exp-routes
At its defaults it draws two routes to e^a as a 8×8 bidiagonal matrix walks up to a defective one. A_δ is upper bidiagonal with diagonal λ, λ+δ, …, so its eigenvalues are exact and its eigenvectors have a closed form. As δ falls the eigendecomposition route's relative error rises to 2.92·10⁶⁵ while scaling and squaring's falls to 4.09·10⁻¹², which is the distance ‖A_δ − A₀‖ = 7·10⁻¹² to the matrix whose exponential is known in closed form. The two agree at the right of the axis, where the matrix really is far from the limit; the divergence is κ(V), which reaches 3.3·10⁸².
exp-routes is one function in lib/figures/funm.js —
matrix functions — the definition that is not a method, and the vector that was wanted. 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.
A_δ is upper bidiagonal with diagonal λ, λ+δ, …, so its eigenvalues are exact and its eigenvectors have a closed form. As δ falls the eigendecomposition route's relative error rises to 2.92·10⁶⁵ while scaling and squaring's falls to 4.09·10⁻¹², which is the distance ‖A_δ − A₀‖ = 7·10⁻¹² to the matrix whose exponential is known in closed form. The two agree at the right of the axis, where the matrix really is far from the limit; the divergence is κ(V), which reaches 3.3·10⁸².
mu: 1
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.
A_δ is upper bidiagonal with diagonal λ, λ+δ, …, so its eigenvalues are exact and its eigenvectors have a closed form. As δ falls the eigendecomposition route's relative error rises to 2.92·10⁶⁵ while scaling and squaring's falls to 4.09·10⁻¹², which is the distance ‖A_δ − A₀‖ = 7·10⁻¹² to the matrix whose exponential is known in closed form. The two agree at the right of the axis, where the matrix really is far from the limit; the divergence is κ(V), which reaches 3.3·10⁸².
mu: 4
The arguments are the ones A function of a matrix is not a function of its entries 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.
A_δ is upper bidiagonal with diagonal λ, λ+δ, …, so its eigenvalues are exact and its eigenvectors have a closed form. As δ falls the eigendecomposition route's relative error rises to 4.57·10⁶⁸ while scaling and squaring's falls to 3.76·10⁻¹², which is the distance ‖A_δ − A₀‖ = 7·10⁻¹² to the matrix whose exponential is known in closed form. The two agree at the right of the axis, where the matrix really is far from the limit; the divergence is κ(V), which reaches 5.4·10⁸⁶.
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.
7 distinct claims across 3 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 size the closed form and the inversions both afford
a superdiagonal the exponential's entries stay representable at
and the eigendecomposition route is wrong by tens of orders of magnitude
at the far end of the family the two routes agree, because the matrix is genuinely that far from the limit
LU is for square matrices
matmul shapes agree
scaling and squaring is accurate to the distance to the limit and no worse
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 92
of 198 generators —
77 print a residual and
15 are exempt with a published reason;
106 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.
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, rankA function of a matrix is not a function of its entries
Everybody learns that f(A) means diagonalise, apply f to the eigenvalues, undiagonalise. That is a definition, not a method. On a matrix seven picometres from a defective one — with exact eigenvalues and eigenvectors from a closed form — the definition returns an answer wrong by sixty-five orders of magnitude, and a method that never mentions an eigenvalue returns the right one.
Structure, and the solver that cannot see itA limit the matrix never reaches
Szegő's theorem gives a Toeplitz family's condition number in closed form — ((1+ρ)/(1−ρ))², which is 81 at ρ = 0.8. The 8×8 section reaches 52% of it, the 128×128 reaches 98.9%, and none of them ever arrives. A statement about a family is not a statement about the matrix in front of you.
Eigenvalues, singular values, rankThe 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, rankThe 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, rankThe gap decides the eigenvector
A symmetric matrix's eigenvalues move by at most the size of the perturbation, whatever the spectrum looks like. Its eigenvectors are governed by a completely different quantity — the distance to the neighbouring eigenvalue — and at a gap of 10⁻⁹ the same perturbation turns them through 27°.
Eigenvalues, singular values, rankThe series that has to be squared back
The Taylor series for the matrix exponential is not wrong — every term is computed correctly — and on Moler and Van Loan's two-by-two its largest term is 5.4 million times the answer it sums to. The method that replaces it scales the matrix down and squares the result back, and both halves of that sentence cost: too few squarings and the approximant is out of range, too many and each one doubles the rounding.