Solution of Ordinary Differential Equation using Runge-Kutta Method | RK4 method for ODE Solution in C

In numerical analysis, the Runge–Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 by the German mathematicians C. Runge and M. W. Kutta.

There are many variants of the Runge-Kutta method, but the most widely used one is the following known as RK-4 method: 

Given 

y' = f(x,y) 

y(xn) = yn 

we compute in turn 

k1 = hf(xn,yn) 

k 2 = hf (xn+h/2,yn+k1/2)

k3 = hf (xn+h/2,yn+k2/2)

k4 = hf(xn + h, yn + k3) 

yn+1 = yn + (k1 + 2*k2 + 2*k3 + k4 )/6

This algorithm can be implemented  using the following source code in C:

Gauss Jordan Method Implementation with C Source Code

In linear algebra, Gaussian Jordan method is an algorithm for solving systems of linear equations. It is usually understood as a sequence of operations performed on the associated matrix of coefficients. This method can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix.

We can implement this method in C using the following source code:

//Implementation of Gauss Jordan Method, @author: +Jivan Nepali, @URL: codeplustech

 #include<stdio.h>  
 #include<conio.h>  
 #include<stdlib.h>  
 #define Max 10  
 int main()  
 {  
   float a[Max][Max+1],t,det=1;  
   int i,j,k,N;  
   printf("Enter the number of unknowwns : ");  
   scanf("%d",&N);  

Modeling & Simulation of Chemical Reaction | Continuous System Simulation | C++ Implementation

Chemical reactions exhibit dynamic equilibrium, which means that a combination reaction is also accomplished by the reverse process of decomposition reaction. At the steady state the rates of the forward and the backward reaction is same.

In addition to the forward reaction where Ch1 and Ch2 react to produce Ch3, there may also be backward reaction, where by, Ch3 decomposes back into Ch1 and Ch2. Let the rate of formation of Ch3 be proportional to the product of the amounts Ch1 and Ch2 present in the mixture and let the rate of decomposition of Ch3 be proportional to its amount in the mixture. 

Let us consider C1, C2  and C3 Amount of Ch1, Ch2 and Ch3 at any instant of time t, the rate of increase of C1, C2& C3 are given by the following differential equations:

dC1/dt = K1C3– K1C1C2

dC2/dt = K2C3– K1C1C2

dC3/dt=2K1C1C2-2K2C3

Where K1and K2 are constants.

Preventing Windows 8 from Automatically Launching Browser on Network Connection | Solutions

With the newer version of Microsoft's Windows Operating System- Windows 8, its users are experiencing newer and more advanced features than the older versions. But, at the same time they may have been facing with some of the strange troubles.

One such issue is that Windows 8 automatically launches its default browser upon the the internet connection. To some users it may be helpful as they need not to open the browser and especially for users in a LAN provided by some institutions such as University, it may be the sign of avialability of the internet connection. But, for most of the users such as those posting in Microsoft's Community Page, it may be really problematic. 

"Windows is launching the following link in Internet Explorer:

http://go.microsoft.com/fwlink/?LinkID=219472&clcid=0x409

which leads to Bing. When I enable my internet connection Bing will popup from time to time. When I disable Internet Explorer via the Windows Features, the link will open in Mozilla Firefox."

So, the steps that I followed for solving this issue are given:

Newton-Raphson Method in C with source codes

Newton-Raphson method is a method for finding successively better roots (zeros) of a real valued function
x: f(x) = 0.

The algorithm can be implemented in C as follows:

#include<stdio.h>
#include<math.h>
#include<conio.h>
#define h pow(10,-6)
#define error pow(10,-6)
double f(double x)
{
 return(x*x-4);
}
double df(double x)
{
 return 2*x;
}

Implementation of Warshall's Algorithm in C++ with Source Codes

Warshall's algorithm is used to find the transitive closure of a graph. It's one of the efficient method to compute closure path that can be produced. Transitive closure is an important application in graph theory in computer science.

The following provides the source codes for implementing the Warshall's algorithm:

 //.............................USE OF WARSHALL'S ALGORITHM TO CREATE TRANSITIVE CLOSURE OF A GRAPH..........

#include<iostream>
#include<conio.h>

using namespace std;
const int num_nodes =10;

int main()
{
    //..............................................................VARIABLE DECLARATION
    int num_nodes,k,n;
    char i,j,res,c;
    int adj[10][10],path[10][10];

C Source Code for Dijkstra's Shortest Path Algorithm

Dijkstra's Shortest Path Algorithm is the popular algorithm for finding shortest/optimal path in compute science due to its widespread application in different branches- computer networks, distributed systems, signal processing etc.

I've implemented the algorithm for finding the minimum distance between any two nodes:

///......................................SHORTEST PATH ALGORITHM << Dijkstra Algorithm >>......................

#include<stdio.h>
#include<conio.h>
#define MAX_NODES 10
#define INFINITY 1000
#define MEMBER 1
#define NON_MEMBER 0

struct arc{
    int adj;
    int weight;
};

int main(){

    int i,j,k,weight[MAX_NODES][MAX_NODES],num_nodes,source,dest,precede[MAX_NODES];
    int distance[MAX_NODES],permanent_nodes_set[MAX_NODES];
    int current,current_distance,n;
    int smalldist,newdist;
    char s,d,node;

    printf("\n\t\t................................................................................\n");
    printf("\t\t\t\tSHORTEST PATH ALGORITHM << Dijkstra Algorithm >>");
    printf("\n\t\t................................................................................\n");
    printf("\n\n\t\t\tAssign 1000 as INFINITY for non-adjacent nodes !!\n\n");
    printf("\nEnter the number of nodes in the GRAPH :");
    scanf("%d",&n);

    for(i=0;i<n;i++)
        for(j=i+1;j<n;j++)
            {
                printf("\nEnter weight for arc[%c][%c] :",65+i,65+j);
                scanf("%d",&weight[i][j]);
                weight[j][i] =weight[i][j];
                weight[i][i] =INFINITY;
            }


    printf("\nEnter the source _node :");scanf(" %c",&s);
    source =(int)(s-97);
    printf("\nEnter the destination_node :");scanf(" %c",&d);
    dest = (int)(d-97);

    printf("\n\t\t\tThe shortest path is :>>\n\n\t\t\t");
    for(i=0;i<n;i++)
    {
        permanent_nodes_set[i] =NON_MEMBER;
        distance[i] =INFINITY;
    }
    permanent_nodes_set[source]=MEMBER;
    distance[source]=0;
    current=source;
    while(current!=dest)
    {
        smalldist =INFINITY;
        current_distance =distance[current];
        for(i=0;i<n;i++)
            if(permanent_nodes_set[i] ==NON_MEMBER)
        {
            newdist =current_distance+weight[current][i];
            if(newdist<distance[i])
            {
                distance[i]=newdist;
                precede[i]=current;
                if(node!=(65+current))
                    printf(" %c ->",65+current);
                node =65+current;

            }
            if(distance[i]<smalldist)
            {
                smalldist =distance[i];
                k =i;
            }

        }

         current =k;
         permanent_nodes_set[current] =MEMBER;

        }

        printf(" %c ->",65+dest);
        printf("\n\n\t\t\tThe shortest distance = %d\n",distance[dest]);

        getch();
        return 0;
}

Priority Queue Implementation: C source codes

A priority queue is data structure in which intrinsic ordering of the elements does determine the results of its basic operations. These are of two types- ascending (e.g. queue) and descending (e.g. stack).
The following source code implements ascending type priority queue in C:

// Ascending Priority Queue Implementation in C

#include<iostream.h>
    #include<conio.h>

    const int MAX=5;

    class pqueue
    {
          int front,rear;
        public:
          struct data
          {
           int val,p,o;
          }d[MAX];

Merge Sort Algorithm with C source Codes

Here, I've presented the merge sort algorithm as the straight merge sort. The basic idea is to break the file into n subfiles of size 1 and merge adjacent disjoint pairs of files and repeating the process until we've single file remaining of size n.
 The C Source codes is as follows:

// Merge sort implementation in C

#include<stdio.h>

void getdata(int arr[],int n)
{
 int i;
  printf("\nEnter the data:");
  for(i=0;i<n;i++)
    {
     scanf("%d",&arr[i]);
    }
}

C Source Codes for Implementing Linked List

The linked list can be implemented using the pointer structure in C so that we can

• add an element
• delete an element
• search for an element
• concatenate two linked list
• Invert a linked list
• Display elements in the list

//Program to demonstrate linked list operations

# include<stdio.h>
# include<conio.h>
# include "malloc.h"
struct node
{
int data;
struct node *link;
};

Implement Infix to postfix conversion using Stack C source codes

We can implement a valid infix expression to a valid postfix expression using stacks in c as shown:

//C PROGRAM TO CONVERT GIVEN VALID INFIX EXPRESSION INTO POSTFIX EXPRESSION USING STACKS.

#include<stdio.h>
#include<math.h>
#include<string.h>
#include<stdlib.h>
#define MAX 20

char stack[MAX];
int istack[MAX];
int top = -1;
char pop();
void push(char item);

Code Generator in C

The following code generates the code itself. The code seems to be simple but it's little bit tricky and is FAQ for programmers in an interview.

#include"stdio.h"

#include"conio.h"

char *s="#include%c#include%cchar *s=%c%s%c;%cvoid main()%c{%cclrscr();-%cprintf(s,10,10,34,s,34,10,10,10,10,10,10);%cgetch();%c}";

void main(){

//clrscr();

printf(s,10,10,34,s,34,10,10,10,10,10,10);

getch();

}

Binary Tree Implementation in C with Tree Traversals

We can implement the binary tree using C so that we can add a node, delete a node, and can perform different tree traversals:

 1. Preorder tree traversal

 2. Inorder tree traversal

 3. Post order tree traversal

/// ..............................................BINARY TREE OPERATIONS AND TREE TRAVERSALS

#include<stdio.h>
#include<conio.h>
#include<malloc.h>

///.................................................................DYNAMIC STRUCTURE FOR BINARY TREE

struct tree
{
     int info;
     struct tree *left_subtree;
     struct tree *right_subtree;
};

Time & Space Complexity of Basic K-means Algorithm

The basic k-means clustering algorithm is a simple algorithm that separates the given data space into different clusters based on centroids calculation using some proximity function. Using this algorithm, we first choose the k- points as initial centroids and then each point is assigned to a cluster with the closest centroid. The algorithm is formally described as follows:

Input: A data set D containing m objects (points) with n attributes in an              Euclidean space

Output: Partitioning of m objects into k-clusters C_1, C₂, C_3, …, C_k, i.e. C_i ⊂ D and C_i ∩ C_j = ᶲ  (for 1 ≤ i, j ≤ k)

DBSCAN Algorithm Implementation in MATLAB

Density Based Spatial Clustering of Applications with Noise (DBSCAN) Algorithm locates regions of high density that are separated from one another by regions of low density. DBSCAN is a center based approach to clustering in which density is estimated for a particular point in the data set by counting the number of points within the specified radius, ɛ, of that point.

The center based approach to density allows us to classify a point as one of the three:

     Core points: These points are in the interior of the dense region
   Border points:These points are not the core points, but fall within the neighborhood of the core points
   Noise points: A noise point is a point that is neither a core point nor a border point.

        The formal definition of DBSCAN algorithm is illustrated below: