projection-geometry
At its defaults it draws the least-squares solution as a projection, with the right angle measured. The column space drawn edge-on as a plane, the data vector above it, and the perpendicular dropped to the plane, with the residual marked at a right angle to it.
projection-geometry 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.
The column space drawn edge-on as a plane, the data vector above it, and the perpendicular dropped to the plane, with the residual marked at a right angle to it.
b: [1.1, 0.4, 1.5]
The arguments are the ones A reduction that changes the order 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.
The column space drawn edge-on as a plane, the data vector above it, and the perpendicular dropped to the plane, with the residual marked at a right angle to 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.
6 distinct claims across 2 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.
‖b‖² = ‖Ax‖² + ‖r‖² agree
and no other point of the plane is closer
and the drawn hypotenuse is ‖b‖ at the same scale
b lies outside the column space, so there is a residual to draw
the drawn legs are in the ratio ‖r‖/‖Ax‖
the residual is perpendicular to the column space
Against the rule
It draws a decomposition and prints its residual. It calls
lstsqQR,
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 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 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.
A reduction that changes the order
A tall-skinny QR computed as a tree of independent block factorisations touches a 512×12 matrix once instead of twelve times, computes a completely different sequence of roundings from the sweep it replaces, and returns ‖AᵀA − RᵀR‖/‖AᵀA‖ = 1.65·10⁻¹⁵ against the sweep's 9.95·10⁻¹⁵. On the same matrix classical Gram–Schmidt returns 4.6·10⁻¹⁰.
Least squares, and the road not to takeThe 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.
Orthogonality, measuredTwo Gram–Schmidts
One argument changes. Classical Gram–Schmidt projects the original column onto each previous direction; modified projects what is left of it. In exact arithmetic the coefficients are identical. In floating point they differ by eight orders of magnitude in the thing that matters.