Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

12 restarts keeping 4 of 8, on a 40×40 matrix

One function in the restart 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 8 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 12 restarts keeping 4 of 8, on a 40×40 matrix. The residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 1.21 to 1.08·10⁻¹³ across 12 cycles and 140 products, and the true error reaches 7.11·10⁻¹⁵. The basis is 8 vectors at every cycle and never grows.

restart-convergence is one function in lib/figures/restart.js — restarting and blocks — a filter on the starting vector, and what one vector cannot see. 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.

12 restarts keeping 4 of 8, on a 40×40 matrixThe residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 1.21 to 1.08·10⁻¹³ across 12 cycles and 140 products, and the true error reaches 7.11·10⁻¹⁵. The basis is 8 vectors at every cycle and never grows.0183654729010812610⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹products with Asizeresidual boundtrue errorbounded memorybasis vectors kept8products with A140worst error in the k wanted7.1·10⁻¹⁵the bound is free and the error is notand the basis never grows

The residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 1.21 to 1.08·10⁻¹³ across 12 cycles and 140 products, and the true error reaches 7.11·10⁻¹⁵. The basis is 8 vectors at every cycle and never grows.

cycles: 12

The arguments are the ones Restarting is a filter passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

12 restarts keeping 4 of 8, on a 40×40 matrixThe residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 1.21 to 1.08·10⁻¹³ across 12 cycles and 140 products, and the true error reaches 7.11·10⁻¹⁵. The basis is 8 vectors at every cycle and never grows.0183654729010812610⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹products with Asizeresidual boundtrue errorbounded memorybasis vectors kept8products with A140worst error in the k wanted7.1·10⁻¹⁵the bound is free and the error is notand the basis never grows

The residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 1.21 to 1.08·10⁻¹³ across 12 cycles and 140 products, and the true error reaches 7.11·10⁻¹⁵. The basis is 8 vectors at every cycle and never grows.

k: 4, p: 4, cycles: 12

The arguments are the ones The same budget, spent five ways passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

12 restarts keeping 4 of 8, on a 40×40 matrixThe residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 1.21 to 1.08·10⁻¹³ across 12 cycles and 140 products, and the true error reaches 7.11·10⁻¹⁵. The basis is 8 vectors at every cycle and never grows.0183654729010812610⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹products with Asizeresidual boundtrue errorbounded memorybasis vectors kept8products with A140worst error in the k wanted7.1·10⁻¹⁵the bound is free and the error is notand the basis never grows

The residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 1.21 to 1.08·10⁻¹³ across 12 cycles and 140 products, and the true error reaches 7.11·10⁻¹⁵. The basis is 8 vectors at every cycle and never grows.

k: 4, p: 1, cycles: 24

The arguments are the ones The same budget, spent five ways passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

24 restarts keeping 4 of 5, on a 40×40 matrixThe residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 0.827 to 6.11·10⁻¹¹ across 24 cycles and 143 products, and the true error reaches 1.24·10⁻¹⁴. The basis is 5 vectors at every cycle and never grows.0183654729010812610⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹products with Asizeresidual boundtrue errorbounded memorybasis vectors kept5products with A143worst error in the k wanted1.2·10⁻¹⁴the bound is free and the error is notand the basis never grows

The residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 0.827 to 6.11·10⁻¹¹ across 24 cycles and 143 products, and the true error reaches 1.24·10⁻¹⁴. The basis is 5 vectors at every cycle and never grows.

k: 4, p: 2, cycles: 16

The arguments are the ones The same budget, spent five ways passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

16 restarts keeping 4 of 6, on a 40×40 matrixThe residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 2.65 to 4.03·10⁻¹² across 16 cycles and 126 products, and the true error reaches 7.11·10⁻¹⁵. The basis is 6 vectors at every cycle and never grows.016324864809611210⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹products with Asizeresidual boundtrue errorbounded memorybasis vectors kept6products with A126worst error in the k wanted7.1·10⁻¹⁵the bound is free and the error is notand the basis never grows

The residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 2.65 to 4.03·10⁻¹² across 16 cycles and 126 products, and the true error reaches 7.11·10⁻¹⁵. The basis is 6 vectors at every cycle and never grows.

k: 4, p: 8, cycles: 8

The arguments are the ones The same budget, spent five ways passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

8 restarts keeping 4 of 12, on a 40×40 matrixThe residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 2.6·10⁻⁴ to 2.37·10⁻¹⁷ across 8 cycles and 152 products, and the true error reaches 1.24·10⁻¹⁴. The basis is 12 vectors at every cycle and never grows.0193857769511413315210⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴products with Asizeresidual boundtrue errorbounded memorybasis vectors kept12products with A152worst error in the k wanted1.2·10⁻¹⁴the bound is free and the error is notand the basis never grows

The residual bound of the worst wanted eigenvalue and its true error, against the number of products with A. The bound falls from 2.6·10⁻⁴ to 2.37·10⁻¹⁷ across 8 cycles and 152 products, and the true error reaches 1.24·10⁻¹⁴. The basis is 12 vectors at every cycle and never grows.

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.

8 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 number of eigenvalues worth asking for

and a basis the restart can afford

and the residual bound has fallen by orders of magnitude

enough cycles to converge and few enough to draw

Jacobi needs a symmetric matrix

matmul shapes agree

on a basis of k + p vectors at every cycle

the 4 wanted eigenvalues are found

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 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

Restarting is a filter

A restart throws away the Ritz values it does not want and begins again from a new starting vector. Written in the eigenbasis, that vector's components have been multiplied by a polynomial with its roots at the discarded values — measured component by component, and agreeing with the polynomial to rounding.

 Eigenvalues, singular values, rank

The same budget, spent five ways

A restarted method has one budget — products with A — and two ways to spend it, in many short cycles or a few long ones. At about a hundred and forty products the answer is the same to a factor of seven whichever split is chosen, and the residual bound the method reports spans ten orders of magnitude across the same five runs.

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