Picard iteration
/ pee-KAR /
Picard iteration is a way to manufacture the solution of an initial value problem by improving a guess over and over until it stops changing. It is both a proof device (it powers Picard-Lindelof) and an honest construction you can carry out by hand on small examples. The trick is to convert the differential equation into an integral equation first, because integrals are gentle and forgiving where derivatives are demanding.
Here is the recipe, step by step. Start from y' = f(x, y), y(x0) = y0 and rewrite it as y(x) = y0 + integral from x0 to x of f(t, y(t)) dt. Now build a sequence of functions: take a crude first guess y0(x) = y0 (the constant). Feed it into the right-hand side to get the next approximation, y1(x) = y0 + integral from x0 to x of f(t, y0(t)) dt. Feed that in to get y2, and keep going: y_{n+1}(x) = y0 + integral from x0 to x of f(t, y_n(t)) dt. Each pass uses the previous approximation inside the integral to spit out a better one. Under the Lipschitz condition these iterates close in on the true solution, and the gap between consecutive ones shrinks geometrically, so convergence is fast.
The point is that this is the contraction-mapping principle wearing work clothes: the map 'plug the function into the integral' is a contraction on a short interval, so its repeated application drives any starting guess toward the single fixed point, which is the solution. The same engine explains why the solution is unique — two different limits are impossible for a contraction. In practice the integrals get messy quickly, so Picard iteration is used more to understand and prove than to compute; numerical methods are the practical tool for getting numbers.
For y' = y, y(0) = 1, start with y0 = 1. Then y1 = 1 + integral from 0 to x of 1 dt = 1 + x; y2 = 1 + integral from 0 to x of (1 + t) dt = 1 + x + x^2/2; y3 = 1 + x + x^2/2 + x^3/6. These are exactly the partial sums of e^x — the iteration is rebuilding the Taylor series of the true solution.
Each pass adds another correct term: Picard iteration is reconstructing e^x term by term toward the true solution.
It is a beautiful proof tool but a poor calculator: the integrals usually become intractable after a few steps, which is why real computation uses numerical methods, not Picard iteration.