Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

‖Aᵏ‖ for a 6×6 matrix whose spectral radius is 0.8, with 2 above the diagonal

One function in the pseudo library, called 9 times across 1 essay. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 173 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 ‖aᵏ‖ for a 6×6 matrix whose spectral radius is 0.8, with 2 above the diagonal. Two curves against the power. The norm of Aᵏ rises to 19800 at step 24 before turning over and decaying to 2.5·10⁻⁵ by step 160; ρᵏ = 0.8ᵏ, drawn beside it, falls from the start. The two horizontal lines are the Kreiss bracket: the peak is at least K = 6757 and at most e·n·K = 1.1·10⁵, both computed from the resolvent norms outside the unit circle and not from the powers at all.

transient-growth is one function in lib/figures/pseudo.js — pseudospectra — where the eigenvalues would be, and what the powers do first. 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.

‖Aᵏ‖ for a 6×6 matrix whose spectral radius is 0.8, with 2 above the diagonalTwo curves against the power. The norm of Aᵏ rises to 19800 at step 24 before turning over and decaying to 2.5·10⁻⁵ by step 160; ρᵏ = 0.8ᵏ, drawn beside it, falls from the start. The two horizontal lines are the Kreiss bracket: the peak is at least K = 6757 and at most e·n·K = 1.1·10⁵, both computed from the resolvent norms outside the unit circle and not from the powers at all.027548110813510⁻⁶10⁻⁴10⁻²110²10⁴10⁶power‖Aᵏ‖Kreiss constant 6760e · n · K‖Aᵏ‖ρᵏtwo routes to one peakspectral radius0.8peak of ‖Aᵏ‖2·10⁴Kreiss constant6757e · n · K1.1·10⁵everything here decays in the endand one of these curves says how much first

Two curves against the power. The norm of Aᵏ rises to 19800 at step 24 before turning over and decaying to 2.5·10⁻⁵ by step 160; ρᵏ = 0.8ᵏ, drawn beside it, falls from the start. The two horizontal lines are the Kreiss bracket: the peak is at least K = 6757 and at most e·n·K = 1.1·10⁵, both computed from the resolvent norms outside the unit circle and not from the powers at all.

m: 2

The arguments are the ones A spectral radius that grows first passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

‖Aᵏ‖ for a 6×6 matrix whose spectral radius is 0.8, with 2 above the diagonalTwo curves against the power. The norm of Aᵏ rises to 19800 at step 24 before turning over and decaying to 2.5·10⁻⁵ by step 160; ρᵏ = 0.8ᵏ, drawn beside it, falls from the start. The two horizontal lines are the Kreiss bracket: the peak is at least K = 6757 and at most e·n·K = 1.1·10⁵, both computed from the resolvent norms outside the unit circle and not from the powers at all.027548110813510⁻⁶10⁻⁴10⁻²110²10⁴10⁶power‖Aᵏ‖Kreiss constant 6760e · n · K‖Aᵏ‖ρᵏtwo routes to one peakspectral radius0.8peak of ‖Aᵏ‖2·10⁴Kreiss constant6757e · n · K1.1·10⁵everything here decays in the endand one of these curves says how much first

Two curves against the power. The norm of Aᵏ rises to 19800 at step 24 before turning over and decaying to 2.5·10⁻⁵ by step 160; ρᵏ = 0.8ᵏ, drawn beside it, falls from the start. The two horizontal lines are the Kreiss bracket: the peak is at least K = 6757 and at most e·n·K = 1.1·10⁵, both computed from the resolvent norms outside the unit circle and not from the powers at all.

m: 0

The arguments are the ones A spectral radius that grows first passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

‖Aᵏ‖ for a 6×6 matrix whose spectral radius is 0.8, with 0 above the diagonalTwo curves against the power. The norm of Aᵏ rises to 1 at step 0 before turning over and decaying to 3.12·10⁻¹⁶ by step 160; ρᵏ = 0.8ᵏ, drawn beside it, falls from the start. The two horizontal lines are the Kreiss bracket: the peak is at least K = 0.9998 and at most e·n·K = 16.31, both computed from the resolvent norms outside the unit circle and not from the powers at all.027548110813510⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹10²power‖Aᵏ‖Kreiss constant 1e · n · K‖Aᵏ‖ρᵏtwo routes to one peakspectral radius0.8peak of ‖Aᵏ‖1Kreiss constant1e · n · K16everything here decays in the endand one of these curves says how much first

Two curves against the power. The norm of Aᵏ rises to 1 at step 0 before turning over and decaying to 3.12·10⁻¹⁶ by step 160; ρᵏ = 0.8ᵏ, drawn beside it, falls from the start. The two horizontal lines are the Kreiss bracket: the peak is at least K = 0.9998 and at most e·n·K = 16.31, both computed from the resolvent norms outside the unit circle and not from the powers at all.

n: 4, m: 2

The arguments are the ones A spectral radius that grows first passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

‖Aᵏ‖ for a 4×4 matrix whose spectral radius is 0.8, with 2 above the diagonalTwo curves against the power. The norm of Aᵏ rises to 252.6 at step 14 before turning over and decaying to 3.27·10⁻⁹ by step 160; ρᵏ = 0.8ᵏ, drawn beside it, falls from the start. The two horizontal lines are the Kreiss bracket: the peak is at least K = 107.2 and at most e·n·K = 1165, both computed from the resolvent norms outside the unit circle and not from the powers at all.027548110813510⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10¹10³power‖Aᵏ‖Kreiss constant 107e · n · K‖Aᵏ‖ρᵏtwo routes to one peakspectral radius0.8peak of ‖Aᵏ‖253Kreiss constant107e · n · K1165everything here decays in the endand one of these curves says how much first

Two curves against the power. The norm of Aᵏ rises to 252.6 at step 14 before turning over and decaying to 3.27·10⁻⁹ by step 160; ρᵏ = 0.8ᵏ, drawn beside it, falls from the start. The two horizontal lines are the Kreiss bracket: the peak is at least K = 107.2 and at most e·n·K = 1165, both computed from the resolvent norms outside the unit circle and not from the powers at all.

n: 10, m: 2, steps: 240

The arguments are the ones A spectral radius that grows first passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

‖Aᵏ‖ for a 10×10 matrix whose spectral radius is 0.8, with 2 above the diagonalTwo curves against the power. The norm of Aᵏ rises to 1.5·10⁸ at step 44 before turning over and decaying to 1.32·10⁻⁴ by step 240; ρᵏ = 0.8ᵏ, drawn beside it, falls from the start. The two horizontal lines are the Kreiss bracket: the peak is at least K = 3.8·10⁷ and at most e·n·K = 10⁹, both computed from the resolvent norms outside the unit circle and not from the powers at all.0408012016020024010⁻⁵10⁻²10¹10⁴10⁷10¹⁰power‖Aᵏ‖Kreiss constant 3.8·10⁷e · n · K‖Aᵏ‖ρᵏtwo routes to one peakspectral radius0.8peak of ‖Aᵏ‖1.5·10⁸Kreiss constant3.8·10⁷e · n · K10⁹everything here decays in the endand one of these curves says how much first

Two curves against the power. The norm of Aᵏ rises to 1.5·10⁸ at step 44 before turning over and decaying to 1.32·10⁻⁴ by step 240; ρᵏ = 0.8ᵏ, drawn beside it, falls from the start. The two horizontal lines are the Kreiss bracket: the peak is at least K = 3.8·10⁷ and at most e·n·K = 10⁹, both computed from the resolvent norms outside the unit circle and not from the powers at all.

m: 3

The arguments are the ones A spectral radius that grows first passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

‖Aᵏ‖ for a 6×6 matrix whose spectral radius is 0.8, with 3 above the diagonalTwo curves against the power. The norm of Aᵏ rises to 1.5·10⁵ at step 24 before turning over and decaying to 1.9·10⁻⁴ by step 160; ρᵏ = 0.8ᵏ, drawn beside it, falls from the start. The two horizontal lines are the Kreiss bracket: the peak is at least K = 50880 and at most e·n·K = 8.3·10⁵, both computed from the resolvent norms outside the unit circle and not from the powers at all.027548110813510⁻⁵10⁻³10⁻¹10¹10³10⁵10⁷power‖Aᵏ‖Kreiss constant 50900e · n · K‖Aᵏ‖ρᵏtwo routes to one peakspectral radius0.8peak of ‖Aᵏ‖1.5·10⁵Kreiss constant5.1·10⁴e · n · K8.3·10⁵everything here decays in the endand one of these curves says how much first

Two curves against the power. The norm of Aᵏ rises to 1.5·10⁵ at step 24 before turning over and decaying to 1.9·10⁻⁴ by step 160; ρᵏ = 0.8ᵏ, drawn beside it, falls from the start. The two horizontal lines are the Kreiss bracket: the peak is at least K = 50880 and at most e·n·K = 8.3·10⁵, both computed from the resolvent norms outside the unit circle and not from the powers at all.

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.

173 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.

and fall at every step, at k = 1 — checked 160 times

a normal matrix's powers never rise above one

a size the repeated products can afford

a superdiagonal between none and the largest drawn

and e·n·K is an upper bound on it

before decaying, as the radius promises

enough steps for the transient to turn over and decay

matmul shapes agree

reaching their largest well away from the start

the Kreiss constant is a lower bound on the peak

the powers rise by an order of magnitude or more before turning over

the spectral radius is 0.8 whatever the superdiagonal is

with a Kreiss constant of one

with a Kreiss constant well above one

Against the rule

It draws a decomposition and prints its residual. It calls spectralRadius, 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.

Eigenvalues, singular values, rank

A spectral radius that grows first

ρ(A) below one guarantees that the powers of A go to zero and says nothing about what they do on the way. Here they rise by a factor of twenty thousand before turning over, and the peak is bracketed above and below by a constant computed from the resolvent norms outside the unit circle — two routes to one number, one through the plane and one through the powers.

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