Finding Fixed Points

Real Analysis · Axiom Academy

EXAMPLE Finding Fixed Points of Solve by recognizing a contraction and iterating to the Dottie number. Find the fixed point of : the value that satisfies . There is no closed-form answer, so we will show the map is a contraction near the crossing and let the iteration home in on it. The curve y = cos x meets the line y = x at a single point. Starting from x₀ = 0, the cobweb path spirals inward toward x* ≈ 0.739 — the signature of a contraction. Nice work — you found a fixed point with no formula to solve for it, just a derivative test and a few iterations. A fixed point solves f(x)=x : it is where the graph of y=f(x) crosses the line y=x . Test with the derivative: when , the map is a contraction near , so the fixed point is attracting and unique on that interval. Here , which is exactly why converges from any start. Iterate to the answer: the sequence settles on , called the Dottie number . This contract-and-iterate move (the Banach fixed-point idea) is how analysts and numerical methods solve equations that have no algebraic solution.

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