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THE GAUSSIAN QUADRATURE
n sample points integrate degree 2n-1 exactly
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Gaussian quadrature approximates an integral by a weighted sum of the function at cleverly-chosen points — and n points, placed at the roots of the Legendre polynomial with matching weights, integrate every polynomial up to degree 2n−1 exactly. That is twice the degree a fixed grid of n points could ever manage: the placement, not just the count, buys the accuracy.
It is the backbone of numerical integration in physics and engineering.
LIT verified live: for n=2…5 the n-point Gauss–Legendre rule reproduces the exact integral of random polynomials of degree ≤2n−1, and is (generally) not exact at degree 2n (window.__gaussianquadrature). FIG no framing; exact to the claimed degree.
It is the backbone of numerical integration in physics and engineering.
LIT verified live: for n=2…5 the n-point Gauss–Legendre rule reproduces the exact integral of random polynomials of degree ≤2n−1, and is (generally) not exact at degree 2n (window.__gaussianquadrature). FIG no framing; exact to the claimed degree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mainframe — the heavy exact numerics a mainframe grinds through, integration done with the fewest evaluations possible. Gaussian quadrature is that frugal exactness. AVAN (AI) built the instrument: the Legendre nodes and weights, the weighted sum, the exact-integral cross-check.
Credit as content: Carl Friedrich Gauss (1814), with Jacobi’s later Legendre-root formulation. The weave: David names the mainframe; I place n points at the Legendre roots and confirm they integrate every polynomial of degree ≤2n−1 exactly.
Credit as content: Carl Friedrich Gauss (1814), with Jacobi’s later Legendre-root formulation. The weave: David names the mainframe; I place n points at the Legendre roots and confirm they integrate every polynomial of degree ≤2n−1 exactly.
3 ONE DIMENSION
The n sample points are not evenly spaced — they sit at the roots of the Legendre polynomial, clustered toward the ends. Each carries a weight; together they pin down 2n unknowns (n nodes + n weights), so 2n−1 degrees are exact.
4 TWO DIMENSIONS · INTERACTIVE
A polynomial and its area on [−1,1]; the Gauss rule with n nodes matches the exact integral for degree ≤2n−1.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the n optimal nodes that integrate degree 2n−1 exactly.
AVAN’s addition (the inverse-companion): choose the sample points and weights optimally — nodes at the roots of the Legendre polynomial — so n points integrate polynomials up to degree 2n−1 exactly, twice what a fixed grid of n points manages. The inverse of ‘fix the grid and add points for accuracy’ is ‘place n points perfectly and get 2n−1 for free.’ Magenta is the evenly-spaced samples that waste the budget; green is the n optimal nodes. Where you sample matters more than how many.
LIT Genuine Gauss-Legendre quadrature (Gauss 1814; Jacobi). Verified live: the n-point rule (nodes at Legendre roots, standard weights) reproduces the exact integral on [-1,1] of random polynomials of degree <=2n-1 for n=2..5, and is not exact at degree 2n (window.__gaussianquadrature.exactToDeg && .notExactAbove).
FIG No framing: the Legendre nodes/weights, the weighted sum, and the exact-integral cross-check run in-browser and are exact to the claimed degree. The AVAN inverse is honest — placing n points at the Legendre roots (choosing 2n unknowns: nodes + weights) makes degree 2n-1 exact, double a fixed grid; magenta is the even samples that waste the budget, green the optimal nodes. Where you sample beats how many.
FIG No framing: the Legendre nodes/weights, the weighted sum, and the exact-integral cross-check run in-browser and are exact to the claimed degree. The AVAN inverse is honest — placing n points at the Legendre roots (choosing 2n unknowns: nodes + weights) makes degree 2n-1 exact, double a fixed grid; magenta is the even samples that waste the budget, green the optimal nodes. Where you sample beats how many.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN