Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

resolvent-map

One function in the pseudo library, called 18 times across 13 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 9 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 σ_min(zi − a) over the complex plane, for a bidiagonal 6×6 matrix with every eigenvalue at 0.8 and 2 above the diagonal. A square of the complex plane shaded by how small σ_min(zI − A) is. Every eigenvalue is at 0.8, marked by the crosshair; the 10⁻³ level reaches out to 1.33, well outside the unit circle drawn through the picture. A perturbation of the matrix at the level of double-precision rounding can put an eigenvalue anywhere in that region.

resolvent-map 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.

σ_min(zI − A) over the complex plane, for a bidiagonal 6×6 matrix with every eigenvalue at 0.8 and 2 above the diagonalA square of the complex plane shaded by how small σ_min(zI − A) is. Every eigenvalue is at 0.8, marked by the crosshair; the 10⁻³ level reaches out to 1.33, well outside the unit circle drawn through the picture. A perturbation of the matrix at the level of double-precision rounding can put an eigenvalue anywhere in that region.darker is smaller: 10⁻¹⁰, 10⁻⁶, 10⁻³, 10⁻¹one spectrum, two matricesspectral radius0.8reach of the 10⁻³ level1.3eigenvalues, all at0.8the circle is |z| = 1and every eigenvalue is well inside it

A square of the complex plane shaded by how small σ_min(zI − A) is. Every eigenvalue is at 0.8, marked by the crosshair; the 10⁻³ level reaches out to 1.33, well outside the unit circle drawn through the picture. A perturbation of the matrix at the level of double-precision rounding can put an eigenvalue anywhere in that region.

m: 2

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.

σ_min(zI − A) over the complex plane, for a bidiagonal 6×6 matrix with every eigenvalue at 0.8 and 2 above the diagonalA square of the complex plane shaded by how small σ_min(zI − A) is. Every eigenvalue is at 0.8, marked by the crosshair; the 10⁻³ level reaches out to 1.33, well outside the unit circle drawn through the picture. A perturbation of the matrix at the level of double-precision rounding can put an eigenvalue anywhere in that region.darker is smaller: 10⁻¹⁰, 10⁻⁶, 10⁻³, 10⁻¹one spectrum, two matricesspectral radius0.8reach of the 10⁻³ level1.3eigenvalues, all at0.8the circle is |z| = 1and every eigenvalue is well inside it

A square of the complex plane shaded by how small σ_min(zI − A) is. Every eigenvalue is at 0.8, marked by the crosshair; the 10⁻³ level reaches out to 1.33, well outside the unit circle drawn through the picture. A perturbation of the matrix at the level of double-precision rounding can put an eigenvalue anywhere in that region.

m: 3

The arguments are the ones A spectral radius that grows first 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.

σ_min(zI − A) over the complex plane, for a bidiagonal 6×6 matrix with every eigenvalue at 0.8 and 3 above the diagonalA square of the complex plane shaded by how small σ_min(zI − A) is. Every eigenvalue is at 0.8, marked by the crosshair; the 10⁻³ level reaches out to 1.54, well outside the unit circle drawn through the picture. A perturbation of the matrix at the level of double-precision rounding can put an eigenvalue anywhere in that region.darker is smaller: 10⁻¹⁰, 10⁻⁶, 10⁻³, 10⁻¹one spectrum, two matricesspectral radius0.8reach of the 10⁻³ level1.5eigenvalues, all at0.8the circle is |z| = 1and every eigenvalue is well inside it

A square of the complex plane shaded by how small σ_min(zI − A) is. Every eigenvalue is at 0.8, marked by the crosshair; the 10⁻³ level reaches out to 1.54, well outside the unit circle drawn through the picture. A perturbation of the matrix at the level of double-precision rounding can put an eigenvalue anywhere in that region.

m: 4

The arguments are the ones A spectral radius that grows first 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.

σ_min(zI − A) over the complex plane, for a bidiagonal 6×6 matrix with every eigenvalue at 0.8 and 4 above the diagonalA square of the complex plane shaded by how small σ_min(zI − A) is. Every eigenvalue is at 0.8, marked by the crosshair; the 10⁻³ level reaches out to 1.69, well outside the unit circle drawn through the picture. A perturbation of the matrix at the level of double-precision rounding can put an eigenvalue anywhere in that region.darker is smaller: 10⁻¹⁰, 10⁻⁶, 10⁻³, 10⁻¹one spectrum, two matricesspectral radius0.8reach of the 10⁻³ level1.7eigenvalues, all at0.8the circle is |z| = 1and every eigenvalue is well inside it

A square of the complex plane shaded by how small σ_min(zI − A) is. Every eigenvalue is at 0.8, marked by the crosshair; the 10⁻³ level reaches out to 1.69, well outside the unit circle drawn through the picture. A perturbation of the matrix at the level of double-precision rounding can put an eigenvalue anywhere in that region.

m: 1

The arguments are the ones Exact along one axis 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.

σ_min(zI − A) over the complex plane, for a bidiagonal 6×6 matrix with every eigenvalue at 0.8 and 1 above the diagonalA square of the complex plane shaded by how small σ_min(zI − A) is. Every eigenvalue is at 0.8, marked by the crosshair; the 10⁻³ level reaches out to 1.12, well outside the unit circle drawn through the picture. A perturbation of the matrix at the level of double-precision rounding can put an eigenvalue anywhere in that region.darker is smaller: 10⁻¹⁰, 10⁻⁶, 10⁻³, 10⁻¹one spectrum, two matricesspectral radius0.8reach of the 10⁻³ level1.1eigenvalues, all at0.8the circle is |z| = 1and every eigenvalue is well inside it

A square of the complex plane shaded by how small σ_min(zI − A) is. Every eigenvalue is at 0.8, marked by the crosshair; the 10⁻³ level reaches out to 1.12, well outside the unit circle drawn through the picture. A perturbation of the matrix at the level of double-precision rounding can put an eigenvalue anywhere in that region.

m: 0

The arguments are the ones The eigenvalues that are not there 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.

σ_min(zI − A) over the complex plane, for a normal 6×6 matrix with every eigenvalue at 0.8A square of the complex plane shaded by how small σ_min(zI − A) is. The dark bands are circles centred on the eigenvalue at 0.8, because for a normal matrix σ_min is exactly the distance to the nearest eigenvalue and the picture says nothing the spectrum did not. The 10⁻³ level is a disc of radius 10⁻³ around the eigenvalue, which is smaller than one cell of this grid.darker is smaller: 10⁻¹⁰, 10⁻⁶, 10⁻³, 10⁻¹one spectrum, two matricesspectral radius0.8reach of the 10⁻³ level0eigenvalues, all at0.8the circle is |z| = 1and every eigenvalue is well inside it

A square of the complex plane shaded by how small σ_min(zI − A) is. The dark bands are circles centred on the eigenvalue at 0.8, because for a normal matrix σ_min is exactly the distance to the nearest eigenvalue and the picture says nothing the spectrum did not. The 10⁻³ level is a disc of radius 10⁻³ around the eigenvalue, which is smaller than one cell of this grid.

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.

9 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 grid the figure has room for and can afford

a size the grid of small SVDs can afford

a superdiagonal between none and the largest drawn

because σ_min is exactly the distance to the nearest eigenvalue

matmul shapes agree

on a normal matrix the 10⁻³ level does not leave the unit disc

the 10⁻³ pseudospectrum reaches outside the unit circle

the spectral radius is 0.8 whatever the superdiagonal is

well beyond every eigenvalue

Against the rule

It draws a decomposition and prints its residual. It calls spectrum, spectralRadius, pseudospectrumGrid, 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 90 of 174 generators — 75 print a residual and 15 are exempt with a published reason; 84 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

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.

 Iterating, instead of factorising

Coarsening in one direction only

Leave the smoother alone and halve only the strong direction, and the smoothing factor is 0.3340 — identical to line relaxation's, at every anisotropy and every weight, to twelve digits. The convergence factors are then a factor of three apart, and at 45° both repairs fail outright.

 Iterating, instead of factorising

Exact along one axis

The tuned diffusion makes the answer exact at every node, and in two dimensions it holds at exactly one flow angle. Five degrees off the grid the relative error goes from 1.2·10⁻¹⁴ to 6.9, and by twenty degrees the scheme is worse than the upwinding it was built to improve on.

 Eigenvalues, singular values, rank

Symmetry is worth more than precision

A symmetric matrix gives up its eigenvalues to full accuracy however ill-conditioned it is. An unsymmetric one can move them by the eighth root of a perturbation, so the rounding involved in merely storing the matrix shifts the spectrum by a hundredth.

 Eigenvalues, singular values, rank

The algorithm the libraries actually run

Factorise, multiply the factors back in the other order, repeat. That description is complete and correct and produces something nobody would use — on a matrix with eigenvalues +1 and −1 it does not converge at all, and the subdiagonal entry does not move by so much as a rounding error.

 Regularisation, and the answer that is chosen

The basis decides what a filter is

The vocabulary of regularisation is spectral — a method keeps a component or discards it, and the weights are a function of the singular value. Row-normalising a symmetric blur so that it preserves a constant makes it 8.6% asymmetric, and that is enough to move GMRES's weights from 7·10⁻¹⁴ off a function of σ to 4.4·10⁻².

 Eigenvalues, singular values, rank

The eigenvalues that are not there

For a normal matrix the resolvent norm is exactly one over the distance to the nearest eigenvalue, so a picture of it carries nothing the spectrum did not. Move one entry above the diagonal and the region a perturbation of 10⁻⁸ can put an eigenvalue into stops being a disc and reaches out past the unit circle, while every eigenvalue stays at 0.8.

 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.

 Structure, and the solver that cannot see it

The matrix that is one row

A circulant of size 16 is sixteen numbers, has no zero entry anywhere, and hands over its entire spectrum in closed form — the discrete Fourier transform of its first column, exactly. An eigensolver spends a sweep of Jacobi rotations over 256 entries arriving at the same answer, and agrees to 1.2·10⁻¹⁵.

 Eigenvalues, singular values, rank

The plane survives what its vectors do not

At a gap of 10⁻⁹ a perturbation of 10⁻⁶ turns the two eigenvectors through half a radian and turns the plane they span through 7.6·10⁻⁸ — a ratio of six million. Ask for the subspace instead of the vectors and a hopeless computation becomes a well-conditioned one, with no change to the arithmetic.

 Iterating, instead of factorising

The spectrum that predicts nothing

For a symmetric matrix the eigenvalues govern how fast an iteration converges. Drop symmetry and they stop governing anything — there is a matrix whose eigenvalues are as evenly spread as eigenvalues can be, on which GMRES makes no progress at all until the last possible step.

 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.

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