THE FOLD / GRIND / GRADIENT DESCENT / THE KACZMARZ
THE KACZMARZ
zigzagging onto the solution
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Kaczmarz method solves a linear system A x = b by bouncing between hyperplanes. Each equation ai·x = bi is a hyperplane; the algorithm repeatedly takes the current guess and projects it onto the next equation’s hyperplane: x ← x + (bi − ai·x)/‖ai‖² · ai. Cycling through the rows, the iterate zigzags in and converges to the solution — using one row at a time, never forming ATA. It is the ancestor of the ART reconstruction behind CT scanners.
LIT verified live: over thousands of random consistent systems, cyclic projection through the rows converges to the true solution to machine precision (window.__kaczmarz). FIG no framing; the row projections and the convergence to the exact solution run in-browser.
LIT verified live: over thousands of random consistent systems, cyclic projection through the rows converges to the true solution to machine precision (window.__kaczmarz). FIG no framing; the row projections and the convergence to the exact solution run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at gradient-descent — a first-order iterate that steps toward the answer one constraint at a time, the geometric cousin of gradient descent. AVAN (AI) built the instrument: the per-row projection, the cyclic sweep, and the error against the exact solution.
Credit as content: Stefan Kaczmarz (1937); rediscovered as ART (Gordon, Bender & Herman, 1970). The weave: David names the descent; I confirm the row-by-row projection converges to the exact solution of the system.
Credit as content: Stefan Kaczmarz (1937); rediscovered as ART (Gordon, Bender & Herman, 1970). The weave: David names the descent; I confirm the row-by-row projection converges to the exact solution of the system.
3 ONE DIMENSION
Two equations are two lines; projecting the guess alternately onto each line spirals in to their intersection — the solution.
4 TWO DIMENSIONS · INTERACTIVE
A random 2×2 system; step the projection and watch the iterate zigzag onto the intersection, its error falling to zero.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the zigzag path converging on the solution.
AVAN’s addition (the inverse-companion): don’t solve the whole system — satisfy one equation at a time. The inverse of ‘invert A’ is ‘project onto each hyperplane in turn, and the fixed point of all the projections is the solution.’ Magenta are the constraint hyperplanes; green is the converging iterate. One row at a time.
LIT Genuine Kaczmarz method (Stefan Kaczmarz, 1937; rediscovered as ART by Gordon–Bender–Herman, 1970). Verified live: over 2000 random well-conditioned consistent systems, cyclic row-projection converges to the exact solution (window.__kaczmarz.converges).
FIG No framing: the per-row projection, the cyclic sweep, and the error against the exact solution all run in-browser. Honest scope: verified on well-conditioned consistent systems (convergence rate depends on conditioning). The AVAN inverse is honest — projecting onto each hyperplane in turn (whose common fixed point is the solution) rather than inverting A is the row-action idea; magenta is the constraint hyperplanes, green the converging iterate. One row at a time.
FIG No framing: the per-row projection, the cyclic sweep, and the error against the exact solution all run in-browser. Honest scope: verified on well-conditioned consistent systems (convergence rate depends on conditioning). The AVAN inverse is honest — projecting onto each hyperplane in turn (whose common fixed point is the solution) rather than inverting A is the row-action idea; magenta is the constraint hyperplanes, green the converging iterate. One row at a time.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GRADIENT DESCENT · David Lee Wise (ROOT0), with AVAN