Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

slice-band

One function in the slice library, called 10 times across 4 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 6 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 share of shifts inside a pair of eigenvalues that count it wrongly, against the pair's separation. Two eigenvalues at 1 and 1 + gap, with six others spread around them, and 100 shifts placed strictly between the pair — where the count must read 3. Down to a separation of 10⁻¹² every shift reads it correctly. At 10⁻¹³ one of 100 does not, at 10⁻¹⁴ 19 do not, and at 10⁻¹⁵ none of them reads it correctly. The boundary sits where n‖A‖u puts it — -4.405 for this matrix — because the floating-point count is the exact count of a matrix within that distance of A. This is the only place in the method where the answer can be wrong**, and it is wrong by a whole eigenvalue when it is: the failure is a miscount, not a small error.

slice-band is one function in lib/figures/slice.js — counting a spectrum — an answer that is an integer, and the band where it is wrong by a whole one. 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 share of shifts inside a pair of eigenvalues that count it wrongly, against the pair's separationTwo eigenvalues at 1 and 1 + gap, with six others spread around them, and 100 shifts placed strictly between the pair — where the count must read 3. Down to a separation of 10⁻¹² every shift reads it correctly. At 10⁻¹³ one of 100 does not, at 10⁻¹⁴ 19 do not, and at 10⁻¹⁵ none of them reads it correctly. The boundary sits where n‖A‖u puts it — -4.405 for this matrix — because the floating-point count is the exact count of a matrix within that distance of A. This is the only place in the method where the answer can be wrong**, and it is wrong by a whole eigenvalue when it is: the failure is a miscount, not a small error.-15-13-11-9-7-5-300.250.50.751log₁₀ separation of the pairshare of shifts counted wronglyn‖A‖uthe only place it failswrong at 10⁻¹²0wrong at 10⁻¹⁴19wrong at 10⁻¹⁵100n‖A‖u-4.4wrong by a whole eigenvalueor not wrong at all

Two eigenvalues at 1 and 1 + gap, with six others spread around them, and 100 shifts placed strictly between the pair — where the count must read 3. Down to a separation of 10⁻¹² every shift reads it correctly. At 10⁻¹³ one of 100 does not, at 10⁻¹⁴ 19 do not, and at 10⁻¹⁵ none of them reads it correctly. The boundary sits where n‖A‖u puts it — -4.405 for this matrix — because the floating-point count is the exact count of a matrix within that distance of A. This is the only place in the method where the answer can be wrong**, and it is wrong by a whole eigenvalue when it is: the failure is a miscount, not a small error.

kappa: 1

The arguments are the ones A preconditioner that need not know the constraint 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.

The share of shifts inside a pair of eigenvalues that count it wrongly, against the pair's separationTwo eigenvalues at 1 and 1 + gap, with six others spread around them, and 100 shifts placed strictly between the pair — where the count must read 3. Down to a separation of 10⁻¹² every shift reads it correctly. At 10⁻¹³ one of 100 does not, at 10⁻¹⁴ 19 do not, and at 10⁻¹⁵ none of them reads it correctly. The boundary sits where n‖A‖u puts it — -4.405 for this matrix — because the floating-point count is the exact count of a matrix within that distance of A. This is the only place in the method where the answer can be wrong**, and it is wrong by a whole eigenvalue when it is: the failure is a miscount, not a small error.-15-13-11-9-7-5-300.250.50.751log₁₀ separation of the pairshare of shifts counted wronglyn‖A‖uthe only place it failswrong at 10⁻¹²0wrong at 10⁻¹⁴19wrong at 10⁻¹⁵100n‖A‖u-4.4wrong by a whole eigenvalueor not wrong at all

Two eigenvalues at 1 and 1 + gap, with six others spread around them, and 100 shifts placed strictly between the pair — where the count must read 3. Down to a separation of 10⁻¹² every shift reads it correctly. At 10⁻¹³ one of 100 does not, at 10⁻¹⁴ 19 do not, and at 10⁻¹⁵ none of them reads it correctly. The boundary sits where n‖A‖u puts it — -4.405 for this matrix — because the floating-point count is the exact count of a matrix within that distance of A. This is the only place in the method where the answer can be wrong**, and it is wrong by a whole eigenvalue when it is: the failure is a miscount, not a small error.

kappa: 100

The arguments are the ones A preconditioner that need not know the constraint 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.

The share of shifts inside a pair of eigenvalues that count it wrongly, against the pair's separationTwo eigenvalues at 1 and 1 + gap, with six others spread around them, and 100 shifts placed strictly between the pair — where the count must read 3. Down to a separation of 10⁻¹² every shift reads it correctly. At 10⁻¹³ one of 100 does not, at 10⁻¹⁴ 0 do not, and at 10⁻¹⁵ none of them reads it correctly. The boundary sits where n‖A‖u puts it — 4807 for this matrix — because the floating-point count is the exact count of a matrix within that distance of A. This is the only place in the method where the answer can be wrong**, and it is wrong by a whole eigenvalue when it is: the failure is a miscount, not a small error.-15-13-11-9-7-5-300.250.50.751log₁₀ separation of the pairshare of shifts counted wronglyn‖A‖uthe only place it failswrong at 10⁻¹²0wrong at 10⁻¹⁴0wrong at 10⁻¹⁵100n‖A‖u4807wrong by a whole eigenvalueor not wrong at all

Two eigenvalues at 1 and 1 + gap, with six others spread around them, and 100 shifts placed strictly between the pair — where the count must read 3. Down to a separation of 10⁻¹² every shift reads it correctly. At 10⁻¹³ one of 100 does not, at 10⁻¹⁴ 0 do not, and at 10⁻¹⁵ none of them reads it correctly. The boundary sits where n‖A‖u puts it — 4807 for this matrix — because the floating-point count is the exact count of a matrix within that distance of A. This is the only place in the method where the answer can be wrong**, and it is wrong by a whole eigenvalue when it is: the failure is a miscount, not a small error.

kappa: 1000000

The arguments are the ones An eigenvalue count that cannot be slightly wrong 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.

The share of shifts inside a pair of eigenvalues that count it wrongly, against the pair's separationTwo eigenvalues at 1 and 1 + gap, with six others spread around them, and 100 shifts placed strictly between the pair — where the count must read 3. Down to a separation of 10⁻¹² every shift reads it correctly. At 10⁻¹³ one of 100 does not, at 10⁻¹⁴ 3 do not, and at 10⁻¹⁵ none of them reads it correctly. The boundary sits where n‖A‖u puts it — 4.859·10⁷ for this matrix — because the floating-point count is the exact count of a matrix within that distance of A. This is the only place in the method where the answer can be wrong**, and it is wrong by a whole eigenvalue when it is: the failure is a miscount, not a small error.-15-13-11-9-7-5-300.250.50.751log₁₀ separation of the pairshare of shifts counted wronglyn‖A‖uthe only place it failswrong at 10⁻¹²0wrong at 10⁻¹⁴3wrong at 10⁻¹⁵100n‖A‖u4.9·10⁷wrong by a whole eigenvalueor not wrong at all

Two eigenvalues at 1 and 1 + gap, with six others spread around them, and 100 shifts placed strictly between the pair — where the count must read 3. Down to a separation of 10⁻¹² every shift reads it correctly. At 10⁻¹³ one of 100 does not, at 10⁻¹⁴ 3 do not, and at 10⁻¹⁵ none of them reads it correctly. The boundary sits where n‖A‖u puts it — 4.859·10⁷ for this matrix — because the floating-point count is the exact count of a matrix within that distance of A. This is the only place in the method where the answer can be wrong**, and it is wrong by a whole eigenvalue when it is: the failure is a miscount, not a small error.

kappa: 10000

The arguments are the ones An eigenvalue count that cannot be slightly wrong 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.

The share of shifts inside a pair of eigenvalues that count it wrongly, against the pair's separationTwo eigenvalues at 1 and 1 + gap, with six others spread around them, and 100 shifts placed strictly between the pair — where the count must read 3. Down to a separation of 10⁻¹² every shift reads it correctly. At 10⁻¹³ one of 100 does not, at 10⁻¹⁴ 30 do not, and at 10⁻¹⁵ none of them reads it correctly. The boundary sits where n‖A‖u puts it — 4.859·10⁵ for this matrix — because the floating-point count is the exact count of a matrix within that distance of A. This is the only place in the method where the answer can be wrong**, and it is wrong by a whole eigenvalue when it is: the failure is a miscount, not a small error.-15-13-11-9-7-5-300.250.50.751log₁₀ separation of the pairshare of shifts counted wronglyn‖A‖uthe only place it failswrong at 10⁻¹²0wrong at 10⁻¹⁴30wrong at 10⁻¹⁵100n‖A‖u4.9·10⁵wrong by a whole eigenvalueor not wrong at all

Two eigenvalues at 1 and 1 + gap, with six others spread around them, and 100 shifts placed strictly between the pair — where the count must read 3. Down to a separation of 10⁻¹² every shift reads it correctly. At 10⁻¹³ one of 100 does not, at 10⁻¹⁴ 30 do not, and at 10⁻¹⁵ none of them reads it correctly. The boundary sits where n‖A‖u puts it — 4.859·10⁵ for this matrix — because the floating-point count is the exact count of a matrix within that distance of A. This is the only place in the method where the answer can be wrong**, and it is wrong by a whole eigenvalue when it is: the failure is a miscount, not a small error.

tries: 200

The arguments are the ones An eigenvalue count that cannot be slightly wrong 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.

The share of shifts inside a pair of eigenvalues that count it wrongly, against the pair's separationTwo eigenvalues at 1 and 1 + gap, with six others spread around them, and 200 shifts placed strictly between the pair — where the count must read 3. Down to a separation of 10⁻¹² every shift reads it correctly. At 10⁻¹³ one of 200 does not, at 10⁻¹⁴ 39 do not, and at 10⁻¹⁵ none of them reads it correctly. The boundary sits where n‖A‖u puts it — -4.405 for this matrix — because the floating-point count is the exact count of a matrix within that distance of A. This is the only place in the method where the answer can be wrong**, and it is wrong by a whole eigenvalue when it is: the failure is a miscount, not a small error.-15-13-11-9-7-5-300.250.50.751log₁₀ separation of the pairshare of shifts counted wronglyn‖A‖uthe only place it failswrong at 10⁻¹²0wrong at 10⁻¹⁴39wrong at 10⁻¹⁵200n‖A‖u-4.4wrong by a whole eigenvalueor not wrong at all

Two eigenvalues at 1 and 1 + gap, with six others spread around them, and 200 shifts placed strictly between the pair — where the count must read 3. Down to a separation of 10⁻¹² every shift reads it correctly. At 10⁻¹³ one of 200 does not, at 10⁻¹⁴ 39 do not, and at 10⁻¹⁵ none of them reads it correctly. The boundary sits where n‖A‖u puts it — -4.405 for this matrix — because the floating-point count is the exact count of a matrix within that distance of A. This is the only place in the method where the answer can be wrong**, and it is wrong by a whole eigenvalue when it is: the failure is a miscount, not a small error.

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.

6 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 scaling of the matrix the double can hold

a shift count the figure can afford

a wide gap is resolved at every shift inside it

and the narrowest at none

Jacobi needs a symmetric matrix

with the boundary where n‖A‖u puts it

Against the rule

It draws a decomposition and prints its residual. It calls jacobiEigSym, countBelow, 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 165 of 306 generators — 150 print a residual and 15 are exempt with a published reason; 141 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.

The matrix a constraint makes

A preconditioner that need not know the constraint

Keep the constraint block exactly and replace the objective block by anything positive definite on the null space. The preconditioned matrix then has 2m eigenvalues at exactly one, and its remaining n − m are the generalised eigenvalues of a pencil in which the constraint does not appear. Sweep its condition number over six decades and they do not move in six digits.

 Eigenvalues, singular values, rank

An eigenvalue count that cannot be slightly wrong

Every spectral computation on this site returns floats with errors in them. Counting eigenvalues below a shift by the signs of an unpivoted elimination returns an integer, and an integer cannot be 6.9999999997 — so the answer is exactly right, or wrong by a whole eigenvalue, and where the second happens is a band of measurable width.

 The matrix a constraint makes

The regularisation that legalises every order

Perturb a saddle-point matrix's two blocks in opposite directions and it acquires a factorisation with a diagonal D under every symmetric permutation — not under a good one, under all of them. Five hundred random orderings, five hundred successes, and a growth factor that spans six orders across them.

 The matrix a constraint makes

The zero that is not a missing entry

A constrained minimisation produces a matrix with a zero block, and the zero is a theorem rather than a sparsity pattern. No pivot order makes it positive definite, no precision changes that, and Cholesky does not fail somewhere on it — it fails at the first constraint row, on a number the problem already contained.

The whole library · All essays · What must fail