Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

interpolation-points

One function in the ratkrylov library, called 13 times across 5 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 11 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 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.

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.10⁻¹110¹10²10³10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵s, on the real axis|H − Hᵣ| ÷ |H|exact where askedpoints4conditions bought8worst at a point5.3·10⁻¹⁶worst away from one8.4·10⁻⁴4 points, 8 conditionsand no bound in between

|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 A basis that is the same subspace and not the same thing 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.

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.10⁻¹110¹10²10³10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵s, on the real axis|H − Hᵣ| ÷ |H|exact where askedpoints4conditions bought8worst at a point5.3·10⁻¹⁶worst away from one8.4·10⁻⁴4 points, 8 conditionsand no bound in between

|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: 6

The arguments are the ones A basis that is the same subspace and not the same thing 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.

A reduced model of order 6, and the 6 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, 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.10⁻¹110¹10²10³10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸s, on the real axis|H − Hᵣ| ÷ |H|exact where askedpoints6conditions bought12worst at a point1.5·10⁻¹⁵worst away from one1.5·10⁻⁷6 points, 12 conditionsand no bound in between

|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.

r: 2

The arguments are the ones Exact at the points that were named 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.

A reduced model of order 2, and the 2 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. 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.10⁻¹110¹10²10³10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²s, on the real axis|H − Hᵣ| ÷ |H|exact where askedpoints2conditions bought4worst at a point10⁻¹⁵worst away from one0.0612 points, 4 conditionsand no bound in between

|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 drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

A reduced model of order 3, and the 3 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. 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.10⁻¹110¹10²10³10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²s, on the real axis|H − Hᵣ| ÷ |H|exact where askedpoints3conditions bought6worst at a point1.5·10⁻¹⁵worst away from one0.00913 points, 6 conditionsand no bound in between

|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 drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.

A reduced model of order 5, and the 5 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, 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.10⁻¹110¹10²10³10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵s, on the real axis|H − Hᵣ| ÷ |H|exact where askedpoints5conditions bought10worst at a point8.9·10⁻¹⁶worst away from one3·10⁻⁵5 points, 10 conditionsand no bound in between

|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.

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.

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 192 of 346 generators — 174 print a residual and 18 are exempt with a published reason; 154 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.

Reduction, and what a model is for

A basis that is the same subspace and not the same thing

The interpolation conditions are conditions on a subspace, so any basis of it will do. The one a derivation writes down reaches a condition number of 7.7·10⁹ by its eighth vector, and the rate at which it gets there is set by a number the user chose with no information.

 The eigenvalue problem that is not linear

An error committed before the arithmetic

Before a nonlinear eigenvalue problem is solved, somebody says where they think the eigenvalues are. That sentence sets the accuracy of everything that follows by five orders, costs nothing to say, and cannot be revised once the approximation built on it is in hand.

 Reduction, and what a model is for

Exact at the points that were named

Balanced truncation asks for nothing and bounds everything, at a cost no large model can pay. The other kind of reduction asks for r numbers, costs r solves, is exact at every one of them — and bounds nothing anywhere else. That trade is the whole of large-scale model reduction.

 Reduction, and what a model is for

Interpolating at the model’s own poles

One choice of interpolation points is not arbitrary — the mirrored poles of the model about to be built. It is a fixed point rather than a guess, and when it is reached it beats a method costing O(n³) — by 0.4 per cent, which is the honest size of the whole contest.

 Reduction, and what a model is for

Where to put the poles of a rational function

Three times in one field the same question arrives from different directions — ADI shifts, rational approximation of a square root, the decay of a Gramian — and it has one answer. Cluster them geometrically towards wherever the function is difficult, and the alternative that looks reasonable costs orders.

The whole library · All essays · What must fail