Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

The summation condition number of six inner products this site already computes

One function in the repro library, called 10 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 4 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 the summation condition number of six inner products this site already computes. κ = Σ|xᵢ| ÷ |Σxᵢ| for six reductions taken from computations elsewhere on this site, at 64 terms: an orthogonality check between two columns of a Q, one component of a residual b − Ax on an ill-conditioned system, a nearly degenerate 2 × 2 determinant, a Hutchinson trace probe zᵀAz, a residual norm rᵀr, and a curvature pᵀAp. The range is 1 to 1.01·10¹⁷. The two at the safe end are sums of squares, where no term can cancel another; the three above 10¹⁰ are the orthogonality check, the residual and the determinant — which is to say every quantity a stopping test, an orthogonality test or an orientation test is written in. The reductions that are reproducible are the ones nobody makes a decision from.

kappa-census is one function in lib/figures/repro.js — one sum, several answers — what a reduction's division costs, and the two policies that remove it. 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 summation condition number of six inner products this site already computesκ = Σ|xᵢ| ÷ |Σxᵢ| for six reductions taken from computations elsewhere on this site, at 64 terms: an orthogonality check between two columns of a Q, one component of a residual b − Ax on an ill-conditioned system, a nearly degenerate 2 × 2 determinant, a Hutchinson trace probe zᵀAz, a residual norm rᵀr, and a curvature pᵀAp. The range is 1 to 1.01·10¹⁷. The two at the safe end are sums of squares, where no term can cancel another; the three above 10¹⁰ are the orthogonality check, the residual and the determinant — which is to say every quantity a stopping test, an orthogonality test or an orientation test is written in. The reductions that are reproducible are the ones nobody makes a decision from.κ of the suma component of b − Ax1.01·10¹⁷ad − bc, near-degenerate3.6·10¹⁶qᵢᵀqⱼ, an orthogonality check7.39·10¹⁵pᵀAp, a curvature409zᵀAz, a trace probe41.8rᵀr, a residual norm1measured, not assumedhighest10¹⁷lowest1above 10¹⁰3terms64sums of squares are safeand nobody decides anything from one

κ = Σ|xᵢ| ÷ |Σxᵢ| for six reductions taken from computations elsewhere on this site, at 64 terms: an orthogonality check between two columns of a Q, one component of a residual b − Ax on an ill-conditioned system, a nearly degenerate 2 × 2 determinant, a Hutchinson trace probe zᵀAz, a residual norm rᵀr, and a curvature pᵀAp. The range is 1 to 1.01·10¹⁷. The two at the safe end are sums of squares, where no term can cancel another; the three above 10¹⁰ are the orthogonality check, the residual and the determinant — which is to say every quantity a stopping test, an orthogonality test or an orientation test is written in. The reductions that are reproducible are the ones nobody makes a decision from.

n: 64

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

The summation condition number of six inner products this site already computesκ = Σ|xᵢ| ÷ |Σxᵢ| for six reductions taken from computations elsewhere on this site, at 64 terms: an orthogonality check between two columns of a Q, one component of a residual b − Ax on an ill-conditioned system, a nearly degenerate 2 × 2 determinant, a Hutchinson trace probe zᵀAz, a residual norm rᵀr, and a curvature pᵀAp. The range is 1 to 1.01·10¹⁷. The two at the safe end are sums of squares, where no term can cancel another; the three above 10¹⁰ are the orthogonality check, the residual and the determinant — which is to say every quantity a stopping test, an orthogonality test or an orientation test is written in. The reductions that are reproducible are the ones nobody makes a decision from.κ of the suma component of b − Ax1.01·10¹⁷ad − bc, near-degenerate3.6·10¹⁶qᵢᵀqⱼ, an orthogonality check7.39·10¹⁵pᵀAp, a curvature409zᵀAz, a trace probe41.8rᵀr, a residual norm1measured, not assumedhighest10¹⁷lowest1above 10¹⁰3terms64sums of squares are safeand nobody decides anything from one

κ = Σ|xᵢ| ÷ |Σxᵢ| for six reductions taken from computations elsewhere on this site, at 64 terms: an orthogonality check between two columns of a Q, one component of a residual b − Ax on an ill-conditioned system, a nearly degenerate 2 × 2 determinant, a Hutchinson trace probe zᵀAz, a residual norm rᵀr, and a curvature pᵀAp. The range is 1 to 1.01·10¹⁷. The two at the safe end are sums of squares, where no term can cancel another; the three above 10¹⁰ are the orthogonality check, the residual and the determinant — which is to say every quantity a stopping test, an orthogonality test or an orientation test is written in. The reductions that are reproducible are the ones nobody makes a decision from.

n: 16

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

The summation condition number of six inner products this site already computesκ = Σ|xᵢ| ÷ |Σxᵢ| for six reductions taken from computations elsewhere on this site, at 16 terms: an orthogonality check between two columns of a Q, one component of a residual b − Ax on an ill-conditioned system, a nearly degenerate 2 × 2 determinant, a Hutchinson trace probe zᵀAz, a residual norm rᵀr, and a curvature pᵀAp. The range is 1 to 9.28·10¹⁶. The two at the safe end are sums of squares, where no term can cancel another; the three above 10¹⁰ are the orthogonality check, the residual and the determinant — which is to say every quantity a stopping test, an orthogonality test or an orientation test is written in. The reductions that are reproducible are the ones nobody makes a decision from.κ of the suma component of b − Ax9.28·10¹⁶ad − bc, near-degenerate3.6·10¹⁶qᵢᵀqⱼ, an orthogonality check8.57·10¹⁵pᵀAp, a curvature86.6zᵀAz, a trace probe10.8rᵀr, a residual norm1measured, not assumedhighest9.3·10¹⁶lowest1above 10¹⁰3terms16sums of squares are safeand nobody decides anything from one

κ = Σ|xᵢ| ÷ |Σxᵢ| for six reductions taken from computations elsewhere on this site, at 16 terms: an orthogonality check between two columns of a Q, one component of a residual b − Ax on an ill-conditioned system, a nearly degenerate 2 × 2 determinant, a Hutchinson trace probe zᵀAz, a residual norm rᵀr, and a curvature pᵀAp. The range is 1 to 9.28·10¹⁶. The two at the safe end are sums of squares, where no term can cancel another; the three above 10¹⁰ are the orthogonality check, the residual and the determinant — which is to say every quantity a stopping test, an orthogonality test or an orientation test is written in. The reductions that are reproducible are the ones nobody makes a decision from.

n: 32

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

The summation condition number of six inner products this site already computesκ = Σ|xᵢ| ÷ |Σxᵢ| for six reductions taken from computations elsewhere on this site, at 32 terms: an orthogonality check between two columns of a Q, one component of a residual b − Ax on an ill-conditioned system, a nearly degenerate 2 × 2 determinant, a Hutchinson trace probe zᵀAz, a residual norm rᵀr, and a curvature pᵀAp. The range is 1 to 1.54·10¹⁷. The two at the safe end are sums of squares, where no term can cancel another; the three above 10¹⁰ are the orthogonality check, the residual and the determinant — which is to say every quantity a stopping test, an orthogonality test or an orientation test is written in. The reductions that are reproducible are the ones nobody makes a decision from.κ of the suma component of b − Ax1.54·10¹⁷ad − bc, near-degenerate3.6·10¹⁶qᵢᵀqⱼ, an orthogonality check1.87·10¹⁶zᵀAz, a trace probe27.9pᵀAp, a curvature6.03rᵀr, a residual norm1measured, not assumedhighest1.5·10¹⁷lowest1above 10¹⁰3terms32sums of squares are safeand nobody decides anything from one

κ = Σ|xᵢ| ÷ |Σxᵢ| for six reductions taken from computations elsewhere on this site, at 32 terms: an orthogonality check between two columns of a Q, one component of a residual b − Ax on an ill-conditioned system, a nearly degenerate 2 × 2 determinant, a Hutchinson trace probe zᵀAz, a residual norm rᵀr, and a curvature pᵀAp. The range is 1 to 1.54·10¹⁷. The two at the safe end are sums of squares, where no term can cancel another; the three above 10¹⁰ are the orthogonality check, the residual and the determinant — which is to say every quantity a stopping test, an orthogonality test or an orientation test is written in. The reductions that are reproducible are the ones nobody makes a decision from.

n: 128

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

The summation condition number of six inner products this site already computesκ = Σ|xᵢ| ÷ |Σxᵢ| for six reductions taken from computations elsewhere on this site, at 128 terms: an orthogonality check between two columns of a Q, one component of a residual b − Ax on an ill-conditioned system, a nearly degenerate 2 × 2 determinant, a Hutchinson trace probe zᵀAz, a residual norm rᵀr, and a curvature pᵀAp. The range is 1 to 8.09·10¹⁷. The two at the safe end are sums of squares, where no term can cancel another; the three above 10¹⁰ are the orthogonality check, the residual and the determinant — which is to say every quantity a stopping test, an orthogonality test or an orientation test is written in. The reductions that are reproducible are the ones nobody makes a decision from.κ of the suma component of b − Ax8.09·10¹⁷ad − bc, near-degenerate3.6·10¹⁶qᵢᵀqⱼ, an orthogonality check3.26·10¹⁶zᵀAz, a trace probe95.7pᵀAp, a curvature73.7rᵀr, a residual norm1measured, not assumedhighest8.1·10¹⁷lowest1above 10¹⁰3terms128sums of squares are safeand nobody decides anything from one

κ = Σ|xᵢ| ÷ |Σxᵢ| for six reductions taken from computations elsewhere on this site, at 128 terms: an orthogonality check between two columns of a Q, one component of a residual b − Ax on an ill-conditioned system, a nearly degenerate 2 × 2 determinant, a Hutchinson trace probe zᵀAz, a residual norm rᵀr, and a curvature pᵀAp. The range is 1 to 8.09·10¹⁷. The two at the safe end are sums of squares, where no term can cancel another; the three above 10¹⁰ are the orthogonality check, the residual and the determinant — which is to say every quantity a stopping test, an orthogonality test or an orientation test is written in. The reductions that are reproducible are the ones nobody makes a decision from.

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.

4 distinct claims across 5 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.

a size the quadratic forms can afford

and the safe end is a sum of squares, at exactly one

every reduction a decision is read from is at the dangerous end

the census spans more than ten decades

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 answer that depends on the machine

The vector that hides it

Every quick demonstration of a parallel sum uses positive numbers, and positive numbers are the one family where the effect is absent. Measured on six inner products this site already computes, the summation condition number runs from exactly 1 to 10¹⁷ — and the safe end is where nobody makes a decision.

 The answer that depends on the machine

Two machines, one certificate

Nothing a solver returns says which of its answers you got. Four things could be reported instead — the summation condition number, the partition count, an exactly accumulated residual and a directed-rounding interval — and each costs about one pass over data the routine already has in hand.

The whole library · All essays · What must fail