Sherman–Morrison against a direct solve, on 20×20 systems whose answer is the identity
At its defaults it draws sherman–morrison against a direct solve, on 20×20 systems whose answer is the identity. A is QΛQᵀ with Λ = (1, …, 1, ε) and the rank-one update takes ε back to 1, so A + uvᵀ is the identity and κ of the problem being solved is 1.000000 at every point on the axis. A direct solve returns 1.1·10⁻¹⁶. The update formula, which is exact algebra, returns 2.5·10⁻⁴ — a slope of 1.00 against κ(A), the matrix that was replaced. Its arithmetic is three solves against that matrix and there is nowhere else the error could have come from.
update-inherits is one function in lib/figures/update.js —
rank-one — the correction that is cheaper than the problem, and what it charges. 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.
A is QΛQᵀ with Λ = (1, …, 1, ε) and the rank-one update takes ε back to 1, so A + uvᵀ is the identity and κ of the problem being solved is 1.000000 at every point on the axis. A direct solve returns 1.1·10⁻¹⁶. The update formula, which is exact algebra, returns 2.5·10⁻⁴ — a slope of 1.00 against κ(A), the matrix that was replaced. Its arithmetic is three solves against that matrix and there is nowhere else the error could have come from.
n: 20
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.
A is QΛQᵀ with Λ = (1, …, 1, ε) and the rank-one update takes ε back to 1, so A + uvᵀ is the identity and κ of the problem being solved is 1.000000 at every point on the axis. A direct solve returns 1.1·10⁻¹⁶. The update formula, which is exact algebra, returns 2.5·10⁻⁴ — a slope of 1.00 against κ(A), the matrix that was replaced. Its arithmetic is three solves against that matrix and there is nowhere else the error could have come from.
n: 12
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.
A is QΛQᵀ with Λ = (1, …, 1, ε) and the rank-one update takes ε back to 1, so A + uvᵀ is the identity and κ of the problem being solved is 1.000000 at every point on the axis. A direct solve returns 1.1·10⁻¹⁶. The update formula, which is exact algebra, returns 3.1·10⁻⁴ — a slope of 0.99 against κ(A), the matrix that was replaced. Its arithmetic is three solves against that matrix and there is nowhere else the error could have come from.
n: 30
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.
A is QΛQᵀ with Λ = (1, …, 1, ε) and the rank-one update takes ε back to 1, so A + uvᵀ is the identity and κ of the problem being solved is 1.000000 at every point on the axis. A direct solve returns 10⁻¹⁶. The update formula, which is exact algebra, returns 8.1·10⁻⁴ — a slope of 1.00 against κ(A), the matrix that was replaced. Its arithmetic is three solves against that matrix and there is nowhere else the error could have come from.
n: 40
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.
A is QΛQᵀ with Λ = (1, …, 1, ε) and the rank-one update takes ε back to 1, so A + uvᵀ is the identity and κ of the problem being solved is 1.000000 at every point on the axis. A direct solve returns 1.9·10⁻¹⁶. The update formula, which is exact algebra, returns 0.0031 — a slope of 1.04 against κ(A), the matrix that was replaced. Its arithmetic is three solves against that matrix and there is nowhere else the error could have come from.
n: 6
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.
A is QΛQᵀ with Λ = (1, …, 1, ε) and the rank-one update takes ε back to 1, so A + uvᵀ is the identity and κ of the problem being solved is 1.000000 at every point on the axis. A direct solve returns 3.4·10⁻¹⁷. The update formula, which is exact algebra, returns 3·10⁻⁴ — a slope of 1.00 against κ(A), the matrix that was replaced. Its arithmetic is three solves against that matrix and there is nowhere else the error could have come from.
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.
13 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 direct solve of it is exact at every ε
a size the repeated orthogonal assemblies can afford
and the update formula is not
LU is for square matrices
matmul shapes agree
the updated matrix is the identity at ε = 0.01
the updated matrix is the identity at ε = 10⁻¹⁰
the updated matrix is the identity at ε = 10⁻¹²
the updated matrix is the identity at ε = 10⁻¹⁴
the updated matrix is the identity at ε = 10⁻⁴
the updated matrix is the identity at ε = 10⁻⁶
the updated matrix is the identity at ε = 10⁻⁸
with a slope of one against the OLD matrix's condition number
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
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.
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.
When the problem arrives againStable once, and three thousand times
A sliding window adds a row and removes one at every step and never looks at the data again. No single step of it amplifies by more than 2.72, no downdate fails, and after three thousand steps the triangular factor in memory is 3.9·10⁻¹⁴ from the matrix it is supposed to be a factor of — six hundred times growth from a per-step bound that says nothing about chains.