├── Adams-Bashforth Method.cpp
├── Bisection_Method.cpp
├── Derivatives Using Forward Difference Formulae.cpp
├── Factorization Method.cpp
├── Gauss-Elimination.cpp
├── Gauss-Jordan Method.cpp
├── Gauss-Seidal Iteration Method.cpp
├── LICENSE
├── Linear Programming—Simplex Method.cpp
├── Method of Group Averages.cpp
├── Method of Least Squares.cpp
├── Method of Moments.cpp
├── Milne’s Method.cpp
├── Modified Euler’s Method.cpp
├── Muller's-Method.cpp
├── Newton-Raphson.cpp
├── Newton’s Divided Difference Formula.cpp
├── Newton’s Forward Interpolation Formula.cpp
├── Power Method.cpp
├── README.md
├── Regula-Falsi.cpp
├── Runge-Kutta Method.cpp
├── Simpson’s Rule.cpp
├── Solution of Heat Equation.cpp
├── Solution of Laplace’s Equation.cpp
├── Solution of Wave Equation.cpp
└── Trapezoidal Rule.cpp


/Adams-Bashforth Method.cpp:
--------------------------------------------------------------------------------
 1 | /*Adams-Bashforth Method*/
 2 | #include<iostream>
 3 | #include<stdio.h>
 4 | #include<malloc.h>
 5 | #include<math.h>
 6 | #include<conio.h>
 7 | using namespace std;
 8 | void main( )
 9 | {
10 |     float *x, *y, *f;
11 | float h;
12 | int i,size,row;
13 | clrscr();
14 | cout <<"enter the size";
15 | cin>>size;
16 | x=new float[size+1];
17 | y=new float[size+1];
18 | f=new float[size+1];
19 | for (i=0;i<size;i++)
20 | {
21 | cout<<"enter the value for x["<<i<<"]";
22 | cin>>x[i];
23 | cout<<"enter the value for y["<<i<<"]";
24 | cin>>y[i];
25 | }
26 | h=x[1]-x[0];
27 | // calculating values [f]
28 | for(i=0;i<4;i++)
29 | {
30 | f[i]=pow(x[i],2)*(1.0+y[i]);
31 | }
32 | cout<<"\nvalues for (x) (y) and (f) are\n";
33 | row=16;
34 | for(i=0;i<4;i++)
35 | {
36 | gotoxy(2,row);cout<<"x=";gotoxy(6,row);cout<<x[i];gotoxy(13,ro
37 | w);cout<<"y"<<i-3;gotoxy(16,row);cout<<"=";gotoxy(18,row);cout
38 | <<y[i];gotoxy(28,row);cout<<"f"<<i-3;gotoxy(32,row);cout<<"=";
39 | gotoxy(35,row);cout<<f[i];
40 | row++;
41 | }
42 | //using predicator
43 | y[size]=y[size-1]+((h/24)*((55*f[size-1])–(59*f[size-
44 | 2])+(37*f[size-
45 | 3])-(9*f[size-4])));
46 | x[size]=1.4;
47 | f[size]=pow(x[size],2)*(1.0+y[size]);
48 | gotoxy(2,row);cout<<"x=";
49 | gotoxy(6,row);cout<<x[size];gotoxy(13,row);cout<<"y1";
50 | gotoxy(16,row);cout<<"="; gotoxy(18,row);cout<<y[size
51 | ];gotoxy(28,row);cout<<"f1";gotoxy(32,row);cout<<"=";
52 | gotoxy(35,row);cout<<f[size];
53 | }


--------------------------------------------------------------------------------
/Bisection_Method.cpp:
--------------------------------------------------------------------------------
 1 | /* Bisection Method */
 2 | #include<iostream>
 3 | #include<iomanip>
 4 | #include<math.h>
 5 | using namespace std;
 6 | float f(float x){
 7 |     return (x*x*x - 4*x -9);
 8 | }
 9 | void bisect(float *x,float a,float b,int *itr){
10 |     *x = (a+b)/2;
11 |     ++(*itr);
12 |     cout<<"Iteration no."<<setw(3)<<*itr
13 |     <<"X = "<<setw(7)<<setprecision(5)
14 |     <<*x<<endl;
15 | }
16 | int main(){
17 |     int itr = 0,maxitr;
18 |     float x,a,b,aerr,x1;
19 |     cout<<"Enter the values of a,b,"
20 |     <<"allowed error, maximum iterations"<<endl;
21 |     cin>> a >> b >> aerr >> maxitr;
22 |     cout<<fixed;
23 |     bisect(&x,a,b,&itr);
24 |     do{
25 |         if(f(a)*f(x) < 0) b = x;
26 |         else a = x;
27 |         bisect(&x1,a,b,&itr);
28 |         if(fabs(x1-x) < aerr){
29 |             cout<<"After "<< itr <<" iteration, root "
30 |             <<"="<< setw(6)<< setprecision(4)
31 |             <<x1 << endl;
32 |             return 0;
33 |         }
34 |         x = x1;
35 |     }while(itr < maxitr);
36 |     cout<<"Solution does not converge,"
37 |     <<"iteration not sufficient"<<endl;
38 |     return 1;
39 | }


--------------------------------------------------------------------------------
/Derivatives Using Forward Difference Formulae.cpp:
--------------------------------------------------------------------------------
 1 | /* derivatives using forward difference*/
 2 | #include<iostream>
 3 | #include<math.h>
 4 | #include<iomanip>
 5 | #include<conio.h>
 6 | using namespace std;
 7 | void main( )
 8 | {
 9 | float *x=NULL, *y=NULL,max;
10 | float *tmp = NULL;
11 | float xval,h,p,x0,y0,yval,sum;
12 | int pos,i;
13 | clrscr( );
14 | cout<<"enter the no of comparisons";
15 | cin>>max;
16 | x=new float[max];
17 | y=new float[max];
18 | cout<<"enter the values in cv table for x and y";
19 | for (i=0;i<max;i++)
20 | {
21 | cout<<"\n value for "<<i<<"x";
22 | cin>>x[i];
23 | cout<<"\n value for "<<i<<"y";
24 | cin>>y[i];
25 | }
26 | cout<<"enter the value of x";
27 | cin>>xval;
28 | for(i=0;i<max;i++)
29 | {
30 | if(x[i]>=xval)
31 | {
32 | pos=i;
33 | break;
34 | }
35 | }
36 | x0=x[pos];
37 | y0=y[pos];
38 | cout<<"\nx0 is "<<x0<<"y0 is "<<y0<<"at"<<pos;
39 | h=x[1]-x[0];
40 | p=(xval-x0)/h;
41 | if(pos<(max))
42 | {
43 | int i,l,j;
44 | // calculating no of elements in array
45 | l=max-pos;
46 | tmp= new float [l* l];
47 | cout<<"\n";
48 | for(i=0;i<l;i++)
49 | {
50 | for(j=0;j<=l;j++)
51 | {
52 | tmp[i*l+j]=0;
53 | }
54 | cout<<"\n";
55 | }
56 | cout<<"\n size of new array" <<l<<"\n";
57 | //copying values of y in array
58 | for(i=0, j=pos;i<l;i++, j++)
59 | {
60 | tmp[i]=y[j];
61 | }
62 | cout<<"\n";
63 | for(i=1;i<l;i++)
64 | {
65 | for(j=0; j<l-i; j++)
66 | {
67 | tmp[i*l+j]=tmp[(i-1)*l+(j+1)]-tmp[(i-1)*l+(j)];
68 | }
69 | }
70 | cout<<"\nvalues are \n";
71 | for(i=0;i<l;i++)
72 | { cout<<x[i+pos]<<"\t";
73 | for(j=0; j<l; j++)
74 | {
75 | cout<<setprecision(3)<<tmp[j*l+i]<<"\t|";
76 | }
77 | cout<<"\n;
78 | }
79 | //appling newtons forward diffenation using first derivates
80 | sum=0;
81 | int k=1;
82 | for(i=1;i<l;i++)
83 | {
84 | sum=sum+((1.0/i)*tmp[i*tmp[i*l+0]*k);
85 | k=-k;
86 | }
87 | cout<<"\n\n first (dy/dx): "<<sum/h;
88 | int v[]={0,0,1.0, 1.0, 11.0/12.0,5.0/6.0,137.0/180.0};
89 | sum=0;
90 | k=1;
91 | for(i=2;i<l;i++)
92 | {
93 | sum=sum+(v[i]*tmp[i*l+0]*k);
94 | k=–k;
95 | }
96 | cout<<"\n\n second (dy/dx): "<<sum/pow(h,2.0);
97 | }


--------------------------------------------------------------------------------
/Factorization Method.cpp:
--------------------------------------------------------------------------------
 1 | /* Factorization method */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | #define N 3
 5 | using namespace std;
 6 | typedef float matrix[N][N];
 7 | matrix l,u,a;
 8 | float b[N],x[N],v[N];
 9 | void urow(int i){
10 |     float s;
11 |     int j,k;
12 |     for (j=i;j<N;j++){
13 |         s = 0;
14 |         for (k=0;k<N-1;k++)
15 |             s += u[k][j]*l[i][k];
16 |             u[i][j] = a[i][j]-s;
17 |     }
18 | }
19 | void lcol(int j){
20 |     float s;
21 |     int i,k;
22 |     for (i=j+1;i<N;i++){
23 |         s = 0;
24 |         for (k=0;k<=j-1;k++)
25 |             s += u[k][j]*l[i][k];
26 |             l[i][j] = (a[i][j]-s)/u[j][j];
27 |     }
28 | }
29 | void printmat(matrix x){
30 |     int i,j;
31 |     for (i=0;i<N;i++){
32 |     for (j=0;j<N;j++)
33 |         cout << setw(8) << setprecision(4) << x[i][j];
34 |         cout << endl;
35 |     }
36 | }
37 | int main(){
38 |     int i,j,m;
39 |     float s;
40 |     cout << "Enter the elements of augmented"
41 |     << " matrix rowwise" << endl;
42 |     for (i=0;i<N;i++){
43 |         for (j=0;j<N;j++)
44 |             cin >> a[i][j];
45 |             cin >> b[i];
46 |     }
47 |     cout << fixed;
48 |     /* now calculating the elements of l and u */
49 |     for (i=0;i<N;i++)
50 |         l[i][i] = 1.0;
51 |     for (m=0;m<N;m++){
52 |         urow(m);
53 |         if (m < N-1) lcol(m);
54 |     }
55 |     /* now printing the elements of l and u */
56 |     cout << setw(14) << "U" << endl; printmat(u);
57 |     cout << setw(14) << "L" << endl; printmat(l);
58 |     /* now solving LV=B
59 |     by forward substitution */
60 |     for (i=0;i<N;i++){
61 |         s = 0;
62 |         for (j=0;j<=i-1;j++)
63 |             s += l[i][j]*v[j];
64 |             v[i] = b[i]-s;
65 |     }
66 |     /* now solving UX=V
67 |     by backward substitution */
68 |     for (i=N-1;i>=0;i--){
69 |         s = 0;
70 |         for (j=i+1;j<N;j++)
71 |             s += u[i][j]*x[j];
72 |             x[i] = (v[i]-s)/u[i][i];
73 |     }
74 |     /* printing the results */
75 |     cout << "The solution is:-" << endl;
76 |     for (i=0;i<N;i++)
77 |         cout << "x[" << setw(3) << i+1 << "] = "
78 |         << setw(6) << setprecision(4)
79 |         << x[i] << endl;
80 |     return 0;
81 | }


--------------------------------------------------------------------------------
/Gauss-Elimination.cpp:
--------------------------------------------------------------------------------
 1 | /* Gauss elimination method */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | #include <math.h>
 5 | #define N 4
 6 | using namespace std;
 7 | int main(){
 8 |     float a[N][N+1],x[N],t,s;
 9 |     int i,j,k;
10 |     cout << "Enter the elements of the"
11 |             "augmented matrix rowwise" << endl;
12 |     cout << fixed;
13 |     for (i=0;i<N;i++)
14 |         for (j=0;j<N+1;j++)
15 |             cin >> a[i][j]);
16 |     for (j=0;j<N-1;j++)
17 |         for (i=j+1;i<N;i++){
18 |             t = a[i][j]/a[j][j];
19 |                 for (k=0;k<N+1;k++)
20 |                     a[i][k] -= a[j][k]*t;
21 |         }
22 |     /* now printing the
23 |     upper triangular matrix */
24 |     cout << "The upper triangular matrix"
25 |             "is:-" << endl;
26 |     for (i=0;i<N;i++){
27 |         for (j=0;j<N+1;j++)
28 |             cout << setw(8) << setprecision(4) << a[i][j];
29 |             cout << endl;
30 |     }
31 |     /* now performing back substitution */
32 |     for (i=N-1;i>=0;i--){
33 |     s = 0;
34 |         for (j=i+1;j<N;j++)
35 |             s += a[i][j]*x[j];
36 |             x[i] = (a[i][N]-s)/a[i][i];
37 |     }
38 |     /* now printing the results */
39 |     cout << "The solution is:- " << endl;
40 |     for (i=0;i<N;i++)
41 |         cout << "x[" << setw(3) << i+1 << "] = "
42 |         << setw(7) << setprecision(4) << x[i] << endl;
43 |     return 0;
44 | }


--------------------------------------------------------------------------------
/Gauss-Jordan Method.cpp:
--------------------------------------------------------------------------------
 1 | /* Gauss jordan method */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | #define N 4
 5 | using namespace std;
 6 | int main(){
 7 |     float a[N][N+1],t;
 8 |     int i,j,k;
 9 |     cout << "Enter the elements of the"
10 |         << "augmented matrix rowwise" << endl;
11 |     for (i=0;i<N;i++)
12 |         for (j=0;j<N+1;j++)
13 |             cin >> a[i][j];
14 |     /* now calculating the values
15 |     of x1,x2,....,xN */
16 |     cout << fixed;
17 |     for (j=0;j<N;j++)
18 |         for (i=0;i<N;i++)
19 |             if (i!=j){
20 |                 t = a[i][j]/a[j][j];
21 |                 for(k=0;k<N+1;k++)
22 |                     a[i][k] -= a[j][k]*t;
23 |             }
24 |     /* now printing the diagonal matrix */
25 |     cout << "The diagonal matrix is:-" << endl;
26 |     for (i=0;i<N;i++){
27 |         for (j=0;j<N+1;j++)
28 |             cout << setw(9) << setprecision(4) << a[i][j];
29 |             cout << endl;
30 |     }
31 |     /* now printing the results */
32 |     cout << "The solution is:- " << endl;
33 |     for (i=0;i<N;i++)
34 |         cout << "x[" << setw(3) << i+1 << "] ="
35 |         << setw(7) << setprecision(4)
36 |         << a[i][N]/a[i][i] << endl;
37 |     return 0;
38 | }


--------------------------------------------------------------------------------
/Gauss-Seidal Iteration Method.cpp:
--------------------------------------------------------------------------------
 1 | /* Gauss Seidal method */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | #include <math.h>
 5 | using namespace std;
 6 | #define N 3 
 7 | int main()
 8 | {
 9 | float a[N][N+1],x[N],aerr,maxerr, t,s,err;
10 | int i,j,itr,maxitr;
11 | /* first initializing the array x */
12 | for (i=0;i<N;i++) x[i]=0;
13 | cout << "Enter the elements of the"
14 | << "augmented matrix rowwise" << endl;
15 | for (i=0;i<N;i++)
16 | for (j=0;j<N+1;j++)
17 | cin >> a[i][j];
18 | cout << "Enter the allowed error,"
19 | << "maximum iterations" << endl;
20 | cin >> aerr >> maxitr;
21 | cout << fixed;
22 | cout << "Iteration" << setw(6) << "x[1]"
23 | << setw(11) << "x[2]"
24 | << setw(11) << "x[3]" << endl;
25 | for (itr=1;itr<=maxitr;itr++)
26 | {
27 | maxerr = 0;
28 | for (i=0;i<N;i++)
29 | {
30 | s = 0;
31 | for (j=0;j<N;j++)
32 | if (j!=i) s += a[i][j]*x[j];
33 | t = (a[i][N]-s)/a[i][i];
34 | err = fabs(x[i]-t);
35 | if (err >> maxerr) maxerr = err;
36 | x[i] = t;
37 | }
38 | cout << setw(5) << itr;
39 | for (i=0;i<N;i++)
40 | cout << setw(11) << setprecision(4) << x[i];
41 | cout << endl;
42 | if (maxerr << aerr)
43 | {
44 | cout << "Converges in" << setw(3) << itr
45 | << "iterations" << endl;
46 | for (i=0;i<N;i++)
47 | cout << "x[" << setw(3) << i+1 << "] = "
48 | << setw(7) << setprecision(4) << x[i]
49 | << endl;
50 | return 0;
51 | }
52 | }
53 | cout << "Solution does not converge,"
54 | << "iterations not sufficient" << endl;
55 | return 1;
56 | }


--------------------------------------------------------------------------------
/LICENSE:
--------------------------------------------------------------------------------
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675 | 


--------------------------------------------------------------------------------
/Linear Programming—Simplex Method.cpp:
--------------------------------------------------------------------------------
 1 | /* Linear programming by simplex method */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | #define ND 2
 5 | #define NS 2
 6 | #define N (ND+NS)
 7 | #define N1 (NS*(N+1))
 8 | using namespace std;
 9 | void init(float x[],int n){
10 |     int i=0;
11 |     for (;i<n;i++) x[i] = 0;
12 | }
13 | int main(){
14 |     int i,j,k,kj,ki,bas[NS];
15 |     float a[NS][N+1],c[N],cb[NS],th[NS],
16 |         x[ND],cj,z,t,b,min,max;
17 |     /* Initializing the arrays to zero */
18 |     init(c,N); init(cb,NS);
19 |     init(th,NS); init(x,ND);
20 |     for (i=0;i<NS;i++) init(a[i],N+1);
21 |     /* Now set coefficients for slack
22 |     variables equal to one */
23 |     for (i=0;i<NS;i++) a[i][i+ND] = 1.0;
24 |     /* Now put the slack variables in the basis */
25 |     for (i=0;i<NS;i++) bas[i] = ND+i;
26 |     /*Now get the constraints
27 |     and the objective function */
28 |     cout << "Enter the constraints" << endl;
29 |     for (i=0;i<NS;i++){
30 |         for (j=0;j<ND;j++)
31 |             cin >> a[i][j];
32 |             cin >> a[i][N];
33 |     }
34 |     cout << "Enter the objective function"
35 |     << endl;
36 |     for (j=0;j<ND;j++)
37 |     cin >> c[j];
38 |     cout << fixed;
39 |     /*Now calculate cj and identify the incoming variable */
40 |     while (1){
41 |         max = 0; kj = 0;
42 |     for (j=0;j<N;j++){
43 |         z = 0;
44 |         for (i=0;i<NS;i++)
45 |             z += cb[i]*a[i][j];
46 |             cj = c[j]-z;
47 |         if(cj > max){
48 |             max = cj; kj = j;}
49 |         }
50 |         /* Apply the optimality test */
51 |         if(max <= 0) break;
52 |         /* Now calculate thetas */
53 |         max = 0;
54 |         for (i=0;i<NS;i++)
55 |             if(a[i][kj]!= 0){
56 |                 th[i] = a[i][N]/a[i][kj];
57 |             if(th[i] > max) max=th[i];
58 |             }
59 |         /* Now check for unbounded soln. */
60 |         if(max <= 0){
61 |             cout << "Unbounded solution";
62 |         return 1;
63 |     }
64 |     /*Now search for the outgoing variable */
65 |     min = max; ki = 0;
66 |     for (i=0;i<NS;i++)
67 |         if ((th[i] < min)&&(th[i]!= 0)){
68 |             min = th[i]; ki = i;
69 |         }
70 |         /*Now a[ki][kj] is the key element*/
71 |         t = a[ki][kj];
72 |         /*Divide the key row by key element*/
73 |         for (j=0;j<N+1;j++) a[ki][j] /= t;
74 |         /*Make all other elements of key column zero */
75 |         for (i=0;i<NS;i++)
76 |         if(i!= ki){
77 |             b = a[i][kj];
78 |             for (k=0;k<N+1;k++)
79 |             a[i][k]-=a[ki][k]*b;
80 |         }
81 |         cb[ki] = c[kj];
82 |         bas[ki] = kj;
83 |     }
84 |     /* Now calculating the optimum value */
85 |     for (i=0;i<NS;i++)
86 |         if ((bas[i] >= 0) && (bas[i]<ND))
87 |             x[bas[i]] = a[i][N];
88 |             z = 0;
89 |     for (i=0;i<ND;i++)
90 |         z += c[i]*x[i];
91 |     for (i=0;i<ND;i++)
92 |         cout << "x[" << setw(3) << i+1 << "] = "
93 |             << setw(7) << setprecision(2)
94 |             << x[i] << endl;
95 |     cout << "Optimal value = "
96 |         << setw(7) << setprecision(2)
97 |         << z << endl;
98 | return 0;
99 | }


--------------------------------------------------------------------------------
/Method of Group Averages.cpp:
--------------------------------------------------------------------------------
 1 | #include<iostream>
 2 | #include<conio.h>
 3 | #include<iomanp>
 4 | using namespace std;
 5 | void main()
 6 | {
 7 | int t[10],n,i,ts1=0,ts2=0;
 8 | float a,b,rs1=0,rs2=0,r[10],x1,y1,x2,y2;
 9 | clrscr();
10 | cout<<"enter the no. of observations"<<endl;
11 | cin>>n;
12 | cout<<"enter the different values of t"<<endl;
13 | for(i=1;i<=n;i++)
14 | {
15 | cin>>t[i];
16 | }
17 | cout<<"\nenter the corresponding values of r"<<endl;
18 | for(i=1;i<=n;i++)
19 | {
20 | cin>>r[i];
21 | }
22 | for(i=1;i<=(n/2);i++)
23 | {
24 | ts1+=t[i];
25 | rs1+=r[i];
26 | }
27 | for(i=((n/2)+1);i<=n;i++)
28 | {
29 | ts2+=t[i];
30 | rs2+=r[i];
31 | }
32 | x1=ts1/(n/2);
33 | y1=rs1/(n/2);
34 | x2=ts2/(n/2);
35 | y2=rs2/(n/2);
36 | b=(y2-y1)/(x2-x1);
37 | a=y1-(b*x1);
38 | cout<<"the value of a&b comes out to be
39 | "<<endl<<"a="<<setw(5)<<setprecision(3)<<a<<"\n"<<"b="<<
40 | setw(5)<<setprecision(3)<<b;
41 | getch();
42 | }


--------------------------------------------------------------------------------
/Method of Least Squares.cpp:
--------------------------------------------------------------------------------
 1 | /* Parabolic fit by least squares */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | using namespace std;
 5 | int main()
 6 | {
 7 | float augm[3][4]={{0,0,0,0},{0,0,0,0},{0,0,0,0}};
 8 | float t,a,b,c,x,y,xsq;
 9 | int i,j,k,n;
10 | cout << "Enter the no. of pairs of"
11 | <<"observed values:" << endl;
12 | cin >> n;
13 | cout << fixed;
14 | augm [0] [0] = n;
15 | for (i=0;i<n;i++)
16 | {
17 | cout << "Pair no. " << i+1 << endl;
18 | cin >> x >> y;
19 | xsq = x*x;
20 | augm[0][1] += x;
21 | augm[0][2] += xsq;
22 | augm[1][2] += x*xsq;
23 | augm[2][2] += xsq*xsq;
24 | augm[0][3] += y;
25 | augm[1][3] += x*y;
26 | augm[2][3] += xsq*y;
27 | }
28 | augm[1][1] = augm[0][2];
29 | augm[2][1] = augm[1][2];
30 | augm[1][0] = augm[0][1];
31 | augm[2][0] = augm[1][1];
32 | cout << "The augmented matrix is:-" << endl;
33 | for (i=0;i<3;i++)
34 | {
35 | for (j=0;j<4;j++)
36 | cout << setw(9) << setprecision (4) << augm[i][j];
37 | cout << endl;
38 | }
39 | /* Now solving for a,b,c
40 | by Gauss Jordan Method */
41 | for (j=0;j<3;j++)
42 | for (i=0;i<3;i++)
43 | if (i!=j)
44 | {
45 | t = augm[i][j]/augm[j][j];
46 | for (k=0;k<4;k++)
47 | augm[i][k] -= augm[j][k]*t;
48 | }
49 | a = augm[0][3]/augm[0][0];
50 | b = augm[1][3]/augm[1][1];
51 | c = augm[2][3]/augm[2][2];
52 | cout << setprecision(4)
53 | << "a = " << setw(8) << a
54 | << "b = " << setw(8) << b
55 | << "c = " << setw(8) << c
56 | << endl;
57 | return 0;
58 | }


--------------------------------------------------------------------------------
/Method of Moments.cpp:
--------------------------------------------------------------------------------
 1 | #include<iostream>
 2 | #include<conio.h>
 3 | #include<iomanip>
 4 | using namespace std;
 5 | void main()
 6 | {
 7 | int x[10],y[10],i,n,yt=0,x1yt=0;
 8 | float a,b,l1,l2,c1,c2,c3,d,d1,d2,m1,m2,h;
 9 | clrscr();
10 | cout<<"enter the no. of observations"<<endl;
11 | cin>>n;
12 | cout<<"enter the different values of x"<<endl;
13 | for(i=1;i<=n;i++)
14 | {
15 | cin>>x[i];
16 | }
17 | cout<<"\nenter the corresponding values of y"<<endl;
18 | for(i=1;i<=n;i++)
19 | {
20 | cin>>y[i];
21 | }
22 | h=x[2]-x[1];
23 | for(i=1;i<=n;i++)
24 | {
25 | yt+=y[i];
26 | x1yt+=x[i]*y[i];
27 | }
28 | m1=h*yt;
29 | m2=h*x1yt;
30 | 11=(-(h/2)+x[1]);
31 | 12=((h/2)+x[n]);
32 | c1=(l2-l1);
33 | c2=((l2*l2)-(l1*l1)/2);
34 | c3=((l2*l2*l2)-(l1*l1*l1*)/3);
35 | cout<<"The observed equations
36 | are"<<endl<<cl<<"a+"<<c2<<"b"<<endl<<"&"<<c2<<"a+"<<c3<<"b";
37 | d=c2/c1;
38 | d1=d*c1;
39 | d2=d*c2;
40 | m1=d*m1;
41 | b=(m2-m1)/(c3-d2);
42 | a=(m1-(d2*b))/d1;
43 | cout<<"\nOn solving these equations we get
44 | a="<<setw(5)<<setprecision(2)<<a<<"
45 | &b="<<setw(5)<<setprecision(2)<<b<<endl;
46 | cout<<"hence the required equation is
47 | y="<<setw(5)<<setprecision(2)<<a<<"+"<<
48 | setw(5)<<setprecision(2)<<b<<"x";
49 | getch();
50 | }


--------------------------------------------------------------------------------
/Milne’s Method.cpp:
--------------------------------------------------------------------------------
 1 | /*Milne predictor corrector*/
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | #include <math.h>
 5 | using namespace std;
 6 | float x[5],y[5],h;
 7 | float f(int i)
 8 | {
 9 | return x[i]-y[i]*y[i];
10 | }
11 | void corect()
12 | {
13 |     y[4] = y[2]+(h/3)*(f(2)+4*f(3)+f(4));
14 | cout << setw(23) << ""
15 | << setprecision(4)
16 | << setw(8) << y[4]
17 | << setw(8) << f(4) << endl;
18 | << "",y[4],f(4)
19 | }
20 | int main()
21 | {
22 | float xr,aerr,yc;
23 | int i;
24 | cout << "Enter the values of x0,xr,h,"
25 | << "allowed error" << endl;
26 | cin >> x[0] >> xr >> h >> aerr;
27 | cout << "Enter the value of y[i], i=0, 3" << endl;
28 | for (i=0;i<=3;i++) cin >> y[i];
29 | cout << fixed;
30 | for (i=1;i<=3;i++) x[i] = x[0]+i*h;
31 | cout << setw(5) << "x" << setw(15) << "Predicted"
32 | << setw(17) << "Corrected" << endl;
33 | cout << setw(11) << "y" << setw(10) << "f"
34 | << setw(7) << "y" << setw(10) << "f" << endl;
35 | while (1)
36 | {
37 | if(x[3] = xr) return 0;
38 | x[4] = x[3]+h;
39 | y[4] = y[0]+
40 | (4*h/3)*(2*(f(1)+f(3))-f(2));
41 | cout << setw(6) << setprecision(2) << x[4]
42 | << setprecision(4)
43 | << setw(8) << y[4]
44 | << setw(8) << f(4) << endl;
45 | corect(1);
46 | while (1)
47 | {
48 | yc = y[4];
49 | corect();
50 | if(fabs(yc-y[4]) <= aerr) break;
51 | }
52 | for (i=0;i<=3;i++)
53 | {
54 |     x[i] = x[i+1];
55 | y[i] = y[i+1];
56 | }
57 | }
58 | }


--------------------------------------------------------------------------------
/Modified Euler’s Method.cpp:
--------------------------------------------------------------------------------
 1 | /* Modified Euler's Method*/
 2 | #include<iostream>
 3 | #include<math.h>
 4 | #include<iomanip>
 5 | #include<conio.h>
 6 | using namespace std;
 7 | void main( )
 8 | {
 9 | clrscr ( );
10 | int i,j;
11 | float x,y,x1=0.0,y1=0.0,h,ms=0.0,flag=0,y2=0.0,t=0.0;
12 | cout<<"\nenter the value of x";
13 | cin>>x;
14 | cout<<"enter the value of y";
15 | cin>>y;
16 | cout<<"enter the height";
17 | cin>>h;
18 | i=7;
19 | j=2;
20 | gotoxy(2,i);
21 | cout<<"x";gotoxy(10,i);cout<<"x+y=y1";gotoxy(28,i);
22 | cout<<"mean slope";gotoxy(45,i);cout<<"old y+.1(mean
23 | slope)new y"; while(x1<x)
24 | {
25 | i++;
26 | do
27 | {
28 | i++;
29 | if(flag==0)
30 | {
31 | y1=x1+y;
32 | gotoxy(2,i);cout<<x1;gotoxy(10,i);
33 | cout<<y1;gotoxy(28,i);cout<<ms;
34 | ms=y1;
35 | y2=y+h*ms;
36 | gotoxy(45,i);cout<<y2;
37 | x1=x1+h;
38 | flag=1;
39 | }
40 | else
41 | {
42 | ms=(y1+(x1+y2))/2.0;
43 | t=y+h*ms;
44 | if(y2==t)
45 | {
46 | y2=y+h*ms;
47 | break;
48 | }
49 | gotoxy(2,i);cout<<x1;gotoxy(10,i);cout<<x1<<"+"<<x2;y2=y+h*ms;
50 | gotoxy(28,i);cout<<ms;gotoxy(45,i);cout<<y2;
51 | }
52 | }while(1);
53 | y=y2;
54 | cout<<"\n";
55 | flag=0;
56 | }
57 | getch( );
58 | }


--------------------------------------------------------------------------------
/Muller's-Method.cpp:
--------------------------------------------------------------------------------
 1 | /* Muller's Method */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | #include <math.h>
 5 | #define I 2
 6 | using namespace std;
 7 | float y(float x){
 8 |     return cos(x)-x*exp(x);
 9 | }
10 | int main(){
11 |     int i,itr,maxitr;
12 |     float x[4],li,di,mu,s,l,aerr;
13 |     cout << "Enter the initial"
14 |     "approximations" << endl;
15 |     for (i = I-2;i<3;i++)
16 |     cin >> x[i];
17 |     cout << "Enter allowed error,"
18 |     "maximum iterations" << endl;
19 |     cin >> aerr >> maxitr;
20 |     cout << fixed;
21 |     for(itr = 1;itr <= maxitr;itr++){
22 |         li = (x[I]-x[I-1])/(x[I-1]-x[I-2]);
23 |         di = (x[I]-x[I-2])/(x[I-1]-x[I-2]);
24 |         mu = y(x[I-2])*li*li
25 |             - y(x[I-1])*di*di
26 |             + y(x[I])*(di+li);
27 |         s = sqrt((mu*mu - 4*y(x[I])*di*li
28 |             *(y(x[I-2])*li-y(x[I-1])
29 |             *di + y(x[I]))));
30 |         if (mu < 0)     l = (2*y(x[I])*di)/(-mu+s);
31 |         else    l = (2*y(x[I])*di)/(-mu-s);
32 |         x[I+1] = x[I]+l*(x[I] - x[I-1]);
33 |         cout << "Iteration no. " << setw(3) << itr
34 |         << "X = " << setw(7) << setprecision(5)
35 |         << x[I+1] << endl;
36 |         if (fabs(x[I+1]-x[I]) < aerr){
37 |         cout << "After" << setw(3) << itr
38 |         << "iterations, the solution is"
39 |         << setw(6) << setprecision(4)
40 |         << x[I+1] << endl;
41 |         return 0;
42 |         }
43 |     for (i=I-2;i<3;i++)
44 |     x[i] = x[i+1];
45 |     }
46 |     cout << "Iterations not sufficient,"
47 |         << "solution does not converge" << endl;
48 |     return 1;
49 | }


--------------------------------------------------------------------------------
/Newton-Raphson.cpp:
--------------------------------------------------------------------------------
 1 | /* Newton Raphson Method */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | #include <math.h>
 5 | using namespace std;
 6 | float f(float x){
 7 |     return x*log10(x)-1.2;
 8 | }
 9 | float df(float x){
10 |     return log10(x) + 0.43429;
11 | }
12 | int main(){
13 |     int itr,maxitr;
14 |     float h,x0,x1,aerr;
15 |     cout << "Enter x0,allowed error,"
16 |     << "maximum iterations" << endl;
17 |     cin >> x0 >> aerr >> maxitr;
18 |     cout << fixed;
19 |     for (itr=1;itr<=maxitr;itr++){
20 |         h = f(x0)/df(x0);
21 |         x1 = x0-h;
22 |         cout << "Iteration no." << setw(3) << itr
23 |         << "X = " << setw(9) << setprecision(6)
24 |         << x1 << endl;
25 |         if (fabs(h) < aerr){
26 |             cout << "After" << setw(3) << itr
27 |             << "iterations, root = "
28 |             << setw(8) << setprecision(6) << x1;
29 |             return 0;
30 |         }
31 |         x0 = x1;
32 |     }
33 |     cout << "Iterations not sufficient,"
34 |     << "solution does not converge" << endl;
35 |     return 1;
36 | }


--------------------------------------------------------------------------------
/Newton’s Divided Difference Formula.cpp:
--------------------------------------------------------------------------------
 1 | #include<iostream>
 2 | #include<conio.h>
 3 | using namespace std;
 4 | void main(){
 5 |     int x[10],y[10],p[10];
 6 |     int k,f,n,i,j=1,f1=1,f2=0;
 7 |     clrscr();
 8 |     cout<<"enter the no. of observations\n";
 9 |     cin>>n;
10 |     cout<<"enter the different values of x\n";
11 |     for(i=1;i<=n;i++){
12 |         cin>>x[i];
13 |     }
14 |     cout<<"enter the corresponding values of y\n";
15 |     for(i=1;i<=n;i++){
16 |         cin>>y[i];
17 |     }
18 |     f=y[1];
19 |     cout<<"enter the value of 'k' in f(k) you want to evaluate\n";
20 |     cin>>k;
21 |     do{
22 |         for(i=1;i<=n-1;i++){
23 |             p[i]=((y(i+1)-y[i])/(x[i+j]-x[i]));
24 |             y[i]=p[i];
25 |         }
26 |         for(i=1;i<=j;i++){
27 |             f1*=(k-x[i]);
28 |         } f2+=(y[1]*f1); f1=1;
29 |         n--;
30 |         j++;
31 |     } while(n!=1); f+=f2; cout<<"f("<<k<<")="<<f; getch();
32 | }


--------------------------------------------------------------------------------
/Newton’s Forward Interpolation Formula.cpp:
--------------------------------------------------------------------------------
 1 | /* Newton's forward interpolation */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | #define MAXN 100
 5 | #define ORDER 4 
 6 | using namespace std;
 7 | int main()
 8 | {
 9 | float ax[MAXN+1],ay[MAXN+1],
10 | diff[MAXN+1][ORDER+1],
11 | nr=1.0,dr=1.0,x,p,h,yp;
12 | int n,i,j,k;
13 | cout << "Enter the value of n" << endl;
14 | cin >> n;
15 | cout << "Enter the values in form x,y" << endl;
16 | for (i=0;i<=n;i++)
17 | cin >> ax[i] >> ay[i];
18 | cout << "Enter the values of x"
19 | << "for which value of y is wanted" << endl;
20 | cin >> x;
21 | cout << fixed;
22 | h=ax[1]–ax[0];
23 | /* now making the diff. table */
24 | /* calculating the 1st order differences */
25 | for (i=0;i<=n-1;i++)
26 | diff[i][1] = ay[i+1]-ay[i];
27 | /* calculating the second & higher order differences.*/
28 | for (j=2;j<=ORDER;j++)
29 | for (i=0;i<=n-j;i++)
30 | diff [i][j] = diff[i+1][j-1]
31 | -diff[i][j-1];
32 | /* now finding x0 */
33 | i=0;
34 | while (!(ax[i] > x)) i++;
35 | /* now ax[i] is x0 & ay[i] is y0 */
36 | i--;
37 | p = (x-ax[i])/h; yp=ay[i];
38 | /* Now carrying out interpolation */
39 | for (k=1;k<=ORDER;k++)
40 | {
41 | nr *= p-k+1; dr *=k;
42 | yp += (nr/dr)*diff[i][k];
43 | }
44 | cout << "When x = "
45 | << setw(6) << setprecision(1)
46 | << x
47 | << " y = "
48 | << setw(6) << setprecision(2)
49 | << yp << endl;
50 | return 0;
51 | }


--------------------------------------------------------------------------------
/Power Method.cpp:
--------------------------------------------------------------------------------
 1 | /* Power method for finding largest eigen value */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | #include <math.h>
 5 | #define N 3
 6 | using namespace std;
 7 | typedef float array[N];
 8 | void findmax(float *max,array x)
 9 | {
10 | int i;
11 | *max = fabs(x[0]);
12 | for (i=1;i<N;i++)
13 | if (fabs(x[i]) > *max)
14 | *max = fabs(x[i]);
15 | }
16 | int main()
17 | {
18 | float a[N][N],x[N],r[N],maxe,
19 | err,errv,aerr,e,s,t;
20 | int i,j,k,itr,maxitr;
21 | cout << "Enter the matrix rowwise" << endl;
22 | for (i=0;i<N;i++)
23 | for (j=0;j<N;j++)
24 | cin << a[i][j];
25 | cout << "Enter the initial approximation"
26 | << "to the eigen vector" << endl;
27 | for (i=0;i<N;i++)
28 | cin >> x[i];
29 | cout << "Enter the allowed error,"
30 | << "maximum iterations" << endl;
31 | cin >> aerr >> maxitr;
32 | cout << fixed;
33 | cout << "Itr no." << setw(11) << "Eigenvalue"
34 | << setw(19) << "EigenVector" << endl;
35 | /*now finding the largest eigenvalue in
36 | the initial approx. to eigen vector */
37 | findmax(&e,x);
38 | /* now starting the iterations */
39 | for (itr=1;itr<=maxitr;itr++)
40 | {
41 | /* loop to multiply the matrices a and x */
42 | for (i=0;i<N;i++)
43 | {
44 | s = 0;
45 | for (k=0;k<N;k++)
46 | s += a[i][k]*x[k];
47 | r[i]=s;
48 | }
49 | findmax(&t,r);
50 | for (i=0;i<N;i++) r[i] /= t;
51 | maxe = 0;
52 | for (i=0;i<N;i++)
53 | {
54 | err = fabs(x[i]-r[i]);
55 | if (err > maxe) maxe = err;
56 | x[i] = r[i];
57 | }
58 | errv = fabs(t-e);
59 | e = t;
60 | cout << setw(4) << itr
61 | << setw(12) << setprecision(4)
62 | << e;
63 | for (i=0;i<N;i++)
64 | cout << setw(9) << setprecision(3)
65 | << x[i];
66 | cout << endl;
67 | if ((errv <= aerr) && (maxe <= aerr))
68 | {
69 | cout << "Converges in" << itr
70 | << "iterations" << endl;
71 | cout << "Largest eigen value ="
72 | << setw(6) << setprecision(2)
73 | << e << endl;
74 | cout << "Eigen Vector:-" << endl;
75 | for (i=0;i<N;i++)
76 | cout << "x[" << setw(3) << i+1 << "] = "
77 | << setw(6) << setprecision(2)
78 | << x[i] << endl;
79 | cout << endl;
80 | return;
81 | }
82 | }
83 | cout << "Solution does not converge,"
84 | << "iterations not sufficient" << endl;
85 | return 1;
86 | }


--------------------------------------------------------------------------------
/README.md:
--------------------------------------------------------------------------------
1 | # Numerical-Algorithm
2 |  In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm
3 | 


--------------------------------------------------------------------------------
/Regula-Falsi.cpp:
--------------------------------------------------------------------------------
 1 | /* Bisection Method */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | #include <math.h>
 5 | using namespace std;
 6 | float f(float x)
 7 | {
 8 |     return (x*x*x - 4*x - 9);
 9 | }
10 | void bisect(float *x,float a,float b,int *itr){
11 |     *x = (a + b)/2;
12 |     ++(*itr);
13 |     cout << "Iteration no." <<setw(3) << *itr
14 |         << "X = " << setw(7) << setprecision(5)
15 |         << *x << endl;
16 | }
17 | int main(){
18 |     int itr = 0, maxitr;
19 |     float x, a, b, aerr, x1;
20 |     cout << "Enter the values of a,b,"
21 |     << "allowed error, maximum iterations" << endl;
22 |     cin >> a >> b >> aerr >> maxitr;
23 |     cout << fixed;
24 |     bisect(&x,a,b,&itr);
25 |     do{
26 |         if (f(a)*f(x) < 0)
27 |         b = x;
28 |         else
29 |         a = x;
30 |         bisect (&x1,a,b,&itr);
31 |         if (fabs(x1-x) < aerr){
32 |             cout << "After" << itr << "iterations, root"
33 |             << "=" << setw(6) << setprecision(4)
34 |             << x1 << endl;
35 |         return 0;
36 |         }
37 |         x = x1;
38 |     } while (itr < maxitr);
39 |     cout << "Solution does not converge,"
40 |     << "iterations not sufficient" << endl;
41 |     return 1;
42 | }


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/Runge-Kutta Method.cpp:
--------------------------------------------------------------------------------
 1 | /* Runge Kutta Method */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | using namespace std;
 5 | float f(float x,float y)
 6 | {
 7 | return x+y*y;
 8 | }
 9 | int main()
10 | {
11 | float x0,y0,h,xn,x,y,k1,k2,k3,k4,k;
12 | cout << "Enter the values of x0,y0,"
13 | << "h,xn" << endl;
14 | cin >> x0 >> y0 >> h >> xn;
15 | x = x0; y = y0;
16 | cout << fixed;
17 | while (1)
18 | {
19 | if (x == xn) break;
20 | k1 = h*f(x,y);
21 | k2 = h*f(x+h/2,y+k1/2);
22 | k3 = h*f(x+h/2,y+k2/2);
23 | k4 = h*f(x+h,y+k3);
24 | k = (k1+(k2+k3)*2+k4)/6;
25 | x += h; y += k;
26 | cout << "When x = " << setprecision(4)"
27 | << setw(8) << x
28 | << " y = " << setw(8) << y << endl;
29 | }
30 | return 0;
31 | }


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/Simpson’s Rule.cpp:
--------------------------------------------------------------------------------
 1 | /* Simpson's rule */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | using namespace std;
 5 | float y(float x){
 6 |     return 1/(1+x*x);
 7 | }
 8 | int main(){
 9 |     float x0,xn,h,s;
10 |     int i,n;
11 |     cout << "Enter x0,xn, no.of subintervals"
12 |         << endl;
13 |     cin >> x0 >> xn >> n;
14 |     cout << fixed;
15 |     h = (xn-x0)/n;
16 |     s = y(x0)+y(xn)+4*y(x0+h);
17 |     for (i=3;i<=n-1;i+=2)
18 |         s += 4*y(x0+i*h)+2*y(x0+(i-1)*h);
19 |     cout << "Value of integral is"
20 |         << setw(6) << setprecision(4)
21 |         << (h/3)*s << endl;
22 |     return 0;
23 | }


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/Solution of Heat Equation.cpp:
--------------------------------------------------------------------------------
 1 | /*Solution of parabolic equations by
 2 | Bendre Schmidt method*/
 3 | #include <iostream>
 4 | #include <iomanip>
 5 | #define XEND 8
 6 | #define TEND 5 
 7 | using namespace std;
 8 | float f(int x){
 9 |     return 4*x-(x*x)/2.0;
10 | }
11 | int main(){
12 |     float u[XEND+1][TEND+1],h=1.0,k=0.125,
13 |         csqr,alpha,ust,uet;
14 |     int i,j;
15 |     cout << "Enter the square of 'c'" << endl;
16 |     cin >> csqr;
17 |     alpha = (csqr*k)/(h*h);
18 |     cout << "Enter the value of u[0,t]" << endl;
19 |     cin >> ust;
20 |     cout <<"Enter the value of u[" << XEND
21 |         << ",t]" << endl;
22 |     cin >> uet;
23 |     cout << fixed;
24 |     for (j=0;j<=TEND;j++)
25 |         u[0][j]=u[XEND][j]=ust;
26 |     for (i=1;i<=XEND-1;i++)
27 |         u[i][0]=f(i);
28 |     for (j=0;j<=TEND-1;j++)
29 |         for (i=1;i<=XEND-1;i++)
30 |             u[i][j+1]=
31 |             alpha*u[i-1][j]
32 |             +(1-2*alpha)*u[i][j]
33 |             +alpha*u[i+1][j];
34 |     cout << "The value of alpha is"
35 |         << setw(4) << setprecision(2)
36 |         << alpha << endl;
37 |     cout << "The values of u[i,j] are:-"
38 |         << endl;
39 |     for (j=0;j<TEND;j++){
40 |         for (i=0;i<=XEND;i++)
41 |             cout << setw(7) << setprecision(4)
42 |                 << u[i][j];
43 |         cout << endl;
44 |     }
45 | return 0;
46 | }


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/Solution of Laplace’s Equation.cpp:
--------------------------------------------------------------------------------
 1 | /* Laplace's Equation */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | #include <math.h>
 5 | #define SQR 4
 6 | typedef float array[SQR+1][SQR+1];
 7 | using namespace std;
 8 | void getrow(int i,array u){
 9 |     int j;
10 |     cout << "Enter the values of u["
11 |     << i << ",j], j=1, " << SQR << endl;
12 |     for (j=1;j<=SQR;j++)
13 |     cin >> u[i][j];
14 | }
15 | void getcol(int j,array u){
16 |     int i;
17 |     cout << "Enter the values of u[i," << j
18 |         << "], i=2," << SQR-1 << endl;
19 |     for (i=2;i<=SQR-1;i++)
20 |         cin >> u[i][j];
21 | }
22 | void printarr(array u,int width,int precision){
23 |     int i,j;
24 |     for (i=1;i<=SQR;i++){
25 |         for (j=1;j<=SQR;j++)
26 |             cout << setw(width) << setprecision(precision)
27 |                 << u[i][j];
28 |         cout << endl;
29 |     }
30 | }
31 | int main (){
32 |     array u;
33 |     float maxerr,aerr,err,t;
34 |     int i,j,itr,maxitr;
35 |     for (i=1;i<=SQR;i++)
36 |         for(j=1;j<=SQR;j++)
37 |             u[i][j]=0;
38 |     cout << "Enter the boundary conditions" << endl;
39 |     getrow(1,u); getrow(SQR,u);
40 |     getcol(1,u); getcol(SQR,u);
41 |     cout << "Enter allowed error,"
42 |         << "maximum iterations" << endl;
43 |     cin >> aerr >> maxitr;
44 |     cout << fixed;
45 |     for (itr=1;itr<=maxitr;itr++){
46 |         maxerr=0;
47 |     for (i=2;i<=SQR-1;i++)
48 |         for(j=2;j<=SQR-1;j++){
49 |             t=(u[i-1][j]+u[i+1][j]+
50 |                 u[i][j+1]+u[i][j-1])/4;
51 |             err=fabs(u[i][j]-t);
52 |             if (err > maxerr)
53 |                 maxerr = err;
54 |             u[i][j]=t;
55 |         }
56 |         cout << "Iteration no. " << itr << endl;
57 |         printarr(u,9,2);
58 |         if (maxerr <= aerr){
59 |             cout << "After " << itr << " iterations"
60 |                 << endl
61 |                 << "The solution:-" << endl;
62 |                 printarr(u,8,1);
63 |                 return 0;
64 |         }
65 |     }
66 |     cout << "Iterations not sufficient." << endl;
67 |     return 1;
68 | }


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/Solution of Wave Equation.cpp:
--------------------------------------------------------------------------------
 1 | /* Solution of Hyperbolic equation */
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | #define XEND 5
 5 | #define TEND 5 
 6 | using namespace std;
 7 | float f(int x){
 8 |     return x*x*(5-x);
 9 | }   
10 | int main(){
11 |     float u[XEND+1][TEND+1],csqr,ust,uet;
12 |     int i,j;
13 |     cout << "Enter the square of 'c'" << endl;
14 |     cin >> csqr;
15 |     cout << "Enter the value of u(0, t)" << endl;
16 |     cin >> ust;
17 |     cout << "Enter the value of u("
18 |         << XEND << ", t)" << endl;
19 |     cin >> uet;
20 |     cout << fixed;
21 |     for (j=0;j<=TEND;j++){
22 |         u[0][j] = ust; u[XEND][j] = uet;
23 |     }
24 |     for (i=1;i<=XEND-1;i++)
25 |         u[i][1] = u[i][0] = f(i);
26 |     for (j=1;j<=TEND-1;j++)
27 |         for (i=1;i<=XEND-1;i++)
28 |             u[i][j+1] = u[i-1][j]+u[i+1][j]
29 |             -u[i][j-1];
30 |     cout << "The values of u(i, j) are:-" << endl;
31 |     for (j=0;j<=TEND;j++){
32 |         for (i=0;i<=XEND;i++)
33 |             cout << setw(6) << setprecision(1)
34 |             << u[i][j];
35 |         cout << end1,
36 |     }
37 |     return 0;
38 | }
39 | 


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/Trapezoidal Rule.cpp:
--------------------------------------------------------------------------------
 1 | /* Trapezoidal rule.*/
 2 | #include <iostream>
 3 | #include <iomanip>
 4 | using namespace std;
 5 | float y(float x){
 6 |     return 1/(1+x*x);
 7 | }
 8 | int main(){
 9 |     float x0,xn,h,s;
10 |     int i,n;
11 |     cout << "Enter x0,xn,no. of subintervals" << endl;
12 |     cin >> x0 >> xn >> n;
13 |     cout << fixed;
14 |     h = (xn-x0)/n;
15 |     s = y(x0)+y(xn);
16 |     for (i=1;i<=n-1;i++)
17 |         s += 2*y(x0+i*h);
18 |     cout << "Value of integral is"
19 |         << setw(6) << setprecision(4)
20 |         << (h/2)*s << endl;
21 |     return 0;
22 | }


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