Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

three-walks

One function in the seqstab library, called 7 times across 3 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 13 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 three accumulations, each against its own quantity, each divided by its own first point. A left-to-right sum against the number of terms, at a fitted slope of 0.486; a chain of rotations against the number of steps, at 0.554; and a conjugate gradient residual recurrence against the largest iterate, at 0.507. The three share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against, drawn here as the upper line, and each comes out at half of it. Nothing is rescaled except the division by each series' own first point, which is what makes three quantities of different sizes comparable in slope and in nothing else.

three-walks is one function in lib/figures/seqstab.js — accumulation — what a carried factor collects, and why it is a walk rather than a sum. 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.

Three accumulations, each against its own quantity, each divided by its own first pointA left-to-right sum against the number of terms, at a fitted slope of 0.486; a chain of rotations against the number of steps, at 0.554; and a conjugate gradient residual recurrence against the largest iterate, at 0.507. The three share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against, drawn here as the upper line, and each comes out at half of it. Nothing is rescaled except the division by each series' own first point, which is what makes three quantities of different sizes comparable in slope and in nothing else.110¹10²10³110¹10²10³the accumulating quantity, relative to its first valuethe error, relative to its first valuethe bounds: slope 1what all three do: slope ½three mechanisms, one exponenta left-to-right sum0.49a chain of rotations0.55a residual recurrence0.51every bound's slope1spread of the three0.067a bound is a sum of the roundingsand the roundings have signs

A left-to-right sum against the number of terms, at a fitted slope of 0.486; a chain of rotations against the number of steps, at 0.554; and a conjugate gradient residual recurrence against the largest iterate, at 0.507. The three share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against, drawn here as the upper line, and each comes out at half of it. Nothing is rescaled except the division by each series' own first point, which is what makes three quantities of different sizes comparable in slope and in nothing else.

decades: 3

The arguments are the ones Stable once, and three thousand times 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.

Three accumulations, each against its own quantity, each divided by its own first pointA left-to-right sum against the number of terms, at a fitted slope of 0.486; a chain of rotations against the number of steps, at 0.554; and a conjugate gradient residual recurrence against the largest iterate, at 0.507. The three share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against, drawn here as the upper line, and each comes out at half of it. Nothing is rescaled except the division by each series' own first point, which is what makes three quantities of different sizes comparable in slope and in nothing else.110¹10²10³110¹10²10³the accumulating quantity, relative to its first valuethe error, relative to its first valuethe bounds: slope 1what all three do: slope ½three mechanisms, one exponenta left-to-right sum0.49a chain of rotations0.55a residual recurrence0.51every bound's slope1spread of the three0.067a bound is a sum of the roundingsand the roundings have signs

A left-to-right sum against the number of terms, at a fitted slope of 0.486; a chain of rotations against the number of steps, at 0.554; and a conjugate gradient residual recurrence against the largest iterate, at 0.507. The three share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against, drawn here as the upper line, and each comes out at half of it. Nothing is rescaled except the division by each series' own first point, which is what makes three quantities of different sizes comparable in slope and in nothing else.

decades: 5

The arguments are the ones Three walks and one bound 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.

Three accumulations, each against its own quantity, each divided by its own first pointA left-to-right sum against the number of terms, at a fitted slope of 0.486; a chain of rotations against the number of steps, at 0.554; and a conjugate gradient residual recurrence against the largest iterate, at 0.507. The three share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against, drawn here as the upper line, and each comes out at half of it. Nothing is rescaled except the division by each series' own first point, which is what makes three quantities of different sizes comparable in slope and in nothing else.110¹10²10³10⁴10⁵110¹10²10³10⁴10⁵the accumulating quantity, relative to its first valuethe error, relative to its first valuethe bounds: slope 1what all three do: slope ½three mechanisms, one exponenta left-to-right sum0.49a chain of rotations0.55a residual recurrence0.51every bound's slope1spread of the three0.067a bound is a sum of the roundingsand the roundings have signs

A left-to-right sum against the number of terms, at a fitted slope of 0.486; a chain of rotations against the number of steps, at 0.554; and a conjugate gradient residual recurrence against the largest iterate, at 0.507. The three share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against, drawn here as the upper line, and each comes out at half of it. Nothing is rescaled except the division by each series' own first point, which is what makes three quantities of different sizes comparable in slope and in nothing else.

decades: 2

The arguments are the ones Three walks and one bound 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.

Three accumulations, each against its own quantity, each divided by its own first pointA left-to-right sum against the number of terms, at a fitted slope of 0.486; a chain of rotations against the number of steps, at 0.554; and a conjugate gradient residual recurrence against the largest iterate, at 0.507. The three share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against, drawn here as the upper line, and each comes out at half of it. Nothing is rescaled except the division by each series' own first point, which is what makes three quantities of different sizes comparable in slope and in nothing else.110¹10²110¹10²the accumulating quantity, relative to its first valuethe error, relative to its first valuethe bounds: slope 1what all three do: slope ½three mechanisms, one exponenta left-to-right sum0.49a chain of rotations0.55a residual recurrence0.51every bound's slope1spread of the three0.067a bound is a sum of the roundingsand the roundings have signs

A left-to-right sum against the number of terms, at a fitted slope of 0.486; a chain of rotations against the number of steps, at 0.554; and a conjugate gradient residual recurrence against the largest iterate, at 0.507. The three share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against, drawn here as the upper line, and each comes out at half of it. Nothing is rescaled except the division by each series' own first point, which is what makes three quantities of different sizes comparable in slope and in nothing else.

decades: 4

The arguments are the ones Three walks and one bound 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.

Three accumulations, each against its own quantity, each divided by its own first pointA left-to-right sum against the number of terms, at a fitted slope of 0.486; a chain of rotations against the number of steps, at 0.554; and a conjugate gradient residual recurrence against the largest iterate, at 0.507. The three share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against, drawn here as the upper line, and each comes out at half of it. Nothing is rescaled except the division by each series' own first point, which is what makes three quantities of different sizes comparable in slope and in nothing else.110¹10²10³10⁴110¹10²10³10⁴the accumulating quantity, relative to its first valuethe error, relative to its first valuethe bounds: slope 1what all three do: slope ½three mechanisms, one exponenta left-to-right sum0.49a chain of rotations0.55a residual recurrence0.51every bound's slope1spread of the three0.067a bound is a sum of the roundingsand the roundings have signs

A left-to-right sum against the number of terms, at a fitted slope of 0.486; a chain of rotations against the number of steps, at 0.554; and a conjugate gradient residual recurrence against the largest iterate, at 0.507. The three share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against, drawn here as the upper line, and each comes out at half of it. Nothing is rescaled except the division by each series' own first point, which is what makes three quantities of different sizes comparable in slope and in nothing else.

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.

13 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 chain of rotations accumulates as the square root of steps taken

a column count the exact Gram matrix stays inside a double for

a left-to-right sum accumulates as the square root of terms added

a residual recurrence accumulates as the square root of the largest iterate

a size the whole run is affordable at

a span of the accumulating quantity to draw over

a spread of scales the integers survive

a window wider than the number of columns, or the factor is singular

an arithmetic whose exact sum still fits a double, so the reference is exact

an eigenvalue small enough to make the iterates large

and the three agree with each other more closely than any agrees with its bound

matmul shapes agree

three independent measurements

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 113 of 238 generators — 98 print a residual and 15 are exempt with a published reason; 125 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.

When the problem arrives again

Stable once, and three thousand times

A sliding window adds a row and removes one at every step and never looks at the data again. No single step of it amplifies by more than 2.72, no downdate fails, and after three thousand steps the triangular factor in memory is 3.9·10⁻¹⁴ from the matrix it is supposed to be a factor of — six hundred times growth from a per-step bound that says nothing about chains.

 Iterating, instead of factorising

The residual the method reports

Conjugate gradients prints a relative residual of 6.9·10⁻²¹. The unit roundoff is 1.1·10⁻¹⁶, so that is not a small residual and not a large one — it is not a residual. The vector the method is holding at that step has ‖b − Ax‖/‖b‖ = 5.1·10⁻¹⁰, and nothing in the run says so.

 The arithmetic underneath

Three walks and one bound

A left-to-right sum, a chain of three thousand rotations and a conjugate gradient residual recurrence share no arithmetic and no vocabulary. Each has a standard bound that is linear in whatever it accumulates against. All three come out at a half — 0.486, 0.554 and 0.507 — and nothing is rescaled.

The whole library · All essays · What must fail