Banach Fixed Point Iteration

Real Analysis · Axiom Academy

Compute 2 by repeating one step Pick a guess, then average it with 2÷guess — that's one step of the ancient Babylonian rule, xₙ₊₁ = (xₙ + 2÷xₙ)÷2. Drag the iteration count and watch each guess slam toward 2, the correct digits doubling every step. Why the error shrinks geometrically The contraction factor k is the whole story. Each step multiplies the distance to the answer by k, so after n steps the error is at most kⁿ times the starting error. Drag k and watch the error ladder fall — small k plummets, k near 1 barely moves. Not every iteration is equal. Newton's method (that 2 rule from Beat 1) is a special contraction whose error squares each step — so the correct digits double . A plain linear contraction only adds a fixed number of digits. Drag the step count and watch the two pull apart. One move — repeat a distance-shrinking step — and it converges to the one answer that stays put. That's the engine behind Newton's method & the Babylonian root , solving equations numerically , Picard iteration proving an ODE has a solution, Google PageRank (power iteration to the web's dominant eigenvector), and fractal image compression (iterating a shrinking map back to the picture). Same theorem, everywhere a guess can be refined.

This is the written version of the interactive lesson above. See the full Real Analysis course.