Improved Euler's Method

Calculus 2 · Axiom Academy

LESSON Improved Euler's Method Averaging the slope at both ends of a step turns a crude line-follower into an accurate one — for almost the same work. 1. Plain Euler: One Slope, One Step Given with a starting point (x_0, y_0) , plain Euler reads the slope only at the start and follows that straight line for the whole step of width h : Step along the starting slope f(x_0, y_0) Take the test problem , whose true solution is y = e^x . With one big step h = 1 , the starting slope is f(0,1) = 1 , so Euler lands at — but the true value is . Watch the straight line peel away from the curving solution: 2. Predict, Then Correct With the Average Improved Euler keeps the same step but uses two slopes. First a throwaway Euler step predicts the endpoint; then we read the slope there and step along the average of the two: On the same problem, the predictor gives , where the slope is f(1, 2) = 2 . Averaging 1 and 2 gives 1.5 , so — far closer to than plain Euler's 2 . This predictor–corrector shape is the heart of the method. Smaller steps sharpen both methods, but Improved Euler pulls away. Here are four steps of h = 0.25 on . Watch the green (improved) path hug the true curve while the red (plain Euler) path sags below it: One slope per step. Cheap, but it lags the curve. Two slopes per step. The error is roughly smaller here. 4. Why the Average Lands Right

This is the written version of the interactive lesson above. See the full Calculus 2 course.