Question

The function ) f (x) = ln(1+ x is to be tabulated at equispaced points in the interval
,2[ using linear interpolation. Find the largest s ]3 tep size h that can be used so that
the error 4
5 10−
≤ × in magnitude

Answer 1

Answer on Question #46685 – Math – Algorithms | Quantitative Methods

The function $f(x) = \ln(1 + x)$ is to be tabulated at equispaced points in the interval 2 using linear interpolation. Find the largest s 3 tep size h that can be used so that

the error 4

5 10-

$\leq \times$ in magnitude

Solution.

$$f(x) = \ln(1 + x)$$

Error for linear interpolation on $[0, h]$: $e \leq \frac{h^2}{8} |maxf''(x)|$

In our case $f''(x) = -\frac{1}{(x+1)^2}$, on $[2, 3] |maxf''(x)| = \frac{1}{9}$, $e = 4 * 10^{-5}$,

$$\frac{h^2}{8} |maxf''(x)| = \frac{h^2}{8} * \frac{1}{9} = \frac{h^2}{72}.$$

So, for $4 * 10^{-5}$, $h \approx 0.05$.

Largest step size equals 0.05.

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