Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

tls-gap

One function in the tls library, called 7 times across 5 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 σ_n(a) − σ_{n+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 σ_n(A) − σ_{n+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: 60

The arguments are the ones The coarse problem is a different problem 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.

What governs a total least-squares problem: the gap σ_n(A) − σ_{n+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: 120

The arguments are the ones When the answer is a choice 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.

What governs a total least-squares problem: the gap σ_n(A) − σ_{n+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 drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

What governs a total least-squares problem: the gap σ_n(A) − σ_{n+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.

m: 24

The arguments are the ones When the matrix is wrong too 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.

What governs a total least-squares problem: the gap σ_n(A) − σ_{n+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.

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.

16 distinct claims across 5 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 — asserted 6 times

the gap closes as the noise rises, at σ = 0.05 — asserted 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 90 of 174 generators — 75 print a residual and 15 are exempt with a published reason; 84 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.

Iterating, instead of factorising

The coarse problem is a different problem

In one dimension the Galerkin coarse operator is the coarse discretisation, entry for entry — this site asserted it. In two dimensions a five-point operator produces a nine-point coarse one, so the recursion solves a different discretisation at every level below the first, and converges at 0.20 a cycle regardless.

 Iterating, instead of factorising

The diffusion that makes the answer exact

Upwinding adds h/2 of artificial diffusion. Central differencing adds none. Add ε·ξ·Pe with ξ = coth(Pe) − 1/Pe and the computed solution is the exact one at every grid point, to 2.4·10⁻¹⁷ — at every Péclet number, on the problem it was derived from and on no other.

 Least squares, and the road not to take

The valley with no bottom

A degree-nine fit's coefficients can be moved by a third of their own size before the residual changes in the sixth significant figure. The arithmetic did not lose those digits. The data never contained them.

 Regularisation, and the answer that is chosen

When 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.

 Least squares, and the road not to take

When the matrix is wrong too

Every least-squares problem on this site 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.

The whole library · All essays · What must fail