Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Every box a search over [−3, 3]² leaves behind, at a stopping width of 0.05

One function in the branchbound library, called 7 times across 1 essay. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 9 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 every box a search over [−3, 3]² leaves behind, at a stopping width of 0.05. A square divided into rectangles of three kinds. 42 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 87 evaluations of the operator.

search-map is one function in lib/figures/branchbound.js — from a verdict to a search — where the box is cut, and what that costs. 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.

Every box a search over [−3, 3]² leaves behind, at a stopping width of 0.05A square divided into rectangles of three kinds. 42 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 87 evaluations of the operator.grey: proved empty · filled: proved to contain exactly one roota covered squareboxes proved empty42boxes proved unique2undecided0operator evaluations87every rectangle carries a proofand the two crosshairs are where the roots are

A square divided into rectangles of three kinds. 42 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 87 evaluations of the operator.

minWidth: 0.05

The arguments are the ones Where the box is cut passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Every box a search over [−3, 3]² leaves behind, at a stopping width of 0.05A square divided into rectangles of three kinds. 42 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 87 evaluations of the operator.grey: proved empty · filled: proved to contain exactly one roota covered squareboxes proved empty42boxes proved unique2undecided0operator evaluations87every rectangle carries a proofand the two crosshairs are where the roots are

A square divided into rectangles of three kinds. 42 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 87 evaluations of the operator.

span: 2

The arguments are the ones Where the box is cut passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Every box a search over [−2, 2]² leaves behind, at a stopping width of 0.05A square divided into rectangles of three kinds. 8 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 19 evaluations of the operator.grey: proved empty · filled: proved to contain exactly one roota covered squareboxes proved empty8boxes proved unique2undecided0operator evaluations19every rectangle carries a proofand the two crosshairs are where the roots are

A square divided into rectangles of three kinds. 8 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 19 evaluations of the operator.

span: 2.5

The arguments are the ones Where the box is cut passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Every box a search over [−2.5, 2.5]² leaves behind, at a stopping width of 0.05A square divided into rectangles of three kinds. 12 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 27 evaluations of the operator.grey: proved empty · filled: proved to contain exactly one roota covered squareboxes proved empty12boxes proved unique2undecided0operator evaluations27every rectangle carries a proofand the two crosshairs are where the roots are

A square divided into rectangles of three kinds. 12 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 27 evaluations of the operator.

span: 4

The arguments are the ones Where the box is cut passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Every box a search over [−4, 4]² leaves behind, at a stopping width of 0.05A square divided into rectangles of three kinds. 26 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 55 evaluations of the operator.grey: proved empty · filled: proved to contain exactly one roota covered squareboxes proved empty26boxes proved unique2undecided0operator evaluations55every rectangle carries a proofand the two crosshairs are where the roots are

A square divided into rectangles of three kinds. 26 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 55 evaluations of the operator.

span: 3.5

The arguments are the ones Where the box is cut passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Every box a search over [−3.5, 3.5]² leaves behind, at a stopping width of 0.05A square divided into rectangles of three kinds. 22 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 47 evaluations of the operator.grey: proved empty · filled: proved to contain exactly one roota covered squareboxes proved empty22boxes proved unique2undecided0operator evaluations47every rectangle carries a proofand the two crosshairs are where the roots are

A square divided into rectangles of three kinds. 22 are proved to contain no root at all; 2 are proved to contain exactly one, and each holds one of the two roots at (±√2, ±√2); 0 are undecided. The whole square is covered, and it took 47 evaluations of the operator.

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.

9 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.

the root at (1.414, 1.414) is inside a verified box — checked 2 times

a square that contains both roots

a stopping width the picture has room for

a subdivision point inside the box

and inside no box proved empty

LU is for square matrices

the search terminates

with both roots verified

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.

The arithmetic underneath

Where the box is cut

A branch-and-bound with an interval operator settles a whole square — two roots proved unique, forty-two regions proved empty, nothing left undecided, in 87 evaluations. Move the roots so one lands on the first bisection and it proves nothing at all, at any depth. Cutting at 0.485 instead of 0.5 finds both, in a quarter of the work.

The whole library · All essays · What must fail