<< Chapter < Page Chapter >> Page >

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

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Programming fundamentals in c++. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10788/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Programming fundamentals in c++' conversation and receive update notifications?

Ask