Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Least squares by QR and by the normal equations in binary32

One function in the lsq library, called 8 times across 3 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 8 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 least squares by qr and by the normal equations in binary32. Relative error of the computed coefficients against epsilon, on log axes. The QR route is a flat line near the bottom; the normal-equations route climbs and then stops, at the epsilon where the cross-product matrix becomes exactly singular.

normal-equations-road is one function in lib/figures/lsq.js — least squares — the projection, the road not to take, and the valley with no bottom. 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.

Least squares by QR and by the normal equations in binary32Relative error of the computed coefficients against epsilon, on log axes. The QR route is a flat line near the bottom; the normal-equations route climbs and then stops, at the epsilon where the cross-product matrix becomes exactly singular.10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10¹ε in Läuchli's matrix (smaller ε, larger κ)relative error in the coefficientsAᵀA exactly singularnormal equationsQRκ from 1.7·10⁸ to 17the cliff is at √u = 2.4·10⁻⁴

Relative error of the computed coefficients against epsilon, on log axes. The QR route is a flat line near the bottom; the normal-equations route climbs and then stops, at the epsilon where the cross-product matrix becomes exactly singular.

bits: 24

The arguments are the ones The licence is not the boundary passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Least squares by QR and by the normal equations in binary32Relative error of the computed coefficients against epsilon, on log axes. The QR route is a flat line near the bottom; the normal-equations route climbs and then stops, at the epsilon where the cross-product matrix becomes exactly singular.10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10¹ε in Läuchli's matrix (smaller ε, larger κ)relative error in the coefficientsAᵀA exactly singularnormal equationsQRκ from 1.7·10⁸ to 17the cliff is at √u = 2.4·10⁻⁴

Relative error of the computed coefficients against epsilon, on log axes. The QR route is a flat line near the bottom; the normal-equations route climbs and then stops, at the epsilon where the cross-product matrix becomes exactly singular.

bits: 16

The arguments are the ones The road that squares the problem passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Least squares by QR and by the normal equations in 16-bitRelative error of the computed coefficients against epsilon, on log axes. The QR route is a flat line near the bottom; the normal-equations route climbs and then stops, at the epsilon where the cross-product matrix becomes exactly singular.10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10¹ε in Läuchli's matrix (smaller ε, larger κ)relative error in the coefficientsAᵀA exactly singularnormal equationsQRκ from 1.7·10⁸ to 17the cliff is at √u = 0.0039

Relative error of the computed coefficients against epsilon, on log axes. The QR route is a flat line near the bottom; the normal-equations route climbs and then stops, at the epsilon where the cross-product matrix becomes exactly singular.

bits: 22

The arguments are the ones The road that squares the problem passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Least squares by QR and by the normal equations in 22-bitRelative error of the computed coefficients against epsilon, on log axes. The QR route is a flat line near the bottom; the normal-equations route climbs and then stops, at the epsilon where the cross-product matrix becomes exactly singular.10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10¹ε in Läuchli's matrix (smaller ε, larger κ)relative error in the coefficientsAᵀA exactly singularnormal equationsQRκ from 1.7·10⁸ to 17the cliff is at √u = 4.9·10⁻⁴

Relative error of the computed coefficients against epsilon, on log axes. The QR route is a flat line near the bottom; the normal-equations route climbs and then stops, at the epsilon where the cross-product matrix becomes exactly singular.

bits: 34

The arguments are the ones The road that squares the problem passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Least squares by QR and by the normal equations in 34-bitRelative error of the computed coefficients against epsilon, on log axes. The QR route is a flat line near the bottom; the normal-equations route climbs and then stops, at the epsilon where the cross-product matrix becomes exactly singular.10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10¹ε in Läuchli's matrix (smaller ε, larger κ)relative error in the coefficientsAᵀA exactly singularnormal equationsQRκ from 1.7·10⁸ to 17the cliff is at √u = 7.6·10⁻⁶

Relative error of the computed coefficients against epsilon, on log axes. The QR route is a flat line near the bottom; the normal-equations route climbs and then stops, at the epsilon where the cross-product matrix becomes exactly singular.

bits: 40

The arguments are the ones The road that squares the problem passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Least squares by QR and by the normal equations in 40-bitRelative error of the computed coefficients against epsilon, on log axes. The QR route is a flat line near the bottom; the normal-equations route climbs and then stops, at the epsilon where the cross-product matrix becomes exactly singular.10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10¹ε in Läuchli's matrix (smaller ε, larger κ)relative error in the coefficientsAᵀA exactly singularnormal equationsQRκ from 1.7·10⁸ to 17the cliff is at √u = 9.5·10⁻⁷

Relative error of the computed coefficients against epsilon, on log axes. The QR route is a flat line near the bottom; the normal-equations route climbs and then stops, at the epsilon where the cross-product matrix becomes exactly singular.

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.

8 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.

and the breakdown is where 1 + ε² rounds to 1, at ε ≈ √u

and the normal equations are far behind before they fail

LU is for square matrices

matmul shapes agree

the normal equations break down somewhere on this range

while QR is accurate across the whole range

κ(AᵀA) = κ(A)² at ε = 0.01 agree

κ(AᵀA) = κ(A)² at ε = 10⁻⁴ agree

Against the rule

It calls a factoriser without drawing a factorisation (lstsqQR, lstsqNormal), so the rule is written down as not applying, with the reason: plots the forward error of two least-squares routes; no factor is displayed

The exemption list is the interesting half of the rule rather than an escape hatch — it is where a decision about a figure had to be argued in one line. residualcheck refuses an exemption that is not doing work, and rejected ten of the fifteen written for the expansion's figures on exactly that ground: a figure whose vertical axis is a residual satisfies the rule by construction, and touching a factoriser does not by itself require an entry.

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.

The answer that depends on the machine

The licence is not the boundary

Cholesky QR is licensed by κ²u ≪ 1, which reaches equality at κ = 9.5·10⁷ in double precision. At 10⁸ the factor it returns is already 0.37 away from orthogonal, and it goes on returning factors as far as 10¹³ — refusing at scattered condition numbers in between, at different ones for eight columns and for six.

 Least squares, and the road not to take

The projection and the right angle

The least-squares solution is the one whose residual is perpendicular to everything the columns can reach. That is not a mnemonic — it is an equation, Aᵀr = 0, and the computed answer satisfies it to 10⁻¹⁶.

 Least squares, and the road not to take

The road that squares the problem

The normal equations are the first method every course teaches and the method no library uses. Forming AᵀA squares the condition number, and below ε = √u it does not degrade — it produces a matrix that is exactly singular, from data that was perfectly usable.

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