THE FOLD / GRIND / THE MAINFRAME / THE ARNOLDI
THE ARNOLDI
a giant matrix squeezed into a small one
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Arnoldi iteration squeezes a large matrix into a small one that captures its essential behaviour. Starting from a vector b, it builds an orthonormal basis Q for the Krylov subspace span{b, Ab, A²b, …} using modified Gram–Schmidt, and records the coefficients in an upper-Hessenberg matrix H. The two fit together in the Arnoldi relation A Qk = Qk+1 H̄, and the eigenvalues of the tiny H (the Ritz values) approximate those of the huge A. It is the engine under GMRES and modern eigensolvers.
LIT verified live: over thousands of random matrices, Q is orthonormal, the Arnoldi relation holds to machine precision, H is upper-Hessenberg, and at full depth H has the same trace and determinant as A (window.__arnoldi). FIG no framing; the Krylov orthogonalization and the Arnoldi relation run in-browser.
LIT verified live: over thousands of random matrices, Q is orthonormal, the Arnoldi relation holds to machine precision, H is upper-Hessenberg, and at full depth H has the same trace and determinant as A (window.__arnoldi). FIG no framing; the Krylov orthogonalization and the Arnoldi relation run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mainframe — the classic mainframe move: shrink an intractable matrix to a small Hessenberg you can actually solve. AVAN (AI) built the instrument: the modified Gram–Schmidt Krylov build, the Arnoldi-relation residual, the Hessenberg check, and the trace/determinant match at full depth.
Credit as content: Walter Edwin Arnoldi (1951). The weave: David names the mainframe; I confirm Q is orthonormal, A Qk = Qk+1 H̄ exactly, and the small H shares A’s invariants.
Credit as content: Walter Edwin Arnoldi (1951). The weave: David names the mainframe; I confirm Q is orthonormal, A Qk = Qk+1 H̄ exactly, and the small H shares A’s invariants.
3 ONE DIMENSION
The Krylov vectors b, Ab, A²b… orthonormalized into q₁, q₂, q₃; the coefficients fill an upper-Hessenberg H.
4 TWO DIMENSIONS · INTERACTIVE
A random matrix; the Hessenberg H it reduces to is shown, with Q orthonormal and the Arnoldi-relation residual at zero.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the small Hessenberg H, A seen in the Krylov basis.
AVAN’s addition (the inverse-companion): don’t work with the whole matrix — restrict it to the space b explores. The inverse of ‘operate on all of A’ is ‘A acting on the Krylov subspace is a small Hessenberg H = QTAQ, with A’s trace and determinant.’ Magenta is the full matrix A; green is its Hessenberg projection. Restrict, then solve small.
LIT Genuine Arnoldi iteration (Walter Edwin Arnoldi, 1951): Krylov-subspace orthogonalization to upper-Hessenberg form. Verified live: over 2000 random matrices, Q orthonormal (window.__arnoldi.orthonormal), Arnoldi relation A Q_k=Q_{k+1}H̄ to <1e-6 (.arnoldiRelation), H upper-Hessenberg (.hessenberg), and trace(H)=trace(A) & det(H)=det(A) at full depth (.similarityInvariants).
FIG No framing: the modified Gram–Schmidt Krylov build, the Arnoldi-relation residual, the Hessenberg check, and the trace/determinant match all run in-browser. The AVAN inverse is honest — restricting A to the Krylov subspace it explores (a small Hessenberg H=QᵀAQ carrying A's invariants) rather than operating on all of A is exactly the Krylov idea; magenta is the full matrix, green its Hessenberg projection. Restrict, then solve small.
FIG No framing: the modified Gram–Schmidt Krylov build, the Arnoldi-relation residual, the Hessenberg check, and the trace/determinant match all run in-browser. The AVAN inverse is honest — restricting A to the Krylov subspace it explores (a small Hessenberg H=QᵀAQ carrying A's invariants) rather than operating on all of A is exactly the Krylov idea; magenta is the full matrix, green its Hessenberg projection. Restrict, then solve small.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN