Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Four candidate singularity tests on three matrices, at n = 40

One function in the determinant library, called 6 times across 1 essay. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 10 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 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 σₘᵢₙ/σₘₐₓ 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.

Four candidate singularity tests on three matrices, at n = 40A 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 σₘᵢₙ/σₘₐₓ 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.does this matrix look nearly singular?green: the test agrees with the truth · red: it does not · the bar under each number is its magnitude, over sixty-two decades|det A||det A|^(1/n)σₘᵢₙ1/κ = σₘᵢₙ/σₘₐₓ0.1·I at n = 40perfectly conditioned10⁻⁴⁰0.10.11κ = 10¹⁰, |det| = 1nearly singular1110·10⁻⁶10·10⁻¹¹Hilbert at n = 8nearly singular2.7·10⁻³³8.5·10⁻⁵1.1·10⁻¹⁰6.6·10⁻¹¹the two counterexamplesκ of the scaled identity1its determinant10⁻⁴⁰κ of the normalised matrix10¹⁰its determinant1det(cA) = cⁿ det(A)so a determinant carries the units n times over

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 σₘᵢₙ/σₘₐₓ 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 The number that decides nothing passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Four candidate singularity tests on three matrices, at n = 40A 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 σₘᵢₙ/σₘₐₓ 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.does this matrix look nearly singular?green: the test agrees with the truth · red: it does not · the bar under each number is its magnitude, over sixty-two decades|det A||det A|^(1/n)σₘᵢₙ1/κ = σₘᵢₙ/σₘₐₓ0.1·I at n = 40perfectly conditioned10⁻⁴⁰0.10.11κ = 10¹⁰, |det| = 1nearly singular1110·10⁻⁶10·10⁻¹¹Hilbert at n = 8nearly singular2.7·10⁻³³8.5·10⁻⁵1.1·10⁻¹⁰6.6·10⁻¹¹the two counterexamplesκ of the scaled identity1its determinant10⁻⁴⁰κ of the normalised matrix10¹⁰its determinant1det(cA) = cⁿ det(A)so a determinant carries the units n times over

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 σₘᵢₙ/σₘₐₓ 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 nobody placed would be a picture no essay asked for and no claim was ever checked against.

Four candidate singularity tests on three matrices, at n = 120A 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 σₘᵢₙ/σₘₐₓ 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.does this matrix look nearly singular?green: the test agrees with the truth · red: it does not · the bar under each number is its magnitude, over sixty-two decades|det A||det A|^(1/n)σₘᵢₙ1/κ = σₘᵢₙ/σₘₐₓ0.1·I at n = 120perfectly conditioned10⁻¹²⁰0.10.11κ = 10¹⁰, |det| = 1nearly singular1110·10⁻⁶10·10⁻¹¹Hilbert at n = 8nearly singular2.7·10⁻³³8.5·10⁻⁵1.1·10⁻¹⁰6.6·10⁻¹¹the two counterexamplesκ of the scaled identity1its determinant10⁻¹²⁰κ of the normalised matrix10¹⁰its determinant1det(cA) = cⁿ det(A)so a determinant carries the units n times over

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 σₘᵢₙ/σₘₐₓ 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: 10

The arguments are the ones The number that decides nothing passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Four candidate singularity tests on three matrices, at n = 10A table of four quantities on three matrices. 0.1·I at n = 10 has a condition number of exactly 1 and a determinant of 10⁻¹⁰; 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 σₘᵢₙ/σₘₐₓ 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.does this matrix look nearly singular?green: the test agrees with the truth · red: it does not · the bar under each number is its magnitude, over sixty-two decades|det A||det A|^(1/n)σₘᵢₙ1/κ = σₘᵢₙ/σₘₐₓ0.1·I at n = 10perfectly conditioned10⁻¹⁰0.10.11κ = 10¹⁰, |det| = 1nearly singular1110·10⁻⁶10·10⁻¹¹Hilbert at n = 8nearly singular2.7·10⁻³³8.5·10⁻⁵1.1·10⁻¹⁰6.6·10⁻¹¹the two counterexamplesκ of the scaled identity1its determinant10⁻¹⁰κ of the normalised matrix10¹⁰its determinant1det(cA) = cⁿ det(A)so a determinant carries the units n times over

A table of four quantities on three matrices. 0.1·I at n = 10 has a condition number of exactly 1 and a determinant of 10⁻¹⁰; 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 σₘᵢₙ/σₘₐₓ 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: 20

The arguments are the ones The number that decides nothing passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Four candidate singularity tests on three matrices, at n = 20A table of four quantities on three matrices. 0.1·I at n = 20 has a condition number of exactly 1 and a determinant of 10⁻²⁰; 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 σₘᵢₙ/σₘₐₓ 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.does this matrix look nearly singular?green: the test agrees with the truth · red: it does not · the bar under each number is its magnitude, over sixty-two decades|det A||det A|^(1/n)σₘᵢₙ1/κ = σₘᵢₙ/σₘₐₓ0.1·I at n = 20perfectly conditioned10⁻²⁰0.10.11κ = 10¹⁰, |det| = 1nearly singular1110·10⁻⁶10·10⁻¹¹Hilbert at n = 8nearly singular2.7·10⁻³³8.5·10⁻⁵1.1·10⁻¹⁰6.6·10⁻¹¹the two counterexamplesκ of the scaled identity1its determinant10⁻²⁰κ of the normalised matrix10¹⁰its determinant1det(cA) = cⁿ det(A)so a determinant carries the units n times over

A table of four quantities on three matrices. 0.1·I at n = 20 has a condition number of exactly 1 and a determinant of 10⁻²⁰; 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 σₘᵢₙ/σₘₐₓ 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: 80

The arguments are the ones The number that decides nothing passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Four candidate singularity tests on three matrices, at n = 80A table of four quantities on three matrices. 0.1·I at n = 80 has a condition number of exactly 1 and a determinant of 10⁻⁸⁰; 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 σₘᵢₙ/σₘₐₓ 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.does this matrix look nearly singular?green: the test agrees with the truth · red: it does not · the bar under each number is its magnitude, over sixty-two decades|det A||det A|^(1/n)σₘᵢₙ1/κ = σₘᵢₙ/σₘₐₓ0.1·I at n = 80perfectly conditioned10⁻⁸⁰0.10.11κ = 10¹⁰, |det| = 1nearly singular1110·10⁻⁶10·10⁻¹¹Hilbert at n = 8nearly singular2.7·10⁻³³8.5·10⁻⁵1.1·10⁻¹⁰6.6·10⁻¹¹the two counterexamplesκ of the scaled identity1its determinant10⁻⁸⁰κ of the normalised matrix10¹⁰its determinant1det(cA) = cⁿ det(A)so a determinant carries the units n times over

A table of four quantities on three matrices. 0.1·I at n = 80 has a condition number of exactly 1 and a determinant of 10⁻⁸⁰; 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 σₘᵢₙ/σₘₐₓ 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 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.

10 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 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 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

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.

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