two-eliminations
At its defaults it draws what each way of eliminating a constraint inherits, over five decades of κ(a). The same system solved twice, at 8 unknowns and 3 constraints with κ(H) = 100. The range-space method forms S = AH⁻¹Aᵀ and inherits κ(S), which rises from 12.99 to 3.981·10¹⁰ — the square of κ(A), for the reason the normal equations square it. The null-space method solves with the reduced Hessian ZᵀHZ, whose condition number is 21.13 at the start of the sweep and 21.13 at the end: it does not contain κ(A) at all. The two forward errors, measured against a solution computed in BigInt rationals, follow their own condition numbers: 5.314·10⁻⁶ against 5.788·10⁻¹² at the far end. Both methods are correct and one of them is usable.
two-eliminations is one function in lib/figures/kkt.js —
the matrix a constraint makes — an inertia known before assembly, and a failure with an address. 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.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
The same system solved twice, at 8 unknowns and 3 constraints with κ(H) = 100. The range-space method forms S = AH⁻¹Aᵀ and inherits κ(S), which rises from 12.99 to 3.981·10¹⁰ — the square of κ(A), for the reason the normal equations square it. The null-space method solves with the reduced Hessian ZᵀHZ, whose condition number is 21.13 at the start of the sweep and 21.13 at the end: it does not contain κ(A) at all. The two forward errors, measured against a solution computed in BigInt rationals, follow their own condition numbers: 5.314·10⁻⁶ against 5.788·10⁻¹² at the far end. Both methods are correct and one of them is usable.
kappaH: 100
The arguments are the ones A condition number sent to infinity passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
The same system solved twice, at 8 unknowns and 3 constraints with κ(H) = 100. The range-space method forms S = AH⁻¹Aᵀ and inherits κ(S), which rises from 12.99 to 3.981·10¹⁰ — the square of κ(A), for the reason the normal equations square it. The null-space method solves with the reduced Hessian ZᵀHZ, whose condition number is 21.13 at the start of the sweep and 21.13 at the end: it does not contain κ(A) at all. The two forward errors, measured against a solution computed in BigInt rationals, follow their own condition numbers: 5.314·10⁻⁶ against 5.788·10⁻¹² at the far end. Both methods are correct and one of them is usable.
m: 2
The arguments are the ones A condition number sent to infinity passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
The same system solved twice, at 8 unknowns and 2 constraints with κ(H) = 100. The range-space method forms S = AH⁻¹Aᵀ and inherits κ(S), which rises from 11.43 to 1.127·10¹¹ — the square of κ(A), for the reason the normal equations square it. The null-space method solves with the reduced Hessian ZᵀHZ, whose condition number is 55.28 at the start of the sweep and 55.28 at the end: it does not contain κ(A) at all. The two forward errors, measured against a solution computed in BigInt rationals, follow their own condition numbers: 2.137·10⁻⁶ against 1.053·10⁻¹¹ at the far end. Both methods are correct and one of them is usable.
kappaH: 10
The arguments are the ones The basis nobody chose on purpose passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
The same system solved twice, at 8 unknowns and 3 constraints with κ(H) = 10. The range-space method forms S = AH⁻¹Aᵀ and inherits κ(S), which rises from 3.169 to 1.585·10¹⁰ — the square of κ(A), for the reason the normal equations square it. The null-space method solves with the reduced Hessian ZᵀHZ, whose condition number is 4.149 at the start of the sweep and 4.149 at the end: it does not contain κ(A) at all. The two forward errors, measured against a solution computed in BigInt rationals, follow their own condition numbers: 2.278·10⁻⁷ against 3.002·10⁻¹² at the far end. Both methods are correct and one of them is usable.
kappaH: 1
The arguments are the ones The zero that is not a missing entry passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
The same system solved twice, at 8 unknowns and 3 constraints with κ(H) = 1. The range-space method forms S = AH⁻¹Aᵀ and inherits κ(S), which rises from 1 to 10·10⁹ — the square of κ(A), for the reason the normal equations square it. The null-space method solves with the reduced Hessian ZᵀHZ, whose condition number is 1 at the start of the sweep and 1 at the end: it does not contain κ(A) at all. The two forward errors, measured against a solution computed in BigInt rationals, follow their own condition numbers: 7.341·10⁻⁷ against 2.025·10⁻¹² at the far end. Both methods are correct and one of them is usable.
kappaH: 1000000
The arguments are the ones Two condition numbers of one matrix passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
The same system solved twice, at 8 unknowns and 3 constraints with κ(H) = 10⁶. The range-space method forms S = AH⁻¹Aᵀ and inherits κ(S), which rises from 1700 to 1.118·10¹² — the square of κ(A), for the reason the normal equations square it. The null-space method solves with the reduced Hessian ZᵀHZ, whose condition number is 7646 at the start of the sweep and 7646 at the end: it does not contain κ(A) at all. The two forward errors, measured against a solution computed in BigInt rationals, follow their own condition numbers: 2.467·10⁻⁶ against 4.419·10⁻¹² at the far end. Both methods are correct and one of them is usable.
What it checked while drawing
Every figure above asserted its own claims on the way to being drawn, and a claim that failed
would have failed the build rather than drawn a wrong picture. Those assertions 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.
9 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.
a constraint no larger than the problem
a finite double, since an infinity is not a rational
a size the exact rational solve can afford
and κ(ZᵀHZ) does not move at all
fewer constraints than unknowns
fewer constraints than unknowns, so something is left to minimise
LU is for square matrices
matmul shapes agree
κ(S) grows like κ(A)² across the sweep
Against the rule
It draws a decomposition and prints its residual. It calls
rangeSpaceSolve, nullSpaceSolve, orthonormalNullSpace, exactKkt,
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 165
of 306 generators —
150 print a residual and
15 are exempt with a published reason;
141 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.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A condition number sent to infinity
An interior-point method manufactures an ill-conditioned matrix on every iteration, deliberately, because the separating of a diagonal is how it discovers which constraints are active. Written one way the answer keeps fifteen digits at a condition number of 3·10¹⁵. Written the other way — the way almost every code writes it — it has none left.
Least squares, and the road not to takeA constraint is a weight at infinity
Stack an equality constraint on top of a least-squares problem with a large weight and the answer approaches the constrained one like 1/τ². The limit is takeable to any accuracy — and how far it can be taken is a property of the solver, not of the problem. One of them stops at the square root of the precision, and one of them does not stop.
The matrix a constraint makesA preconditioner that need not know the constraint
Keep the constraint block exactly and replace the objective block by anything positive definite on the null space. The preconditioned matrix then has 2m eigenvalues at exactly one, and its remaining n − m are the generalised eigenvalues of a pencil in which the constraint does not appear. Sweep its condition number over six decades and they do not move in six digits.
Eigenvalues, singular values, rankAn eigenvalue count that cannot be slightly wrong
Every spectral computation on this site returns floats with errors in them. Counting eigenvalues below a shift by the signs of an unpivoted elimination returns an integer, and an integer cannot be 6.9999999997 — so the answer is exactly right, or wrong by a whole eigenvalue, and where the second happens is a band of measurable width.
Sparsity, and what elimination costsAn ordering that does not wait for the numbers
A sparse factorisation's memory is decided by an ordering computed from the graph, and its stability by pivots computed from the values, and the two decisions fight. On one family of matrices they do not — the ordering can be chosen for fill alone, and the fill the symbolic phase predicts is the fill the factorisation produces — exactly, not as a bound.
Orthogonality, measuredThe 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.
Randomised, and the guarantee that changes kindThe half of a problem a sketch may touch
A sketch guarantees that a norm is preserved to within a factor. An equality constraint is a statement that a quantity is zero, and no multiplicative guarantee says anything about zero. Sketch a constrained problem written as a weighted one and the constraint is not destroyed — it is demoted, from a violation of 1/τ² to one of ε/τ, exactly half the exponent.
The matrix a constraint makesThe regularisation that legalises every order
Perturb a saddle-point matrix's two blocks in opposite directions and it acquires a factorisation with a diagonal D under every symmetric permutation — not under a good one, under all of them. Five hundred random orderings, five hundred successes, and a growth factor that spans six orders across them.
The matrix a constraint makesThe zero that is not a missing entry
A constrained minimisation produces a matrix with a zero block, and the zero is a theorem rather than a sparsity pattern. No pivot order makes it positive definite, no precision changes that, and Cholesky does not fail somewhere on it — it fails at the first constraint row, on a number the problem already contained.
The matrix a constraint makesThree eigenvalues, and two are the golden ratio
Precondition a saddle-point system by the block diagonal of its own two definite pieces and the preconditioned matrix has exactly three distinct eigenvalues — 1, and the two roots of λ² − λ − 1. A minimal polynomial of degree three means three steps, at every conditioning, and the preconditioner nobody can afford turns out to be the statement the affordable ones are measured against.
Two errors, and whose fault they areTwo condition numbers of one matrix
κ₂ is a worst case over perturbations of a given norm, and a normwise perturbation may put its whole budget on the smallest entry. The componentwise number is a worst case over perturbations proportional to the entries, which is what a backward-stable factorisation actually makes. On one matrix they are 3·10¹³ and 13.3, and the error obeys the second.
The matrix a constraint makesTwo ways to remove a constraint
A constrained system can be reduced by eliminating the multipliers or by eliminating the constrained directions. Both give the same answer in exact arithmetic and inherit different condition numbers — one of them squares the constraint's, and the other does not contain it at all.
When the problem arrives againWhat survives one step of the barrier
An interior-point method solves the same system dozens of times with the same pattern and different numbers, and exactly p entries change between one step and the next. The pattern is reusable for ever. The factorisation is reusable for none of them, and the threshold that says so is a reduction factor of about a per cent against schedules that use ten.