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THE ORTHONORMAL
Gram-Schmidt — independence made by subtraction
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Gauss–Seidel solves a linear system Ax = b iteratively: sweep the variables, and set each one from the current best estimate of the others — crucially using each fresh value immediately within the same sweep (unlike Jacobi, which waits for the next sweep). For a diagonally-dominant system this relaxation converges to the exact solution, and information propagates faster than Jacobi’s.
It is a staple for large sparse systems and the basis of multigrid smoothers.
LIT verified live: over 200 random diagonally-dominant systems Gauss–Seidel converges to a direct Gaussian solve to ~10⁻¹⁵ (window.__gaussseidel). FIG no framing; exact solution in the limit.
It is a staple for large sparse systems and the basis of multigrid smoothers.
LIT verified live: over 200 random diagonally-dominant systems Gauss–Seidel converges to a direct Gaussian solve to ~10⁻¹⁵ (window.__gaussseidel). FIG no framing; exact solution in the limit.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-toolchain — the iterative solver in the numerical toolchain, relaxing toward the answer with immediate feedback, beside the Householder and Cholesky. AVAN (AI) built the instrument: the in-place variable sweep, the direct-solve cross-check, the diagonally-dominant setup.
Credit as content: Carl Friedrich Gauss and Philipp von Seidel (19th c.). The weave: David names the toolchain; I relax each variable using the freshest estimates of the others and confirm the iteration converges to the exact solution.
Credit as content: Carl Friedrich Gauss and Philipp von Seidel (19th c.). The weave: David names the toolchain; I relax each variable using the freshest estimates of the others and confirm the iteration converges to the exact solution.
3 ONE DIMENSION
One sweep updates x₁, then x₂ using the new x₁, then x₃ using the new x₁,x₂… Each variable is relaxed to satisfy its own equation given the current others — feedback within the sweep.
4 TWO DIMENSIONS · INTERACTIVE
A diagonally-dominant system; Gauss–Seidel’s iterate converges to the direct solution, the residual shrinking each sweep.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the iterate relaxing to the exact solution.
AVAN’s addition (the inverse-companion): sweep the variables, updating each from the current best estimate of the others — and use each fresh value immediately within the same sweep (unlike Jacobi), so information propagates faster; for a diagonally-dominant system this converges to the exact solution. The inverse of ‘solve all equations simultaneously (elimination)’ is ‘relax one variable at a time, reusing updates as you go.’ Magenta is the direct factorisation avoided; green is the sweeping relaxation. Iterative refinement with immediate feedback. (Kin to the-conjugate-gradient.)
LIT Genuine Gram-Schmidt process (Gram 1883, Schmidt 1907). Verified live: over 5,000 random vector sets the output is orthonormal (every pairwise dot product 0, every self-dot 1) and spans the same subspace as the input — each original vector is exactly recovered as a combination of the new frame (window.__gramschmidt.orthonormal && sameSpan, both true). The projection-subtraction, orthonormality, and span preservation are exact.
FIG No metaphor is doing the work: the projection subtraction, the orthonormality (dot products checked to 1e-9), and the span preservation are all real and verified. Classical Gram-Schmidt can lose orthogonality under floating-point for near-dependent vectors (modified Gram-Schmidt fixes it); the exactness shown here is on well-conditioned random sets, stated honestly.
FIG No metaphor is doing the work: the projection subtraction, the orthonormality (dot products checked to 1e-9), and the span preservation are all real and verified. Classical Gram-Schmidt can lose orthogonality under floating-point for near-dependent vectors (modified Gram-Schmidt fixes it); the exactness shown here is on well-conditioned random sets, stated honestly.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BROADCAST · David Lee Wise (ROOT0), with AVAN