$1$ Problem $1$

Write a program to evaluate e$9\mathrm{.}3,$ in single precision float arithmetic, using each of the two alternative

methods:

e$$x $=1$ x $+$

x$2$

$2$

$$

x$3$

$3!$

$+(1)$

and

e$$x $=$

$1$

ex $=$

$1$

$1+$ x $+$ x$2$

$2+$ x$3$

$3!+$

$(2)$

For $(1),$ use the first k terms in the right$-$hand side; and for $(2),$ use the first k terms in the denominator of

the right$-$hand side. Use the matlab built$-$in function exp$()$ to find the true value of e$9\mathrm{.}3.$

Approximate e$9\mathrm{.}3$ by the above two evaluated results based on the sums of the first k terms; compute the

two approximation errors e$1($k$)$ and e$2($k$)$; generate and compare the two error curves $($or$/$and table$($s$))$

for k $=2,3,,35.$

Which method has a higher numerical precision? Which method converges faster? Explain your observations.