determinant-sweep
At its defaults it draws log₁₀|det h_n| by three routes, to n = 30. The exact rational determinant of the Hilbert matrix falls to 10^-518 at n = 30. The accumulated logarithm of the pivots follows it to n = 13 and then departs, reaching 10^-352 — wrong by 166 orders of magnitude and still an ordinary-looking number. The product of the pivots is a separate curve that ends at n = 29, where it underflows to exactly zero.
determinant-sweep 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.
The exact rational determinant of the Hilbert matrix falls to 10^-518 at n = 30. The accumulated logarithm of the pivots follows it to n = 13 and then departs, reaching 10^-352 — wrong by 166 orders of magnitude and still an ordinary-looking number. The product of the pivots is a separate curve that ends at n = 29, where it underflows to exactly zero.
nMax: 30
The arguments are the ones A factorisation with nothing to pivot for 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 exact rational determinant of the Hilbert matrix falls to 10^-518 at n = 30. The accumulated logarithm of the pivots follows it to n = 13 and then departs, reaching 10^-352 — wrong by 166 orders of magnitude and still an ordinary-looking number. The product of the pivots is a separate curve that ends at n = 29, where it underflows to exactly zero.
nMax: 26
The arguments are the ones A norm that overflows before it is a norm 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 exact rational determinant of the Hilbert matrix falls to 10^-387 at n = 26. The accumulated logarithm of the pivots follows it to n = 13 and then departs, reaching 10^-289 — wrong by 98 orders of magnitude and still an ordinary-looking number. The product of the pivots is a separate curve that ends at n = 26, where it underflows to exactly zero.
nMax: 22
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.
The exact rational determinant of the Hilbert matrix falls to 10^-274 at n = 22. The accumulated logarithm of the pivots follows it to n = 13 and then departs, reaching 10^-225 — wrong by 49 orders of magnitude and still an ordinary-looking number. The product of the pivots is a separate curve that ends at n = 22, where it underflows to exactly zero.
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.
4 distinct claims across 4 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 size the exact rational determinant is affordable at
and they do so before the product stops being representable
LU is for square matrices
the float routes depart from the exact determinant inside the sweep
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.
A factorisation with nothing to pivot for
Cholesky's growth factor is not bounded by one. It is equal to one, at every size and every condition number, and the two-line reason is why the algorithm needs no pivoting at all — not "usually gets away without it". Its only failure is the square root of a non-positive number, which is exactly the test for definiteness, and in floating point that test moves with the precision.
The arithmetic underneathA norm that overflows before it is a norm
The vector of sixteen thousands has a Euclidean norm of 4,000, which fp16 represents exactly. Written as the square root of the sum of squares it returns infinity, because squaring doubles the exponent — and the expression costs half the format's range on the one computation every iterative method performs at every step.
Two errors, and whose fault they areAn answer that is known
Almost every demonstration of numerical error estimates the error by computing the same thing more carefully. The Hilbert matrix does not need that: its inverse is a closed form in integers, so the true answer is available exactly and the error is measured rather than approximated.
Iterating, instead of factorisingAn operator with no entries
At the sizes where linear algebra is expensive the matrix does not exist. What exists is a subroutine that returns Av. Every Krylov method survives that unchanged; every algorithm that reads an entry disappears. And the derivative such a code computes is accurate to ten digits instead of sixteen, which turns out to cost nothing at all.
The arithmetic underneathOne exponent for thirty-two numbers
Share the exponent across a block and the cost per value drops from eight bits to 6.25, and the accuracy improves — up to about three octaves of spread inside a block. Past that a single outlier deletes the thirty-one values beside it, and the 2-norm barely notices.
Eigenvalues, singular values, rankThe algorithm the libraries actually run
Factorise, multiply the factors back in the other order, repeat. That description is complete and correct and produces something nobody would use — on a matrix with eigenvalues +1 and −1 it does not converge at all, and the subdiagonal entry does not move by so much as a rounding error.
Elimination, and the swapThe inverse that is never formed
x = A⁻¹b is how the solution of a linear system is written and it is not how it is computed. The usual reason given is cost — three times the arithmetic. The real reason is that one of the two routes is backward stable and the other is not, and at κ = 10¹⁴ they differ by twelve orders of magnitude in the number that says whose fault a wrong answer is.
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.
The arithmetic underneathThe numbers below the smallest one
Below the smallest normal number the spacing stops halving and stays put, all the way to zero. That is what gradual underflow is, and the thing it buys is the sentence every algorithm assumes without being told — x minus y is zero only when x equals y.
The arithmetic underneathThe other half of a format
fp16 and tf32 have the same eleven significand bits and their largest numbers are 65,504 and 3.4·10³⁸. For two phases this site simulated the significand alone, so it was obliged to report them as the same format — which is a claim, and a false one.
The arithmetic underneathWhat a float can hold
The representable numbers are not a fine fuzz spread evenly over the line. They are evenly spaced inside each power-of-two interval and twice as far apart in the next one up, and almost everything else in this subject is a consequence of that one fact.