Condition Number

Quantify the worst local amplification of admissible relative input perturbations into relative output changes for a specified mathematical problem, point, and choice of norms.

Version

v3 · 2026-09-19 · History

Domain-specific #

1530

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomain

Numerical Analysis → Mathematics

Aliases

Conditioning, Problem condition number, Relative condition number

Core Idea

A condition number is a numerical-analysis measure of how strongly a mathematical problem amplifies small changes in its data. The problem is represented by a solution map that sends admissible input data to an answer. At a specified input, one compares the size of a small perturbation of the data with the resulting change in the answer, then takes a local worst case over permitted perturbation directions. A relative condition number compares relative changes; an absolute condition number compares absolute changes. John Rice's general theory made the problem-and-data formulation explicit and separated the sensitivity of the mathematical task from the behavior of any particular implementation.

How would you explain it like I'm…

The Wobbliness Score

Some questions are wobbly: if you change what you start with just a tiny bit, the answer changes a lot. Other questions are steady: small changes make only small changes to the answer. A condition number is a score for how wobbly a question is — it's about the question itself, not about who or what is solving it.

How Much Small Errors Grow

When we solve a math problem with real-world numbers, those numbers are often a little off. A Condition Number tells how much a small error in the starting numbers can grow into an error in the answer. A small condition number means the problem is steady; a large one means it's touchy, and small errors can blow up. It describes the problem, not the method used to solve it: a good method can still give a shaky answer on a touchy problem, and a bad method can mess up even a steady one. A big number is a warning, not an exact count of how wrong the answer will be.

Problem Sensitivity to Input Changes

A Condition Number measures how strongly a mathematical problem magnifies small changes in its input data. You think of the problem as a map from input data to answer, nudge the input slightly in the worst possible allowed direction, and compare the size of the change in the answer to the size of the nudge; this can be done in absolute or relative terms. Conditioning belongs to the problem, while stability belongs to the algorithm: a stable algorithm can still give a large error on an ill-conditioned problem, and a well-conditioned problem can't save an unstable algorithm. For solving a linear system Ax = b, the familiar condition number is ‖A‖‖A⁻¹‖, which depends on the chosen norm. A large value warns about possible accuracy loss but isn't an exact count of digits lost.

 

A condition number, in numerical analysis, quantifies how strongly a problem amplifies small perturbations of its data. The problem is modeled as a solution map from admissible data to answers; at a specified input, one compares the size of a small data perturbation with the resulting change in the answer and takes the local worst case over permitted perturbation directions. Relative and absolute condition numbers compare relative or absolute changes respectively. John Rice's general theory made this problem-and-data framing explicit and separated the sensitivity of the mathematical task from any implementation. Conditioning is a property of the posed problem, including its representation, scaling, metric, and evaluation point; stability is a property of an algorithm, where a backward-stable method returns the exact answer to a nearby problem. Hence a stable algorithm can give large forward error on an ill-conditioned problem, and good conditioning cannot rescue an unstable method. For a differentiable map, the local condition number is governed by the derivative; for a nonsingular system Ax = b with fixed A and normwise perturbations, it is κ(A) = ‖A‖‖A⁻¹‖, which in the Euclidean norm equals the ratio of largest to smallest singular value. A large value warns about attainable accuracy but is neither an exact count of lost digits nor a guarantee that every perturbation is maximally amplified.

Scope of Application

Condition numbers are literal wherever a mathematical answer depends on uncertain or finite-precision input and the local amplification can be specified independently of an algorithm.

• Linear systems. Bounding sensitivity of a solution to changes in a matrix or right-hand side.
• Least squares. Distinguishing residual, solution, normwise, and componentwise sensitivity.
• Eigenvalue problems. Measuring how eigenvalues or invariant subspaces respond to perturbations.
• Polynomial roots. Explaining why nearby coefficients can produce widely displaced roots.
• Function evaluation. Using derivatives and scaling to assess local relative sensitivity.
• Optimization. Measuring sensitivity of solutions or values under a declared regularity regime.
• Geometric numerical problems. Defining condition relative to distance from ill-posed instances.
• Scientific computing. Combining data uncertainty, condition estimation, and backward-error analysis.

Clarity

State the solution map, base point, perturbed data, admissible perturbation set, input and output norms, and whether the quantity is absolute or relative. Write the limiting supremum or the theorem that reduces it to a derivative or operator norm. If the output can vanish, explain how relative scaling is handled. For matrices, name the norm and whether A, b, or both vary. Distinguish the exact condition number from an estimate.

Manages Complexity

The abstraction compresses infinitely many nearby problem instances into one local amplification scale. That scale lets analysts decide whether observed error is explained by the problem itself, by the algorithm, or by both. It guides precision selection, data-quality requirements, preconditioning, and reformulation. The compression discards direction-specific behavior, finite-perturbation nonlinearities, probability structure, and correlations among data errors. A single scalar may therefore be pessimistic, and structured or componentwise condition numbers can be much more informative than an unstructured normwise value.

Abstract Reasoning

1. Specify the mathematical solution map rather than naming only a matrix or algorithm.
2. Fix the data point and identify which components are uncertain.
3. Choose admissible perturbations that preserve any relevant structure.
4. Choose input and output size rules and declare absolute or relative scaling.
5. Form the ratio of output change to input change.
6. Take a local limit and worst case over allowed directions.

Knowledge Transfer

The strict parent is Perturbation. A condition number begins with a reference instance, applies a small admissible disturbance, observes response, and extracts the leading local amplification. Perturbation applies to physical, statistical, and dynamical systems without producing a condition number. Condition Number adds a solution map, paired size measures, a limiting worst-direction ratio, and the numerical-analysis distinction between problem conditioning and algorithm stability. Sensitivity Analysis in Operations Research is a close surface but its accepted identity is post-optimality analysis of solved optimization models, so it is not the literal parent of a general numerical condition number.

Relationships to Other Abstractions

Current abstraction Condition Number Domain-specific

Parents (1) — outgoing typed relations

• Condition Number is a kind of Perturbation Prime

  Perturbation is the strict parent by composition: a condition number presupposes a specified small departure from a reference input and evaluates the leading response.

Hierarchy paths (2) — routes to 2 parentless roots

• Condition Number → is a kind of Perturbation → presupposes Observability

Show alternative path (1)

Neighborhood in Abstraction Space

Condition Number sits in a sparse region of the domain-specific corpus (88^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

• Directional Derivative — 0.81
• Well-posed problem — 0.81
• Neuman–Sándor Mean — 0.80
• Error analysis (mathematics) — 0.80
• Signal Quantization — 0.79

Computed from structural-signature embeddings · 2026-10-25