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The second output line illustrated in Table 3.4 displays the same type of items as the first line, except that the approximation to ex requires the two of two terms of the approximating polynomial. Notice also that the first item on the second line, the value obtained by the exp() function, is the same as the first item on the first line. This means that this item does not have to recalculated; the value calculated for the first line can simply be displayed a second line. Once the data for the second line have been calculated, a single cout statement can again be used to display the required values.

Finally, only the second and third items on the last two output lines shown in Figure 1 need to be recalculated because the first item on these lines is the same as previously calculated for the first line. Thus, for this problem, the complete algorithm described in pseudocode is:

Display a prompt for the input value of x.

Read the input value.

Display the heading lines.

Calculate the exponential value of x using the exp() function.

Calculate the first approximation.

Calculate the first difference.

Print the first output line.

Calculate the second approximation.

Calculate the second difference.

Print the second output line.

Calculate the third approximation.

Calculate the third difference.

Print the third output line.

Calculate the fourth approximation.

Calculate the fourth difference.

Print the fourth output line.

To ensure that we understand the processing used in the algorithm, we will do a hand calculation. The result of this calculation can then be used to verify the result produced by the program that we write. For test purposes, we use a value of 2 for x, which causes the following approximations:

Using the first term of the polynomial, the approximation is

e^2 = 1

Using the first two terms of the polynomial, the approximation is

e^2 = 1 + 2/1 = 3

Using the first three terms of the polynomial, the approximation is

e^2 = 3 + 2^2/2 = 5

Using the first four terms of the polynomial, the approximation is

e^2 = 5 + 2^3/6 = 6.3333

Notice that the first four terms of the polynomial, it was not necessary to recalculate the value of the first three terms; instead, we used the previously calculated value.

Step 3: code the algorithm

The following program represents a description of the selected algorithm in C++.

// This program approximates the function e raised to the x power

// using one, two, three, and four terms of an approximating polynomial.

#include<iostream.h>

#include<iomanip.h>

#include<math.h>

int main()

{

double x, funcValue, approx, difference;

cout<<“\n Enter a value of x: “;

cin>>x;

// print two title lines

cout<<“ e to the x Approximation Difference\n”

cout<<“------------ --------------------- --------------\n”;

funcValue = exp(x);

// calculate the first approximation

approx = 1;

difference = abs(funcValue – approx);

cout<<setw(10)<<setiosflags(iso::showpoint)<<funcValue

<<setw(18)<<approx

<<setw(18)<<difference<<endl;

// calculate the first approximation

approx = 1;

difference = abs(funcValue – approx);

cout<<setw(10)<<setiosflags(iso::showpoint)<<funcValue

<<setw(18)<<approx

<<setw(18)<<difference<<endl;

// calculate the second approximation

approx = approx + x;

difference = abs(funcValue – approx);

cout<<setw(10)<<setiosflags(iso::showpoint)<<funcValue

<<setw(18)<<approx

<<setw(18)<<difference<<endl;

// calculate the third approximation

approx = approx + pow(x,2)/2.0;

difference = abs(funcValue – approx);

cout<<setw(10)<<setiosflags(iso::showpoint)<<funcValue

<<setw(18)<<approx

<<setw(18)<<difference<<endl;

// calculate the fourth approximation

approx = approx + pow(x,3)/6.0;

difference = abs(funcValue – approx);

cout<<setw(10)<<setiosflags(iso::showpoint)<<funcValue

<<setw(18)<<approx

<<setw(18)<<difference<<endl;

return 0;

}

In reviewing the program, notice that the input value of x is obtained first. The two title lines are then printed prior to any calculations being made. The value of ex is then computed using the exp() library function and assigned to the variable funcValue. This assignment permits this value to be used in the four difference calculations and displayed four times without the need for recalculation.

Since the approximation to the ex is “built up” using more and more terms of the approximating polynomial, only the new term for each approximation is calculated and added to the previous approximation. Finally, to permit the same variables to be reused, the values in them are immediately printed before the next approximation is made.

Step 4: test and correct the program

The following is the sample run produced by the above program is:

A sample run produced by the above program

The first two columns of output data produced by the sample run agree with our hand calculation. A hand check of the last column verifies that it also correctly contains the difference in values between the first two columns.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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Source:  OpenStax, Programming fundamentals in c++. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10788/1.1
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