Trapezoid method, fixed-point iteration

Hi,

**problem:**

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.

**attempt:**

(a)

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.

(b)

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.

Thanks!