Fitting the same degree-11 polynomial in two bases

At its defaults it draws fitting the same degree-11 polynomial in two bases. On the left, the data and two fitted curves that lie on top of each other. On the right, the condition numbers of the two design matrices, ten orders of magnitude apart.

basis-conditioning is one function in lib/figures/lsq.js — least squares — the projection, the road not to take, and the valley with no bottom. 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.

On the left, the data and two fitted curves that lie on top of each other. On the right, the condition numbers of the two design matrices, ten orders of magnitude apart.

Three bases for the polynomials of each degree, on a logarithmic axis. The monomial design matrix reaches 1.5·10¹⁷ by degree 48; Chebyshev polynomials reach 1.55·10⁷; the basis built from the sample points by Arnoldi's process is orthonormal on them and its condition number is 1 to rounding at every degree. The dashed line is 1/u, past which a double cannot hold the matrix's conditioning at all.

On the left, the data and two fitted curves that lie on top of each other. On the right, the condition numbers of the two design matrices, ten orders of magnitude apart.

The largest distance, at the sample points, between the least-squares curve computed in monomials or in Chebyshev polynomials and the same fit computed in the basis Arnoldi builds from the points, on a logarithmic axis against the degree. With noise of 0.001 the residual is 8.3·10⁻⁴, and each basis's curve is off by about its condition number times the unit roundoff times that residual — the dashed line of the same colour. At degree 48 the monomial curve is off by 0.004 and the Chebyshev curve by 2.4·10⁻¹⁵.

The largest distance, at the sample points, between the least-squares curve computed in monomials or in Chebyshev polynomials and the same fit computed in the basis Arnoldi builds from the points, on a logarithmic axis against the degree. With noise of 0.1 the residual is 0.083, and each basis's curve is off by about its condition number times the unit roundoff times that residual — the dashed line of the same colour. At degree 48 the monomial curve is off by 0.4 and the Chebyshev curve by 2.9·10⁻¹⁵.

The largest distance, at the sample points, between the least-squares curve computed in monomials or in Chebyshev polynomials and the same fit computed in the basis Arnoldi builds from the points, on a logarithmic axis against the degree. With noise of 0.001 the residual is 8.5·10⁻⁴, and each basis's curve is off by about its condition number times the unit roundoff times that residual — the dashed line of the same colour. At degree 48 the monomial curve is off by 6.6·10⁻⁴ and the Chebyshev curve by 2.9·10⁻¹³.

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.

39 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.

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