Tea Time Numerical Analysis · Problem 1.1.24

Preliminaries — Leon Brin

Mathematics — Numerical Analysis · algorithmic-roundoff-error

Problem

Calculate (i)

\tilde{p}^*

using a calculator or computer, (ii) the absolute error,

|\tilde{p}^* - p|

, and (iii) the relative error,

\frac{|\tilde{p}^*-p|}{|p|}

. Then use the given value of

\tilde{p}

to compute (iv) the algorithmic error,

|\tilde{p}-p|

and (v) the round-off error,

|\tilde{p}^* - \tilde{p}|

. ((a)) Let

f(x) = x^4+7x^3-63x^2-295x+350

and let

p = f'(-2)

. The value

\tilde{p} = \dfrac{f(-2+10^{-7})-f(-2-10^{-7})}{2(10)^{-7}}

is a good approximation of

p

\tilde{p}

is exactly

8.99999999999999

. ((b)) Let

f'(x) = e^x \sin(10x)

and

f(0) = 0

and let

p = f(1)

. It can be shown that

p = \frac{1}{101}e(\sin 10 - 10\cos 10) + \frac{10}{101}

. Euler's method produces the approximation

\tilde{p} = \frac{1}{10}\sum_{i=1}^{10} e^{i/10}\sin i

. Accurate to 28 significant digits,

\tilde{p}

0.2071647018159241499410798569

. ((c)) Let

a_0 = \frac{5+\sqrt{5}}{8}

and

a_{n+1} = 4a_n(1-a_n)

, and consider

p = a_{51}

. It can be shown that

p = a_{51} = \frac{5-\sqrt{5}}{8}

. The most direct algorithm for calculating

a_{51}

is to calculate

a_1, a_2, a_3, \ldots a_{51}

in succession, according to the given recursion relation. Use this algorithm to compute

\tilde{p}^*

and

\tilde{p}