UNIT 4 • INITIAL VALUE PROBLEMS (ODE)

Milne's Method
Predictor-Corrector

Milne's predictor-corrector method solves $\frac{dy}{dx} = f(x, y)$ using four earlier points: it predicts $y_{n+1}$ with an open formula, then corrects it with Simpson's rule.

MILNE'S FORMULASNEEDS 4 STARTING POINTS

\begin{gathered} \text{Predictor: } y_{n+1}^P = y_{n-3} + \frac{4h}{3}\left(2f_{n-2} - f_{n-1} + 2f_n\right) \\ \text{Corrector: } y_{n+1}^C = y_{n-1} + \frac{h}{3}\left(f_{n-1} + 4f_n + f_{n+1}\right) \end{gathered}

1. Start UpGet

y_1, y_2, y_3

by RK4 and work out

f_0

f_3

2. PredictUse the predictor for

y_4^P

, then

f_4 = f(x_4, y_4^P)

3. CorrectPut

f_4

into the corrector; repeat until it stops changing.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Solve $\frac{dy}{dx} = x + y$ , $y(0) = 1$ with $h = 0.1$ , and find $y(0.4)$ by Milne's method.

STEP 01 • STARTING VALUES BY RK4

y_1 = 1.110342, \quad y_2 = 1.242805, \quad y_3 = 1.399717

f_1 = 1.210342, \quad f_2 = 1.442805, \quad f_3 = 1.699717

STEP 02 • PREDICT

y_4^P = 1 + \frac{0.4}{3}\left(2 \times 1.210342 - 1.442805 + 2 \times 1.699717\right) = 1.583642

STEP 03 • CORRECT

f_4 = 0.4 + 1.583642 = 1.983642

y_4^C = 1.242805 + \frac{0.1}{3}\left(1.442805 + 4 \times 1.699717 + 1.983642\right) = 1.583649

The exact value $2e^{0.4} - 1.4 = 1.583649$ agrees to six decimal places.

FINAL CONCLUSION

y(0.4) ≈ 1.583649

y(0.4) \approx 1.583649

The calculator below opens on this problem and shows every predictor and corrector value.

LABORATORY

Interactive Milne Calculator

LABORATORY

Milne Predictor-Corrector Engine

dy/dx = f(x, y)x₀y₀ = y(x₀)

Step hFind y at x =

Common Questions

Why does Milne's method need starting values?

It is a multi-step method: each new value uses the four before it. So the first values

y_1, y_2, y_3

must come from a one-step method, usually Runge-Kutta 4.

What do the predictor and corrector do?

The predictor (an open formula) makes a first guess

y_{n+1}^P

from earlier slopes. The corrector (Simpson's rule) uses the slope at that guess to improve it. Comparing the two gives a check on the accuracy.

Is Milne's method always stable?

No. For some equations (for example

y' = -y

over long ranges) its errors can grow. Adams-Bashforth-Moulton is the safer general choice.

PreviousAdams-Bashforth-Moulton

Unit 4•37 of 63

NextMethod of Least Squares

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 4 • INITIAL VALUE PROBLEMS (ODE)

Milne's Method
Predictor-Corrector

Milne's predictor-corrector method solves $\frac{dy}{dx} = f(x, y)$ using four earlier points: it predicts $y_{n+1}$ with an open formula, then corrects it with Simpson's rule.

MILNE'S FORMULASNEEDS 4 STARTING POINTS

\begin{gathered} \text{Predictor: } y_{n+1}^P = y_{n-3} + \frac{4h}{3}\left(2f_{n-2} - f_{n-1} + 2f_n\right) \\ \text{Corrector: } y_{n+1}^C = y_{n-1} + \frac{h}{3}\left(f_{n-1} + 4f_n + f_{n+1}\right) \end{gathered}

1. Start UpGet

y_1, y_2, y_3

by RK4 and work out

f_0

f_3

2. PredictUse the predictor for

y_4^P

, then

f_4 = f(x_4, y_4^P)

3. CorrectPut

f_4

into the corrector; repeat until it stops changing.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Solve $\frac{dy}{dx} = x + y$ , $y(0) = 1$ with $h = 0.1$ , and find $y(0.4)$ by Milne's method.

STEP 01 • STARTING VALUES BY RK4

y_1 = 1.110342, \quad y_2 = 1.242805, \quad y_3 = 1.399717

f_1 = 1.210342, \quad f_2 = 1.442805, \quad f_3 = 1.699717

STEP 02 • PREDICT

y_4^P = 1 + \frac{0.4}{3}\left(2 \times 1.210342 - 1.442805 + 2 \times 1.699717\right) = 1.583642

STEP 03 • CORRECT

f_4 = 0.4 + 1.583642 = 1.983642

y_4^C = 1.242805 + \frac{0.1}{3}\left(1.442805 + 4 \times 1.699717 + 1.983642\right) = 1.583649

The exact value $2e^{0.4} - 1.4 = 1.583649$ agrees to six decimal places.

FINAL CONCLUSION

y(0.4) ≈ 1.583649

y(0.4) \approx 1.583649

The calculator below opens on this problem and shows every predictor and corrector value.

LABORATORY

Interactive Milne Calculator

LABORATORY

Milne Predictor-Corrector Engine

dy/dx = f(x, y)x₀y₀ = y(x₀)

Step hFind y at x =

Common Questions

Why does Milne's method need starting values?

It is a multi-step method: each new value uses the four before it. So the first values

y_1, y_2, y_3

must come from a one-step method, usually Runge-Kutta 4.

What do the predictor and corrector do?

The predictor (an open formula) makes a first guess

y_{n+1}^P

from earlier slopes. The corrector (Simpson's rule) uses the slope at that guess to improve it. Comparing the two gives a check on the accuracy.

Is Milne's method always stable?

No. For some equations (for example

y' = -y

over long ranges) its errors can grow. Adams-Bashforth-Moulton is the safer general choice.

PreviousAdams-Bashforth-Moulton

Unit 4•37 of 63

NextMethod of Least Squares

Milne's Method Predictor-Corrector

Step-by-Step Worked Example

y(0.4) ≈ 1.583649

Interactive Milne Calculator

Milne Predictor-Corrector Engine

Common Questions

NETZ Engine

Milne's Method Predictor-Corrector

Step-by-Step Worked Example

y(0.4) ≈ 1.583649

Interactive Milne Calculator

Milne Predictor-Corrector Engine

Common Questions

Milne's Method
Predictor-Corrector

Milne's Method
Predictor-Corrector