Numerical Programs In C
Note: The code of all method is Develop in "C" is compiled with GNU GCC compiler. However, these codes are compatible with all other operating systems.
$ git clone https://github.com/KaizIqbal/Numeric_Pro.git
$ cd Numeric_Pro/
In All Programs stand with '.c' Have One Function Called "fxy()".It Used For Equation
float fxy(float x,float y)
{
return(x*y); //x*y is Basic Equation
}
Runge-Kutta method for approximating the solution of the initial value problem y'(x) = f(x,y);
y(x0) = y0 which evaluates the integrand,f(x,y), twice for each step. For step i+1,
y(i+1) = y(i) + 1/4(k1 + 3*k2),
Where
k1 = h*f(x(i) ,y(i)),
k2 = h*f(x(i) + 2*h/3 ,y(i) + 2*k1/3),
and x(i) = x(0+i) * h.
Algorithm For RUNGE-KUTTA SECOND ORDER FORMULA METHOD
[FUNCTION fxy]
function fxy[x,y]{
return (dy/dx)
}
[Step-1] [Initialization]
int i <- 1
int count <- 1
float x0,xn,y0,h,x_array[n],y_array[n],k1,k2
[Step-2] [Get value Of x0,xn,h,y1]
write(Enter x0 : )
read(x0)
write(Enter xn : )
read(xn)
write(Enter y0 : )
read(y0)
write(Enter h :)
read(h)
[Step-3] [Set initial Value To Array]
x_array(i)<-x0
y_array(i)<-y1
[Step-4] [Find x0 to xn Values and Count It]
for(i<-1 to x_array(i)<=xn){
x_array(i+1) <- x_array(i)+h;
count++;
}
[Step-5] [Find y_array Value To apply Runge-Kutta Second Order Formula]
for(i<-1 to i<=count){
/* Runge-Kutta Second Order Formula */
k1 <- h*fxy[x_array(i),y_array(i)];
k2 <- h*fxy[x_array(i)+((2*h)/3),y_array(i)+((2*k1)/3)];
y_array(i+1) <- y_array(i)+(k1+(3*k2));
}
[Step-6] [Display]
for(i<-1 to i<=count){
write(x_array(i) y_array(i))
}
[Step-7] [Exit]
End.
$ cd Runge_Kutta_2ndOrder/
$ ls
$ gcc RK_2nd.c
$ ./a.out
This method is a third order Runge-Kutta method for approximating the solution of the initial value problem y'(x) = f(x,y);
y(x0) = y0 which evaluates the integrand,f(x,y), three times per step. For step i+1,
y(i+1) = y(i) + 1/6 * (k1 + 4k2 + k3),
Where
k1 = h*f(x(i) ,y(i)),
k2 = h*f(x(i) + h/2 ,y(i) + k1/2),
k3 = h*f(xi + h ,y(i) - k1 + 2*k2),
and x(i) = x(0+i) * h.
Algorithm For RUNGE-KUTTA THIRD ORDER FORMULA METHOD
[FUNCTION fxy]
function fxy[x,y]{
return (dy/dx)
}
[Step-1] [Initialization]
int i <- 1
int count <- 1
float x0,xn,y0,h,x_array[n],y_array[n],k1,k2,k3
[Step-2] [Get value Of x0,xn,h,y1]
write(Enter x0 : )
read(x0)
write(Enter xn : )
read(xn)
write(Enter y0 : )
read(y0)
write(Enter h :)
read(h)
[Step-3] [Set initial Value To Array]
x_array(i)<-x0
y_array(i)<-y1
[Step-4] [Find x0 to xn Values and Count It]
for(i<-1 to x_array(i)<=xn){
x_array(i+1) <- x_array(i)+h;
count++;
}
[Step-5] [Find y_array Value To apply Runge-Kutta Third Order Formula]
for(i<-1 to i<=count){
/* Runge-Kutta Third Order Formula */
k1 <- h*fxy[x_array(i),y_array(i)];
k2 <- h*fxy[x_array(i)+(h/2),y_array(i)+(k1/2)];
k3 <- h*fxy[x_array(i)+h,y_array(i)-k1+(2*k2)];
y_array(i+1) <- y_array(i)+(1/6)*(k1+(4*k2)+k3);
}
[Step-6] [Display]
for(i<-1 to i<=count){
write(x_array(i) y_array(i))
}
[Step-7] [Exit]
End.
$ cd Runge_Kutta_3rdOrder/
$ ls
$ gcc Rk_3rd.c
$ ./a.out
This method is the classical fourth order Runge-Kutta method for approximating the solution of the initial value problem y'(x) = f(x,y);
y(x0) = y0 which evaluates the integrand,f(x,y), four times per step. For step i+1,
y(i+1) = y(i) + 1/6(k1 + 2*k2 + 2*k3 + k4 ),
Where
k1 = h*f(x(i) ,y(i)),
k2 = h*f(x(i) + h/2 ,y(i) + k1/2 ),
k3 = h*f(x(i) + h/2 ,y(i) + k2/2 ),
k4 = h*f(x(i) + h ,y(i) + k3 ),
and x(i) = x(0+i) * h.
Algorithm For RUNGE-KUTTA FOURTH ORDER FORMULA METHOD
[FUNCTION fxy]
function fxy[x,y]{
return (dy/dx)
}
[Step-1] [Initialization]
int i <- 1
int count <- 1
float x0,xn,y0,h,x_array[n],y_array[n],k1,k2,k3,k4
[Step-2] [Get value Of x0,xn,h,y1]
write(Enter x0 : )
read(x0)
write(Enter xn : )
read(xn)
write(Enter y0 : )
read(y0)
write(Enter h :)
read(h)
[Step-3] [Set initial Value To Array]
x_array(i)<-x0
y_array(i)<-y1
[Step-4] [Find x0 to xn Values and Count It]
for(i<-1 to x_array(i)<=xn){
x_array(i+1) <- x_array(i)+h;
count++;
}
[Step-5] [Find y_array Value To apply Runge-Kutta Fourth Order Formula]
for(i<-1 to i<=count){
/* Runge-Kutta Fourth Order Formula */
k1=h*fxy[x_array(i),y_array(i)];
k2=h*fxy[x_array(i)+(h/2),y_array(i)+(k1/2)];
k3=h*fxy[x_array(i)+(h/2),y_array(i)+(k2/2)];
k4=h*fxy[x_array(i)+h,y_array(i)+k3];
y_array(i+1)=y_array(i)+(1/6)*(k1+(2*k2)+(2*k3)+k4);
}
[Step-6] [Display]
for(i<-1 to i<=count){
write(x_array(i) y_array(i))
}
[Step-7] [Exit]
End.
$ cd Runge_Kutta_4thOrder/
$ ls
$ gcc RK_4th.c
$ ./a.out
The solution for Euler’s numerical method is generated by iterating on the two formulas:
x(i+1) = x(i)+h
y(i+1) = y(i)+h*f(x(i), y(i))
Where h=Length of subdivisions
Algorithm For EULER'S METHOD
[FUNCTION fxy]
function fxy[x,y]{
return (dy/dx)
}
[Step-1] [Initialization]
int i <- 1
int count <- 1
float y1,y2,x1,x0,xn,h,x_array[n],y_array[n]
[Step-2] [Get value Of x0,xn,h,y1]
write(Enter x0 : )
read(x0)
write(Enter xn : )
read(xn)
write(Enter h : )
read(h)
write(Enter y1 :)
read(y1)
[Step-3] [Set initial Value To Array]
x_array(i) <- x0
y_array(i) <- y1
[Step-4] [Find x0 to xn Values and Count It]
for(i<-1 to x_array(i)<=xn){
x_array(i+1) <- x_array(i)+h;
count++;
}
[Step-5] [Find y_array Value To apply Euler's Formula]
for(i<-1 to i<=count){
/* Eulers Formula */
y_array(i+1) <- y_array(i)+h*fxy[x_array(i),y_array(i)];
}
[Step-6] [Display]
for(i<-1 to i<=count){
write(x_array(i) y_array(i))
}
[Step-7] [Exit]
End.
$ cd Eulers_Method/
$ ls
$ gcc Euler.c
$ ./a.out
The solution for Euler’s numerical method is generated by iterating on the three formulas:
x(i+1) = x(i)+h
<!-- Forward Euler's Method -->
y(i+1) = y(i) + h*f(x(i), y(i))
<!-- Modified Euler's Method -->
y(i+1) = y(i) + h/2(f[x(i), y(i)] + f[x(i+1), y(i+1)])
Where h=Length of subdivisions
Algorithm For MODIFIED EULER'S METHOD
[FUNCTION fxy]
function fxy[x,y]{
return (dy/dx)
}
[Step-1] [Initialization]
int i <- 1
int count <- 1
float y1,y2,x1,x0,xn,h,x_array[n],y_array[n]
[Step-2] [Get value Of x0,xn,h,y1]
write(Enter x0 : )
read(x0)
write(Enter xn : )
read(xn)
write(Enter h : )
read(h)
write(Enter y1 :)
read(y1)
[Step-3] [Set initial Value To Array]
x_array(i) <- x0
y_array(i) <- y1
[Step-4] [Find x0 to xn Values and Count It]
for(i<-1 to x_array(i)<=xn){
x_array(i+1) <- x_array(i)+h;
count++;
}
[Step-5] [Find y_array Value To apply Euler's Formula]
for(i<-1 to i<=count){
/* Eulers Formula */
y_array(i+1) <- y_array(i)+h*fxy[x_array(i),y_array(i)];
/* Eulers Modified Formula */
y_array(i+1) <- y_array(i)+(h/2)*fxy[x_array(i),y_array(i)]+fxy[x_array(i+1),y_array(i+1)];
}
[Step-6] [Display]
for(i<-1 to i<=count){
write(x_array(i) y_array(i))
}
[Step-7] [Exit]
End.
$ cd Eulers_Modified_Method/
$ ls
$ gcc Eulers_Mod.c
$ ./a.out
".dat" File Is output Holder File Thats Created in Programs.
int main(){
FILE *fp;
: :
: :
fp=fopen("rk_demo_out.dat","w"); //.dat File Created Here
: :
: :
fclose(fp);
return 0;
}