Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

log₁₀|det Hₙ| by three routes, to n = 30

One function in the determinant library, called 7 times across 3 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 4 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

At its defaults it draws log₁₀|det hₙ| 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.

log₁₀|det Hₙ| by three routes, to n = 30The 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.26101418222630-550-450-350-250-150-50nlog₁₀|det Hₙ|exact rationalΣ log|uᵢᵢ|Π uᵢᵢthree routes, one theoremexact at the largest n-518accumulated logarithm-352decades of disagreement166smallest pivot at that n1.2·10⁻¹⁷every pivot is a normal numberat every size on this axis

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: 14

The arguments are the ones An answer that is known passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

log₁₀|det Hₙ| by three routes, to n = 14The exact rational determinant of the Hilbert matrix falls to 10^-107 at n = 14. The accumulated logarithm of the pivots follows it to n = 13 and then departs, reaching 10^-107 — wrong by 1 orders of magnitude and still an ordinary-looking number. The product of the pivots is a separate curve that ends at n = 14, where it underflows to exactly zero.2468101214-150-100-500nlog₁₀|det Hₙ|exact rationalΣ log|uᵢᵢ|Π uᵢᵢthree routes, one theoremexact at the largest n-107accumulated logarithm-107decades of disagreement0.53smallest pivot at that n2·10⁻¹⁷every pivot is a normal numberat every size on this axis

The exact rational determinant of the Hilbert matrix falls to 10^-107 at n = 14. The accumulated logarithm of the pivots follows it to n = 13 and then departs, reaching 10^-107 — wrong by 1 orders of magnitude and still an ordinary-looking number. The product of the pivots is a separate curve that ends at n = 14, where it underflows to exactly zero.

nMax: 22

The arguments are the ones An answer that is known passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

log₁₀|det Hₙ| by three routes, to n = 22The 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.25811141720-300-250-200-150-100-500nlog₁₀|det Hₙ|exact rationalΣ log|uᵢᵢ|Π uᵢᵢthree routes, one theoremexact at the largest n-274accumulated logarithm-225decades of disagreement49smallest pivot at that n4·10⁻¹⁷every pivot is a normal numberat every size on this axis

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.

nMax: 30

The arguments are the ones An answer that is known passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

log₁₀|det Hₙ| by three routes, to n = 30The 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.26101418222630-550-450-350-250-150-50nlog₁₀|det Hₙ|exact rationalΣ log|uᵢᵢ|Π uᵢᵢthree routes, one theoremexact at the largest n-518accumulated logarithm-352decades of disagreement166smallest pivot at that n1.2·10⁻¹⁷every pivot is a normal numberat every size on this axis

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: 18

The arguments are the ones An answer that is known passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

log₁₀|det Hₙ| by three routes, to n = 18The exact rational determinant of the Hilbert matrix falls to 10^-181 at n = 18. The accumulated logarithm of the pivots follows it to n = 13 and then departs, reaching 10^-166 — wrong by 15 orders of magnitude and still an ordinary-looking number. The product of the pivots is a separate curve that ends at n = 18, where it underflows to exactly zero.258111417-200-150-100-500nlog₁₀|det Hₙ|exact rationalΣ log|uᵢᵢ|Π uᵢᵢthree routes, one theoremexact at the largest n-181accumulated logarithm-166decades of disagreement15smallest pivot at that n1.5·10⁻¹⁸every pivot is a normal numberat every size on this axis

The exact rational determinant of the Hilbert matrix falls to 10^-181 at n = 18. The accumulated logarithm of the pivots follows it to n = 13 and then departs, reaching 10^-166 — wrong by 15 orders of magnitude and still an ordinary-looking number. The product of the pivots is a separate curve that ends at n = 18, where it underflows to exactly zero.

nMax: 26

The arguments are the ones An answer that is known passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

log₁₀|det Hₙ| by three routes, to n = 26The 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.261014182226-400-300-200-1000nlog₁₀|det Hₙ|exact rationalΣ log|uᵢᵢ|Π uᵢᵢthree routes, one theoremexact at the largest n-387accumulated logarithm-289decades of disagreement98smallest pivot at that n3.1·10⁻¹⁸every pivot is a normal numberat every size on this axis

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.

What it checked while drawing

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks 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 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 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 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 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.

Two errors, and whose fault they are

An 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.

 Eigenvalues, singular values, rank

Small compared to what

This site's own singular value routine has carried a sentence since the month it was written — that one-sided Jacobi computes the small singular values to high relative accuracy and the standard method does not. It has never been measured here, because measuring it needs a σ that is known rather than computed. A bidiagonal matrix and a Sturm count in exact rationals supply one.

 Two errors, and whose fault they are

The 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 whole library · All essays · What must fail