singularity-tests
At its defaults it draws four candidate singularity tests on three matrices, at n = 40. A table of four quantities on three matrices. 0.1·I at n = 40 has a condition number of exactly 1 and a determinant of 10⁻⁴⁰; a matrix with κ = 10¹⁰ normalised to |det| = 1 has a determinant of one. The determinant and its nth root give the wrong verdict on both, the smallest singular value gives the wrong verdict on the normalised matrix because it carries the units, and only σ_min/σ_max is right on all three. A test is read as saying *singular* when its number is below 10⁻⁶; green cells agree with the true verdict and red ones do not.
singularity-tests is one function in lib/figures/determinant.js —
the determinant — the scalar that answers none of the questions it is asked. 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.
A table of four quantities on three matrices. 0.1·I at n = 40 has a condition number of exactly 1 and a determinant of 10⁻⁴⁰; a matrix with κ = 10¹⁰ normalised to |det| = 1 has a determinant of one. The determinant and its nth root give the wrong verdict on both, the smallest singular value gives the wrong verdict on the normalised matrix because it carries the units, and only σ_min/σ_max is right on all three. A test is read as saying *singular* when its number is below 10⁻⁶; green cells agree with the true verdict and red ones do not.
n: 40
The arguments are the ones An estimate that can be fooled 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.
A table of four quantities on three matrices. 0.1·I at n = 40 has a condition number of exactly 1 and a determinant of 10⁻⁴⁰; a matrix with κ = 10¹⁰ normalised to |det| = 1 has a determinant of one. The determinant and its nth root give the wrong verdict on both, the smallest singular value gives the wrong verdict on the normalised matrix because it carries the units, and only σ_min/σ_max is right on all three. A test is read as saying *singular* when its number is below 10⁻⁶; green cells agree with the true verdict and red ones do not.
n: 120
The arguments are the ones The number that decides nothing 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.
A table of four quantities on three matrices. 0.1·I at n = 120 has a condition number of exactly 1 and a determinant of 10⁻¹²⁰; a matrix with κ = 10¹⁰ normalised to |det| = 1 has a determinant of one. The determinant and its nth root give the wrong verdict on both, the smallest singular value gives the wrong verdict on the normalised matrix because it carries the units, and only σ_min/σ_max is right on all three. A test is read as saying *singular* when its number is below 10⁻⁶; green cells agree with the true verdict and red ones do not.
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.
a multiple of the identity is perfectly conditioned at every size
a scaling below one, so the determinant underflows rather than overflows
a size the SVD of the small cases can afford
and Hilbert is the case where both numbers agree that something is wrong
and is ill-conditioned by ten orders of magnitude
and its determinant is exactly cⁿ
LU is for square matrices
matmul shapes agree
the constructed matrix has κ = 10000000000
the normalised matrix has a determinant of one
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.
An estimate that can be fooled
Nobody computes a condition number, because forming an inverse costs more than the solve did. Every library estimates it instead, from four or five products with a factorisation already in hand. The estimate is exactly right on four random matrices out of five — and there is a matrix, three distinct entries wide, on which it returns a twentieth of the truth.
Eigenvalues, singular values, rankRank is a decision
A floating-point matrix does not have a rank. It has a spectrum of singular values, and somewhere in that spectrum is a place where the values stop being signal and start being noise. Deciding where is a judgement, and the evidence for it is a gap.
Eigenvalues, singular values, rankThe cheap rank and what it cannot see
Almost nobody computes singular values to decide a rank. The standard substitute is QR with column pivoting, read off the diagonal of R — and there is a triangular matrix on which the greedy rule makes no interchange at all, has no better column available at any step, and reports a matrix eight orders of magnitude further from singular than it is.
Two errors, and whose fault they areThe condition number is an amplifier
κ is usually introduced as a definition and then quoted. It is a measurement: perturb the input by a known amount, look at how much the output moves, and the largest ratio you can find is the number.
Two errors, and whose fault they areThe number that decides nothing
The determinant is the first scalar anybody attaches to a matrix and the last one worth consulting. A tenth of the identity has a determinant of 10⁻⁶⁰ and a condition number of exactly one. The Hilbert matrix's determinant stops being right at n = 13 and stops being a number at n = 29, and nothing in between reports either.
Two errors, and whose fault they areThe units the matrix is measured in
One linear system, written twice. The rows of the second are the rows of the first in different units, the solution is identical to the last bit, and the condition number has moved by eight orders of magnitude. One of those two numbers is a fact about the problem and the other is a fact about the notation.
Regularisation, and the answer that is chosenWhen the answer is a choice
A backward-stable least-squares solve of this problem returns an answer whose relative error is 5.5·10⁸. Nothing went wrong. The singular values decay exponentially with no gap anywhere in them, the data does not determine the answer, and something outside the data has to choose — which is the computation rather than a preliminary to it.
Regularisation, and the answer that is chosenWhere the answer stops being in the data
The Picard condition finds the index where a noisy right-hand side stops carrying signal, from the data alone, with no knowledge of the answer. It lands at 32 where the truncation that actually minimises the error is 28 — and at 45 where the best is 38. It overshoots at every noise level from 1% to 0.001%, and it overshoots for a reason.