Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

complex-pair-plane

One function in the francis library, called 3 times across 3 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 5 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 the spectrum in the plane, with the pair at 1 ± 2.00i. A complex plane with the real axis marked. Open circles show the eigenvalues the matrix was built from and filled dots show the ones the algorithm returned; two of them sit symmetrically above and below the real axis.

complex-pair-plane is one function in lib/figures/francis.js — francis — the real schur form and the two shifts that are never formed. 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.

The spectrum in the plane, with the pair at 1 ± 2.00iA complex plane with the real axis marked. Open circles show the eigenvalues the matrix was built from and filled dots show the ones the algorithm returned; two of them sit symmetrically above and below the real axis.-5-3-1135-202real partimaginary partbuilt incomputedthe real line — where a real shift lives‖A − ZTZᵀ‖/‖A‖4.1·10⁻¹⁵‖ZᵀZ − I‖4.2·10⁻¹⁵worst eigenvalue error1.1·10⁻¹⁴departure from normality10⁻¹⁸5×5 real matrix, 1 conjugate pairthe answer is not on the axis

A complex plane with the real axis marked. Open circles show the eigenvalues the matrix was built from and filled dots show the ones the algorithm returned; two of them sit symmetrically above and below the real axis.

imag: 2.5

The arguments are the ones A condition number for one eigenvalue passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

The spectrum in the plane, with the pair at 1 ± 2.50iA complex plane with the real axis marked. Open circles show the eigenvalues the matrix was built from and filled dots show the ones the algorithm returned; two of them sit symmetrically above and below the real axis.-5-3-1135-202real partimaginary partbuilt incomputedthe real line — where a real shift lives‖A − ZTZᵀ‖/‖A‖4·10⁻¹⁵‖ZᵀZ − I‖4.4·10⁻¹⁵worst eigenvalue error8.9·10⁻¹⁵departure from normality10⁻¹⁸5×5 real matrix, 1 conjugate pairthe answer is not on the axis

A complex plane with the real axis marked. Open circles show the eigenvalues the matrix was built from and filled dots show the ones the algorithm returned; two of them sit symmetrically above and below the real axis.

imag: 2

The arguments are the ones The form a real matrix can reach passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

The spectrum in the plane, with the pair at 1 ± 2.00iA complex plane with the real axis marked. Open circles show the eigenvalues the matrix was built from and filled dots show the ones the algorithm returned; two of them sit symmetrically above and below the real axis.-5-3-1135-202real partimaginary partbuilt incomputedthe real line — where a real shift lives‖A − ZTZᵀ‖/‖A‖4.1·10⁻¹⁵‖ZᵀZ − I‖4.2·10⁻¹⁵worst eigenvalue error1.1·10⁻¹⁴departure from normality10⁻¹⁸5×5 real matrix, 1 conjugate pairthe answer is not on the axis

A complex plane with the real axis marked. Open circles show the eigenvalues the matrix was built from and filled dots show the ones the algorithm returned; two of them sit symmetrically above and below the real axis.

imag: 1.5

The arguments are the ones Two shifts that are never formed passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

The spectrum in the plane, with the pair at 1 ± 1.50iA complex plane with the real axis marked. Open circles show the eigenvalues the matrix was built from and filled dots show the ones the algorithm returned; two of them sit symmetrically above and below the real axis.-5-3-1135-202real partimaginary partbuilt incomputedthe real line — where a real shift lives‖A − ZTZᵀ‖/‖A‖2.1·10⁻¹⁵‖ZᵀZ − I‖3.7·10⁻¹⁵worst eigenvalue error4.4·10⁻¹⁵departure from normality10⁻¹⁸5×5 real matrix, 1 conjugate pairthe answer is not on the axis

A complex plane with the real axis marked. Open circles show the eigenvalues the matrix was built from and filled dots show the ones the algorithm returned; two of them sit symmetrically above and below the real axis.

What it checked while drawing

Every figure above asserted its own claims on the way to being drawn, and a claim that failed would have failed the build rather than drawn a wrong picture. Those assertions 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.

5 distinct claims across 4 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 = ZTZᵀ

and it found the spectrum the matrix was built from

matmul shapes agree

the double shift converged

the single real shift does not converge on a complex pair

Against the rule

It draws a decomposition and prints its residual. It calls qrAlgorithm, realSchur, 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 52 of 99 generators — 37 print a residual and 15 are exempt with a published reason; 47 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 condition number for one eigenvalue

In the symmetric case every eigenvalue has condition number exactly one. In this four-by-four matrix two of them have condition number 100.005 and the other two have exactly 1, and the number belongs to the eigenvalue rather than to the matrix.

 Eigenvalues, singular values, rank

The form a real matrix can reach

A real matrix with complex eigenvalues has no real triangular form, and the reason is one line — a real triangular matrix has a real diagonal, and a similarity does not move the spectrum. What it has instead is triangular except for one two-by-two block per conjugate pair, and the count is decided by the matrix rather than by where the iteration stopped.

 Eigenvalues, singular values, rank

Two shifts that are never formed

The double shift is defined as a factorisation of (A − μI)(A − μ̄I), which nobody computes. What is computed is the first column of that product — three numbers — and the bulge those three numbers create, pushed down the subdiagonal by n − 2 reflectors until it falls off the bottom.

The whole library · All essays · What must fail