12. Numerical Methods

8/2/2019 12. Numerical Methods

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NUMERICAL METHODS

Programs


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1. Successive approximation method

Algorithm

1. Start

2. Define the iterative equation

3. Input x0, initial guess

4. Input the accuracy

5. Input the max iterations suggested, n

6. For i =1 to n, increment in steps of 1a. x1= f(x0)

b. Error=|(x1-x0)/x1|

c. x0 =x1

d. If (error


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void main()

{ float f(float );

float x0,x1, error, acc;

int i, n;

cin>>x0>>acc>>n;

for(i=0; i


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2. Newton Raphson method

1. Start

2. Input x0, acc

3. K=0

4. x[k+1] = x[k] - f(x[k]) / f1(x[k])

5. Error=fabs ((x[k+1] - x[k]) / x[k+1])

6. If (error>= accy)

k=k+1; go to step 4;

7. Print x[k+1]

8. stop


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void main()

{

double x[200], accy;

int k = 0;

double f(double a);

double f1(double a);

void where(void);

where();

cout x[0];cout accy;

x[1] = x[0] - f(x[0]) / f1(x[0]);



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cout


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// Subroutine to evaluate the given function

double f(double a)

{double y;

y = a * a * a - 3 * a * a + a + 1;

return(y);

}

// Subroutine to evaluate the derivative of the given function

double f1(double a)

{

double y;

y = 3 * a * a - 6 * a + 1;return(y);

}



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//Subroutine to determine the intervals where the roots lie

void where(void)

{

float y0, y1, j;

for(j = -50; j


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3. Regula Falsi method

1. Read x0, x1, error, n

2. f0 =f(x0)

3. f1= f(x1)

4. For (i = 1 to n) in steps of 1 do

5. x2= (x0*f1-x1*f0)/(f1-f0)

6. f2= f(x2)

7. If |f2|0) x0=x2

else x1=x2

End for loop

9. stop


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3. Regula Falsi methodvoid main()

{ int n, i;

float x0,x1,x2,error;cin>>x0>>x1;

cin>>error;

float f0, f1, f2;

f0=f(x0);

f1=f(x1);

do

{ x2=(x0*f1- x1*f0)/(f1- f0);

f2=f(x2);

if ((f0*f2)>0)

{ x0 = x2;

f0 = f2;

}

else

{ x1=x2;

f1 = f2;}

} while (f2>error);

}


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4. Fitting a straight line

void main()

{float x[20], y[20];

int i, n;

float s1, s2, s3, s4;

cout n;

for( i = 0; i < n; i++)

{

cout


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5. Fitting a parabolavoid main()

{

float x[20], y[20], a[5][5];

int i, n;

float s1, s2, s3, s4, s5, s6, s7;

void guass(float a[5][5]);

cout n;

for( i = 0; i < n; i++)

{

cout


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a[0][0] = (float) n;

a[0][1] = s1;

a[0][2] = s3;a[0][3] = s2;

a[1][0] = s1;

a[1][1] = s3;

a[1][2] = s4;

a[1][3] = s6;a[2][0] = s3;

a[2][1] = s4;

a[2][2] = s5;

a[2][3] = s7;

guass(a);

}

//Solution using Guass eliminationmethod

void guass(float a[5][5]){

float x[5], ratio;

int n = 3, i, j, k;

for(k = 0; k < n - 1; k++)

for(i = k + 1; i < n; i++){

ratio = a[i][k] / a[k][k];

for(j = 0; j < n + 1; j++)

a[i][j] -= ratio * a[k][j];

}

5. Fitting a parabola


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5. Fitting a parabola

//Back substitution

x[n-1] = a[n - 1][n] / a[n - 1][n-1];

for(k = n - 2; k >= 0; k--)

{

x[k] = a[k][n];

for(j = k + 1;j < n; j++)

x[k] -= a[k][j] * x[j];

x[k] /= a[k][k];

}

cout


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6. Trapezoidal rulevoid main()

{

float a, b, h, x0, xn, xi;

double sum;

int n, i;

double funct(float a);

cout a >>b;

cout >n;

h = (b - a) / n;

x0 = a;

xn = b;

sum = funct (x0) + funct (xn);

cout


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7. Simpsons rulevoid main()

{

float a, b, h, x0, xn, xi;

double sum;int n, i;

double funct(float a);

cout a >>b;

cout n;

h = (b - a) / n;

x0 = a;

xn = b;

sum = funct (x0) + funct (xn);

cout


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double funct(float a)

{

double b;

b = exp (-a * a / 2);

return (b);

}

7. Simpsons rule


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8. Gauss elimination method

void main()

{

float a[20][20], ratio, x[20];int i, j, k, n;

cout >n;

cout a[i][j];//Triangularisation

for(k = 0; k < n - 1; k++)

for(i = k + 1; i < n; i++)

{

ratio = a[i][k] / a[k][k];

for(j = 0; j < n + 1; j++)

a[i][j] -= ratio * a[k][j];

}

//Back substitution

x[n-1] = a[n - 1][n] / a[n - 1][n-1];

for(k = n - 2; k >= 0; k--)

{

x[k] = a[k][n];

for(j = k + 1;j < n; j++)

x[k] -= a[k][j] * x[j];

x[k] /= a[k][k];

}

for(i = 0; i < n; i++)cout


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9. Gauss quadraturevoid main()

{

double c[10], x[10];double sum = 0;

int n, i;

double funct(double a);

cout n;

switch(n)

{case 2: c[0] = c[1] = 1;

x[0] = -0.57735;

x[1] = 0.57735;

break;

case 3 : c[0] = c[2] = 0.55556;

c[1] = 0.88889;

x[0] = -0.77460;x[1] = 0;

x[2] = 0.77460;

break;

case 4 :c[0] = c[3] = 0.34785;c[1] = c[2] = 0.65215;

x[0] = -0.86114;

x[1] = -0.33998;

x[2] = 0.33998;

x[3] = 0.86114;

break;


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case 5 :c[0] = c[4] = 0.23693;

c[1] = c[3] = 0.47863;

c[2] = 0.56889;

x[0] = -0.90618;

x[1] = -0.53847;

x[2] = 0;

x[3] = 0.53847;

x[4] = 0.90618;

break;

}

for(i = 0; i < n; i++)

sum += c[i] * funct(x[i]);

cout


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10. Gauss- Siedel Iterationvoid main()

{

float a[10][10],x[10],y[10];

int i,j,n;

clrscr();

/* inputting the coefficients*/

coutn;

cout


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/* display of coefficients */

cout

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12. Numerical Methods