Bisection Method

f@x_D := Cos@xD;x0 = 0.0; x1 = 2.0; n = 14;

f1 = Sign@f@x1DD;If@f@x0D * f1 > 0,Print@"IVP not satisfied"D,For@i = 1, i n, i++, p = Hx0 + x1L 2;Print@i, " th iteration value is: ", pD;pSign = Sign@f@pDD;If@pSign * f1 < 0, x0 = p, x1 = p; f1 = pSignD;Print@"The new interval is: ", "H", x0, ",", x1, "L"D;Print@".........................................."D;D;Print@"Final Approximation is ", pDD

1 th iteration value is: 1.

The new interval is: H1.,2.L

..........................................

2 th iteration value is: 1.5

The new interval is: H1.5,2.L

..........................................

3 th iteration value is: 1.75

The new interval is: H1.5,1.75L

..........................................

4 th iteration value is: 1.625

The new interval is: H1.5,1.625L

..........................................

5 th iteration value is: 1.5625

The new interval is: H1.5625,1.625L

..........................................

6 th iteration value is: 1.59375

The new interval is: H1.5625,1.59375L

..........................................

7 th iteration value is: 1.57813

The new interval is: H1.5625,1.57813L

..........................................

8 th iteration value is: 1.57031

The new interval is: H1.57031,1.57813L

..........................................

9 th iteration value is: 1.57422

The new interval is: H1.57031,1.57422L

..........................................

10 th iteration value is: 1.57227

The new interval is: H1.57031,1.57227L

..........................................

11 th iteration value is: 1.57129

The new interval is: H1.57031,1.57129L

..........................................

12 th iteration value is: 1.5708

The new interval is: H1.57031,1.5708L

..........................................

13 th iteration value is: 1.57056

The new interval is: H1.57056,1.5708L

..........................................

14 th iteration value is: 1.57068

The new interval is: H1.57068,1.5708L

..........................................

Final Approximation is 1.57068

Both error tolerance and max condition

In[39]:= Clear@fD;f@x_D := Cos@xD;x0 = 0.0; x1 = 2.0; n = 20;

f1 = Sign@f@x1DD;If@f@x0D * f1 > 0,Print@"IVP not satisfied"D,For@i = 1, i n, i++,p = Hx0 + x1L 2.0;Print@i, " th iteration value is ", pD;pSign = Sign@f@pDD;Print@"The error is ", Abs@x1 - x0D 2.0D;If@Abs@x1 - x0D < 2 * 0.0001, Break@DD;If@pSign * f1 < 0, x0 = p, x1 = p; f1 = pSignD;Print@"The root enclosing interval is: ", "H", x0, ",", x1, "L"D;Print@"....................................................."D;D;Print@"....................................................."D;Print@"Final approximation is: ", pD;D

1 th iteration value is 1.

The error is 1.

The root enclosing interval is: H1.,2.L

.....................................................

2 th iteration value is 1.5

The error is 0.5

2 Numerical Methods and Programming.nb

The root enclosing interval is: H1.5,2.L

.....................................................

3 th iteration value is 1.75

The error is 0.25

The root enclosing interval is: H1.5,1.75L

.....................................................

4 th iteration value is 1.625

The error is 0.125

The root enclosing interval is: H1.5,1.625L

.....................................................

5 th iteration value is 1.5625

The error is 0.0625

The root enclosing interval is: H1.5625,1.625L

.....................................................

6 th iteration value is 1.59375

The error is 0.03125

The root enclosing interval is: H1.5625,1.59375L

.....................................................

7 th iteration value is 1.57813

The error is 0.015625

The root enclosing interval is: H1.5625,1.57813L

.....................................................

8 th iteration value is 1.57031

The error is 0.0078125

The root enclosing interval is: H1.57031,1.57813L

.....................................................

9 th iteration value is 1.57422

The error is 0.00390625

The root enclosing interval is: H1.57031,1.57422L

.....................................................

10 th iteration value is 1.57227

The error is 0.00195313

The root enclosing interval is: H1.57031,1.57227L

.....................................................

11 th iteration value is 1.57129

The error is 0.000976563

The root enclosing interval is: H1.57031,1.57129L

.....................................................

12 th iteration value is 1.5708

Numerical Methods and Programming.nb 3

The error is 0.000488281

The root enclosing interval is: H1.57031,1.5708L

.....................................................

13 th iteration value is 1.57056

The error is 0.000244141

The root enclosing interval is: H1.57056,1.5708L

.....................................................

14 th iteration value is 1.57068

The error is 0.00012207

The root enclosing interval is: H1.57068,1.5708L

.....................................................

15 th iteration value is 1.57074

The error is 0.0000610352

.....................................................

Final approximation is: 1.57074

Only Error tolerance

In[56]:= Clear@f, x0, x1, i, p, f1D;In[97]:= f@x_D := Cos@xD;

x0 = 0.0; x1 = 2.0;

f1 = Sign@f@x1DD;If@f@x0D * f1 > 0, Print@"IVP not satisfied"D,

For@i = 1, Abs@x1 - x0D > 2 * 0.0001, i++,p = Hx1 + x0L 2.0;Print@i, " th iteration approximation is ",p, " in interval ", "H", x0, ",", x1, "L"D;

Print@"Error bound is ", Abs@x1 - x0D 2.0D;Print@"..............................................................."D;pSign = Sign@f@pDD;If@pSign * f1 < 0, x0 = p, x1 = p, f1 = pSignD;D;Print@"Final approximation is: ", p, " with error ", Abs@x1 - x0D 2.0DD;

1 th iteration approximation is 1. in interval H0.,2.L

Error bound is 1.

...............................................................

2 th iteration approximation is 1.5 in interval H1.,2.L

Error bound is 0.5

...............................................................

3 th iteration approximation is 1.75 in interval H1.5,2.L

Error bound is 0.25

...............................................................

4 Numerical Methods and Programming.nb

4 th iteration approximation is 1.625 in interval H1.5,1.75L

Error bound is 0.125

...............................................................

5 th iteration approximation is 1.5625 in interval H1.5,1.625L

Error bound is 0.0625

...............................................................

6 th iteration approximation is 1.59375 in interval H1.5625,1.625L

Error bound is 0.03125

...............................................................

7 th iteration approximation is 1.57813 in interval H1.5625,1.59375L

Error bound is 0.015625

...............................................................

8 th iteration approximation is 1.57031 in interval H1.5625,1.57813L

Error bound is 0.0078125

...............................................................

9 th iteration approximation is 1.57422 in interval H1.57031,1.57813L

Error bound is 0.00390625

...............................................................

10 th iteration approximation is 1.57227 in interval H1.57031,1.57422L

Error bound is 0.00195313

...............................................................

11 th iteration approximation is 1.57129 in interval H1.57031,1.57227L

Error bound is 0.000976563

...............................................................

12 th iteration approximation is 1.5708 in interval H1.57031,1.57129L

Error bound is 0.000488281

...............................................................

13 th iteration approximation is 1.57056 in interval H1.57031,1.5708L

Error bound is 0.000244141

...............................................................

14 th iteration approximation is 1.57068 in interval H1.57056,1.5708L

Error bound is 0.00012207

...............................................................

Final approximation is: 1.57068 with error 0.0000610352

Newtons Method

Numerical Methods and Programming.nb 5

In[122]:= x0 = 1.0;

Nmax = 5; eps = 0.0001;

f@x_D := Cos@xD;For@i = 1, i Nmax, i++,

x1 = N@x0 - Hf@x0DL Hf'@x0DLD;If @Abs@x1 - x0D < eps, Break@D,Print@i, " th iteration is: ", x1D;Print@"Estimated error is: ", Abs@x1 - x0DD;Print@"__________________________________"D;x0 = x1D;

D;Print@"Final approximation is: ", x1D1 th iteration is: 1.64209

Estimated error is: 0.642093

__________________________________

2 th iteration is: 1.57068

Estimated error is: 0.0714173

__________________________________

3 th iteration is: 1.5708

Estimated error is: 0.00012105

__________________________________

Final approximation is: 1.5708

Newtons method with error tolerance only

In[216]:= x0 = 1.0;

eps = 0.0001;

f@x_D := Cos@xD;x1 = N@x0 - f@x0D f'@x0DD;Print@1, " th iteration approximation is: ", x1D;For@i = 2, Abs@x1 - x0D > eps, i++,

x0 = x1;

x1 = N@x0 - f@x0D f'@x0DD;Print@i, " th iteration approximation is: ", x1DD;

Print@"Final approximation is: ", x1D1 th iteration approximation is: 1.64209

2 th iteration approximation is: 1.57068

3 th iteration approximation is: 1.5708

4 th iteration approximation is: 1.5708

Final approximation is: 1.5708

Gauss Jacobi Method

4x1+x2+x3=2

x1+5x2+2x3=-6

x1+2x2+3x3=-4

6 Numerical Methods and Programming.nb

In[152]:= A = 885, 1, 2

1 th iteration approximation is:

2.

-1.55556

4.71429

2 th iteration approximation is:

0.425397

-2.98413

4.55556

3 th iteration approximation is:

0.774603

-3.43845

3.92245

4 th iteration approximation is:

1.11871

-3.04067

3.84253

5 th iteration approximation is:

1.07112

-2.89044

4.00534

6 th iteration approximation is:

0.975953

-2.97867

4.04146

7 th iteration approximation is:

0.979148

-3.02644

4.00266

8 th iteration approximation is:

1.00422

-3.00813

3.98947

9 th iteration approximation is:

1.00584

-2.99391

3.99828

10 th iteration approximation is:

0.99947

-2.99729

4.00257

11 th iteration approximation is:

0.998428

-3.00132

4.0007

12 th iteration approximation is:

0.999985

-3.00083

3.9994

13 th iteration approximation is:

1.00041

-2.99974

3.99976

14 th and final Approximation is:

1.00004

-2.99976

4.00013

810.0007, -13.9974, -33.0004 0, Print@"IVP not satisfied."D,For@i = 1, i nMax, i++,p = N@x1 - f@x1D * HHx1 - x0L Hf@x1D - f@x0DLL, 7D;Print@i, " th approximation is: ", pD;pSign = Sign@f@pDD;If@pSign * f1 < 0, x0 = p, x1 = p; f1 = pSignDD

D

1 th approximation is: 1.1

2 th approximation is: 1.15174

3 th approximation is: 1.17684

4 th approximation is: 1.18863

5 th approximation is: 1.19408

6 th approximation is: 1.19658

7 th approximation is: 1.19773

8 th approximation is: 1.19825

9 th approximation is: 1.19849

10 th approximation is: 1.1986

Regular Falsi Method with stopping condition

In[358]:= f@x_D := x^3 + 2 x^2 - 3 x - 1;x0 = 1.0; x1 = 2.0; n = 20;

f1 = Sign@f@x1DD;errList = 8 0, Print@"Ivp not satisfied"D,For@i = 1, i 2, i++,p = x1 - HHx1 - x0L Hf@x1D - f@x0DLL * f@x1D;Print@i, " th approximation is: ", pD;errList = Append@errList, pD;pSign = Sign@f@pDD;If@pSign * f1 < 0, x0 = p, x1 = p; f1 = pSignDD;For@i = 3, i n, i++,p = x1 - HHx1 - x0L Hf@x1D - f@x0DLL * f@x1D;Print@i, " th approximation is: ", pD;errList = Append@errList, pD; = HerrList@@-1DD - errList@@-2DDL HerrList@@-2DD - errList@@-3DDL;If@Abs@ H - 1LD * Abs@HerrList@@-1DD - errList@@-2DDLD < eps, Break@DD;pSign = Sign@f@pDD;If@pSign * f1 < 0, x0 = p, x1 = p; f1 = pSignDD;D

10 Numerical Methods and Programming.nb

1 th approximation is: 1.1

2 th approximation is: 1.15174

3 th approximation is: 1.17684

4 th approximation is: 1.18863

5 th approximation is: 1.19408

6 th approximation is: 1.19658

7 th approximation is: 1.19773

8 th approximation is: 1.19825

9 th approximation is: 1.19849

10 th approximation is: 1.1986

11 th approximation is: 1.19865

Basic Computations

In[369]:= f@n_D := If@n 1, Return@1D, N@1 n + f@n - 1DDDIn[379]:= a = Input@"Enter the the n"D;

f@aDOut[380]= 1.5

In[388]:=

a = 3

Out[388]= 3

In[389]:= sum = 0.0;

For@i = 1, i a, i++, sum = sum + 1 iDPrint@sumD1.83333

In[396]:=

y = Input@"Enter an array which you want to sort ?"D;Print@"The array is ", yD;For@i = 1, i < 10, i++,

If@y@@iDD > y@@i + 1DD,For@j = i, j 1, j--,If@y@@i + 1DD > y@@jDD, Break@DDD;

y = Insert@y, y@@i + 1DD, j + 1D;y = Delete@y, i + 2D

D;D;

Print@"The sorted array is ", yDThe array is 810, 7, 2, 9, 8, 6, 3, 4, 5, 1