Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Petrov–Galerkin — where it appears

A projection using two different subspaces, one to build the reduced object and one to test against. For a reduced model the second subspace buys the derivative conditions that a one-sided projection does not.

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

10⁻¹110¹10²10³10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵s, on the real axis|H − Hᵣ| ÷ |H|exact where askedpoints4conditions bought8worst at a point5.3·10⁻¹⁶worst away from one8.4·10⁻⁴4 points, 8 conditionsand no bound in between

Exact at the points that were named

Balanced truncation asks for nothing and bounds everything, at a cost no large model can pay. The other kind of reduction asks for r numbers, costs r solves, is exact at every one of them — and bounds nothing anywhere else. That trade is the whole of large-scale model reduction.

reduction · moment matching 2345678110²10⁴10⁶10⁸10¹⁰vectors in the basisκ₂ of the basisas derivedsolves spread outone subspace, two spanning setseight moments at one point7.7·10⁹eight points, spread1growth per vector661/u4.5·10¹⁵the same subspaceand only one of them usable

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

Named alongside it

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

Condition numberKrylov subspaceMoment matchingRational krylovA-priori boundBalanced truncationBasisFlop countHermite interpolationOrthogonalityPower methodReorthogonalisation

All concepts