Numerical Linear Algebra
where the algebra stops being the arithmetic
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Downdating a Cholesky factor against refactorising it, as the removed row's leverage approaches one

One function in the update library, called 7 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 14 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 downdating a cholesky factor against refactorising it, as the removed row's leverage approaches one. A rank-one term is removed from a 6×6 Gram matrix by hyperbolic rotations and, separately, by factorising the downdated matrix from scratch. The downdate's residual rises from 2.1·10⁻¹⁶ at h = 0.3 to 3.5·10⁻¹⁰ at h = 1 − 10⁻⁷, a slope of 1.04 against 1/(1 − h). The refactorisation is flat at 8.1·10⁻¹⁷, and the matrix both are producing has a condition number of 4.3 at every point on the axis — so the difficulty belongs to the route and not to the answer.

downdate-leverage is one function in lib/figures/update.js — rank-one — the correction that is cheaper than the problem, and what it charges. 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.

Downdating a Cholesky factor against refactorising it, as the removed row's leverage approaches oneA rank-one term is removed from a 6×6 Gram matrix by hyperbolic rotations and, separately, by factorising the downdated matrix from scratch. The downdate's residual rises from 2.1·10⁻¹⁶ at h = 0.3 to 3.5·10⁻¹⁰ at h = 1 − 10⁻⁷, a slope of 1.04 against 1/(1 − h). The refactorisation is flat at 8.1·10⁻¹⁷, and the matrix both are producing has a condition number of 4.3 at every point on the axis — so the difficulty belongs to the route and not to the answer.110¹10²10³10⁴10⁵10⁶10⁷10⁻¹⁷10⁻¹⁶10⁻¹⁵10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸1/(1 − h), the leverage of the removed row‖R̄ᵀR̄ − (G − aaᵀ)‖ / ‖G − aaᵀ‖downdatedrefactorisedat h = 1 − 10⁻⁷κ of the downdated matrix4.3κ of the matrix downdated9.3·10⁶rotation's amplification344downdate residual3.5·10⁻¹⁰a hyperbolic rotation is not orthogonaland that is exactly what it is for

A rank-one term is removed from a 6×6 Gram matrix by hyperbolic rotations and, separately, by factorising the downdated matrix from scratch. The downdate's residual rises from 2.1·10⁻¹⁶ at h = 0.3 to 3.5·10⁻¹⁰ at h = 1 − 10⁻⁷, a slope of 1.04 against 1/(1 − h). The refactorisation is flat at 8.1·10⁻¹⁷, and the matrix both are producing has a condition number of 4.3 at every point on the axis — so the difficulty belongs to the route and not to the answer.

p: 8

The arguments are the ones Influence is decided before the data passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Downdating a Cholesky factor against refactorising it, as the removed row's leverage approaches oneA rank-one term is removed from a 8×8 Gram matrix by hyperbolic rotations and, separately, by factorising the downdated matrix from scratch. The downdate's residual rises from 2.7·10⁻¹⁶ at h = 0.3 to 4.6·10⁻¹⁰ at h = 1 − 10⁻⁷, a slope of 1.02 against 1/(1 − h). The refactorisation is flat at 1.5·10⁻¹⁶, and the matrix both are producing has a condition number of 4.9 at every point on the axis — so the difficulty belongs to the route and not to the answer.110¹10²10³10⁴10⁵10⁶10⁷10⁻¹⁷10⁻¹⁶10⁻¹⁵10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸1/(1 − h), the leverage of the removed row‖R̄ᵀR̄ − (G − aaᵀ)‖ / ‖G − aaᵀ‖downdatedrefactorisedat h = 1 − 10⁻⁷κ of the downdated matrix4.9κ of the matrix downdated2·10⁷rotation's amplification399downdate residual4.6·10⁻¹⁰a hyperbolic rotation is not orthogonaland that is exactly what it is for

A rank-one term is removed from a 8×8 Gram matrix by hyperbolic rotations and, separately, by factorising the downdated matrix from scratch. The downdate's residual rises from 2.7·10⁻¹⁶ at h = 0.3 to 4.6·10⁻¹⁰ at h = 1 − 10⁻⁷, a slope of 1.02 against 1/(1 − h). The refactorisation is flat at 1.5·10⁻¹⁶, and the matrix both are producing has a condition number of 4.9 at every point on the axis — so the difficulty belongs to the route and not to the answer.

p: 6

The arguments are the ones The observation that cannot be removed passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Downdating a Cholesky factor against refactorising it, as the removed row's leverage approaches oneA rank-one term is removed from a 6×6 Gram matrix by hyperbolic rotations and, separately, by factorising the downdated matrix from scratch. The downdate's residual rises from 2.1·10⁻¹⁶ at h = 0.3 to 3.5·10⁻¹⁰ at h = 1 − 10⁻⁷, a slope of 1.04 against 1/(1 − h). The refactorisation is flat at 8.1·10⁻¹⁷, and the matrix both are producing has a condition number of 4.3 at every point on the axis — so the difficulty belongs to the route and not to the answer.110¹10²10³10⁴10⁵10⁶10⁷10⁻¹⁷10⁻¹⁶10⁻¹⁵10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸1/(1 − h), the leverage of the removed row‖R̄ᵀR̄ − (G − aaᵀ)‖ / ‖G − aaᵀ‖downdatedrefactorisedat h = 1 − 10⁻⁷κ of the downdated matrix4.3κ of the matrix downdated9.3·10⁶rotation's amplification344downdate residual3.5·10⁻¹⁰a hyperbolic rotation is not orthogonaland that is exactly what it is for

A rank-one term is removed from a 6×6 Gram matrix by hyperbolic rotations and, separately, by factorising the downdated matrix from scratch. The downdate's residual rises from 2.1·10⁻¹⁶ at h = 0.3 to 3.5·10⁻¹⁰ at h = 1 − 10⁻⁷, a slope of 1.04 against 1/(1 − h). The refactorisation is flat at 8.1·10⁻¹⁷, and the matrix both are producing has a condition number of 4.3 at every point on the axis — so the difficulty belongs to the route and not to the answer.

p: 3

The arguments are the ones The observation that cannot be removed passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Downdating a Cholesky factor against refactorising it, as the removed row's leverage approaches oneA rank-one term is removed from a 3×3 Gram matrix by hyperbolic rotations and, separately, by factorising the downdated matrix from scratch. The downdate's residual rises from 2·10⁻¹⁶ at h = 0.3 to 4.6·10⁻¹⁰ at h = 1 − 10⁻⁷, a slope of 0.96 against 1/(1 − h). The refactorisation is flat at 0, and the matrix both are producing has a condition number of 2.3 at every point on the axis — so the difficulty belongs to the route and not to the answer.110¹10²10³10⁴10⁵10⁶10⁷10⁻¹⁷10⁻¹⁶10⁻¹⁵10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸1/(1 − h), the leverage of the removed row‖R̄ᵀR̄ − (G − aaᵀ)‖ / ‖G − aaᵀ‖downdatedrefactorisedat h = 1 − 10⁻⁷κ of the downdated matrix2.3κ of the matrix downdated1.3·10⁷rotation's amplification557downdate residual4.6·10⁻¹⁰a hyperbolic rotation is not orthogonaland that is exactly what it is for

A rank-one term is removed from a 3×3 Gram matrix by hyperbolic rotations and, separately, by factorising the downdated matrix from scratch. The downdate's residual rises from 2·10⁻¹⁶ at h = 0.3 to 4.6·10⁻¹⁰ at h = 1 − 10⁻⁷, a slope of 0.96 against 1/(1 − h). The refactorisation is flat at 0, and the matrix both are producing has a condition number of 2.3 at every point on the axis — so the difficulty belongs to the route and not to the answer.

p: 4

The arguments are the ones The observation that cannot be removed passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Downdating a Cholesky factor against refactorising it, as the removed row's leverage approaches oneA rank-one term is removed from a 4×4 Gram matrix by hyperbolic rotations and, separately, by factorising the downdated matrix from scratch. The downdate's residual rises from 1.9·10⁻¹⁶ at h = 0.3 to 6.9·10⁻¹⁰ at h = 1 − 10⁻⁷, a slope of 1.13 against 1/(1 − h). The refactorisation is flat at 8.5·10⁻¹⁷, and the matrix both are producing has a condition number of 3.5 at every point on the axis — so the difficulty belongs to the route and not to the answer.110¹10²10³10⁴10⁵10⁶10⁷10⁻¹⁷10⁻¹⁶10⁻¹⁵10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸1/(1 − h), the leverage of the removed row‖R̄ᵀR̄ − (G − aaᵀ)‖ / ‖G − aaᵀ‖downdatedrefactorisedat h = 1 − 10⁻⁷κ of the downdated matrix3.5κ of the matrix downdated2.1·10⁷rotation's amplification657downdate residual6.9·10⁻¹⁰a hyperbolic rotation is not orthogonaland that is exactly what it is for

A rank-one term is removed from a 4×4 Gram matrix by hyperbolic rotations and, separately, by factorising the downdated matrix from scratch. The downdate's residual rises from 1.9·10⁻¹⁶ at h = 0.3 to 6.9·10⁻¹⁰ at h = 1 − 10⁻⁷, a slope of 1.13 against 1/(1 − h). The refactorisation is flat at 8.5·10⁻¹⁷, and the matrix both are producing has a condition number of 3.5 at every point on the axis — so the difficulty belongs to the route and not to the answer.

p: 10

The arguments are the ones The observation that cannot be removed passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Downdating a Cholesky factor against refactorising it, as the removed row's leverage approaches oneA rank-one term is removed from a 10×10 Gram matrix by hyperbolic rotations and, separately, by factorising the downdated matrix from scratch. The downdate's residual rises from 1.7·10⁻¹⁶ at h = 0.3 to 6·10⁻¹⁰ at h = 1 − 10⁻⁷, a slope of 0.96 against 1/(1 − h). The refactorisation is flat at 1.3·10⁻¹⁶, and the matrix both are producing has a condition number of 6.7 at every point on the axis — so the difficulty belongs to the route and not to the answer.110¹10²10³10⁴10⁵10⁶10⁷10⁻¹⁷10⁻¹⁶10⁻¹⁵10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸1/(1 − h), the leverage of the removed row‖R̄ᵀR̄ − (G − aaᵀ)‖ / ‖G − aaᵀ‖downdatedrefactorisedat h = 1 − 10⁻⁷κ of the downdated matrix6.7κ of the matrix downdated3.5·10⁷rotation's amplification377downdate residual6·10⁻¹⁰a hyperbolic rotation is not orthogonaland that is exactly what it is for

A rank-one term is removed from a 10×10 Gram matrix by hyperbolic rotations and, separately, by factorising the downdated matrix from scratch. The downdate's residual rises from 1.7·10⁻¹⁶ at h = 0.3 to 6·10⁻¹⁰ at h = 1 − 10⁻⁷, a slope of 0.96 against 1/(1 − h). The refactorisation is flat at 1.3·10⁻¹⁶, and the matrix both are producing has a condition number of 6.7 at every point on the axis — so the difficulty belongs to the route and not to the answer.

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.

14 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 downdate produces a factor at h = 0.3000000 — checked 8 times

a number of columns the repeated searches can afford

and the matrix it produces is well conditioned

as the first power of 1/(1 − h)

enough rows for a design matrix

matmul shapes agree

while the downdate loses digits as the leverage approaches one

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.

Least squares, and the road not to take

Influence is decided before the data

The diagonal of the hat matrix sums to the number of columns and the response appears nowhere in it, so a fit has exactly p units of influence to hand out among m observations. The same row at h = 0.5 is a ten-fold outlier on one design and a boundary case on another, and which of those it is was settled before a single measurement was taken.

 Least squares, and the road not to take

The observation that cannot be removed

Removing a rank-one term from a Cholesky factor needs a rotation that is not orthogonal, and the number under its square root is 1 − h, where h is the leverage of the row being removed. The algorithm's breakdown condition and the statistician's warning are the same quantity, arrived at from opposite ends, and neither field states it in the other's language.

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