Concept

Basis — where it appears

A set of vectors spanning a subspace, of which many exist for the same subspace. Which one is used decides nothing in exact arithmetic and decides whether a projection can be computed at all in floating point.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A basis that is the same subspace and not the same thing

The interpolation conditions are conditions on a subspace, so any basis of it will do. The one a derivation writes down reaches a condition number of 7.7·10⁹ by its eighth vector, and the rate at which it gets there is set by a number the user chose with no information.

reduction · moment matching

The basis nobody chose on purpose

A method that eliminates a constraint has to pick a basis for its null space, and every basis is correct. Their condition numbers are eight orders apart, the reduced problem inherits the square, and the choice is usually made by a one-line rule nobody thought of as a numerical decision.

orthogonality · null space basis

Named alongside it

The objects these essays reach for when they reach for this one.

Condition number Column pivoting Constrained minimisation Krylov subspace Moment matching Null space Null space basis Orthogonality Orthonormal basis Petrov–Galerkin Power method QR factorisation

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