The trapezoid method for solving ordinary differential equations is,
Since this method is implicit, we must solve for in every step. If we use fixed-point iteration , we will have convergence if for .
(a) Give the expression for .
(b) Give a condition on such that the fixed-point iterations converges at each step.
For each iteration , we have to run fixed-point iteration enough times to make sure we get the wanted accuracy.
We take a guess at , calculate and see how big is. If it's close enough to , we stop the fixed-point iteration.
I'm not really sure what to make of this question...
Fixed-point methods converge if .
I do not see much sense in my last expression.