LAB MANUAL. ANTC Lab (ME-321- F) DEPARTMENT OF APPLIED SCIENCE AND HUMINITIES

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1 LAB MANUAL ANTC Lab (ME-321- F) DEPARTMENT OF APPLIED SCIENCE AND HUMINITIES 1

2 Check list for Lab Manual S. No. Particulars Page Number 1 Mission and Vision 3 2 Guidelines for the student 4 3 List of Programs as per University 5 4 Sample copy of File

3 Mission To develop BRCM College of Engineering & Technology into a Center of Excellence By : Providing State of the art Laboratories, Workshops, Research and instructional facilities Encouraging students to delve into technical pursuits beyond their academic curriculum. Facilitating Post graduate teaching and research Creating an environment for complete personality development of students. Assisting in the best possible placement Vision To Nurture and Harness talent for empowerment towards self actualization in all technical domains both existing for the future 3

4 Guidelines for the Students: 1. Students should be regular and come prepared for the lab practice. 2. In case a student misses a class, it is his/her responsibility to complete that missed experiment(s). 3. Students should bring the observation book, lab journal and lab manual. Prescribed textbook and class notes can be kept ready for reference if required. 4. They should implement the given Program individually. 5. While conducting the experiments students should see that their programs would meet the following criteria: Programs should be interactive with appropriate prompt messages, error messages if any, and descriptive messages for outputs. Programs should perform input validation (Data type, range error, etc.) and give appropriate error messages and suggest corrective actions. Comments should be used to give the statement of the problem and every function should indicate the purpose of the function, inputs and outputs Statements within the program should be properly indented Use meaningful names for variables and functions. Make use of Constants and type definitions wherever needed. 6. Once the experiment(s) get executed, they should show the program and results to the instructors and copy the same in their observation book. 7. Questions for lab tests and exam need not necessarily be limited to the questions in the manual, but could involve some variations and / or combinations of the questions. 4

5 LIST OF PROGRAMS(University Syllabus) ANTC Lab (ME-321-F ) S.NO PROGRAM 1 Solution of Non-linear equation in single variable using the method of successive bisection. 2 Solution of an algebraic equation by using Simpson s method. 3 Solution of Non-Linear equation in single variable using the Newton Raphson method. 4 Solution of a system of simultaneous algebraic equations using the Gaussian elimination procedure 5 Numerical solution of an ordinary differential equation using the Euler s method. 6 Numerical solution of an ordinary differential equation using the Runge Kutta 4th order method 7 Solution of an algebraic equation by using trapezoidal rule. 8 Program of Lagrange s Interpolation Formula. 9 Solution of Non-Linear equation in single variable using the regula-falsi method 10 Solution of a system of simultaneous algebraic equations using the Gauss-Seidel iterative method. 5

6 Sample copy of file Program 1: Write a program to find solution of Non-linear equation in single variable using the method of successive bisection. #include<conio.h> float f(float x) return x*x*x-4*x-9; void bisect(float *x, float a, float b, int, *itr) *x=(a+b)/2; ++(*itr); int main() int itr,maxitr; float x,a,b,aerr,x1; clrscr(); printf("enter the values of a,b,aerr,maximum no of iterations"); scanf("%f %f %f %d",&a,&b,&aerr,&maxitr); bisect(&x,a,b,&itr); do if(f(a)*f(x)<0) b=x; else a=x; bisect(&x1,a,b,&itr); if(fabs(x1-x)<aerr) printf("the root is: %f",x1); return 0; x=x1; while(itr==maxitr); printf("the root does not converge in given no of iterations"); return 1; 6

7 Program 2: Write a program to find the solution of an algebraic equation by using Simpson s method. /*simpson's rule*/ #include<math.h> float y(float x) return 1/(1+x*x); main() float x0,xn,h,s; int i,n; printf("enter x0,xn,number of subinterval"); scanf("%f%f%d",&x0,&xn,&n); h=(xn-x0)/n; s=y(x0)+y(xn)+4*y(x0+h); for(i=3;i<=n-1;i+=2) s+=4*y(x0+i*h)+2*y(x0+(i-1)*h); printf("value of integral is %6.4f/n",(h/3)*s); return 0; Program 3 : Write a program to find the solution of Non-Linear equation in single variable using the Newton Raphson method. #include<conio.h> #include<math.h> float f(float x) return x*sin(x)+cos(x); float df(float x) return x*cos(x); 7

8 int main() int itr,maxitr; float h,x0,x1,aerr; clrscr(); printf("\n\n\t enter x0, allowed error and maximum no. of iterations"); scanf("%f %f %d",&x0,&aerr,&maxitr); for(itr=1;itr<=maxitr;itr++) h=f(x0)/df(x0); x1=x0-h; printf("\n\t iteration no.= %d \n x= %f \n",itr,x1); if(fabs(h)<aerr) printf("\n\t after %d iteration \n root= %f \n",itr, x1); return 0; x0=x1; printf("\n\f the solution does not converge"); return 1; Program 4 : Write the program to find the solution of a system of simultaneous algebraic equations using the Gaussian elimination procedure. /*Gauss Elimination */ # include <stdio.h> # include <conio.h> # include <math.h> # define MAX 10 void main() int i,j,n,k; float mat[max][max],x[max],temp,pivot,sum=0; clrscr(); printf("\t\t\t GAUSS ELIMINITION METHOD\n"); printf("enter No of Equtions : "); scanf("%d",&n); printf("enter Coefficients of Eqution \n"); for(i=1;i<=n;i++) for(j=1;j<=n;j++) scanf("%f",&mat[i][j]); 8

9 printf("enter Constant value\n"); for(i=1;i<=n;i++) scanf("%f",&mat[i][n+1]); x[i]=mat[i][n+1]; for(i=2;i<=n;i++) for(j=i;j<=n;j++) pivot=mat[j][i-1]/mat[i-1][i-1]; for(k=i-1;k<=n+1;k++) mat[j][k]=mat[j][k]-pivot*mat[i-1][k]; printf("eliminated matrix as :- \n"); for(i=1;i<=n;i++) for(j=1;j<=n+1;j++) printf("\t%.2f",mat[i][j]); printf("\n"); for(i=1;i<=n;i++) if(mat[i][i]==0) printf("since diagonal element become zero\n Hence solution is not possible\n"); exit(1); printf("solution : \n"); for(i=0;i sum=0; for(j=n;j>n-i;j--) sum=sum+mat[n-i][j]; x[n-i]=(mat[n-i][n+1]-sum*x[n])/mat[n-i][n-i]; printf("x%d = %4.2f\n",n-i,x[n-i]); getch(); 9

10 Program 5: Write the program to find Numerical solution of an ordinary differential equation using the Euler s meth #include<conio.h> #include<math.h> #define f(x,y) (x+y) void main() float h,x0,y0,y1; int i,n; clrscr(); printf("\n\n\t enter the value x0,y0,n"); scanf("%f%f%d",&x0,&y0,&n); h=(x0-y0)/n; for(i=1;i<=n;i++) y1=y0+h*f(x0,y0); y0=y1; x0=x0+h; printf("\n\t when x=%f y=%f",x0,y0); getch(); Program 6 : Write a program to find the numerical solution of an ordinary differential equation using the Runge Kutta 4th order method #include<math.h> float f(float x,float y) return x+y*y; main() float x0,y0,h,xn,x,y,k1,k2,k3,k4,k printf("enter the values of x0,y0,"h,xn\n"); scanf("%f %f %f %f",&x0,&y0,&h,&xn); x=x0;y=y0 10

11 while(1) if (x==xn) break; k1=h*f(x,y); k2=h*f(x+h/2,y+k1/2); k3=h*f(x+h/2,y+k2/2); k4=h*f(x+h,y+k3); k=(k1+2*(k2+k3)+k4)/6; x+=h; y+=k; printf("when x=%8.4f"y=%8.4f\n",x,y); Program 7 : Write a program to solve of an algebraic equation by using trapezoidal rule. #include<conio.h> float y(float x) return 1/(1+x*x); void main() float x0,xn,h,s; int i,n; puts("enter x0,xn,no.of subintervals"); scanf("%f%f%d",&x0,&xn,&n); h=(xn-x0)/n; s=y(x0)+y(xn); for(i=1;i<=n;i++) s+=2*y(x0+i*h); printf("value Of interval is %6.4f\n",(h/2)*s); 11

12 Program 8 : Write a program of Lagrange s Interpolation Formula #include<conio.h> #include<math.h> #define max 100 main() float ax[max+1],ay[max+1],nr,dr,x,y=0; int i,j,n; printf("enter the value of n\n"); scanf("%d",&n); printf("enter the set of values\n"); for(i=0;i<=n;i++) scanf("%f%f",&a x[i],&ay[i]); printf("enter the value of x for which value of y is wanted"); scanf("%f" &x); for(i=0;i<=n;i++) nr=dr=1 for(i=0;j<=n;j++) if(j!=i) nr*=x-ax[j]; dr*=ax[i]-ax[j]; y+=(nr/dr)=ay[i]; printf("when x=%4.1f & y=7.1f\n",x,y); Program 9 : Write a program to find the solution of Non-Linear equation in single variable using the regula-falsi method. /* regula falsi method */ #include<math.h> float f(float x) return cos(x)-x*exp(x); void regula (float*x,float x0,float x1,float fx0,float fx1,int*itr) *x=x0-((x1-x0)/(fx1-fx0))*fx0; ++(*itr); printf("iteration number %3d x=%7.5f\n",*itr,*x); 12

13 main() int itr=0,maxitr; float x0,x1,x2,x3,aerr; printf("enter the value for x0,x1,allowed error,maxium iteration\n"); scanf("%f %f %f %d",&x0,&x1,&aerr,maxitr); regula(&x2,x0,x1,f(x0),f(x1),&itr); do if (f(x0)*f(x2)<0) x1=x2; else x0=x2; regula(&x3,x0,x1,f(x0),f(x1),&itr); if (fabs(x3-x2)<aerr) printf("after %d iteration,root=%6.4f\n",itr,x3); return 0; x2=x3; while(itr<maxitr); printf("solution does not converge.iteration not sufficient\n"); return 1; Program 10 : Write a program using arrays to find solution of a system of simultaneous algebraic equations using the Gauss-Seidel iterative method. #include<math.h> #define N 3 main() float a[n][n+1],x[n],aerr,maxerr,t,s,err; int i,j,itr,maxitr; for(i=0;i<n;i++) x[i]=0; printf("enter the element of the augmented matrix rowwise\n"; for(i=0;i<n;i++) for(j=0;j<n+1;j++) 13

14 scanf("%f",&a[i][j]); printf("enter the allowed error,maximum iteration\n"); scanf("%f %d",&aerr,&maxitr); printf("iterationx[1] x[2] x[3]\n"); for(itr=1;itr<=maxitr;itr++) maxerr=0; for(i=0;i<n;i++) s=0; for(j=0;j<n;j++) if(j!=i) s+=a[i][j]*x[j]; t=(a[i][n]-s)/a[i][i]; err=fabs(x[i]-t); if(err>maxerr) maxerr=err; x[i]=t; printf(%5d",itr); for(i=0;i<n;i++) printf( %9.4f,x[i]); printf \n ); if(maxerr<aerr) printf( converges in%3d iterations\n,itr); for(i=0;i<n;i++) printf( x[%3d]=%7.4f\n,i+1,x[i]); return 0; printf( solution does not converge iteration not sufficient\n ); return 1; 14

NUMERICAL METHODS AND COMPUTATIONAL PROGRAMMING LAB MATH-204-F

LAB MANUAL NUMERICAL METHODS AND COMPUTATIONAL PROGRAMMING LAB MATH-204-F LIST OF EXPERIMENTS NUMERICAL METHODS OF COMPUTATIONAL PROGRAMMING LAB MATH-204-F S. No. NAME OF EXPERIMENTS 1. Solution of Non-linear