nearest-orthogonal
At its defaults it draws ‖a − q‖_f for four orthogonal matrices, on an 8×8 matrix with κ = 10. The polar factor is 1.8554 from A. QR with its column signs fixed is 2.1265 — 15 per cent further. QR as Householder returns it, with 7 of 8 columns negated, is 3.8226, which is further than the best of two hundred orthogonal matrices drawn at random. The strip beneath the bars is those two hundred draws, whose best is 2.6928; the polar factor is to the left of all of them, which is the minimisation being checked rather than assumed.
nearest-orthogonal is one function in lib/figures/polar.js —
the polar factor — the nearest orthogonal matrix, and two ways to it without an svd. 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 polar factor is 1.8554 from A. QR with its column signs fixed is 2.1265 — 15 per cent further. QR as Householder returns it, with 7 of 8 columns negated, is 3.8226, which is further than the best of two hundred orthogonal matrices drawn at random. The strip beneath the bars is those two hundred draws, whose best is 2.6928; the polar factor is to the left of all of them, which is the minimisation being checked rather than assumed.
logKappa: 1
The arguments are the ones A reflection cannot stop being one 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 polar factor is 1.8554 from A. QR with its column signs fixed is 2.1265 — 15 per cent further. QR as Householder returns it, with 7 of 8 columns negated, is 3.8226, which is further than the best of two hundred orthogonal matrices drawn at random. The strip beneath the bars is those two hundred draws, whose best is 2.6928; the polar factor is to the left of all of them, which is the minimisation being checked rather than assumed.
logKappa: 4
The arguments are the ones The nearest orthogonal 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 polar factor is 2.5188 from A. QR with its column signs fixed is 2.8670 — 14 per cent further. QR as Householder returns it, with 5 of 8 columns negated, is 3.1329, which is further than the best of two hundred orthogonal matrices drawn at random. The strip beneath the bars is those two hundred draws, whose best is 2.7435; the polar factor is to the left of all of them, which is the minimisation being checked rather than assumed.
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.
10 distinct claims across 3 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.
the constructed matrix has κ = 10 — asserted 2 times
a conditioning between nearly orthogonal and badly stretched
a power of ten rather than an exponent literal
a size the two hundred random draws can afford
and it is at least as near as the sign-fixed QR factor
and nearer than the best of two hundred random orthogonal matrices
matmul shapes agree
the polar factor is orthogonal to rounding
while the unfixed factorisation reconstructs A perfectly
Against the rule
It draws a decomposition and prints its residual. It calls
qrHouseholder,
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 92
of 198 generators —
77 print a residual and
15 are exempt with a published reason;
106 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 reflection cannot stop being one
Householder QR holds orthogonality at 10⁻¹⁵ whatever the condition number of the matrix, and Gram–Schmidt does not. The reason is not that it is more careful. It is that its Q is built from unit vectors, and rounding a unit vector gives a different reflection rather than a broken one.
Orthogonality, measuredAn iteration that only multiplies
Newton's iteration for the polar factor needs an inverse every step. Newton–Schulz needs only matrix products — nothing that reads an entry, nothing that pivots — and it converges if and only if every singular value is below √3. At 1.73205 it converges and at 1.73206 it returns an orthogonal matrix that is not the answer, with a residual of 5·10⁻¹⁶ and nothing to say so.
Orthogonality, measuredOrthogonal is a number
"Q is orthogonal" is a claim about a measurable quantity, ‖QᵀQ − I‖, and on the eight-by-eight Hilbert matrix two standard algorithms return 10⁻¹⁵ and 1 for it. The one that returns 1 still reconstructs the matrix perfectly, which is why nothing warns you.
Eigenvalues, singular values, rankThe best approximation there is
The error of the best rank-k approximation is not bounded by the next singular value. It is equal to it. That is an unusually sharp theorem, and it makes the theorem itself usable as an independent check on the computation.
Orthogonality, measuredThe nearest orthogonal matrix
Every field that has to clean up a drifted rotation reaches for QR, and QR does not answer the question. The nearest orthogonal matrix is the orthogonal factor of the polar decomposition — nearer by about a tenth, and, more to the point, the same matrix whatever order the columns were written in. QR's answer changes completely.