Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

The arrowhead matrix, eliminated from each end

One function in the sparse library, called 7 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 4 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 arrowhead matrix, eliminated from each end. Three sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.

arrowhead-two-ways is one function in lib/figures/sparse.js — sparsity — fill counted two ways, and the ordering that decides the memory. 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 arrowhead matrix, eliminated from each endThree sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.the matrix43 entriestip eliminated first253 entriestip eliminated last43 entries‖A − LLᵀ‖/‖A‖, tip first1.4·10⁻¹⁶‖A − LLᵀ‖/‖A‖, tip last0dense factor is n(n+1)/2 = 253 · sparse factor is 2n − 1 = 43one row swapped to the endnothing numerical chose between them

Three sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.

n: 8

The arguments are the ones A correction cheaper than the problem passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The arrowhead matrix, eliminated from each endThree sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.the matrix15 entriestip eliminated first36 entriestip eliminated last15 entries‖A − LLᵀ‖/‖A‖, tip first3.1·10⁻¹⁷‖A − LLᵀ‖/‖A‖, tip last0dense factor is n(n+1)/2 = 36 · sparse factor is 2n − 1 = 15one row swapped to the endnothing numerical chose between them

Three sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.

n: 22

The arguments are the ones Two ends of the same arrow passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The arrowhead matrix, eliminated from each endThree sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.the matrix43 entriestip eliminated first253 entriestip eliminated last43 entries‖A − LLᵀ‖/‖A‖, tip first1.4·10⁻¹⁶‖A − LLᵀ‖/‖A‖, tip last0dense factor is n(n+1)/2 = 253 · sparse factor is 2n − 1 = 43one row swapped to the endnothing numerical chose between them

Three sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.

n: 10

The arguments are the ones Two ends of the same arrow passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The arrowhead matrix, eliminated from each endThree sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.the matrix19 entriestip eliminated first55 entriestip eliminated last19 entries‖A − LLᵀ‖/‖A‖, tip first1.1·10⁻¹⁶‖A − LLᵀ‖/‖A‖, tip last0dense factor is n(n+1)/2 = 55 · sparse factor is 2n − 1 = 19one row swapped to the endnothing numerical chose between them

Three sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.

n: 16

The arguments are the ones Two ends of the same arrow passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The arrowhead matrix, eliminated from each endThree sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.the matrix31 entriestip eliminated first136 entriestip eliminated last31 entries‖A − LLᵀ‖/‖A‖, tip first8.2·10⁻¹⁷‖A − LLᵀ‖/‖A‖, tip last0dense factor is n(n+1)/2 = 136 · sparse factor is 2n − 1 = 31one row swapped to the endnothing numerical chose between them

Three sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.

n: 28

The arguments are the ones Two ends of the same arrow passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The arrowhead matrix, eliminated from each endThree sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.the matrix55 entriestip eliminated first406 entriestip eliminated last55 entries‖A − LLᵀ‖/‖A‖, tip first1.7·10⁻¹⁶‖A − LLᵀ‖/‖A‖, tip last9.8·10⁻¹⁷dense factor is n(n+1)/2 = 406 · sparse factor is 2n − 1 = 55one row swapped to the endnothing numerical chose between them

Three sparsity plots. The first shows an arrowhead matrix with a dense first row and column. The second shows its Cholesky factor, completely dense. The third shows the factor obtained after moving the dense row to the end, which has no fill at all.

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.

4 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.

and both factorisations are exact

matmul shapes agree

tip first fills the factor completely

tip last creates no fill at all

Against the rule

It draws a decomposition and prints its residual. It calls cholesky, 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 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.

Least squares, and the road not to take

A correction cheaper than the problem

Sherman and Morrison's formula updates a solved system for a rank-one change to the matrix, at 4n² operations instead of (2/3)n³. It is exact algebra. On a problem whose updated matrix is the identity — condition number one, the easiest system there is — it returns a forward error of 2.5·10⁻⁴ where a direct solve returns 10⁻¹⁶.

 Sparsity, and what elimination costs

Two ends of the same arrow

One matrix, one row moved from the front of the elimination order to the back, and the factor goes from completely dense to no fill at all. Both factorisations are exact to rounding, and nothing numerical chose between them.

The whole library · All essays · What must fail