Numerical Linear Algebra
where the algebra stops being the arithmetic
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Error of a difference quotient for J(x)v against ε, on the Bratu problem at n = 64

One function in the matfree library, called 6 times across 1 essay. 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 error of a difference quotient for j(x)v against ε, on the bratu problem at n = 64. The Jacobian of this problem is a formula, so the error of each quotient is measured against a derivative that is exact rather than against a better quotient. The forward difference falls with a slope of 1.01 — first order — reaches 1.28·10⁻¹⁰ at ε = 10^-6, and rises again with a slope of -1.00 as cancellation takes over. The central difference falls with a slope of 2.00 and bottoms at 1.11·10⁻¹² for two residual evaluations instead of one. Neither gets near the unit roundoff at any ε.

difference-floor is one function in lib/figures/matfree.js — matrix-free — what survives when the matrix is a subroutine, and what it costs. 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.

Error of a difference quotient for J(x)v against ε, on the Bratu problem at n = 64The Jacobian of this problem is a formula, so the error of each quotient is measured against a derivative that is exact rather than against a better quotient. The forward difference falls with a slope of 1.01 — first order — reaches 1.28·10⁻¹⁰ at ε = 10^-6, and rises again with a slope of -1.00 as cancellation takes over. The central difference falls with a slope of 2.00 and bottoms at 1.11·10⁻¹² for two residual evaluations instead of one. Neither gets near the unit roundoff at any ε.10⁻¹⁷10⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1εrelative error in J(x)vforwardcentralcancellationtruncationagainst a derivative that is exactforward floor1.3·10⁻¹⁰central floor1.1·10⁻¹²truncation slope, forward1truncation slope, central2no ε reaches the roundoffand the analytic derivative is free of the choice

The Jacobian of this problem is a formula, so the error of each quotient is measured against a derivative that is exact rather than against a better quotient. The forward difference falls with a slope of 1.01 — first order — reaches 1.28·10⁻¹⁰ at ε = 10^-6, and rises again with a slope of -1.00 as cancellation takes over. The central difference falls with a slope of 2.00 and bottoms at 1.11·10⁻¹² for two residual evaluations instead of one. Neither gets near the unit roundoff at any ε.

n: 64

The arguments are the ones An operator with no entries passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Error of a difference quotient for J(x)v against ε, on the Bratu problem at n = 64The Jacobian of this problem is a formula, so the error of each quotient is measured against a derivative that is exact rather than against a better quotient. The forward difference falls with a slope of 1.01 — first order — reaches 1.28·10⁻¹⁰ at ε = 10^-6, and rises again with a slope of -1.00 as cancellation takes over. The central difference falls with a slope of 2.00 and bottoms at 1.11·10⁻¹² for two residual evaluations instead of one. Neither gets near the unit roundoff at any ε.10⁻¹⁷10⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1εrelative error in J(x)vforwardcentralcancellationtruncationagainst a derivative that is exactforward floor1.3·10⁻¹⁰central floor1.1·10⁻¹²truncation slope, forward1truncation slope, central2no ε reaches the roundoffand the analytic derivative is free of the choice

The Jacobian of this problem is a formula, so the error of each quotient is measured against a derivative that is exact rather than against a better quotient. The forward difference falls with a slope of 1.01 — first order — reaches 1.28·10⁻¹⁰ at ε = 10^-6, and rises again with a slope of -1.00 as cancellation takes over. The central difference falls with a slope of 2.00 and bottoms at 1.11·10⁻¹² for two residual evaluations instead of one. Neither gets near the unit roundoff at any ε.

n: 16

The arguments are the ones An operator with no entries passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Error of a difference quotient for J(x)v against ε, on the Bratu problem at n = 16The Jacobian of this problem is a formula, so the error of each quotient is measured against a derivative that is exact rather than against a better quotient. The forward difference falls with a slope of 1.01 — first order — reaches 2.94·10⁻¹⁰ at ε = 10^-6, and rises again with a slope of -0.98 as cancellation takes over. The central difference falls with a slope of 2.00 and bottoms at 5.73·10⁻¹³ for two residual evaluations instead of one. Neither gets near the unit roundoff at any ε.10⁻¹⁷10⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1εrelative error in J(x)vforwardcentralcancellationtruncationagainst a derivative that is exactforward floor2.9·10⁻¹⁰central floor5.7·10⁻¹³truncation slope, forward1truncation slope, central2no ε reaches the roundoffand the analytic derivative is free of the choice

The Jacobian of this problem is a formula, so the error of each quotient is measured against a derivative that is exact rather than against a better quotient. The forward difference falls with a slope of 1.01 — first order — reaches 2.94·10⁻¹⁰ at ε = 10^-6, and rises again with a slope of -0.98 as cancellation takes over. The central difference falls with a slope of 2.00 and bottoms at 5.73·10⁻¹³ for two residual evaluations instead of one. Neither gets near the unit roundoff at any ε.

n: 128

The arguments are the ones An operator with no entries passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Error of a difference quotient for J(x)v against ε, on the Bratu problem at n = 128The Jacobian of this problem is a formula, so the error of each quotient is measured against a derivative that is exact rather than against a better quotient. The forward difference falls with a slope of 1.00 — first order — reaches 3.45·10⁻¹¹ at ε = 10^-5, and rises again with a slope of -0.99 as cancellation takes over. The central difference falls with a slope of 1.98 and bottoms at 3.19·10⁻¹³ for two residual evaluations instead of one. Neither gets near the unit roundoff at any ε.10⁻¹⁷10⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1εrelative error in J(x)vforwardcentralcancellationtruncationagainst a derivative that is exactforward floor3.4·10⁻¹¹central floor3.2·10⁻¹³truncation slope, forward1truncation slope, central2no ε reaches the roundoffand the analytic derivative is free of the choice

The Jacobian of this problem is a formula, so the error of each quotient is measured against a derivative that is exact rather than against a better quotient. The forward difference falls with a slope of 1.00 — first order — reaches 3.45·10⁻¹¹ at ε = 10^-5, and rises again with a slope of -0.99 as cancellation takes over. The central difference falls with a slope of 1.98 and bottoms at 3.19·10⁻¹³ for two residual evaluations instead of one. Neither gets near the unit roundoff at any ε.

n: 32

The arguments are the ones An operator with no entries passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Error of a difference quotient for J(x)v against ε, on the Bratu problem at n = 32The Jacobian of this problem is a formula, so the error of each quotient is measured against a derivative that is exact rather than against a better quotient. The forward difference falls with a slope of 1.01 — first order — reaches 1.2·10⁻¹⁰ at ε = 10^-6, and rises again with a slope of -0.98 as cancellation takes over. The central difference falls with a slope of 2.00 and bottoms at 6.52·10⁻¹³ for two residual evaluations instead of one. Neither gets near the unit roundoff at any ε.10⁻¹⁷10⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1εrelative error in J(x)vforwardcentralcancellationtruncationagainst a derivative that is exactforward floor1.2·10⁻¹⁰central floor6.5·10⁻¹³truncation slope, forward1truncation slope, central2no ε reaches the roundoffand the analytic derivative is free of the choice

The Jacobian of this problem is a formula, so the error of each quotient is measured against a derivative that is exact rather than against a better quotient. The forward difference falls with a slope of 1.01 — first order — reaches 1.2·10⁻¹⁰ at ε = 10^-6, and rises again with a slope of -0.98 as cancellation takes over. The central difference falls with a slope of 2.00 and bottoms at 6.52·10⁻¹³ for two residual evaluations instead of one. Neither gets near the unit roundoff at any ε.

n: 200

The arguments are the ones An operator with no entries passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Error of a difference quotient for J(x)v against ε, on the Bratu problem at n = 200The Jacobian of this problem is a formula, so the error of each quotient is measured against a derivative that is exact rather than against a better quotient. The forward difference falls with a slope of 1.00 — first order — reaches 2.24·10⁻¹¹ at ε = 10^-5, and rises again with a slope of -1.00 as cancellation takes over. The central difference falls with a slope of 1.84 and bottoms at 1.9·10⁻¹³ for two residual evaluations instead of one. Neither gets near the unit roundoff at any ε.10⁻¹⁷10⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1εrelative error in J(x)vforwardcentralcancellationtruncationagainst a derivative that is exactforward floor2.2·10⁻¹¹central floor1.9·10⁻¹³truncation slope, forward1truncation slope, central1.8no ε reaches the roundoffand the analytic derivative is free of the choice

The Jacobian of this problem is a formula, so the error of each quotient is measured against a derivative that is exact rather than against a better quotient. The forward difference falls with a slope of 1.00 — first order — reaches 2.24·10⁻¹¹ at ε = 10^-5, and rises again with a slope of -1.00 as cancellation takes over. The central difference falls with a slope of 1.84 and bottoms at 1.9·10⁻¹³ for two residual evaluations instead of one. Neither gets near the unit roundoff at any ε.

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.

7 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 Bratu parameter below the fold, where the problem has a solution

a size the sweep of residual evaluations can afford

and past the floor both rise as 1/ε, which is the cancellation

and the central difference does better, for a second evaluation

and the central quotient's is second order

the best forward difference is far above the unit roundoff

the forward quotient's truncation error is first order in ε

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.

Iterating, instead of factorising

An operator with no entries

At the sizes where linear algebra is expensive the matrix does not exist. What exists is a subroutine that returns Av. Every Krylov method survives that unchanged; every algorithm that reads an entry disappears. And the derivative such a code computes is accurate to ten digits instead of sixteen, which turns out to cost nothing at all.

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