Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Wilkinson's 7×7 matrix and its upper triangular factor

One function in the elim library, called 6 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 wilkinson's 7×7 matrix and its upper triangular factor. The matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to 64 — which is 2^6, exactly the growth bound partial pivoting permits at this size, and it is reached with 0 row interchanges and a residual of 0.

wilkinson-growth is one function in lib/figures/elim.js — elimination — the swap, the growth factor, and the matrix with a known answer. 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.

Wilkinson's 7×7 matrix and its upper triangular factorThe matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to 64 — which is 2^6, exactly the growth bound partial pivoting permits at this size, and it is reached with 0 row interchanges and a residual of 0.1·····1-11····1-1-11···1-1-1-11··1-1-1-1-11·1-1-1-1-1-111-1-1-1-1-1-11A1·····1·1····2··1···4···1··8····1·16·····132······64U‖PA − LU‖/‖A‖0growth factor64the 2ⁿ⁻¹ bound64Row interchanges performed: 0.Partial pivoting had nothing to choose.every entry is 0, 1 or −1the bound is attained here

The matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to 64 — which is 2^6, exactly the growth bound partial pivoting permits at this size, and it is reached with 0 row interchanges and a residual of 0.

n: 7

The arguments are the ones Elimination is a sequence of choices passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Wilkinson's 7×7 matrix and its upper triangular factorThe matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to 64 — which is 2^6, exactly the growth bound partial pivoting permits at this size, and it is reached with 0 row interchanges and a residual of 0.1·····1-11····1-1-11···1-1-1-11··1-1-1-1-11·1-1-1-1-1-111-1-1-1-1-1-11A1·····1·1····2··1···4···1··8····1·16·····132······64U‖PA − LU‖/‖A‖0growth factor64the 2ⁿ⁻¹ bound64Row interchanges performed: 0.Partial pivoting had nothing to choose.every entry is 0, 1 or −1the bound is attained here

The matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to 64 — which is 2^6, exactly the growth bound partial pivoting permits at this size, and it is reached with 0 row interchanges and a residual of 0.

n: 4

The arguments are the ones The bound that is never attained passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Wilkinson's 4×4 matrix and its upper triangular factorThe matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to 8 — which is 2^3, exactly the growth bound partial pivoting permits at this size, and it is reached with 0 row interchanges and a residual of 0.1··1-11·1-1-111-1-1-11A1··1·1·2··14···8U‖PA − LU‖/‖A‖0growth factor8the 2ⁿ⁻¹ bound8Row interchanges performed: 0.Partial pivoting had nothing to choose.every entry is 0, 1 or −1the bound is attained here

The matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to 8 — which is 2^3, exactly the growth bound partial pivoting permits at this size, and it is reached with 0 row interchanges and a residual of 0.

n: 9

The arguments are the ones The bound that is never attained passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Wilkinson's 9×9 matrix and its upper triangular factorThe matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to 256 — which is 2^8, exactly the growth bound partial pivoting permits at this size, and it is reached with 0 row interchanges and a residual of 0.1·······1-11······1-1-11·····1-1-1-11····1-1-1-1-11···1-1-1-1-1-11··1-1-1-1-1-1-11·1-1-1-1-1-1-1-111-1-1-1-1-1-1-1-11A1·······1·1······2··1·····4···1····8····1···16·····1··32······1·64·······1128········256U‖PA − LU‖/‖A‖0growth factor256the 2ⁿ⁻¹ bound256Row interchanges performed: 0.Partial pivoting had nothing to choose.every entry is 0, 1 or −1the bound is attained here

The matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to 256 — which is 2^8, exactly the growth bound partial pivoting permits at this size, and it is reached with 0 row interchanges and a residual of 0.

n: 11

The arguments are the ones The bound that is never attained passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Wilkinson's 11×11 matrix and its upper triangular factorThe matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to 1024 — which is 2^10, exactly the growth bound partial pivoting permits at this size, and it is reached with 0 row interchanges and a residual of 0.1·········1-11········1-1-11·······1-1-1-11······1-1-1-1-11·····1-1-1-1-1-11····1-1-1-1-1-1-11···1-1-1-1-1-1-1-11··1-1-1-1-1-1-1-1-11·1-1-1-1-1-1-1-1-1-111-1-1-1-1-1-1-1-1-1-11A1·········1·1········2··1·······4···1······8····1·····16·····1····32······1···64·······1··128········1·256·········1512··········1024U‖PA − LU‖/‖A‖0growth factor1024the 2ⁿ⁻¹ bound1024Row interchanges performed: 0.Partial pivoting had nothing to choose.every entry is 0, 1 or −1the bound is attained here

The matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to 1024 — which is 2^10, exactly the growth bound partial pivoting permits at this size, and it is reached with 0 row interchanges and a residual of 0.

n: 13

The arguments are the ones The bound that is never attained passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Wilkinson's 13×13 matrix and its upper triangular factorThe matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to 4096 — which is 2^12, exactly the growth bound partial pivoting permits at this size, and it is reached with 0 row interchanges and a residual of 0.1···········1-11··········1-1-11·········1-1-1-11········1-1-1-1-11·······1-1-1-1-1-11······1-1-1-1-1-1-11·····1-1-1-1-1-1-1-11····1-1-1-1-1-1-1-1-11···1-1-1-1-1-1-1-1-1-11··1-1-1-1-1-1-1-1-1-1-11·1-1-1-1-1-1-1-1-1-1-1-111-1-1-1-1-1-1-1-1-1-1-1-11A1···········1·1··········2··1·········4···1········8····1·······16·····1······32······1·····64·······1····128········1···256·········1··512··········1·1024···········12048············4096U‖PA − LU‖/‖A‖0growth factor4096the 2ⁿ⁻¹ bound4096Row interchanges performed: 0.Partial pivoting had nothing to choose.every entry is 0, 1 or −1the bound is attained here

The matrix on the left has ones on the diagonal, minus ones below it and a column of ones at the right. On the right, its U factor, whose last column doubles down the rows to 4096 — which is 2^12, exactly the growth bound partial pivoting permits at this size, and it is reached with 0 row interchanges and a residual of 0.

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 whose two grids fit the canvas stacked, if not abreast

and all of the growth is in the final column agree

and the growth is exactly the bound agree

LU is for square matrices

matmul shapes agree

partial pivoting swaps no row of Wilkinson's matrix

the factorisation is still exact to rounding

Against the rule

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

Elimination, and the swap

Elimination is a sequence of choices

Gaussian elimination is taught as a procedure with no decisions in it. There is one decision at every step — which row to use — and every stability property the algorithm has comes from making it well.

 Elimination, and the swap

The bound that is never attained

Partial pivoting's stability guarantee permits the entries to double at every step — a factor of 5.5·10¹¹ at n = 40. The measured growth on random matrices of that size is about three. The gap is eleven orders of magnitude, and the guarantee is still worth having.

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