Numerical stability

Numerical stability describes how well an algorithm limits the effects of rounding errors in finite-precision computation. A stable algorithm produces results close to the exact solution of a nearby problem.

Backward error analysis: An analysis method that asks whether a computed result exactly solves a slightly perturbed input problem. It gives a precise test for whether an algorithm’s rounding errors amount to a small input perturbation.
Numerical conditioning: The sensitivity of a mathematical problem’s solution to small changes in its input. A poorly conditioned problem can defeat accuracy even when its algorithm is stable.
Gaussian elimination: A sequence of row operations used to solve systems of linear equations or factor matrices. Pivoting variants illustrate how algorithm design can control rounding-error growth.
Round-off error: The difference introduced when a number is rounded to a finite-precision representation. It is the perturbation whose propagation defines the central concern of numerical stability.
Forward error: The difference between a computed result and the exact result for the stated input. It measures the output discrepancy that numerical stability seeks to control.
Numerical instability: A tendency of an algorithm to amplify rounding errors into large computational errors. It names the failure that numerical stability is intended to avoid.
LU decomposition: A factorization of a matrix into lower- and upper-triangular factors. Its computed factors can be judged by whether they represent a nearby matrix accurately.
Truncation error: The error caused by replacing an exact mathematical process with a finite approximation. It differs from rounding error, though both can affect a numerical algorithm’s result.
Condition number: A measure of how strongly a problem’s solution changes when its input changes slightly. It separates sensitivity inherent in the problem from error introduced by the algorithm.
Exact arithmetic: Arithmetic that represents and manipulates values without rounding approximation. It removes finite-precision rounding, the main error source addressed by stability analysis.

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