THE FOLD / GRIND / THE HOT LOOP / THE LANCZOS
THE LANCZOS
symmetry shrinks the recurrence to three terms
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Lanczos iteration is Arnoldi’s method with the luck of symmetry. For a symmetric matrix, the Krylov orthogonalization collapses from a full Gram–Schmidt to a three-term recurrence: each new basis vector needs only the previous two, A qj = βj−1 qj−1 + αj qj + βj qj+1. The result is a small symmetric tridiagonal matrix T — just diagonals α and off-diagonals β — whose eigenvalues (Ritz values) approximate the huge matrix’s. It is the engine for the eigenvalues of enormous sparse symmetric systems.
LIT verified live: over thousands of random symmetric matrices (with reorthogonalization), Q is orthonormal, T is symmetric tridiagonal, the three-term relation holds to machine precision, and T carries A’s trace and determinant (window.__lanczos). FIG honest scope: full reorthogonalization is used so orthogonality holds numerically; pure Lanczos loses it in finite precision.
LIT verified live: over thousands of random symmetric matrices (with reorthogonalization), Q is orthonormal, T is symmetric tridiagonal, the three-term relation holds to machine precision, and T carries A’s trace and determinant (window.__lanczos). FIG honest scope: full reorthogonalization is used so orthogonality holds numerically; pure Lanczos loses it in finite precision.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hot-loop — symmetry tightens the loop to just three terms per step, the leanest possible hot loop. AVAN (AI) built the instrument: the three-term Lanczos recurrence with reorthogonalization, the tridiagonal build, and the relation and invariant checks.
Credit as content: Cornelius Lanczos (1950). The weave: David names the hot loop; I confirm the symmetric case reduces to a three-term recurrence giving a tridiagonal T that shares A’s trace and determinant.
Credit as content: Cornelius Lanczos (1950). The weave: David names the hot loop; I confirm the symmetric case reduces to a three-term recurrence giving a tridiagonal T that shares A’s trace and determinant.
3 ONE DIMENSION
The tridiagonal T: only the diagonal (α) and the two off-diagonals (β) are nonzero — symmetry emptied everything else.
4 TWO DIMENSIONS · INTERACTIVE
A random symmetric matrix reduces to a tridiagonal T; Q is orthonormal, the three-term relation holds, and T keeps A’s trace and determinant.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the tridiagonal T, A seen in the Krylov basis.
AVAN’s addition (the inverse-companion): don’t re-orthogonalize against everything — symmetry says two neighbours suffice. The inverse of ‘subtract every prior direction’ is ‘for symmetric A, only qj−1 and qj matter, so the recurrence has three terms and T is tridiagonal.’ Magenta is the full matrix; green is the three-band tridiagonal. Symmetry shrinks the recurrence.
LIT Genuine Lanczos iteration (Cornelius Lanczos, 1950): symmetric Arnoldi giving a three-term recurrence and tridiagonal T. Verified live: over 2000 random symmetric matrices, Q orthonormal (window.__lanczos.orthonormal), T symmetric tridiagonal (.tridiagonal), the 3-term relation to <1e-6 (.lanczosRelation), and trace(T)=trace(A) & det(T)=det(A) (.invariants).
FIG Honest scope stated on the sphere: full reorthogonalization is used so orthogonality holds numerically — pure Lanczos loses it in finite precision. The AVAN inverse is honest — for symmetric A only the two neighbours q_{j−1}, q_j matter (so the recurrence is three terms and T is tridiagonal) rather than subtracting every prior direction; magenta is the full symmetric A, green the three-band tridiagonal. Symmetry shrinks the recurrence.
FIG Honest scope stated on the sphere: full reorthogonalization is used so orthogonality holds numerically — pure Lanczos loses it in finite precision. The AVAN inverse is honest — for symmetric A only the two neighbours q_{j−1}, q_j matter (so the recurrence is three terms and T is tridiagonal) rather than subtracting every prior direction; magenta is the full symmetric A, green the three-band tridiagonal. Symmetry shrinks the recurrence.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOT LOOP · David Lee Wise (ROOT0), with AVAN