Condition number

A condition number measures how strongly a problem’s output can change in response to small changes in its input. A large value signals sensitivity; a small value signals stability.

Relative error: The magnitude of an error divided by the magnitude of the quantity being measured. Relative changes in input and output define the usual condition number.
Matrix condition number: A measure of how sensitively a linear system’s solution responds to perturbations in its matrix or right-hand side. It is the standard condition number used to assess linear-system sensitivity.
Numerical stability: A property of an algorithm whose computed results remain close to exact results for nearby data. Conditioning belongs to the problem; stability describes how an algorithm handles it.
Condition number of a function: The local relative sensitivity of a function’s output to relative changes in its input. This definition provides the general model behind problem-specific condition numbers.
Perturbation theory: The study of how solutions change when a mathematical problem is slightly modified. Condition numbers summarize the sensitivity that perturbation analysis investigates.
Linear system: A set of linear equations whose unknowns must satisfy all equations simultaneously. Solving linear systems is a central setting where conditioning predicts solution sensitivity.
Backward error: The smallest input perturbation that makes a computed result exact. It helps distinguish a stable computation from an ill-conditioned problem.
Differential calculus: The branch of mathematics that studies derivatives and local rates of change. Derivatives yield local condition numbers for smooth functions.
Jacobian matrix: A matrix of first partial derivatives that describes how a multivariable function changes locally. Its norm measures local sensitivity for differentiable problems.
Least squares: A method that estimates unknowns by minimizing the sum of squared residuals. The design matrix’s condition number reveals sensitivity in fitted coefficients.

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