This document contains MATLAB code for numerical methods and curve fitting techniques. It includes code for root finding methods like bisection and Newton Raphson, solving systems of linear equations using Gaussian elimination and Gauss-Seidel methods, and curve fitting linear, quadratic, and power equations to data. The code is accompanied by explanations and is from Dr. D.P. Bhaskar of Sanjivani College of Engineering, Kopargaon.

1. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
1
Numerical Methods by MATLAB
Mechanical Engineering
Prepared by
Dr.D.P.Bhaskar
Department of Mechanical Engineering
Sanjivani College of Engineering, Kopargaon
Maharastra
(An Autonomous Institute SP Pune University)

2. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
2
Unit 1 Roots of Equation
%Bisection Method
clc
clear all
f=inline('x‐cos(x)'); acc=0.001;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for i=1:100
x1=input('Enter new x1');
x2=input('Enter new x2');
if(f(x1)*f(x2)<0) break
end
end
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐n');
fprintf('I x1 x2 x3 Acc Error)');
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐');
for i=1:100
x3=(x1+x2)/2;
b=abs(x1‐x2); c=(f(x1)*f(x3));
fprintf('n%d %.4f %.4f %.4f %.4f %.4f ',i,x1,x2,x3,b,c);
if(b<acc) break
else if(c>0) x1=x3; else x2=x3; end
end
end
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐');
fprintf('n Root=%f',x3);

3. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
3
%Newton Raphson Method
clc
clear all
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
f=inline('x.^3‐cos(x)*cos(x)');
fd=inline('2*cos(x)*sin(x) + 3*x.^2');
acc=0.001; x1=1;
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐n');
fprintf('I x1 f(x1) fd(x1) x2 Acc');
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐');
abs(f(x1))
abs(fd(x1))
for i=1:100
x2=x1‐f(x1)/fd(x1);
b=abs(x1‐x2);
fprintf('n%d %.4f %.4f %.4f %.4f %.4f',i,x1, f(x1), fd(x1),x2,b);
if(b <acc) break
else x1=x2;
end
end
fprintf('‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐');
fprintf('n Root=%f',x2);

4. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
4
Unit 2 Simultaneous Equation
%2.0 GEM
clc
clear all
n=3; m=n+1;
a=[1 2 2 7;2 ‐4 1 ‐5;1 1 2 5];
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for k=1:(n‐1) %Ip=Jp=1 2 3
for i=k+1:n
PVT= a(k,k); TL= a(i,k); p=TL/PVT;
for j=k:m
a(i,j)=a(i,j)‐p*a(k,j);
end
end
end
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for i=n:‐1:1
sum=0.0;
for j=i+1:n
sum=sum+a(i,j)*x(j);
end
x(i)=(a(i,n+1)‐sum)/a(i,i);
end
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for i=1:n
fprintf('x(%d)=%0.2fn',i,x(i));
end

5. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
5
% GEM with Partial Pivoting
clc
clear all
n=3; m=n+1;
a=[1 2 2 7;2 ‐4 1 ‐5;1 1 2 5];
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for k = 1:n‐1
for i = k+1:n
if (abs(a(k,k)) < abs(a(i,k)))
a([k i],:) = a([i k],:);
end
end
end
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for k=1:(n‐1) %Ip=Jp=1 2 3
for i=k+1:n
PVT= a(k,k); TL= a(i,k); p=TL/PVT;
for j=k:m
a(i,j)=a(i,j)‐p*a(k,j);
end
end
end
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for i=n:‐1:1
sum=0.0;
for j=i+1:n
sum=sum+a(i,j)*x(j);
end
x(i)=(a(i,n+1)‐sum)/a(i,i);
end
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for i=1:n
fprintf('x(%d)=%0.2fn',i,x(i));
end
%SOLVER
clc
clear all
a=[1 2 2 ;2 ‐4 1 ;1 1 2];
b=[4;5;6];
x=inv(x)*b

6. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
6
%GAUSS SEIDAL METHOD
clc
clear all
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
f1=inline('(13‐4*y‐3*z)/2');
f2=inline('(16‐3*x‐z)/‐6');
f3=inline('(9‐x‐3*y)/2');
x1=0; y1=0; z1=0; acc=0.001;
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐');
fprintf('n I X Y Z ');
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐');
for i=1:100
x2=f1(y1,z1); y2=f2(x2,z1); z2=f3(x2,y2);
fprintf('n %d %f %f %fn',i,x2,y2,z2);
a=abs(x1‐x2); b=abs(y1‐y2); c=abs(z1‐z2);
if (a<acc && b<acc && c<acc) break
else x1=x2; z1=z2; y1=y2;
end
end
fprintf('‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐');

7. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
7
Unit 3 Curve Fitting
%3.0 Curve fitting F Line Yf=C0+C1(X)
clc
clear all
%‐‐‐‐‐‐‐‐‐‐‐‐‐Direct input of Data‐‐‐‐‐‐‐‐‐‐‐
x=[1 2 3 4];
y=[2 6 12 20];
n=length(x);
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐OR‐User UInput of data‐‐‐‐‐
% n=input('Enter n=');
% for i=1:n
% fprintf('Enter x(%d)=',i);
% x(i)=input('');
% fprintf('Enter y(%d)=',i);
% y(i)=input('');
% end
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
s1=sum(x); s2=sum(x.^2); s3=sum(y);
s4=sum(x.*y);
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
d=[n s1;
s1 s2];
d1=[s3 s1;
s4 s2];
d2=[n s3;
s1 s4];
d=det(d); d1=det(d1); d2=det(d2);
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
c0=d1/d; c1=d2/d;
fprintf('BEST FIT ,Yn=%0.3f+%0.3f(X)n',c0,c1)
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
yn=c0+c1*x;
plot(x,y,'*',x,yn)
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐

8. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
8
% Quadratic FITTING Yf=C0+C1(X)+C2(x^2);
clc
clear all
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
x=[1 2 3 4 5 6 7 8 9];
y=[2 6 7 8 10 11 11 10 9];
n=9;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
x2=x.^2; x3=x.^3; x4=x.^4; xy=x.*y; x2y=(x.^2).*y;
s1=sum(x); s2=sum(x2); s3=sum(x3); s4=sum(x4); s5=sum(y); s6=sum(xy); s7=sum(x2y);
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
d=[n s1 s2;
s1 s2 s3
s2 s3 s4];
d1=[s5 s1 s2;
s6 s2 s3
s7 s3 s4];
d2=[n s5 s2;
s1 s6 s3
s2 s7 s4];
d3=[n s1 s5;
s1 s2 s6
s2 s3 s7];
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
d=det(d); d1=det(d1); d2=det(d2); d3=det(d3); c0=d1/d; c1=d2/d; c2=d3/d;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
fprintf('BEST FIT ,Yn=%0.3f+%0.3f(X)+%0.3f(x^2) /n',c0,c1,c2)
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
yn=c0+c1*x+c2*x2;
plot(x,y,'*',x,yn,'r')
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐

9. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
9
%Power Equation
clc
clear all
%‐‐‐‐‐‐‐‐‐‐‐‐‐Power Equation‐‐‐‐‐‐‐‐‐‐‐
%PV^Gamma=C
p=[0.5 1.0 1.5 2.0 2.5 3.0];
v=[1.62 1.0 0.75 0.62 0.52 0.46];
n=6;
y=log10(p);
x=log10(v);
xy=x.*y;
x2=x.^2;
s1=sum(x); s2=sum(x2); s3=sum(y); s4=sum(xy);
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
d=n*s2‐s1*s1;
d1=s3*s2‐s4*s1;
d2=n*s4‐s1*s3;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
c0=d1/d; c1=d2/d;
gamma=‐c1;
c=exp(c0);
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
fprintf('PV^%0.3f=%f/n',gamma,c)
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
%plot(p,v);
for i=1:n
fprintf('%f %f %f %f %f %fn',v(i),p(i),x(i),y(i),x2(i),xy(i));
end

10. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
10
%Exponential
clc
clear all
%‐‐‐‐‐‐‐‐‐‐‐‐‐Exponential‐‐‐‐‐‐‐‐‐‐‐
%y=ae^bx
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
x=[2 4 6 8];
y=[25 38 56 84];
n=length(x);
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
y1=log(y);
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
x2=x.^2; xy1=x.*y1;
s1=sum(x); s2=sum(x2); s3=sum(y1); s4=sum(xy1);
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
d=n*s2‐s1*s1;
d1=s3*s2‐s4*s1;
d2=n*s4‐s1*s3;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
c0=d1/d; c1=d2/d;
a=exp(c0);
b=c1;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
fprintf('y=%0.3f e^(%0.3fx) n',a,b)
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for i=1:n
fprintf('%f %f %f %f %f n',x(i),y(i),y1(i),x2(i),xy1(i));
end
%Solver
clear all
x=[19 25 30 36 40 45 50];
y=[76 77 79 80 82 83 85];
d=1; %d=1 (line) d=2(parabola)
c=polyfit(x,y,d)
% yn = polyval(c, x);
yn=c(2)+c(1)*x;
plot(x,y,'*',x,yn)

11. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
11
Unit 4 ODE & PDE
%EULERS METHOD
clc
clear all
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
f=inline('x*x*y+y*y');
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
x1=input('ENTER x1='); y1=input('ENTER y1=');
xn=input('ENTER xn='); n=input('ENTER n=');
h=(xn‐x1)/n;
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ ');
fprintf('n X Y ');
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ ');
fprintf('n %0.3f %0.3f',x1,y1);
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for i=1:n
x2=x1+h;
y2=y1+h*f(x1,y1);
fprintf('n %0.3f %0.3f',x2,y2);
x1=x2; y1=y2;
end
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ ');

12. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
12
%RK4
clc
clear all
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
f=inline('x*x*y+y*y');%(Sample function)
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
x1=input('ENTER x1='); y1=input('ENTER y1='); % BCns
xn=input('ENTER xn='); n=input('ENTER n=');
h=(xn‐x1)/n;
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ ');
fprintf('n X Y ');
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ ');
fprintf('n %0.3f %0.3f',x1,y1);
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for i=1:n
s1=h*f(x1,y1); s2=h*f(x1+h/2,y1+s1/2);
s3=h*f(x1+h/2,y1+s2/2); s4=h*f(x1+h,y1+s3);
s=(s1+2*s2+2*s3+s4)/6;
y2=y1+s; x2=x1+h;
fprintf('n %0.3f %0.3f',x2,y2);
x1=x2; y1=y2;
end
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ ');
%SOLVER
clc
clear all
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
f=inline('x+y');
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
x1=0; xn=0.2; y1=2; h=0.1;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
[x,y]=ode23(f,[x1:h:xn],y1)
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐

13. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
13
%Simultaneous ODE %RK2
clc
clear all
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
f1=inline('x+(y*z)');%(Sample function)
f2=inline('(x*x)‐(y*y)');%(Sample function)
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
x1=0; y1=1; z1=0; xn=0.2;% BCns Direct Input
n=2; % Number of strips Direct Input
h=(xn‐x1)/n;
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ ');
fprintf('n X Y ');
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ ');
fprintf('n %0.3f %0.3f',x1,y1);
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for i=1:n
s1=h*f1(x1,y1,z1); k1=h*f2(x1,y1,z1);
s2=h*f1(x1+h,y1+s1,z1+k1); k2=h*f2(x1+h,y1+s1,z1+k1);
s=(s1+s2)/2; k=(k1+k2)/2;
y2=y1+s; z2=z1+k; x2=x1+h;
fprintf('n %0.3f %0.3f %0.3f',x2,y2,z2);
x1=x2; y1=y2; z1=z2;
end

14. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
14
%Elliptical Laplace Equation
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
clc
clear all
rh=0;lh=100;top=100;bot=0;itr=4;
n=3;m=3;h=1;k=1;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for i=1:n+1
for j=1:m+1
a(i,j)=0;
end
end
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for i=2:(n)
a(i,1)=bot; a(i,m+1)=top;
end
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for j=2:(m)
a(1,j)=lh; a(m+1,j)=rh;
end
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for k=1:itr
for i=2:n
for j=2:m
a(i,j)=(a(i‐1,j)+ a(i+1,j)+a(i,j‐1)+ a(i,j+1))/4;
end
end
end
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
disp(a)

15. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
15
Unit 5 Interpolation
%LAGRANGES Method
clc
clear all
xp=1.1;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
x=[1 1.2 1.3 1.5];
y=[1 1.0954 1.1402 1.2247];
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐OR‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
n=length(x);
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
sum=0;
for i=1:n
term=y(i);
for j=1:n
if(i~=j)
term=term*(xp‐x(j))/(x(i)‐x(j))
end
end
sum=sum+term
end
fprintf('x=%.3f,yp=%.5f',xp,sum)
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
% n=input('Enter n=');
% for i=1:n
% fprintf('Enter x(%d)=',i);
% x(i)=input('');
% fprintf('Enter y(%d)=',i);
% y(i)=input('');
% end

16. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
16
%Newton Forward Method
clc
clear all
x=[0 1 2 3 4 5]; y=[1 2 9 28 65 126];
% x=input('Enter [x]='); y=input('Enter [y]=');
n=length(x); h=x(2)‐x(1);
xp=input('xp=');
%‐‐‐‐‐‐‐‐‐‐‐‐‐Table Calculations‐‐‐
for j=1:(n‐1)
for i=1:(n‐j)
if(j==1) f(i,j)=y(i+1)‐y(i);
else f(i,j)=f(i+1,j‐1)‐f(i,j‐1);
end
end
end
fprintf('n‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐NF TABLE ‐‐‐‐‐‐‐‐')
for i=1:n
fprintf('n %0.2f t %0.2f',x(i),y(i));
for j=1:(n‐i)
fprintf('tt %0.2f',f(i,j));
end
end
fprintf('‐‐‐‐‐‐‐‐‐‐‐‐‐‐TABLE ENDS‐‐‐‐‐‐‐')
sum=y(1);
for i=1:(n‐1)
prod=f(1,i);
for j=1:i
prod=prod*(xp‐x(j))/(j*h);
end
sum=sum+prod;
end
fprintf(' YP=%f',sum);

17. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
17
Unit 6 Integration
%TRAP
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
clc
clear all
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
f=inline('x.^3');
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
a=input('Enter Lower Limit,a='); b=input('Enter Upper Limit,b=');
ns=1; % depends on method
na=input('Enter Number of application ,na=');
n=na*ns;
h=(b‐a)/n;
sum=0;
x1=a;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for i=1:na
sum=sum+f(x1)*1+f(x1+h)*1; % depends on method
x1=(x1+h);
end
Area=sum*h/2; % depends on method
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
fprintf('n Area of integration=%0.2f',Area)
% Solver
clc
clear all
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
f =inline('x.^3');
a=2;b=3;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
Q = quad(f,a,b)

18. NM [DR D.P.BHASKAR, SANJIVANI COLLEGE OF ENGINEERING KOPARGAON]
18
% Double Trap Integration
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
clc
clear all
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
f=inline('x+y');
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
ns=1;
a=1;b=3;c=0;d=2;na=2;ma=2;
n=na*ns;m=ns*ma;
h=(b‐a)/n; k=(d‐c)/m;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
y1=c;
for j=1:m+1
x1=a;
sum=0;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
for i=1:na
sum=sum+f(x1,y1)+f((x1+h),y1);
x1=(x1+h);
end
s(j)= h/2*sum;
y1=y1+k;
end
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
sum=0;
for j=2:ma
sum=sum+s(j);
end
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
sum=sum*2;
sum=sum+s(1)+s(m+1);
sum=sum*k/2
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
fprintf('Volume=%f',sum);
%Solver Double Integration (dblquad)
clc
clear all
f =inline('x+y')
a=1;b=3;c=0;d=2;
%‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐
Volume=dblquad(f,a,b,c,d);
fprintf(' Volume of integration=%0.2f',Volume);