Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Two routes to e^A as a 8×8 bidiagonal matrix walks up to a defective one

One function in the funm library, called 12 times across 2 essays. 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 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.

Two routes to e^A as a 8×8 bidiagonal matrix walks up to a defective oneA(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 3.3·10⁸².10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁻¹⁴10⁻¹10¹²10²⁵10³⁸10⁵¹10⁶⁴δ, the gap between consecutive eigenvaluesrelative error in eᴬan answer with no correct digitsV f(Λ) V⁻¹scaling and squaring‖A(δ) − A₀‖exact eigenvalues throughoutκ(V) at the smallest δ3.3·10⁸²eigen route2.9·10⁶⁵scaling and squaring4.1·10⁻¹²distance to the limit7·10⁻¹²the eigenvalues are the diagonaland they are exact at every stop

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 function of a matrix is not a function of its entries passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two routes to e^A as a 8×8 bidiagonal matrix walks up to a defective oneA(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 3.3·10⁸².10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁻¹⁴10⁻¹10¹²10²⁵10³⁸10⁵¹10⁶⁴δ, the gap between consecutive eigenvaluesrelative error in eᴬan answer with no correct digitsV f(Λ) V⁻¹scaling and squaring‖A(δ) − A₀‖exact eigenvalues throughoutκ(V) at the smallest δ3.3·10⁸²eigen route2.9·10⁶⁵scaling and squaring4.1·10⁻¹²distance to the limit7·10⁻¹²the eigenvalues are the diagonaland they are exact at every stop

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 nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two routes to e^A as a 8×8 bidiagonal matrix walks up to a defective oneA(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 5.4·10⁸⁶.10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁻¹⁴110¹⁴10²⁸10⁴²10⁵⁶δ, the gap between consecutive eigenvaluesrelative error in eᴬan answer with no correct digitsV f(Λ) V⁻¹scaling and squaring‖A(δ) − A₀‖exact eigenvalues throughoutκ(V) at the smallest δ5.4·10⁸⁶eigen route4.6·10⁶⁸scaling and squaring3.8·10⁻¹²distance to the limit7·10⁻¹²the eigenvalues are the diagonaland they are exact at every stop

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.

mu: 0.25

The arguments are the ones A function of a matrix is not a function of its entries passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two routes to e^A as a 8×8 bidiagonal matrix walks up to a defective oneA(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 2.5·10⁶¹ while scaling and squaring's falls to 4.18·10⁻¹², 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 2·10⁷⁸.10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁻¹⁴10⁻¹10¹²10²⁵10³⁸10⁵¹δ, the gap between consecutive eigenvaluesrelative error in eᴬan answer with no correct digitsV f(Λ) V⁻¹scaling and squaring‖A(δ) − A₀‖exact eigenvalues throughoutκ(V) at the smallest δ2·10⁷⁸eigen route2.5·10⁶¹scaling and squaring4.2·10⁻¹²distance to the limit7·10⁻¹²the eigenvalues are the diagonaland they are exact at every stop

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 2.5·10⁶¹ while scaling and squaring's falls to 4.18·10⁻¹², 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 2·10⁷⁸.

mu: 0.5

The arguments are the ones A function of a matrix is not a function of its entries passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two routes to e^A as a 8×8 bidiagonal matrix walks up to a defective oneA(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 2.96·10⁶³ while scaling and squaring's falls to 4.15·10⁻¹², 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 2.6·10⁸⁰.10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁻¹⁴10⁻¹10¹²10²⁵10³⁸10⁵¹10⁶⁴δ, the gap between consecutive eigenvaluesrelative error in eᴬan answer with no correct digitsV f(Λ) V⁻¹scaling and squaring‖A(δ) − A₀‖exact eigenvalues throughoutκ(V) at the smallest δ2.6·10⁸⁰eigen route3·10⁶³scaling and squaring4.2·10⁻¹²distance to the limit7·10⁻¹²the eigenvalues are the diagonaland they are exact at every stop

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 2.96·10⁶³ while scaling and squaring's falls to 4.15·10⁻¹², 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 2.6·10⁸⁰.

mu: 2

The arguments are the ones A function of a matrix is not a function of its entries passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Two routes to e^A as a 8×8 bidiagonal matrix walks up to a defective oneA(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 1.81·10⁶⁷ while scaling and squaring's falls to 3.96·10⁻¹², 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 4.2·10⁸⁴.10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁻¹⁴110¹⁴10²⁸10⁴²10⁵⁶δ, the gap between consecutive eigenvaluesrelative error in eᴬan answer with no correct digitsV f(Λ) V⁻¹scaling and squaring‖A(δ) − A₀‖exact eigenvalues throughoutκ(V) at the smallest δ4.2·10⁸⁴eigen route1.8·10⁶⁷scaling and squaring4·10⁻¹²distance to the limit7·10⁻¹²the eigenvalues are the diagonaland they are exact at every stop

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 1.81·10⁶⁷ while scaling and squaring's falls to 3.96·10⁻¹², 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 4.2·10⁸⁴.

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 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 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.

Eigenvalues, singular values, rank

A 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.

 Eigenvalues, singular values, rank

The 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.

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