Fixed-point lab

Every method draws a line through (x, T(x)) with some slope s, then jumps to where that line meets y = x: x⁺ = x + (T(x) − x) / (1 − s). The methods differ only in the slope.

Model slope

Click or drag on the plot to choose the starting point. Keyboard: space or → to step, ← to go back, R to reset.

Error |x_k − x*| for every method from the same start

The unknown is now a whole curve. For the damped pendulum, x = T[x] reads x(t) = x₀ + ∫₀ᵗ f(x(s)) ds. Picard applies T to the current curve. Newton replaces T by its linearization along the current curve and solves that linear ODE.

Predict. There is no discount factor here, so what makes Picard converge? Lengthen the horizon: why does the error climb before it falls, and how does that differ from Newton?

Horizon

Initial angle

Iterates: angle θ(t) as one point in a space of curves

Solution of the discretized equation Current iterate, older iterates paler

Largest error along the curve

Picard Newton

Last time the reservoir needed a terminal value, and we made one up. Now ask for the terminal value that reproduces itself: v = B v, where (B v)(x) is one full year of backward induction ending on γ v. The unknown is the function v(x), the value of storage on 1 January if the plant runs forever.

Predict. With a one-year horizon and no terminal value, what happens to the water by December? As γ goes from 0.8 to 0.99, how many more years does value iteration need, and how many more Newton steps?

Annual discount γ

Years of backward induction k

Value iteration: years to reach v* within 1 $

Policy iteration: Newton steps to reach v*

Years until the greedy policy loses less than 1 k$

Terminal water price implied by v* at 1100 hm³ (the slider last time said 50)

Value of storage on 1 January (M$)

v_k = B^k 0, value iteration Policy iteration iterates v*

Sup-norm distance to v* (M$)

Value iteration, ‖v_k − v*‖ Loss of its greedy policy Policy iteration

Solving x = T(x): the same idea on four unknowns

Error |xk − x*| for every method from the same start

Iterates: angle θ(t) as one point in a space of curves

Largest error along the curve

Value of storage on 1 January (M$)

Sup-norm distance to v* (M$)

Error |x_k − x*| for every method from the same start