Two Krylov methods for one quadratic, priced by the numbers they store

At its defaults it draws two krylov methods for one quadratic, priced by the numbers they store. A chain of 40 masses. Arnoldi on the 80 × 80 linearisation stores m vectors of length 80 and returns m Ritz values; the second-order recurrence stores m vectors of length 40 and its projection is taken onto the QUADRATIC, which has 2m eigenvalues for an m × m problem. So at every subspace size the second route holds half as many numbers and returns twice as many approximations — a factor of four in Ritz values per number stored, exactly and by construction. Plotted against storage, its curve is the lower one at every point: at 240 numbers it is 0.01379 from the dominant eigenvalue where the linearised route needs 480 to reach 0.07294. Both converge slowly — twenty-six vectors buy three digits — because the chain's spectrum is an arc with no isolated extreme, and that is the honest half: this is a comparison of prices rather than a fast method.

second-order-basis is one function in lib/figures/soar.js — a krylov space with no matrix — half the storage, twice the ritz values, and a basis that collapses. 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.

A chain of 40 masses. Arnoldi on the 80 × 80 linearisation stores m vectors of length 80 and returns m Ritz values; the second-order recurrence stores m vectors of length 40 and its projection is taken onto the QUADRATIC, which has 2m eigenvalues for an m × m problem. So at every subspace size the second route holds half as many numbers and returns twice as many approximations — a factor of four in Ritz values per number stored, exactly and by construction. Plotted against storage, its curve is the lower one at every point: at 240 numbers it is 0.01379 from the dominant eigenvalue where the linearised route needs 480 to reach 0.07294. Both converge slowly — twenty-six vectors buy three digits — because the chain's spectrum is an arc with no isolated extreme, and that is the honest half: this is a comparison of prices rather than a fast method.

A chain of 40 masses. Arnoldi on the 80 × 80 linearisation stores m vectors of length 80 and returns m Ritz values; the second-order recurrence stores m vectors of length 40 and its projection is taken onto the QUADRATIC, which has 2m eigenvalues for an m × m problem. So at every subspace size the second route holds half as many numbers and returns twice as many approximations — a factor of four in Ritz values per number stored, exactly and by construction. Plotted against storage, its curve is the lower one at every point: at 240 numbers it is 0.01379 from the dominant eigenvalue where the linearised route needs 480 to reach 0.07294. Both converge slowly — twenty-six vectors buy three digits — because the chain's spectrum is an arc with no isolated extreme, and that is the honest half: this is a comparison of prices rather than a fast method.

A chain of 16 masses. Arnoldi on the 32 × 32 linearisation stores m vectors of length 32 and returns m Ritz values; the second-order recurrence stores m vectors of length 16 and its projection is taken onto the QUADRATIC, which has 2m eigenvalues for an m × m problem. So at every subspace size the second route holds half as many numbers and returns twice as many approximations — a factor of four in Ritz values per number stored, exactly and by construction. Plotted against storage, its curve is the lower one at every point: at 96 numbers it is 0.01231 from the dominant eigenvalue where the linearised route needs 192 to reach 0.07122. Both converge slowly — twenty-six vectors buy three digits — because the chain's spectrum is an arc with no isolated extreme, and that is the honest half: this is a comparison of prices rather than a fast method.

A chain of 56 masses. Arnoldi on the 112 × 112 linearisation stores m vectors of length 112 and returns m Ritz values; the second-order recurrence stores m vectors of length 56 and its projection is taken onto the QUADRATIC, which has 2m eigenvalues for an m × m problem. So at every subspace size the second route holds half as many numbers and returns twice as many approximations — a factor of four in Ritz values per number stored, exactly and by construction. Plotted against storage, its curve is the lower one at every point: at 336 numbers it is 0.01393 from the dominant eigenvalue where the linearised route needs 672 to reach 0.07166. Both converge slowly — twenty-six vectors buy three digits — because the chain's spectrum is an arc with no isolated extreme, and that is the honest half: this is a comparison of prices rather than a fast method.

A chain of 24 masses. Arnoldi on the 48 × 48 linearisation stores m vectors of length 48 and returns m Ritz values; the second-order recurrence stores m vectors of length 24 and its projection is taken onto the QUADRATIC, which has 2m eigenvalues for an m × m problem. So at every subspace size the second route holds half as many numbers and returns twice as many approximations — a factor of four in Ritz values per number stored, exactly and by construction. Plotted against storage, its curve is the lower one at every point: at 144 numbers it is 0.01305 from the dominant eigenvalue where the linearised route needs 288 to reach 0.06995. Both converge slowly — twenty-six vectors buy three digits — because the chain's spectrum is an arc with no isolated extreme, and that is the honest half: this is a comparison of prices rather than a fast method.

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.

11 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.

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.