The computed angle against the true one, two formulations
At its defaults it draws the computed angle against the true one, two formulations. Two curves of computed angle against true angle on logarithmic axes. One follows the diagonal all the way down; the other leaves it and flattens at a fixed level.
angle-floor is one function in lib/figures/subspace.js —
subspaces — the gap, the eigenvector, and the plane that survives both. 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.
Two curves of computed angle against true angle on logarithmic axes. One follows the diagonal all the way down; the other leaves it and flattens at a fixed level.
n: 8
The arguments are the ones Cancellation takes the answer, not a digit passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two curves of computed angle against true angle on logarithmic axes. One follows the diagonal all the way down; the other leaves it and flattens at a fixed level.
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.
30 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.
the sine route returns the angle it was given at θ = 0.01 — checked 3 times
a dimension the bases can be drawn from
and somewhere on the sweep it overstates the angle by three decades
matmul shapes agree
so at a small enough angle it is wrong by orders of magnitude, in one direction or the other
the floor of the arccosine route against √(2u) agree
the sine route returns the angle it was given at θ = 10⁻¹⁰
the sine route returns the angle it was given at θ = 10⁻¹¹
the sine route returns the angle it was given at θ = 10⁻¹²
the sine route returns the angle it was given at θ = 10⁻¹³
the sine route returns the angle it was given at θ = 10⁻¹⁴
the sine route returns the angle it was given at θ = 10⁻⁴
the sine route returns the angle it was given at θ = 10⁻⁵
the sine route returns the angle it was given at θ = 10⁻⁶
the sine route returns the angle it was given at θ = 10⁻⁷
the sine route returns the angle it was given at θ = 10⁻⁸
the sine route returns the angle it was given at θ = 10⁻⁹
the sine route returns the angle it was given at θ = 3.2·10⁻¹⁰
the sine route returns the angle it was given at θ = 3.2·10⁻¹¹
the sine route returns the angle it was given at θ = 3.2·10⁻¹²
the sine route returns the angle it was given at θ = 3.2·10⁻¹³
the sine route returns the angle it was given at θ = 3.2·10⁻¹⁴
the sine route returns the angle it was given at θ = 3.2·10⁻⁴
the sine route returns the angle it was given at θ = 3.2·10⁻⁵
the sine route returns the angle it was given at θ = 3.2·10⁻⁶
the sine route returns the angle it was given at θ = 3.2·10⁻⁷
the sine route returns the angle it was given at θ = 3.2·10⁻⁸
the sine route returns the angle it was given at θ = 3.2·10⁻⁹
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.
Cancellation takes the answer, not a digit
Subtracting two nearly equal numbers is exact. That is what makes it dangerous — the subtraction introduces no error at all, it exposes error the operands were already carrying, and the exposure can consume every significant figure at once.
Orthogonality, measuredOrthogonal is a number
"Q is orthogonal" is a claim about a measurable quantity, ‖QᵀQ − I‖, and on the eight-by-eight Hilbert matrix two standard algorithms return 10⁻¹⁵ and 1 for it. The one that returns 1 still reconstructs the matrix perfectly, which is why nothing warns you.
Eigenvalues, singular values, rankTwo matrices and one problem
Ax = λBx is what a finite element model, a structural vibration and a constrained optimisation actually produce, and it is not the one-matrix problem with a change of variables. Everybody is told not to form B⁻¹A because it is not symmetric. That is true, the departure from symmetry is about one, and it is not what decides the accuracy.