfr.html)

Outward rounding and interval inclusion

For a real variable x in the closed interval X=[-2,2], both occurrences of x in x*x take the same value. The square range is therefore square(X)=[0,4]. Interval multiplication instead encloses products of independently chosen operands from X: X*X=[-4,4]. For example, choosing -2 and 2 gives -4, although a real square cannot be negative. Thus interval evaluation of x*x can overestimate the actual square range by losing the dependence between repeated occurrences of the variable. The product interval still contains every actual square value.

Conditions and limits:
- These exact mathematical ranges include their endpoints.
- This example establishes overestimation for this interval and expression; it does not quantify the effect for other expressions.
- Computational containment depends on exact arithmetic or suitable directed rounding and supported operations.
- Containment alone establishes neither the narrowest enclosure nor statistical confidence.

Source:
- [Boost Interval Arithmetic Library — Introduction and Overview](https://www.boost.org/doc/libs/latest/libs/numeric/interval/doc/interval.htm)

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