Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

The filter description, and the orthogonality it rests on

One function in the krylovreg 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 6 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 filter description, and the orthogonality it rests on. Two quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 20.

filter-identity-gap is one function in lib/figures/krylovreg.js — iterative regularisation — a step count as the parameter, and the filter it applies. 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 filter description, and the orthogonality it rests onTwo quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 20.2610141822263010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹stepsizeleast error: 20‖QᵀQ − I‖filter disagreementan identity with an expiry datedisagreement at step 1210⁻¹³disagreement at step 300.35least error at step20exact while the basis is orthogonaland false where the method is best

Two quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 20.

steps: 30

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

The filter description, and the orthogonality it rests onTwo quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 20.2610141822263010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹stepsizeleast error: 20‖QᵀQ − I‖filter disagreementan identity with an expiry datedisagreement at step 1210⁻¹³disagreement at step 300.35least error at step20exact while the basis is orthogonaland false where the method is best

Two quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 20.

steps: 40

The arguments are the ones An expiry date the noise does not move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The filter description, and the orthogonality it rests onTwo quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 20.26101418222630343810⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹stepsizeleast error: 20‖QᵀQ − I‖filter disagreementan identity with an expiry datedisagreement at step 1210⁻¹³disagreement at step 400.014least error at step20exact while the basis is orthogonaland false where the method is best

Two quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 20.

steps: 40, noise: 0.1

The arguments are the ones An expiry date the noise does not move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The filter description, and the orthogonality it rests onTwo quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 3.26101418222630343810⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹stepsizeleast error: 3‖QᵀQ − I‖filter disagreementan identity with an expiry datedisagreement at step 121.5·10⁻¹²disagreement at step 400.036least error at step3exact while the basis is orthogonaland false where the method is best

Two quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 3.

steps: 40, noise: 0.03

The arguments are the ones An expiry date the noise does not move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The filter description, and the orthogonality it rests onTwo quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 7.26101418222630343810⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹stepsizeleast error: 7‖QᵀQ − I‖filter disagreementan identity with an expiry datedisagreement at step 121.6·10⁻¹³disagreement at step 400.028least error at step7exact while the basis is orthogonaland false where the method is best

Two quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 7.

steps: 40, noise: 0.007

The arguments are the ones An expiry date the noise does not move passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The filter description, and the orthogonality it rests onTwo quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 24.26101418222630343810⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹stepsizeleast error: 24‖QᵀQ − I‖filter disagreementan identity with an expiry datedisagreement at step 122.7·10⁻¹³disagreement at step 400.021least error at step24exact while the basis is orthogonaland false where the method is best

Two quantities against the step count on a logarithmic vertical axis. The disagreement between the measured filter factors and the polynomial predicted from the recurrence stays at the level of rounding for the first dozen steps and then rises through ten orders of magnitude, tracking the loss of orthogonality in the Lanczos basis underneath it. The vertical line marks the step at which the error is least, which is step 24.

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.

6 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.

a size the SVD is affordable at

and have parted company by the end

enough steps for the identity to break inside the figure

Jacobi needs a symmetric matrix

matmul shapes agree

the two routes agree at the early steps

Against the rule

It draws a decomposition and prints its residual. It calls cgls, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

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.

Methods that were designed apart

A parameter that counts steps

The regularisation field's knob is a positive real number chosen by one of three rules. The iterative field's is an integer nobody called a knob — where to stop. On the same problem the best step is 20 and the best λ is 0.025, and they reach 0.1426 and 0.1406.

 Methods that were designed apart

An expiry date the noise does not move

The polynomial description of conjugate gradients leaves the level of rounding at step 17 or 18 on this operator, at every noise level from 10% to 0.1%. The step worth stopping at moves from 3 to 44 across the same range. They coincide at about 1% noise, which is where the coincidence was first read, and it is a fact about the noise rather than about the method.

The whole library · All essays · What must fail