Why does the lecturer recommend halving each interval when refining the step size?

Conditions

• Refinement is by successive bisection of existing subintervals.
• Benefit comes from reusing previously computed function values.

Reasoning, step by step

1. Start from a set of computed function values on a current partition.
2. Refine by dividing each interval by 2.
3. Reuse the old partition points' function values in the new, finer partition.
4. Reduce repeated function evaluations, which are often the dominant cost.

Example

The lecturer recommends dividing each interval by 2 at each refinement step because earlier function values can be reused in the next computation. This reduces redundant evaluations of $f(x)$, which the lecturer says often dominate the runtime.

Common misconceptions

• Assuming that decreasing step size requires recomputing all function values from scratch.
• Thinking that halving intervals increases the computational cost unnecessarily.

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