Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

What the solver reported, and what it achieved

One function in the mixedcg 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 11 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 what the solver reported, and what it achieved. Final relative residual and final relative error against the working precision, on a logarithmic vertical axis. The residual sits at the stopping tolerance at every precision — the method converged, by every test available to it. The error runs from 1.4·10⁻¹³ at 53 bits to 0.00597 at 8, tracking the unit roundoff, which is drawn beside it.

residual-blindness is one function in lib/figures/mixedcg.js — precision inside an iteration — what may be rounded, and what may not. 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.

What the solver reported, and what it achievedFinal relative residual and final relative error against the working precision, on a logarithmic vertical axis. The residual sits at the stopping tolerance at every precision — the method converged, by every test available to it. The error runs from 1.4·10⁻¹³ at 53 bits to 0.00597 at 8, tracking the unit roundoff, which is drawn beside it.614223038465410⁻¹³10⁻¹⁰10⁻⁷10⁻⁴working significand bitsrelative sizeerrorresidualuthe gap the solver cannot seeerror ÷ residual at 24 bits1.5·10⁶error ÷ residual at 16 bits2.3·10⁷error ÷ residual at 8 bits2.4·10¹⁰every convergence test passesand the answer is wrong in proportion to u

Final relative residual and final relative error against the working precision, on a logarithmic vertical axis. The residual sits at the stopping tolerance at every precision — the method converged, by every test available to it. The error runs from 1.4·10⁻¹³ at 53 bits to 0.00597 at 8, tracking the unit roundoff, which is drawn beside it.

k: 6

The arguments are the ones The observation that cannot be removed passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

What the solver reported, and what it achievedFinal relative residual and final relative error against the working precision, on a logarithmic vertical axis. The residual sits at the stopping tolerance at every precision — the method converged, by every test available to it. The error runs from 4.91·10⁻¹³ at 53 bits to 0.00501 at 8, tracking the unit roundoff, which is drawn beside it.614223038465410⁻¹³10⁻¹⁰10⁻⁷10⁻⁴working significand bitsrelative sizeerrorresidualuthe gap the solver cannot seeerror ÷ residual at 24 bits2.3·10⁵error ÷ residual at 16 bits5.2·10⁷error ÷ residual at 8 bits3.9·10¹⁰every convergence test passesand the answer is wrong in proportion to u

Final relative residual and final relative error against the working precision, on a logarithmic vertical axis. The residual sits at the stopping tolerance at every precision — the method converged, by every test available to it. The error runs from 4.91·10⁻¹³ at 53 bits to 0.00501 at 8, tracking the unit roundoff, which is drawn beside it.

k: 10

The arguments are the ones The reading that never moves passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

What the solver reported, and what it achievedFinal relative residual and final relative error against the working precision, on a logarithmic vertical axis. The residual sits at the stopping tolerance at every precision — the method converged, by every test available to it. The error runs from 1.4·10⁻¹³ at 53 bits to 0.00597 at 8, tracking the unit roundoff, which is drawn beside it.614223038465410⁻¹³10⁻¹⁰10⁻⁷10⁻⁴working significand bitsrelative sizeerrorresidualuthe gap the solver cannot seeerror ÷ residual at 24 bits1.5·10⁶error ÷ residual at 16 bits2.3·10⁷error ÷ residual at 8 bits2.4·10¹⁰every convergence test passesand the answer is wrong in proportion to u

Final relative residual and final relative error against the working precision, on a logarithmic vertical axis. The residual sits at the stopping tolerance at every precision — the method converged, by every test available to it. The error runs from 1.4·10⁻¹³ at 53 bits to 0.00597 at 8, tracking the unit roundoff, which is drawn beside it.

k: 14

The arguments are the ones The reading that never moves passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

What the solver reported, and what it achievedFinal relative residual and final relative error against the working precision, on a logarithmic vertical axis. The residual sits at the stopping tolerance at every precision — the method converged, by every test available to it. The error runs from 2.75·10⁻¹² at 53 bits to 0.0126 at 8, tracking the unit roundoff, which is drawn beside it.614223038465410⁻¹²10⁻⁹10⁻⁶10⁻³working significand bitsrelative sizeerrorresidualuthe gap the solver cannot seeerror ÷ residual at 24 bits2.7·10⁵error ÷ residual at 16 bits4.8·10⁷error ÷ residual at 8 bits1.3·10¹⁰every convergence test passesand the answer is wrong in proportion to u

Final relative residual and final relative error against the working precision, on a logarithmic vertical axis. The residual sits at the stopping tolerance at every precision — the method converged, by every test available to it. The error runs from 2.75·10⁻¹² at 53 bits to 0.0126 at 8, tracking the unit roundoff, which is drawn beside it.

k: 8

The arguments are the ones The reading that never moves passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

What the solver reported, and what it achievedFinal relative residual and final relative error against the working precision, on a logarithmic vertical axis. The residual sits at the stopping tolerance at every precision — the method converged, by every test available to it. The error runs from 5.26·10⁻¹⁴ at 53 bits to 0.00804 at 8, tracking the unit roundoff, which is drawn beside it.614223038465410⁻¹³10⁻¹⁰10⁻⁷10⁻⁴working significand bitsrelative sizeerrorresidualuthe gap the solver cannot seeerror ÷ residual at 24 bits4.1·10⁵error ÷ residual at 16 bits3.6·10⁷error ÷ residual at 8 bits1.9·10¹⁰every convergence test passesand the answer is wrong in proportion to u

Final relative residual and final relative error against the working precision, on a logarithmic vertical axis. The residual sits at the stopping tolerance at every precision — the method converged, by every test available to it. The error runs from 5.26·10⁻¹⁴ at 53 bits to 0.00804 at 8, tracking the unit roundoff, which is drawn beside it.

k: 12

The arguments are the ones The reading that never moves passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

What the solver reported, and what it achievedFinal relative residual and final relative error against the working precision, on a logarithmic vertical axis. The residual sits at the stopping tolerance at every precision — the method converged, by every test available to it. The error runs from 9.06·10⁻¹³ at 53 bits to 0.0114 at 8, tracking the unit roundoff, which is drawn beside it.614223038465410⁻¹³10⁻¹⁰10⁻⁷10⁻⁴working significand bitsrelative sizeerrorresidualuthe gap the solver cannot seeerror ÷ residual at 24 bits5.9·10⁵error ÷ residual at 16 bits1.2·10⁸error ÷ residual at 8 bits2.7·10¹⁰every convergence test passesand the answer is wrong in proportion to u

Final relative residual and final relative error against the working precision, on a logarithmic vertical axis. The residual sits at the stopping tolerance at every precision — the method converged, by every test available to it. The error runs from 9.06·10⁻¹³ at 53 bits to 0.0114 at 8, tracking the unit roundoff, which is drawn beside it.

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.

11 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.

the reported residual says converged at 32 bits — checked 5 times

a grid the dense reference solve is affordable on

and the error tracks u across the range

and the true error is orders of magnitude above it

LU is for square matrices

matmul shapes agree

the incomplete Cholesky factorisation exists on the model problem

Against the rule

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

Least squares, and the road not to take

The observation that cannot be removed

Removing a rank-one term from a Cholesky factor needs a rotation that is not orthogonal, and the number under its square root is 1 − h, where h is the leverage of the row being removed. The algorithm's breakdown condition and the statistician's warning are the same quantity, arrived at from opposite ends, and neither field states it in the other's language.

 The answer that depends on the machine

The reading that never moves

Across thirty runs — five grids from 36 to 196 unknowns, six working precisions from 53 significand bits down to 8 — the residual conjugate gradients stops on stays between 1.10·10⁻¹³ and 9.95·10⁻¹³. Over the same thirty runs the error of the answer spans a factor of 2.39·10¹¹, and the step count more than doubles. The one number the run publishes is the only one that responds to neither axis.

The whole library · All essays · What must fail