Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

What governs a total least-squares problem: the gap σₙ(A) − σₙ₊₁([A b]), not κ(A)

One function in the tls library, called 3 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 16 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 what governs a total least-squares problem: the gap σₙ(a) − σₙ₊₁([a b]), not κ(a). Three curves against the noise level, both axes logarithmic, as medians over 12 seeds of a 60×3 fit. The gap between the smallest singular value of A and the smallest of the augmented matrix closes from 2.1 to 0.108, and the total least-squares error rises with it from 0.0067 to 1.69 — a factor of 252. κ(A) moves from 3.66 to 2.48 across the same sweep and predicts none of it.

tls-gap is one function in lib/figures/tls.js — errors in the matrix too — where the residual orders two methods backwards. 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.

What governs a total least-squares problem: the gap σₙ(A) − σₙ₊₁([A b]), not κ(A)Three curves against the noise level, both axes logarithmic, as medians over 12 seeds of a 60×3 fit. The gap between the smallest singular value of A and the smallest of the augmented matrix closes from 2.1 to 0.108, and the total least-squares error rises with it from 0.0067 to 1.69 — a factor of 252. κ(A) moves from 3.66 to 2.48 across the same sweep and predicts none of it.10⁻¹10⁻³10⁻²10⁻¹110¹noise level, relative to the datagap, error, and condition numberκ(A)TLS errorthe gapone of these predicts the errorgap, at the least noise2.1gap, at the most0.11error, ratio across the sweep252κ(A), ratio across the sweep1.5the condition number is nearly flatand the error moves by two orders

Three curves against the noise level, both axes logarithmic, as medians over 12 seeds of a 60×3 fit. The gap between the smallest singular value of A and the smallest of the augmented matrix closes from 2.1 to 0.108, and the total least-squares error rises with it from 0.0067 to 1.69 — a factor of 252. κ(A) moves from 3.66 to 2.48 across the same sweep and predicts none of it.

m: 24

The arguments are the ones The two numbers a caller has passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

What governs a total least-squares problem: the gap σₙ(A) − σₙ₊₁([A b]), not κ(A)Three curves against the noise level, both axes logarithmic, as medians over 12 seeds of a 24×3 fit. The gap between the smallest singular value of A and the smallest of the augmented matrix closes from 1.4 to 0.239, and the total least-squares error rises with it from 0.0117 to 0.722 — a factor of 62. κ(A) moves from 3.54 to 2.46 across the same sweep and predicts none of it.10⁻¹10⁻³10⁻²10⁻¹110¹noise level, relative to the datagap, error, and condition numberκ(A)TLS errorthe gapone of these predicts the errorgap, at the least noise1.4gap, at the most0.24error, ratio across the sweep62κ(A), ratio across the sweep1.4the condition number is nearly flatand the error moves by two orders

Three curves against the noise level, both axes logarithmic, as medians over 12 seeds of a 24×3 fit. The gap between the smallest singular value of A and the smallest of the augmented matrix closes from 1.4 to 0.239, and the total least-squares error rises with it from 0.0117 to 0.722 — a factor of 62. κ(A) moves from 3.54 to 2.46 across the same sweep and predicts none of it.

m: 120

The arguments are the ones The two numbers a caller has passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

What governs a total least-squares problem: the gap σₙ(A) − σₙ₊₁([A b]), not κ(A)Three curves against the noise level, both axes logarithmic, as medians over 12 seeds of a 120×3 fit. The gap between the smallest singular value of A and the smallest of the augmented matrix closes from 2.93 to 0.415, and the total least-squares error rises with it from 0.00509 to 1.02 — a factor of 201. κ(A) moves from 3.71 to 2.42 across the same sweep and predicts none of it.10⁻¹10⁻³10⁻²10⁻¹110¹noise level, relative to the datagap, error, and condition numberκ(A)TLS errorthe gapone of these predicts the errorgap, at the least noise2.9gap, at the most0.42error, ratio across the sweep201κ(A), ratio across the sweep1.5the condition number is nearly flatand the error moves by two orders

Three curves against the noise level, both axes logarithmic, as medians over 12 seeds of a 120×3 fit. The gap between the smallest singular value of A and the smallest of the augmented matrix closes from 2.93 to 0.415, and the total least-squares error rises with it from 0.00509 to 1.02 — a factor of 201. κ(A) moves from 3.71 to 2.42 across the same sweep and predicts none of it.

m: 240

The arguments are the ones When the matrix is wrong too passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

What governs a total least-squares problem: the gap σₙ(A) − σₙ₊₁([A b]), not κ(A)Three curves against the noise level, both axes logarithmic, as medians over 12 seeds of a 240×3 fit. The gap between the smallest singular value of A and the smallest of the augmented matrix closes from 4.1 to 0.373, and the total least-squares error rises with it from 0.00434 to 1.48 — a factor of 342. κ(A) moves from 3.73 to 2.45 across the same sweep and predicts none of it.10⁻¹10⁻³10⁻²10⁻¹110¹noise level, relative to the datagap, error, and condition numberκ(A)TLS errorthe gapone of these predicts the errorgap, at the least noise4.1gap, at the most0.37error, ratio across the sweep342κ(A), ratio across the sweep1.5the condition number is nearly flatand the error moves by two orders

Three curves against the noise level, both axes logarithmic, as medians over 12 seeds of a 240×3 fit. The gap between the smallest singular value of A and the smallest of the augmented matrix closes from 4.1 to 0.373, and the total least-squares error rises with it from 0.00434 to 1.48 — a factor of 342. κ(A) moves from 3.73 to 2.45 across the same sweep and predicts none of it.

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.

16 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.

every problem at σ = 0.02 has a total least-squares solution — checked 6 times

the gap closes as the noise rises, at σ = 0.05 — checked 5 times

a number of observations the sweep can afford

and the error grows by far more than κ does

and the error rises with it

enough seeds for the gap to be a median rather than a draw

while κ(A) barely moves across the same sweep

Against the rule

It draws a decomposition and prints its residual. It calls compare, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

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.

Least squares, and the road not to take

The two numbers a caller has

Choosing between the two least-squares methods is a statement about where the noise is, and the two quantities a caller can compute are both blind to it. The residual separates the answers by 0.14 per cent where their accuracies differ by 14, and κ(A) falls from 3.54 to 2.46 across a sweep in which the error rises by a factor of sixty-two.

 Least squares, and the road not to take

When the matrix is wrong too

Every least-squares problem here has assumed A is exact and b is not, and moved b onto the column space of A. Where both were measured, the smallest correction that makes the system consistent moves the matrix as well — and on the problems where that answer is more accurate, it has the larger residual, by construction rather than by luck.

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