A reduced model of order 4, and the 4 places it is exact
At its defaults it draws a reduced model of order 4, and the 4 places it is exact. |H(s) − Hᵣ(s)| ÷ |H(s)| along the real axis, for a rational-Krylov reduction of a 24-state model at the interpolation points 1.5, 5, 16, 50. At each of them the curve falls to 5.3·10⁻¹⁶ — the reduced function passes through the original, and its derivative does too, because the projection is two-sided. Between and beyond them it reaches 8.37·10⁻⁴, and there is no bound on it: the method buys 8 exact conditions for 4 solves and offers nothing anywhere else. That is the trade against balanced truncation, which asks for nothing and bounds everything at a cost of two Lyapunov solves.
interpolation-points is one function in lib/figures/ratkrylov.js —
matching the function where you choose — interpolation points, the basis they need, and the fixed point. 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.
|H(s) − Hᵣ(s)| ÷ |H(s)| along the real axis, for a rational-Krylov reduction of a 24-state model at the interpolation points 1.5, 5, 16, 50. At each of them the curve falls to 5.3·10⁻¹⁶ — the reduced function passes through the original, and its derivative does too, because the projection is two-sided. Between and beyond them it reaches 8.37·10⁻⁴, and there is no bound on it: the method buys 8 exact conditions for 4 solves and offers nothing anywhere else. That is the trade against balanced truncation, which asks for nothing and bounds everything at a cost of two Lyapunov solves.
r: 4
The arguments are the ones Exact at the points that were named passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
|H(s) − Hᵣ(s)| ÷ |H(s)| along the real axis, for a rational-Krylov reduction of a 24-state model at the interpolation points 1.5, 5, 16, 50. At each of them the curve falls to 5.3·10⁻¹⁶ — the reduced function passes through the original, and its derivative does too, because the projection is two-sided. Between and beyond them it reaches 8.37·10⁻⁴, and there is no bound on it: the method buys 8 exact conditions for 4 solves and offers nothing anywhere else. That is the trade against balanced truncation, which asks for nothing and bounds everything at a cost of two Lyapunov solves.
r: 2
The arguments are the ones Exact at the points that were named passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
|H(s) − Hᵣ(s)| ÷ |H(s)| along the real axis, for a rational-Krylov reduction of a 24-state model at the interpolation points 1.5, 5. At each of them the curve falls to 10⁻¹⁵ — the reduced function passes through the original, and its derivative does too, because the projection is two-sided. Between and beyond them it reaches 0.0607, and there is no bound on it: the method buys 4 exact conditions for 2 solves and offers nothing anywhere else. That is the trade against balanced truncation, which asks for nothing and bounds everything at a cost of two Lyapunov solves.
r: 3
The arguments are the ones Exact at the points that were named passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
|H(s) − Hᵣ(s)| ÷ |H(s)| along the real axis, for a rational-Krylov reduction of a 24-state model at the interpolation points 1.5, 5, 16. At each of them the curve falls to 1.5·10⁻¹⁵ — the reduced function passes through the original, and its derivative does too, because the projection is two-sided. Between and beyond them it reaches 0.00915, and there is no bound on it: the method buys 6 exact conditions for 3 solves and offers nothing anywhere else. That is the trade against balanced truncation, which asks for nothing and bounds everything at a cost of two Lyapunov solves.
r: 5
The arguments are the ones Exact at the points that were named passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
|H(s) − Hᵣ(s)| ÷ |H(s)| along the real axis, for a rational-Krylov reduction of a 24-state model at the interpolation points 1.5, 5, 16, 50, 160. At each of them the curve falls to 8.9·10⁻¹⁶ — the reduced function passes through the original, and its derivative does too, because the projection is two-sided. Between and beyond them it reaches 2.97·10⁻⁵, and there is no bound on it: the method buys 10 exact conditions for 5 solves and offers nothing anywhere else. That is the trade against balanced truncation, which asks for nothing and bounds everything at a cost of two Lyapunov solves.
r: 6
The arguments are the ones Exact at the points that were named passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
|H(s) − Hᵣ(s)| ÷ |H(s)| along the real axis, for a rational-Krylov reduction of a 24-state model at the interpolation points 1.5, 5, 16, 50, 160, 500. At each of them the curve falls to 1.5·10⁻¹⁵ — the reduced function passes through the original, and its derivative does too, because the projection is two-sided. Between and beyond them it reaches 1.51·10⁻⁷, and there is no bound on it: the method buys 12 exact conditions for 6 solves and offers nothing anywhere else. That is the trade against balanced truncation, which asks for nothing and bounds everything at a cost of two Lyapunov solves.
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.
11 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 degree the state dimension can carry
a grid fine enough to have modes and coarse enough to draw
a number of points the basis can carry
a projection that exists
an actuator and a sensor on the grid
an interpolation point that adds a direction
and does not, away from them
LU is for square matrices
matmul shapes agree
the projection is biorthogonal as normalised
the reduced model matches at every point it was given
Against the rule
It draws a decomposition and prints its residual. It calls
transferBySolve, interpolatoryReduction,
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.