Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

The singular values of B, and the number of infinite eigenvalues the exact polynomial says there are

One function in the pencil library, called 6 times across 1 essay. 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 the singular values of b, and the number of infinite eigenvalues the exact polynomial says there are. The pencil has 2 algebraic constraints, so 2 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 4, so n − 4 = 2 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 4.27, 2.6, 0.918, 0.416, 1.13·10⁻¹⁶, 4.1·10⁻¹⁵⁷: the gap is a factor of 2.75·10¹⁴⁰ and it falls at position 5, where the boundary between the 4 nonzero values and the 2 zero ones is at 4 — so the largest gap is INSIDE the null block and a rule that cut there would report rank 5 where the polynomial says 4. It is still a decision, and it is the same decision this site's rank essay is about.

degree-deficit is one function in lib/figures/pencil.js — pencils — two matrices, an eigenvalue with no value, and a problem with no 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.

The singular values of B, and the number of infinite eigenvalues the exact polynomial says there areThe pencil has 2 algebraic constraints, so 2 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 4, so n − 4 = 2 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 4.27, 2.6, 0.918, 0.416, 1.13·10⁻¹⁶, 4.1·10⁻¹⁵⁷: the gap is a factor of 2.75·10¹⁴⁰ and it falls at position 5, where the boundary between the 4 nonzero values and the 2 zero ones is at 4 — so the largest gap is INSIDE the null block and a rule that cut there would report rank 5 where the polynomial says 4. It is still a decision, and it is the same decision this site's rank essay is about.σ14.27σ22.6σ30.918σ40.416σ51.13·10⁻¹⁶σ610⁻¹⁸n = 6, and det(A − λB) has degree 4an integer, and a judgementdegree of det(A − λB), exactly4infinite eigenvalues, from the degree2singular values below the cut2largest gap in the spectrum2.7·10¹⁴⁰a degree cannot be nearly threeand a singular value can be nearly zero

The pencil has 2 algebraic constraints, so 2 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 4, so n − 4 = 2 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 4.27, 2.6, 0.918, 0.416, 1.13·10⁻¹⁶, 4.1·10⁻¹⁵⁷: the gap is a factor of 2.75·10¹⁴⁰ and it falls at position 5, where the boundary between the 4 nonzero values and the 2 zero ones is at 4 — so the largest gap is INSIDE the null block and a rule that cut there would report rank 5 where the polynomial says 4. It is still a decision, and it is the same decision this site's rank essay is about.

k: 3, n: 7

The arguments are the ones The largest gap is inside the null space passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The singular values of B, and the number of infinite eigenvalues the exact polynomial says there areThe pencil has 3 algebraic constraints, so 3 of its 7 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 4, so n − 4 = 3 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 4.87, 2.55, 1.08, 0.782, 8.04·10⁻¹⁷, 0, 0: the gap is a factor of ∞ and it falls at position 5, where the boundary between the 4 nonzero values and the 3 zero ones is at 4 — so the largest gap is INSIDE the null block and a rule that cut there would report rank 5 where the polynomial says 4. It is still a decision, and it is the same decision this site's rank essay is about.σ14.87σ22.55σ31.08σ40.782σ58.04·10⁻¹⁷σ610⁻¹⁸σ710⁻¹⁸n = 7, and det(A − λB) has degree 4an integer, and a judgementdegree of det(A − λB), exactly4infinite eigenvalues, from the degree3singular values below the cut3largest gap in the spectrum∞a degree cannot be nearly threeand a singular value can be nearly zero

The pencil has 3 algebraic constraints, so 3 of its 7 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 4, so n − 4 = 3 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 4.87, 2.55, 1.08, 0.782, 8.04·10⁻¹⁷, 0, 0: the gap is a factor of ∞ and it falls at position 5, where the boundary between the 4 nonzero values and the 3 zero ones is at 4 — so the largest gap is INSIDE the null block and a rule that cut there would report rank 5 where the polynomial says 4. It is still a decision, and it is the same decision this site's rank essay is about.

k: 1

The arguments are the ones The largest gap is inside the null space passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The singular values of B, and the number of infinite eigenvalues the exact polynomial says there areThe pencil has 1 algebraic constraint, so 1 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 5, so n − 5 = 1 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 3.25, 2.36, 1.04, 0.835, 0.3, 5.4·10⁻¹⁵⁸: the gap is a factor of 5.55·10¹⁵⁶ and it falls at position 5, where the boundary between the 5 nonzero values and the 1 zero ones is at 5, so the two agree and the decision is comfortable here. It is still a decision, and it is the same decision this site's rank essay is about.σ13.25σ22.36σ31.04σ40.835σ50.3σ610⁻¹⁸n = 6, and det(A − λB) has degree 5an integer, and a judgementdegree of det(A − λB), exactly5infinite eigenvalues, from the degree1singular values below the cut1largest gap in the spectrum5.6·10¹⁵⁶a degree cannot be nearly threeand a singular value can be nearly zero

The pencil has 1 algebraic constraint, so 1 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 5, so n − 5 = 1 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 3.25, 2.36, 1.04, 0.835, 0.3, 5.4·10⁻¹⁵⁸: the gap is a factor of 5.55·10¹⁵⁶ and it falls at position 5, where the boundary between the 5 nonzero values and the 1 zero ones is at 5, so the two agree and the decision is comfortable here. It is still a decision, and it is the same decision this site's rank essay is about.

k: 2

The arguments are the ones The largest gap is inside the null space passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The singular values of B, and the number of infinite eigenvalues the exact polynomial says there areThe pencil has 2 algebraic constraints, so 2 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 4, so n − 4 = 2 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 4.27, 2.6, 0.918, 0.416, 1.13·10⁻¹⁶, 4.1·10⁻¹⁵⁷: the gap is a factor of 2.75·10¹⁴⁰ and it falls at position 5, where the boundary between the 4 nonzero values and the 2 zero ones is at 4 — so the largest gap is INSIDE the null block and a rule that cut there would report rank 5 where the polynomial says 4. It is still a decision, and it is the same decision this site's rank essay is about.σ14.27σ22.6σ30.918σ40.416σ51.13·10⁻¹⁶σ610⁻¹⁸n = 6, and det(A − λB) has degree 4an integer, and a judgementdegree of det(A − λB), exactly4infinite eigenvalues, from the degree2singular values below the cut2largest gap in the spectrum2.7·10¹⁴⁰a degree cannot be nearly threeand a singular value can be nearly zero

The pencil has 2 algebraic constraints, so 2 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 4, so n − 4 = 2 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 4.27, 2.6, 0.918, 0.416, 1.13·10⁻¹⁶, 4.1·10⁻¹⁵⁷: the gap is a factor of 2.75·10¹⁴⁰ and it falls at position 5, where the boundary between the 4 nonzero values and the 2 zero ones is at 4 — so the largest gap is INSIDE the null block and a rule that cut there would report rank 5 where the polynomial says 4. It is still a decision, and it is the same decision this site's rank essay is about.

k: 3

The arguments are the ones The largest gap is inside the null space passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The singular values of B, and the number of infinite eigenvalues the exact polynomial says there areThe pencil has 3 algebraic constraints, so 3 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 3, so n − 3 = 3 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 1.73, 1, 1, 0, 0, 0: the gap is a factor of ∞ and it falls at position 3, where the boundary between the 3 nonzero values and the 3 zero ones is at 3, so the two agree and the decision is comfortable here. It is still a decision, and it is the same decision this site's rank essay is about.σ11.73σ21σ31σ410⁻¹⁸σ510⁻¹⁸σ610⁻¹⁸n = 6, and det(A − λB) has degree 3an integer, and a judgementdegree of det(A − λB), exactly3infinite eigenvalues, from the degree3singular values below the cut3largest gap in the spectrum∞a degree cannot be nearly threeand a singular value can be nearly zero

The pencil has 3 algebraic constraints, so 3 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 3, so n − 3 = 3 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 1.73, 1, 1, 0, 0, 0: the gap is a factor of ∞ and it falls at position 3, where the boundary between the 3 nonzero values and the 3 zero ones is at 3, so the two agree and the decision is comfortable here. It is still a decision, and it is the same decision this site's rank essay is about.

k: 4

The arguments are the ones The largest gap is inside the null space passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

The singular values of B, and the number of infinite eigenvalues the exact polynomial says there areThe pencil has 4 algebraic constraints, so 4 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 2, so n − 2 = 4 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 3.16, 1, 4.28·10⁻¹⁷, 0, 0, 0: the gap is a factor of ∞ and it falls at position 3, where the boundary between the 2 nonzero values and the 4 zero ones is at 2 — so the largest gap is INSIDE the null block and a rule that cut there would report rank 3 where the polynomial says 2. It is still a decision, and it is the same decision this site's rank essay is about.σ13.16σ21σ34.28·10⁻¹⁷σ410⁻¹⁸σ510⁻¹⁸σ610⁻¹⁸n = 6, and det(A − λB) has degree 2an integer, and a judgementdegree of det(A − λB), exactly2infinite eigenvalues, from the degree4singular values below the cut4largest gap in the spectrum∞a degree cannot be nearly threeand a singular value can be nearly zero

The pencil has 4 algebraic constraints, so 4 of its 6 eigenvalues are infinite. Two routes say so and they are not the same kind of statement. The exact route computes det(A − λB) in BigInt rationals and finds it has degree 2, so n − 2 = 4 eigenvalues are at infinity — an integer, arrived at without rounding. The float route counts the singular values of B below a cut, and the values run 3.16, 1, 4.28·10⁻¹⁷, 0, 0, 0: the gap is a factor of ∞ and it falls at position 3, where the boundary between the 2 nonzero values and the 4 zero ones is at 2 — so the largest gap is INSIDE the null block and a rule that cut there would report rank 3 where the polynomial says 2. It is still a decision, and it is the same decision this site's rank essay is about.

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 pencil small enough for the exact determinant

a regular pencil

an integer entry, which is what makes the exact route exact

and whose B has exactly that many singular values below the cut

at least one constraint, or there is no deficit to draw

matmul shapes agree

whose polynomial loses one degree per constraint

Against the rule

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

Eigenvalues, singular values, rank

The largest gap is inside the null space

The rule recommended for counting a pencil's infinite eigenvalues is to cut at the largest gap in the singular values of B. On integer pencils, with no perturbation anywhere and an exact answer available from the characteristic polynomial, it returns the wrong count on nine of twenty-five — because the singular values that are mathematically zero come back spread over a hundred and forty orders of magnitude, and the largest ratio in the list is between two of them.

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