Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

tsqr-ladder

One function in the cost library, called 4 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 four factorisations of one 512×12 matrix, κ = 7151. A ranked bar chart of the departure from orthogonality, on a logarithmic scale. Classical Gram–Schmidt loses 4.6·10⁻¹⁰, modified 1.5·10⁻¹², Householder 2·10⁻¹⁴. The reduction tree, which touches the matrix once instead of 12 times, is of the same size as Householder's — and its error does not grow with the depth of the tree.

tsqr-ladder is one function in lib/figures/cost.js — cost — where the flop count stopped predicting the time. 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.

Four factorisations of one 512×12 matrix, κ = 7151A ranked bar chart of the departure from orthogonality, on a logarithmic scale. Classical Gram–Schmidt loses 4.6·10⁻¹⁰, modified 1.5·10⁻¹², Householder 2·10⁻¹⁴. The reduction tree, which touches the matrix once instead of 12 times, is of the same size as Householder's — and its error does not grow with the depth of the tree.classical Gram–Schmidt4.62·10⁻¹⁰modified Gram–Schmidt1.49·10⁻¹²Householder, one sweep2.03·10⁻¹⁴reduction tree, 16 leaves1.48·10⁻¹⁵departure from orthogonality, logarithmicthe tree, at four depths‖AᵀA − RᵀR‖/‖AᵀA‖, depth 13.4·10⁻¹⁵‖AᵀA − RᵀR‖/‖AᵀA‖, depth 24.3·10⁻¹⁵‖AᵀA − RᵀR‖/‖AᵀA‖, depth 31.7·10⁻¹⁵‖AᵀA − RᵀR‖/‖AᵀA‖, depth 41.5·10⁻¹⁵the same algebra, four timestwo of them are products of reflections

A ranked bar chart of the departure from orthogonality, on a logarithmic scale. Classical Gram–Schmidt loses 4.6·10⁻¹⁰, modified 1.5·10⁻¹², Householder 2·10⁻¹⁴. The reduction tree, which touches the matrix once instead of 12 times, is of the same size as Householder's — and its error does not grow with the depth of the tree.

rows: 512

The arguments are the ones A reduction that changes the order 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.

Four factorisations of one 512×12 matrix, κ = 7151A ranked bar chart of the departure from orthogonality, on a logarithmic scale. Classical Gram–Schmidt loses 4.6·10⁻¹⁰, modified 1.5·10⁻¹², Householder 2·10⁻¹⁴. The reduction tree, which touches the matrix once instead of 12 times, is of the same size as Householder's — and its error does not grow with the depth of the tree.classical Gram–Schmidt4.62·10⁻¹⁰modified Gram–Schmidt1.49·10⁻¹²Householder, one sweep2.03·10⁻¹⁴reduction tree, 16 leaves1.48·10⁻¹⁵departure from orthogonality, logarithmicthe tree, at four depths‖AᵀA − RᵀR‖/‖AᵀA‖, depth 13.4·10⁻¹⁵‖AᵀA − RᵀR‖/‖AᵀA‖, depth 24.3·10⁻¹⁵‖AᵀA − RᵀR‖/‖AᵀA‖, depth 31.7·10⁻¹⁵‖AᵀA − RᵀR‖/‖AᵀA‖, depth 41.5·10⁻¹⁵the same algebra, four timestwo of them are products of reflections

A ranked bar chart of the departure from orthogonality, on a logarithmic scale. Classical Gram–Schmidt loses 4.6·10⁻¹⁰, modified 1.5·10⁻¹², Householder 2·10⁻¹⁴. The reduction tree, which touches the matrix once instead of 12 times, is of the same size as Householder's — and its error does not grow with the depth of the tree.

rows: 256, cols: 12

The arguments are the ones A reduction that changes the order 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.

Four factorisations of one 256×12 matrix, κ = 7069A ranked bar chart of the departure from orthogonality, on a logarithmic scale. Classical Gram–Schmidt loses 5·10⁻¹⁰, modified 1.8·10⁻¹², Householder 1.2·10⁻¹⁴. The reduction tree, which touches the matrix once instead of 12 times, is of the same size as Householder's — and its error does not grow with the depth of the tree.classical Gram–Schmidt5.03·10⁻¹⁰modified Gram–Schmidt1.8·10⁻¹²Householder, one sweep1.25·10⁻¹⁴reduction tree, 16 leaves1.69·10⁻¹⁵departure from orthogonality, logarithmicthe tree, at four depths‖AᵀA − RᵀR‖/‖AᵀA‖, depth 14.4·10⁻¹⁵‖AᵀA − RᵀR‖/‖AᵀA‖, depth 21.6·10⁻¹⁵‖AᵀA − RᵀR‖/‖AᵀA‖, depth 31.4·10⁻¹⁵‖AᵀA − RᵀR‖/‖AᵀA‖, depth 41.7·10⁻¹⁵the same algebra, four timestwo of them are products of reflections

A ranked bar chart of the departure from orthogonality, on a logarithmic scale. Classical Gram–Schmidt loses 5·10⁻¹⁰, modified 1.8·10⁻¹², Householder 1.2·10⁻¹⁴. The reduction tree, which touches the matrix once instead of 12 times, is of the same size as Householder's — and its error does not grow with the depth of the tree.

rows: 512, cols: 12

The arguments are the ones Two Gram–Schmidts 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.

Four factorisations of one 512×12 matrix, κ = 7151A ranked bar chart of the departure from orthogonality, on a logarithmic scale. Classical Gram–Schmidt loses 4.6·10⁻¹⁰, modified 1.5·10⁻¹², Householder 2·10⁻¹⁴. The reduction tree, which touches the matrix once instead of 12 times, is of the same size as Householder's — and its error does not grow with the depth of the tree.classical Gram–Schmidt4.62·10⁻¹⁰modified Gram–Schmidt1.49·10⁻¹²Householder, one sweep2.03·10⁻¹⁴reduction tree, 16 leaves1.48·10⁻¹⁵departure from orthogonality, logarithmicthe tree, at four depths‖AᵀA − RᵀR‖/‖AᵀA‖, depth 13.4·10⁻¹⁵‖AᵀA − RᵀR‖/‖AᵀA‖, depth 24.3·10⁻¹⁵‖AᵀA − RᵀR‖/‖AᵀA‖, depth 31.7·10⁻¹⁵‖AᵀA − RᵀR‖/‖AᵀA‖, depth 41.5·10⁻¹⁵the same algebra, four timestwo of them are products of reflections

A ranked bar chart of the departure from orthogonality, on a logarithmic scale. Classical Gram–Schmidt loses 4.6·10⁻¹⁰, modified 1.5·10⁻¹², Householder 2·10⁻¹⁴. The reduction tree, which touches the matrix once instead of 12 times, is of the same size as Householder's — and its error does not grow with the depth of the tree.

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.

11 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.

the tree with 2 leaves is backward stable — asserted 4 times

a tall matrix the norms can be formed for

and a skinny one

and modified Gram–Schmidt sits between them

and the error does not grow with the depth

each row block is at least as tall as it is wide

matmul shapes agree

while classical Gram–Schmidt is not

Against the rule

It draws a decomposition and prints its residual. It calls gramSchmidt, tsqr, 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.

Where the flop count stopped predicting the time

A reduction that changes the order

A tall-skinny QR computed as a tree of independent block factorisations touches a 512×12 matrix once instead of twelve times, computes a completely different sequence of roundings from the sweep it replaces, and returns ‖AᵀA − RᵀR‖/‖AᵀA‖ = 1.65·10⁻¹⁵ against the sweep's 9.95·10⁻¹⁵. On the same matrix classical Gram–Schmidt returns 4.6·10⁻¹⁰.

 Orthogonality, measured

Two Gram–Schmidts

One argument changes. Classical Gram–Schmidt projects the original column onto each previous direction; modified projects what is left of it. In exact arithmetic the coefficients are identical. In floating point they differ by eight orders of magnitude in the thing that matters.

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