Problem $1$

In numerical methods, there are often many ways to make a calculation that gives the same answer but at very

different efficiencies $($i$.$e$.,$ one method is faster than another$).$ In this problem we will consider one such

calculation. In molecular dynamics we often need to compute the so$-$called Lenard$-$Jones Potential: a

mathematically simply model approximating the interactions strength between two neutral $($uncharged$)$

particles. If the Euclidean distance between the two particles is r then $=1$

$6+1$

$12,=2+2+2$

$,=12,=12,=12.$

Let us assume that two particles are located one at $(1,0,3)$ and another at $(2,-2,-1)$ and that two methods are

available for the calculation of LJ$,$ namely

Method $1$

$(1)$ Calculate $=2+2+2$

$(2)$ Calculate $=1$

$6+1$

$12$

Methods $2$

$(1)$ Calculate $2=2+2+2$

$(2)$ Calculate $1$

$2$

$(3)$ Calculate $1$

$6=1$

$2$

$1$

$2$

$1$

$2$

$(4)$ Calculate $1$

$12=1$

$6$

$1$

$6$

$(5)$ Calculate $=1$

$6+1$

$12$

Write a Matlab code that determines the amount of time it takes for Method $1$ and Method

$2$ to do the calculation $10,102,103,104,105,106,107,108$ times. Look at how the calculation times increase with

the number of times you do the calculation. For estimating the time, you should use the Matlab commands tic and

toc. For example, if you write the following in Matlab

tic;

for i$=1$:$1000000$

end

elapsedTime$=$toc

after running these $4$ lines of code you should get something like:

$>>$ elapsedTime $=$

$0\mathrm{.}0176$

which gives you the time it took do perform the loop in seconds. Finally, produce a log$-$log plot showing the

time that is required to perform the calculations for the two methods versus the number of calculations.