Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

orthogonality-collapse

One function in the lanczos library, called 3 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 7 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 ‖qᵀq − i‖ at every step of a lanczos run, n = 40. The departure from orthogonality of the Lanczos basis, against the step, on a logarithmic vertical axis. It sits at the level of rounding for 13 steps and then climbs by a factor of about seventeen a step for ten steps running, twelve orders of magnitude, before saturating. The dashed line marks the step at which the first Ritz value converged, which is step 11.

orthogonality-collapse is one function in lib/figures/lanczos.js — lanczos — three terms, lost orthogonality, and an eigenvalue that arrives twice. 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.

‖QᵀQ − I‖ at every step of a Lanczos run, n = 40The departure from orthogonality of the Lanczos basis, against the step, on a logarithmic vertical axis. It sits at the level of rounding for 13 steps and then climbs by a factor of about seventeen a step for ten steps running, twelve orders of magnitude, before saturating. The dashed line marks the step at which the first Ritz value converged, which is step 11.051015202530354010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹step‖QᵀQ − I‖first Ritz value converges√uno reorthogonalisationfullthe two events are one eventstep orthogonality crosses √u14step the first Ritz value converges11consecutive steps at ≥5× growth10a drift would grow like a square rootthis grows like a geometric series and then stops

The departure from orthogonality of the Lanczos basis, against the step, on a logarithmic vertical axis. It sits at the level of rounding for 13 steps and then climbs by a factor of about seventeen a step for ten steps running, twelve orders of magnitude, before saturating. The dashed line marks the step at which the first Ritz value converged, which is step 11.

n: 40

The arguments are the ones An eigenvalue that arrives twice 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.

‖QᵀQ − I‖ at every step of a Lanczos run, n = 40The departure from orthogonality of the Lanczos basis, against the step, on a logarithmic vertical axis. It sits at the level of rounding for 13 steps and then climbs by a factor of about seventeen a step for ten steps running, twelve orders of magnitude, before saturating. The dashed line marks the step at which the first Ritz value converged, which is step 11.051015202530354010⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹step‖QᵀQ − I‖first Ritz value converges√uno reorthogonalisationfullthe two events are one eventstep orthogonality crosses √u14step the first Ritz value converges11consecutive steps at ≥5× growth10a drift would grow like a square rootthis grows like a geometric series and then stops

The departure from orthogonality of the Lanczos basis, against the step, on a logarithmic vertical axis. It sits at the level of rounding for 13 steps and then climbs by a factor of about seventeen a step for ten steps running, twelve orders of magnitude, before saturating. The dashed line marks the step at which the first Ritz value converged, which is step 11.

n: 24

The arguments are the ones An eigenvalue that arrives twice 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.

‖QᵀQ − I‖ at every step of a Lanczos run, n = 24The departure from orthogonality of the Lanczos basis, against the step, on a logarithmic vertical axis. It sits at the level of rounding for 13 steps and then climbs by a factor of about seventeen a step for ten steps running, twelve orders of magnitude, before saturating. The dashed line marks the step at which the first Ritz value converged, which is step 11.0481216202410⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹step‖QᵀQ − I‖first Ritz value converges√uno reorthogonalisationfullthe two events are one eventstep orthogonality crosses √u14step the first Ritz value converges11consecutive steps at ≥5× growth10a drift would grow like a square rootthis grows like a geometric series and then stops

The departure from orthogonality of the Lanczos basis, against the step, on a logarithmic vertical axis. It sits at the level of rounding for 13 steps and then climbs by a factor of about seventeen a step for ten steps running, twelve orders of magnitude, before saturating. The dashed line marks the step at which the first Ritz value converged, which is step 11.

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.

7 distinct claims across 3 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 dense similarity is affordable at

and reorthogonalising prevents all of it

and they happen together

both events happen inside n steps

Jacobi needs a symmetric matrix

matmul shapes agree

the loss is geometric over a run of steps

Against the rule

It draws a decomposition and prints its residual. It calls lanczos, 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 52 of 99 generators — 37 print a residual and 15 are exempt with a published reason; 47 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.

Eigenvalues, singular values, rank

An eigenvalue that arrives twice

A matrix with forty distinct eigenvalues, handed to Lanczos for eighty steps, returns twenty-five extra copies of thirteen of them — the largest arriving five times. Every copy is accurate to 1.9·10⁻⁸ relative. No arithmetic error was made, nothing overflowed, and a caller counting eigenvalues gets the wrong multiplicity from a computation in which no individual number is wrong.

 Iterating, instead of factorising

An orthogonalisation nobody calls one

Conjugate gradients are derived as a minimisation and behave as an orthogonalisation, which is why the finite-termination property in every textbook is not a property the method has in floating point.

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