This document contains C++ code examples using numerical methods like Newton-Raphson, successive approximation, and secant methods to find the real roots of various equations. It includes 6 questions/problems with sample code to find roots of equations like x^4 - 11x = 8, e^x - 3x^2 = 0, x^3 - 2x - 3 = 0, and sin(x) + 3cos(x) = 2. For each problem, the code provided the numerical method used, input/output examples, and calculations to iteratively approximate the root to within a set tolerance.

1. TUTORIAL 7 SJEM2231: STRUCTURED PROGRAMMING (C++)
QUESTION 1
Write a C++ program by using Newton-Raphson method to find the
real root near 2 of the equation x4 – 11x= 8
USING DO-WHILE LOOP(FIRST WAY)
#include <iostream>
#include <cmath>
using namespace std;
#define f(x) pow(x,4) - 11*x - 8
#define ff(x) 4*pow(x,3) - 11
void main ()
{
double guess, x1, fguess,ffguess;
cout<<"Enter an initial guess of the solution: ";
cin>> guess;
do
{
fguess= f(guess);
ffguess =ff(guess);
x1= guess - (fguess/ ffguess);
cout << "n" <<guess << "tt" << x1<< "tt" << fguess <<"tt" << ffguess
<< "nn";
guess=x1;
//new approx. becomes previous and it is approximation for the next iteration

2. }while ( fabs(fguess) > 0.00001);
cout<<"The root is "<< x1<<endl;
}
//Output:
Enter an initial guess of the solution: 24.0911
24.0911 18.072 336569 55917.1
18.072 13.5607 106459 23598.1
13.5607 10.1825 33659.1 9963.79
10.1825 7.65875 10630.4 4212.07
7.65875 5.78392 3348.33 1785.94
5.78392 4.41096 1047.53 762.974
4.41096 3.44181 322.039 332.29
3.44181 2.82066 94.4696 152.088
2.82066 2.5125 24.2728 78.7663
2.5125 2.43218 4.21212 52.4422
2.43218 2.42704 0.239179 46.5503
2.42704 2.42702 0.000935689 46.1863
2.42702 2.42702 1.45057e-008 46.1849
The root is 2.42702

3. TUTORIAL 7 SJEM2231: STRUCTURED PROGRAMMING (C++)
USING DO-WHILE LOOP(SECOND WAY)
#include <cmath>
#include <iostream>
#include <iomanip>
using namespace std;
double func(double x);
double diff(double x);
double newton(double x);
int main ()
{
double a=2,b,r,iter=1;
cout << "NEWTON-RAPHSON METHOD" << endl;
cout << "x^4-11x-8 [near 2]n" << endl;
cout << "itxttf(x)ttf'(x)" <<endl;
do
{
r=newton(a);
cout << setprecision(0) << iter << "t"<< setprecision(5) << fixed << a << "tt"
<< func(a) << "tt" << diff(a) << endl;
b=a;
a=r;
iter++;
} while (fabs((a-b)/b) > 0.001);

4. cout << "nThe root is " << a << endl;
return 0;
}
double func(double x) {return (double)pow(x,4)-11*x-8;}
double diff(double x) {return (double)4*pow(x,3)-11;}
double newton(double x) {return (double)x-(func(x)/diff(x));}
//Output:
NEWTON-RAPHSON METHOD
x^4-11x-8 [near 2]
i x f(x) f'(x)
1 2.00000 -14.00000 21.00000
2 2.66667 13.23457 64.85185
3 2.46259 1.68798 48.73623
4 2.42796 0.04324 46.25104
The root is 2.42702

5. TUTORIAL 7 SJEM2231: STRUCTURED PROGRAMMING (C++)
ALTERNATIVE METHOD FOR QUESTION 1 USING FOR LOOP
#include <iostream>
#include <cmath>
using namespace std;
#define f(x) pow(x,4) - 11*x - 8
#define ff(x) 4*pow(x,3) - 11
void main ()
{
double guess, x1, fguess,ffguess;
cout<<"Enter an initial guess of the solution: ";
cin>> guess;
for(int i=0 ; i<=10; i++)
{
fguess=f(guess);
ffguess =ff(guess);
x1= guess - (fguess/ ffguess);
if (fabs(x1- guess)/guess < 0.00001)
break;
else
guess=x1;
}
cout<<"The root is "<< x1<<endl;
}

6. //Output:
Enter an initial guess of the solution: 24.0911
The root is 2.42704
QUESTION 2
Write a C++ function program to find the root of the equation
f(x)=ex – 3x2 to an accuracy of 5 digits.
USING DO-WHILE LOOP(FIRST WAY)
#include <iostream>
#include <cmath>
using namespace std;
double f( double x)
{
double fx;
fx=exp(x) - 3*pow(x,2);
return fx;
}
double ff(double x)
{
double ffx;
ffx=exp(x) - 6*x;
return ffx;
}
void main ()
{
double guess, x1, fguess,ffguess;

7. TUTORIAL 7 SJEM2231: STRUCTURED PROGRAMMING (C++)
cout<<"Enter an initial guess of the solution: ";
cin>> guess;
do
{
fguess= f(guess);
ffguess =ff(guess);
x1= guess - (fguess/ ffguess);
cout << "n" <<guess << "t" << x1<< "t" << fguess <<"t" << ffguess <<
"nn";
guess=x1;
//new approx becomes previous and it is approximation for the next iteration
} while ( fabs(fguess) > 0.000001);
cout<<"The root is "<< x1<<endl;
}
//Output:
Enter an initial guess of the solution: 1
1 0.914155 -0.281718 -3.28172
0.914155 0.910018 -0.0123726 -2.99026
0.910018 0.910008 -3.00348e-005 -2.97574
0.910008 0.910008 -1.79075e-010 -2.9757
The root is 0.910008

8. USING DO-WHILE LOOP(SECOND WAY)
#include <cmath>
#include <iostream>
#include <iomanip>
using namespace std;
double func(double x);
double diff(double x);
double newton(double x);
int main ()
{
double a=1,b,r;
int iter=1;
cout << "NEWTON-RAPHSON METHOD" << endl;
cout << "e^x-3x^2n" << endl;
cout << "itxttf(x)ttf'(x)" <<endl;
do
{
r=newton(a);
cout << iter << "t"<< setprecision(5) << fixed << a << "tt" << func(a) << "t"
<< diff(a) << endl;
b=a;
a=r;
iter++;
} while (fabs((a-b)/b) > 0.001);

9. TUTORIAL 7 SJEM2231: STRUCTURED PROGRAMMING (C++)
cout << "nThe root is " << a << endl;
return 0;
}
double func(double x) {return (double)exp(x)-3*pow(x,2);}
double diff(double x) {return (double)exp(x)-6*x;}
double newton(double x) {return (double)x-(func(x)/diff(x));}
//Output:
NEWTON-RAPHSON METHOD
e^x-3x^2
i x f(x) f'(x)
1 1.00000 -0.28172 -3.28172
2 0.91416 -0.01237 -2.99026
3 0.91002 -0.00003 -2.97574
The root is 0.91001
QUESTION 3
Write a C++ Function program to find the smallest positive root of
the equation x3– 2x – 3 =0. using Successive Approximation Method
#include <cmath>
#include <iostream>
#include <iomanip>

10. using namespace std;
double f(double x);
double g(double x);
int main ()
{
double xo=0,xn=1;
cout << "SUCCESSIVE APPROXIMATION METHOD" << endl;
cout << "X^3-2X-3n" << endl;
while (f(xo)*f(xn) > 0)
{
xn=xo;
xo++;
}
while (fabs(xn-xo) > 0.000001)
{
xo=xn;
xn=g(xo);
cout << xo << "t" << xn << endl;
}
cout << "nThe root is " << setprecision(6) << xn << endl;
return 0;
}
double f(double x) {return (double)pow(x,3)-2*x-3;}
double g(double x) {return (double)pow(2*x+3,0.333333);}

11. TUTORIAL 7 SJEM2231: STRUCTURED PROGRAMMING (C++)
//Output:
SUCCESSIVE APPROXIMATION METHOD
X^3-2X-3
1 1.70998
1.70998 1.85856
1.85856 1.88681
1.88681 1.89208
1.89208 1.89306
1.89306 1.89325
1.89325 1.89328
1.89328 1.89329
1.89329 1.89329
1.89329 1.89329
The root is 1.89329
QUESTION 4
Write a C++ function to find the positive root of the equation
cos(x)- 3x +5=0 using Successive Approximation Method
#include <cmath>
#include <iostream>
#include <iomanip>
using namespace std;
double f(double x) {return (double)cos(x)-3*x+5;}
double g(double x) {return (double)(cos(x)+5)/3;}

12. int main ()
{
double xo=0,xn=1;
cout << "SUCCESSIVE APPROXIMATION METHOD" << endl;
cout << "cos(x)-3x+5n" << endl;
while (f(xo)*f(xn) > 0)
{
xn=xo;
xo++;
}
while (fabs(xn-xo) > 0.000001)
{
xo=xn;
xn=g(xo);
cout << xo << "t" << xn << endl;
}
cout << "nThe root is " << setprecision(6) << xn << endl;
return 0;
}
//Output:
SUCCESSIVE APPROXIMATION METHOD
cos(x)-3x+5

13. TUTORIAL 7 SJEM2231: STRUCTURED PROGRAMMING (C++)
1 1.84677
1.84677 1.57584
1.57584 1.66499
1.66499 1.63532
1.63532 1.64517
1.64517 1.6419
1.6419 1.64299
1.64299 1.64262
1.64262 1.64274
1.64274 1.6427
1.6427 1.64272
1.64272 1.64271
1.64271 1.64271
1.64271 1.64271
The root is 1.64271
QUESTION 5
Write a C++ program to find the root of the equation sin(x) +
3cos(x) =2 using Secant method. Use initial approximations 0 and 1.5
#include <cmath>
#include <iostream>
#include <iomanip>
using namespace std;
double f(double x)
{

14. return (double)sin(x)+3*cos(x)-2;
}
double secant(double x0, double x1)
{
return (double)((x0*f(x1))-(x1*f(x0)))/(f(x1)-f(x0));
}
int main ()
{
double x0,x1=0,x2=1.5;
cout << "SECANT METHOD" << endl;
cout << "sin(x)+3cos(x)-2n" << endl;
do
{
x0=x1;
x1=x2;
x2=secant(x0,x1);
cout << setprecision(6) << fixed << x0 << "t" << x1 << "t" << x2 << endl;
}while (fabs(x2-x1) > 0.00001);
cout << "nThe root is " << setprecision(5) << fixed << x2 << endl;
return 0;
}

15. TUTORIAL 7 SJEM2231: STRUCTURED PROGRAMMING (C++)
//Output:
SECANT METHOD
sin(x)+3cos(x)-2
0.000000 1.500000 0.837851
1.500000 0.837851 1.160351
0.837851 1.160351 1.218120
1.160351 1.218120 1.207622
1.218120 1.207622 1.207827
1.207622 1.207827 1.207828
The root is 1.20783
QUESTION 6
Use Secant method to find the real root of the equation x3- 8x =5
#include <cmath>
#include <iostream>
#include <iomanip>
using namespace std;
double f(double x)
{
return (double)pow(x,3)-8*x-5;
}

16. double secant(double x0, double x1)
{
return (double)((x0*f(x1))-(x1*f(x0)))/(f(x1)-f(x0));
}
void main ()
{
double x0,x1=0,x2=1;
cout << "SECANT METHOD" << endl;
cout << "x^3-8x-5n" << endl;
while (f(x1)*f(x2) > 0)
{
x1=x2;
x2++;
}
do
{
x0=x1;
x1=x2;
x2=secant(x0,x1);
cout << setprecision(6) << fixed << x0 << "t" << x1 << "t" << x2 << endl;
}
while (fabs(x2-x1) > 0.00001);
cout << "nThe root is " << setprecision(5) << fixed << x2 << endl;
}

17. TUTORIAL 7 SJEM2231: STRUCTURED PROGRAMMING (C++)
//Output:
SECANT METHOD
x^3-8x-5
3.000000 4.000000 3.068966
4.000000 3.068966 3.090738
3.068966 3.090738 3.100570
3.090738 3.100570 3.100431
3.100570 3.100431 3.100432
The root is 3.10043