orthogonalisation-order
At its defaults it draws reversing the columns of a 6×6 matrix: what each orthogonalisation returns. Both grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is: ‖U₁P − U₂‖_F = 1.53·10⁻¹⁵. QR's is not: ‖Q₁P − Q₂‖_F = 2.353 on matrices whose own Frobenius norm is 2.449, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.
orthogonalisation-order 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.
Both grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is: ‖U₁P − U₂‖_F = 1.53·10⁻¹⁵. QR's is not: ‖Q₁P − Q₂‖_F = 2.353 on matrices whose own Frobenius norm is 2.449, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.
n: 6
The arguments are the ones Exact along one axis 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.
Both grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is: ‖U₁P − U₂‖_F = 1.53·10⁻¹⁵. QR's is not: ‖Q₁P − Q₂‖_F = 2.353 on matrices whose own Frobenius norm is 2.449, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.
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.
8 distinct claims across 2 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 conditioning the construction covers
a power of ten rather than an exponent literal
a size a grid of differences can print
and QR's is a different matrix
by a fraction of its own size rather than a perturbation
matmul shapes agree
the constructed matrix has κ = 10
the polar factor of a column-permuted matrix is the permuted polar factor
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
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.
Exact along one axis
The tuned diffusion makes the answer exact at every node, and in two dimensions it holds at exactly one flow angle. Five degrees off the grid the relative error goes from 1.2·10⁻¹⁴ to 6.9, and by twenty degrees the scheme is worse than the upwinding it was built to improve on.
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.
Iterating, instead of factorisingThe direction the diffusion does not go
Streamline diffusion adds τbbᵀ, a rank-one tensor that annihilates every direction across the flow. That is the design. The price is 18, 50 and 48 nodes where the computed solution leaves the interval the equation guarantees — and half a coefficient of crosswind diffusion halves the error at every angle while costing exactly nothing where the scheme was exact.
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.
Orthogonality, measuredTwo Gram–Schmidts
One argument changes. Classical Gram–Schmidt projects the original column onto each previous direction; modified projects what is left of it. In exact arithmetic the coefficients are identical. In floating point they differ by eight orders of magnitude in the thing that matters.