Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

transfer-response

One function in the lti library, called 12 times across 7 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 8 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 transfer function of a 24-state model, by a solve and by a formula. |H(iω)| for the second-difference operator on 24 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 24 × 24 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 1.05·10⁻¹⁴ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 3.85·10⁻¹⁵ at every frequency. A state dimension of 24 and a function of one variable: the gap between those is what the field is about.

transfer-response is one function in lib/figures/lti.js — a model that is a rational function — the degree that is a rank, by two routes that disagree. 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 transfer function of a 24-state model, by a solve and by a formula|H(iω)| for the second-difference operator on 24 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 24 × 24 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 1.05·10⁻¹⁴ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 3.85·10⁻¹⁵ at every frequency. A state dimension of 24 and a function of one variable: the gap between those is what the field is about.10⁻¹110¹10²10³10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹frequency ω|H(iω)|one function, two routesstates24relative, ω ≤ 10010⁻¹⁴against the terms3.9·10⁻¹⁵‖H‖∞0.0924 statesone function of one variable

|H(iω)| for the second-difference operator on 24 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 24 × 24 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 1.05·10⁻¹⁴ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 3.85·10⁻¹⁵ at every frequency. A state dimension of 24 and a function of one variable: the gap between those is what the field is about.

n: 24

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.

The transfer function of a 24-state model, by a solve and by a formula|H(iω)| for the second-difference operator on 24 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 24 × 24 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 1.05·10⁻¹⁴ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 3.85·10⁻¹⁵ at every frequency. A state dimension of 24 and a function of one variable: the gap between those is what the field is about.10⁻¹110¹10²10³10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹frequency ω|H(iω)|one function, two routesstates24relative, ω ≤ 10010⁻¹⁴against the terms3.9·10⁻¹⁵‖H‖∞0.0924 statesone function of one variable

|H(iω)| for the second-difference operator on 24 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 24 × 24 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 1.05·10⁻¹⁴ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 3.85·10⁻¹⁵ at every frequency. A state dimension of 24 and a function of one variable: the gap between those is what the field is about.

n: 8

The arguments are the ones A model that is a rational function 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 transfer function of a 8-state model, by a solve and by a formula|H(iω)| for the second-difference operator on 8 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 8 × 8 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 3.48·10⁻¹⁵ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 7.17·10⁻¹⁶ at every frequency. A state dimension of 8 and a function of one variable: the gap between those is what the field is about.10⁻¹110¹10²10³10⁻¹⁰10⁻⁹10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹frequency ω|H(iω)|one function, two routesstates8relative, ω ≤ 1003.5·10⁻¹⁵against the terms7.2·10⁻¹⁶‖H‖∞0.0748 statesone function of one variable

|H(iω)| for the second-difference operator on 8 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 8 × 8 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 3.48·10⁻¹⁵ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 7.17·10⁻¹⁶ at every frequency. A state dimension of 8 and a function of one variable: the gap between those is what the field is about.

n: 60

The arguments are the ones A model that is a rational function 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 transfer function of a 60-state model, by a solve and by a formula|H(iω)| for the second-difference operator on 60 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 60 × 60 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 4.67·10⁻¹⁴ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 1.44·10⁻¹⁴ at every frequency. A state dimension of 60 and a function of one variable: the gap between those is what the field is about.10⁻¹110¹10²10³10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹frequency ω|H(iω)|one function, two routesstates60relative, ω ≤ 1004.7·10⁻¹⁴against the terms1.4·10⁻¹⁴‖H‖∞0.09260 statesone function of one variable

|H(iω)| for the second-difference operator on 60 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 60 × 60 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 4.67·10⁻¹⁴ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 1.44·10⁻¹⁴ at every frequency. A state dimension of 60 and a function of one variable: the gap between those is what the field is about.

n: 12

The arguments are the ones A model that is a rational function 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 transfer function of a 12-state model, by a solve and by a formula|H(iω)| for the second-difference operator on 12 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 12 × 12 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 2.96·10⁻¹⁵ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 1.23·10⁻¹⁵ at every frequency. A state dimension of 12 and a function of one variable: the gap between those is what the field is about.10⁻¹110¹10²10³10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1frequency ω|H(iω)|one function, two routesstates12relative, ω ≤ 1003·10⁻¹⁵against the terms1.2·10⁻¹⁵‖H‖∞0.1212 statesone function of one variable

|H(iω)| for the second-difference operator on 12 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 12 × 12 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 2.96·10⁻¹⁵ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 1.23·10⁻¹⁵ at every frequency. A state dimension of 12 and a function of one variable: the gap between those is what the field is about.

n: 40

The arguments are the ones A model that is a rational function 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 transfer function of a 40-state model, by a solve and by a formula|H(iω)| for the second-difference operator on 40 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 40 × 40 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 2.18·10⁻¹⁴ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 9.62·10⁻¹⁵ at every frequency. A state dimension of 40 and a function of one variable: the gap between those is what the field is about.10⁻¹110¹10²10³10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹frequency ω|H(iω)|one function, two routesstates40relative, ω ≤ 1002.2·10⁻¹⁴against the terms9.6·10⁻¹⁵‖H‖∞0.09340 statesone function of one variable

|H(iω)| for the second-difference operator on 40 points with one actuator and one sensor. The curve is C(iωI − A)⁻¹B, one factorisation of a complex 40 × 40 matrix per frequency, and the crosses are Σ(cᵀvₖ)(vₖᵀb)/(iω − λₖ) with the eigenvalues and eigenvectors written down in closed form. The two share the coefficient matrices and no code. Below ω = 100 they agree to 2.18·10⁻¹⁴ relatively; above it the modal sum is computing a number far below its own largest term, because the residues sum to cᵀb and the sensor is not at the actuator, so the relative comparison stops meaning what it looks like. Measured against the size of the terms being added, the two agree to 9.62·10⁻¹⁵ at every frequency. A state dimension of 40 and a function of one variable: the gap between those is what the field is about.

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.

8 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 division by a complex number that is not zero

a formula and a factorisation are the same function

a grid fine enough to have modes and cheap enough to draw

a grid fine enough to have modes and coarse enough to draw

a system with a known eigenbasis

an actuator and a sensor on the grid

and agree against the size of what is added

LU is for square matrices

Against the rule

It draws a decomposition and prints its residual. It calls transferBySolve, hinfNorm, 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.

 Reduction, and what a model is for

A model that is a rational function

A state matrix has a hundred thousand rows and the thing anyone wants from it is a function of one complex variable. The number that says how much of that size was ever the complexity is a rank — and the rank a derivation writes down cannot be computed, while one built from samples alone can.

 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

The bound that is known in advance

Almost every error on this site is measured after the fact. Balanced truncation has one that is computable before the reduced model exists, in a norm of a function rather than of a residual — and on ordinary problems it is not an upper bound that is loose. It is attained.

 Reduction, and what a model is for

The product nobody had to form

The Hankel singular values are the square roots of the eigenvalues of PQ. Form that product and half of them stop existing, at a floor this site can predict from one number — and the fix is the one the least-squares field has had since its first essay, arriving in a place with no least-squares problem in it.

 Reduction, and what a model is for

Why a Gramian can be truncated at all

Every method in this field rests on one fact nobody states the reason for — the eigenvalues of a Gramian fall off a cliff. The equation defining it has a rank-one right-hand side and no low-rank structure anywhere — and the answer's decay is a rational approximation problem with a closed-form rate.

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