The constraint preconditioner's spectrum, over six decades of the constraint's condition number

At its defaults it draws the constraint preconditioner's spectrum, over six decades of the constraint's condition number. P = [[G, Aᵀ], [A, 0]] with G an approximation to H — here a well-conditioned matrix that is not H at all. The theorem says P⁻¹K has eigenvalue 1 with multiplicity 2m = 8 and n − m = 6 others, which are the generalised eigenvalues of the pencil (ZᵀHZ, ZᵀGZ) for any basis Z of the null space of A. A does not appear in that list, and the measurement is the flat lines: κ(A) crosses six decades along the horizontal axis and the 6 eigenvalues move by 7.14·10⁻⁶ relative, which is the arithmetic. A preconditioner for a constrained problem can decline to know anything about the constraint, because the constraint has already been inverted exactly inside it. The one quantity that does move is the drift of the eigenvalues the theorem puts at exactly one: 6.65·10⁻⁹ at κ(A) = 1 and 6.04·10⁻⁵ at 10⁶, which is κ(A) times the unit roundoff.

constraint-spectrum is one function in lib/figures/blockprec.js — preconditioning a saddle point — three eigenvalues, and a spectrum with no constraint in it. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

P = [[G, Aᵀ], [A, 0]] with G an approximation to H — here a well-conditioned matrix that is not H at all. The theorem says P⁻¹K has eigenvalue 1 with multiplicity 2m = 8 and n − m = 6 others, which are the generalised eigenvalues of the pencil (ZᵀHZ, ZᵀGZ) for any basis Z of the null space of A. A does not appear in that list, and the measurement is the flat lines: κ(A) crosses six decades along the horizontal axis and the 6 eigenvalues move by 7.14·10⁻⁶ relative, which is the arithmetic. A preconditioner for a constrained problem can decline to know anything about the constraint, because the constraint has already been inverted exactly inside it. The one quantity that does move is the drift of the eigenvalues the theorem puts at exactly one: 6.65·10⁻⁹ at κ(A) = 1 and 6.04·10⁻⁵ at 10⁶, which is κ(A) times the unit roundoff.

P = [[G, Aᵀ], [A, 0]] with G an approximation to H — here a well-conditioned matrix that is not H at all. The theorem says P⁻¹K has eigenvalue 1 with multiplicity 2m = 8 and n − m = 6 others, which are the generalised eigenvalues of the pencil (ZᵀHZ, ZᵀGZ) for any basis Z of the null space of A. A does not appear in that list, and the measurement is the flat lines: κ(A) crosses six decades along the horizontal axis and the 6 eigenvalues move by 7.14·10⁻⁶ relative, which is the arithmetic. A preconditioner for a constrained problem can decline to know anything about the constraint, because the constraint has already been inverted exactly inside it. The one quantity that does move is the drift of the eigenvalues the theorem puts at exactly one: 6.65·10⁻⁹ at κ(A) = 1 and 6.04·10⁻⁵ at 10⁶, which is κ(A) times the unit roundoff.

P = [[G, Aᵀ], [A, 0]] with G an approximation to H — here a well-conditioned matrix that is not H at all. The theorem says P⁻¹K has eigenvalue 1 with multiplicity 2m = 8 and n − m = 6 others, which are the generalised eigenvalues of the pencil (ZᵀHZ, ZᵀGZ) for any basis Z of the null space of A. A does not appear in that list, and the measurement is the flat lines: κ(A) crosses six decades along the horizontal axis and the 6 eigenvalues move by 1.64·10⁻⁶ relative, which is the arithmetic. A preconditioner for a constrained problem can decline to know anything about the constraint, because the constraint has already been inverted exactly inside it. The one quantity that does move is the drift of the eigenvalues the theorem puts at exactly one: 2.57·10⁻⁹ at κ(A) = 1 and 3.3·10⁻⁴ at 10⁶, which is κ(A) times the unit roundoff.

P = [[G, Aᵀ], [A, 0]] with G an approximation to H — here a well-conditioned matrix that is not H at all. The theorem says P⁻¹K has eigenvalue 1 with multiplicity 2m = 8 and n − m = 6 others, which are the generalised eigenvalues of the pencil (ZᵀHZ, ZᵀGZ) for any basis Z of the null space of A. A does not appear in that list, and the measurement is the flat lines: κ(A) crosses six decades along the horizontal axis and the 6 eigenvalues move by 3.02·10⁻⁶ relative, which is the arithmetic. A preconditioner for a constrained problem can decline to know anything about the constraint, because the constraint has already been inverted exactly inside it. The one quantity that does move is the drift of the eigenvalues the theorem puts at exactly one: 1.93·10⁻⁸ at κ(A) = 1 and 3.11·10⁻⁵ at 10⁶, which is κ(A) times the unit roundoff.

P = [[G, Aᵀ], [A, 0]] with G an approximation to H — here a well-conditioned matrix that is not H at all. The theorem says P⁻¹K has eigenvalue 1 with multiplicity 2m = 8 and n − m = 6 others, which are the generalised eigenvalues of the pencil (ZᵀHZ, ZᵀGZ) for any basis Z of the null space of A. A does not appear in that list, and the measurement is the flat lines: κ(A) crosses six decades along the horizontal axis and the 6 eigenvalues move by 1.17·10⁻⁵ relative, which is the arithmetic. A preconditioner for a constrained problem can decline to know anything about the constraint, because the constraint has already been inverted exactly inside it. The one quantity that does move is the drift of the eigenvalues the theorem puts at exactly one: 5.85·10⁻⁹ at κ(A) = 1 and 1.94·10⁻⁵ at 10⁶, which is κ(A) times the unit roundoff.

P = [[G, Aᵀ], [A, 0]] with G an approximation to H — here a well-conditioned matrix that is not H at all. The theorem says P⁻¹K has eigenvalue 1 with multiplicity 2m = 6 and n − m = 5 others, which are the generalised eigenvalues of the pencil (ZᵀHZ, ZᵀGZ) for any basis Z of the null space of A. A does not appear in that list, and the measurement is the flat lines: κ(A) crosses six decades along the horizontal axis and the 5 eigenvalues move by 3.84·10⁻⁹ relative, which is the arithmetic. A preconditioner for a constrained problem can decline to know anything about the constraint, because the constraint has already been inverted exactly inside it. The one quantity that does move is the drift of the eigenvalues the theorem puts at exactly one: 1.77·10⁻⁸ at κ(A) = 1 and 1.22·10⁻⁷ at 10⁶, which is κ(A) times the unit roundoff.

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

11 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

It draws a decomposition and prints its residual. It calls constraintRun, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

Across the library: the rule bites on 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.