THE FOLD / SPAWN / THE TOOLCHAIN / THE GIVENS
THE GIVENS
QR by plane rotations — a rotation per entry
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Givens rotations factor a matrix into an orthonormal Q and upper-triangular R by zeroing out below-diagonal entries one at a time. Each Givens rotation is a 2×2 plane rotation that annihilates a single element while preserving lengths (it’s orthogonal). Because each rotation touches only two rows, it is ideal for sparse matrices and for updating a factorisation incrementally.
It is the QR method of choice for sparse and streaming problems.
LIT verified live: over 300 random matrices Q·R reconstructs A to ~10⁻¹⁵, R is upper-triangular, and QᵀQ is the identity (window.__givens). FIG no framing; exact factorisation.
It is the QR method of choice for sparse and streaming problems.
LIT verified live: over 300 random matrices Q·R reconstructs A to ~10⁻¹⁵, R is upper-triangular, and QᵀQ is the identity (window.__givens). FIG no framing; exact factorisation.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-toolchain — the numerical-linear-algebra tool that factors by rotations, beside the Householder QR (reflections) and the Cholesky. AVAN (AI) built the instrument: the per-entry plane rotation, the accumulation into Q, the reconstruction / upper-triangular / orthonormality checks.
Credit as content: Wallace Givens (1958). The weave: David names the toolchain; I rotate away one below-diagonal entry at a time and accumulate the rotations into Q, confirming Q·R = A with Q orthonormal.
Credit as content: Wallace Givens (1958). The weave: David names the toolchain; I rotate away one below-diagonal entry at a time and accumulate the rotations into Q, confirming Q·R = A with Q orthonormal.
3 ONE DIMENSION
A Givens rotation spins a 2-vector (a, b) in its plane until the second component is zero: the angle with cos = a/r, sin = b/r sends (a,b) to (r, 0). One entry annihilated, lengths unchanged.
4 TWO DIMENSIONS · INTERACTIVE
A matrix A and its Q, R via Givens rotations; Q·R reconstructs A, R is upper-triangular, QᵀQ is the identity.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the orthonormal Q built from a rotation per entry.
AVAN’s addition (the inverse-companion): zero out one below-diagonal entry at a time with a 2×2 rotation (a Givens rotation), each a plane rotation that annihilates a single element while preserving lengths — ideal for sparse matrices, since it touches only two rows. The inverse of ‘subtract projections column by column (Gram–Schmidt)’ is ‘rotate away one entry at a time.’ Magenta is Gram–Schmidt’s subtractions; green is the plane rotations that zero entries. A rotation per entry builds Q — the sparse-friendly cousin of Householder’s reflections. (Kin to the-householder-qr and the-orthonormal.)
LIT Genuine Givens rotations QR (Givens 1958). Verified live: the plane-rotation factorization gives Q*R = A to max error ~1e-15, R with zero below-diagonal entries, and Q^T*Q equal to the identity, across 300 random matrices (window.__givens.reconstructs && .upperTri && .orthonormal).
FIG No framing: the per-entry plane rotation, the accumulation into Q, and the reconstruction + upper-triangular + orthonormality checks run in-browser and hold to floating precision. The AVAN inverse is honest — each 2x2 rotation annihilates one below-diagonal entry while preserving lengths, touching only two rows (sparse-friendly); magenta is Gram-Schmidt's subtractions, green the plane rotations. The rotation-based cousin of Householder's reflections. Kin to the-householder-qr and the-orthonormal.
FIG No framing: the per-entry plane rotation, the accumulation into Q, and the reconstruction + upper-triangular + orthonormality checks run in-browser and hold to floating precision. The AVAN inverse is honest — each 2x2 rotation annihilates one below-diagonal entry while preserving lengths, touching only two rows (sparse-friendly); magenta is Gram-Schmidt's subtractions, green the plane rotations. The rotation-based cousin of Householder's reflections. Kin to the-householder-qr and the-orthonormal.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE TOOLCHAIN · David Lee Wise (ROOT0), with AVAN