MA206.01 STATISTICAL AND NUMERICAL METHODS

Syllabus Attendence Data Representation and Analysis Probability Probability Exercise

MA741 (M.TECH)

Practical-1:Use of MATLAB Commands Internal Test Marks Proposed Practical List

BS307 BIOSTATISTICS

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MATLAB Code and Algorithms

Dear Student, Here you can find MATLAB code of some numerical techniques. You can modify the code if you find error in execution!!

BISECTION Algorithm:

>> x = [-3:0.1:3];
>> y = fnroot(x);
>> plot(x,y);
>> grid on;
>> ylabel('x^3 - x -1');
>> xlabel('x');
>> title('Graph of f(x) cuts x-axis');
>> ylabel('f(x)=x^3 - x -1');

function y = fnroot(x)

y = x.^3-x-1;

end

function [ c,err ] = bisectf(f,a,b,E)

fa = feval(f,a);

fb = feval(f,b);

for i=1 : 30

c(i) =(a+b)/2;

fc = feval(f,c(i));

err(i)= abs(a-b);

if abs(fc) < E

    break

else if fa * fc < 0

    b = c(i);

    fb = fc;

else 

    a = c(i);

    fa = fc; 

    end

    

end  

end

    

end

>> [c ,err] = bisectf('fnroot',0,2,0.005);

>> [c' err']

ans =

    1.0000    2.0000

    1.5000    1.0000

    1.2500    0.5000

    1.3750    0.2500

    1.3125    0.1250

    1.3438    0.0625

    1.3281    0.0313

    1.3203    0.0156

    1.3242    0.0078

NEWTON RAPHSON Algorithm

function y = fnroot(x)
y = x.^3-x-1;
end

function y = df(x)
y = 3*x*x-1;

end

function [ n,c ] = ntr(f,df,a,E)
fa = feval(f,a);
dfa = feval(df,a);
for i=1 : 30
c(i) = a - fa/dfa;

fa = feval(f,c(i));
n(i) = i;
if abs(fa) < E
    break
else if abs(a - c(i))< E
        break
else
dfa = feval(df,c(i));
a = c(i);
    end
end

end 
end  

>> [n,root]=ntr('fnroot','df',1,0.0005);
>> [n' root']

ans =

    1.0000    1.5000
    2.0000    1.3478
    3.0000    1.3252
    4.0000    1.3247


NEWTON'S FORWARD DIFFERENCE Interpolation Algorithm (File Name Forward_D.m)

x = input('Equidistant values of x:');
y = input('Values of y for each x:');
n = length(x);
d = zeros(n);
for i=1:n
    d(i,1)=y(i);
end
for j=2:n
    for i=1:n-(j-1)
    d(i,j)=d(i+1,j-1)-d(i,j-1);
    end
end
D = [x',d];
fprintf('The Forward Difference Table\n');
disp(D);
%disp(d);
a = input('Enter value of x to find y:');
h = x(2)-x(1);
product =1;
sum = d(1,1);
p = (a-x(1))/h;
for j=2:n
product = product * (p-(j-2))/(j-1);
sum = sum + product*d(1,j);
end

>> Forward_D
Equidistant values of x:[0 1 2 3]
Values of y for each x:[1 0 1 10]
The Forward Difference Table
     0     1    -1     2     6
     1     0     1     8     0
     2     1     9     0     0
     3    10     0     0     0

Enter value of x to find y:4

The Interpolating value y(4) is 33.000000

Solution to Linear System of Equation AX = b 
Gauss Jacobi Algorithm

>> A = [10 -2 -1 -1;-2 10 -1 -1;-1 -1 10 -2;-1 -1 -2 10];
>> b = [3;15;27;-9];
>> A

A =

    10    -2    -1    -1
    -2    10    -1    -1
    -1    -1    10    -2
    -1    -1    -2    10

>> b

b =

     3
    15
    27
    -9
>>

function x  = jacobinew( A,b,M,E)
n = length(b);
fprintf('The Solution to sytem of equation AX = b is \n');
for i=1:n
if i==1
    B(i,:)=[A(i,2),A(i,3:n)]*(-1/A(i,i));
else if i==n
    B(i,:)=[A(i,1:n-1)]*(-1/A(i,i)); 
else 
   B(i,:)=[A(i,1:i-1),A(i,i+1:n)]*(-1/A(i,i));    
end
end
end
for i=1:n
C(i)=b(i)*(1/A(i,i));
end
C = C';
x = zeros(n,1);
u = zeros(n,1);
y = zeros(n);

temp = 0;

for j=2:M
for i=1:n
 if i==1
y(i,j)= B(i,:)*u(2:n)+C(i); 
    
x(i)=B(i,:)*u(2:n)+C(i);

 else if i==n

  y(i,j)=B(i,:)*u(1:n-1)+C(i);

  x(i)=B(i,:)*u(1:n-1)+C(i);

     else
         w = [u(1:i-1);u(i+1:n)];

         y(i,j)=B(i,:)*w+C(i);

         x(i)=B(i,:)*w+C(i);
     end
 end
end
u = x;
disp(x')
  for i=1:n
  if abs(y(i,j)-y(i,j-1)) > E
      temp = temp + 1;   
  end    
  end
  if temp ==0
    break;
  end 
  temp = 0;
end

end

>> x = jacobinew(A,b,15,0.001);
The Solution to sytem of equation AX = b is 
    0.3000    1.5000    2.7000   -0.9000

    0.7800    1.7400    2.7000   -0.1800

    0.9000    1.9080    2.9160   -0.1080

    0.9624    1.9608    2.9592   -0.0360

    0.9845    1.9848    2.9851   -0.0158

    0.9939    1.9938    2.9938   -0.0060

    0.9975    1.9975    2.9976   -0.0025

    0.9990    1.9990    2.9990   -0.0010

    0.9996    1.9996    2.9996   -0.0004

>> x = jacobinew(A,b,15,0.0001);
The Solution to sytem of equation AX = b is 
    0.3000    1.5000    2.7000   -0.9000

    0.7800    1.7400    2.7000   -0.1800

    0.9000    1.9080    2.9160   -0.1080

    0.9624    1.9608    2.9592   -0.0360

    0.9845    1.9848    2.9851   -0.0158

    0.9939    1.9938    2.9938   -0.0060

    0.9975    1.9975    2.9976   -0.0025

    0.9990    1.9990    2.9990   -0.0010

    0.9996    1.9996    2.9996   -0.0004

    0.9998    1.9998    2.9998   -0.0002

    0.9999    1.9999    2.9999   -0.0001

>> x = jacobinew(A,b,15,0.00001);
The Solution to sytem of equation AX = b is 
    0.3000    1.5000    2.7000   -0.9000

    0.7800    1.7400    2.7000   -0.1800

    0.9000    1.9080    2.9160   -0.1080

    0.9624    1.9608    2.9592   -0.0360

    0.9845    1.9848    2.9851   -0.0158

    0.9939    1.9938    2.9938   -0.0060

    0.9975    1.9975    2.9976   -0.0025

    0.9990    1.9990    2.9990   -0.0010

    0.9996    1.9996    2.9996   -0.0004

    0.9998    1.9998    2.9998   -0.0002

    0.9999    1.9999    2.9999   -0.0001

    1.0000    2.0000    3.0000   -0.0000

    1.0000    2.0000    3.0000   -0.0000

    1.0000    2.0000    3.0000   -0.0000


>>

Gauss Seidel Algorithm

function x = gseidel(A,b,M,E)
n = length(b);
fprintf('The Solution to system of equation AX = b is \n');
for i=1:n
if i==1
    B(i,:)=[A(i,2:n)]*(-1/A(i,i));
else if i==n
        B(i,:)=[A(i,1:n-1)]*(-1/A(i,i));
    else
        B(i,:)=[A(i,1:i-1),A(i,i+1:n)]*(-1/A(i,i));
    end
end
end
for i=1:n
C(i)=b(i)/A(i,i);
end
C = C';
x =zeros(n,1);
y = zeros(n);
temp = 0;
for j= 2:M
for i=1:n
if i==1  
 x(i) = B(i,:)*x(2:n)+C(i);
else if i==n
  x(i)=B(i,:)*x(1:n-1)+C(i);
    else      
  x(i)=B(i,:)*[x(1:i-1);x(i+1:n)]+C(i);
    end
end
y(i,j) = x(i);
end
for i=1:n
  if abs(y(i,j)-y(i,j-1)) > E
      temp = temp + 1;     
  end 
end
if temp ==0

    break;
  end
disp(x');
temp = 0;
end

end
>> x = gseidel(A,b,15,0.001);
The Solution to system of equation AX = b is 
    0.3000    1.5600    2.8860   -0.1368

    0.8869    1.9523    2.9566   -0.0248

    0.9836    1.9899    2.9924   -0.0042

    0.9968    1.9982    2.9987   -0.0008

    0.9994    1.9997    2.9998   -0.0001

>> x = gseidel(A,b,15,0.00001);
The Solution to system of equation AX = b is 
    0.3000    1.5600    2.8860   -0.1368

    0.8869    1.9523    2.9566   -0.0248

    0.9836    1.9899    2.9924   -0.0042

    0.9968    1.9982    2.9987   -0.0008

    0.9994    1.9997    2.9998   -0.0001

    0.9999    1.9999    3.0000   -0.0000

    1.0000    2.0000    3.0000   -0.0000

    1.0000    2.0000    3.0000   -0.0000


>>

Numerical Integration Algorithm

function t = f(x)
t = (1+x).^(-1);
end

function I = simpson13(y,n,h)
sum13 = y(1)+y(n+1);
for i=2:n
if rem(i,2)==0
    sum13 = sum13 + 4*y(i);
else
    sum13 = sum13 + 2*y(i);
end
I = (h/3)*sum13;

end

function I = Simpson38(y,n,h)
sum38 = y(1)+y(n+1);
for i=2:n
    if rem(i-1,3)==0
        sum38 = sum38 + 2*y(i);
    else
        sum38 = sum38 + 3*y(i);
    end  
end
I = (3*h/8)*sum38;

end

function I = trapz(y,n,h)
sumtr = y(1)+y(n+1);
for i=2:n
sumtr = sumtr + 2*y(i);
end
I = (h/2)*sumtr;

end

function Itg = NumInteg(f)
a = input('Enter lower limit of integration:');
b = input('Enter Upper limit of integration:');
n = input('Number of Intervals:');
h = (b-a)/n;
x = a:h:b;
y = feval(f,x);
fprintf('Table values of (x,y) is\n');
[x' y']
fprintf('The Function is f(x) = 1/(1+x)\n');
if mod(n,2)== 0
    fprintf('Simpson 1/3 Rule:\n');
    Itg = simpson13(y,n,h);
elseif mod(n,3) == 0
    fprintf('Simpson 3/8 Rule:\n');
    Itg = Simpson38(y,n,h);
else
   fprintf('Trapezoidal Rule:\n');
   Itg =  trapz(y,n,h);

end

>> I = NumInteg('f')
Enter lower limit of integration:0
Enter Upper limit of integration:1
Number of Intervals:4
Table values of (x,y) is

ans =

         0    1.0000
    0.2500    0.8000
    0.5000    0.6667
    0.7500    0.5714
    1.0000    0.5000

The Function is f(x) = 1/(1+x)
Simpson 1/3 Rule:

I =

    0.6933

>> I = NumInteg('f')
Enter lower limit of integration:0
Enter Upper limit of integration:1
Number of Intervals:6
Table values of (x,y) is

ans =

         0    1.0000
    0.1667    0.8571
    0.3333    0.7500
    0.5000    0.6667
    0.6667    0.6000
    0.8333    0.5455
    1.0000    0.5000

The Function is f(x) = 1/(1+x)
Simpson 1/3 Rule:

I =

    0.6932

>> I = NumInteg('f')
Enter lower limit of integration:0
Enter Upper limit of integration:1
Number of Intervals:7
Table values of (x,y) is

ans =

         0    1.0000
    0.1429    0.8750
    0.2857    0.7778
    0.4286    0.7000
    0.5714    0.6364
    0.7143    0.5833
    0.8571    0.5385
    1.0000    0.5000

The Function is f(x) = 1/(1+x)
Trapezoidal Rule:

I =

    0.6944


>> 

 Dear Student, you can download here Simple Algorithms of some numerical techniques that you know. You can use them to write a simple C/C++ program or MATLAB Program.

Basic Mathematics

Dear Student, you can find here the discussions on topics of Elementary Mathematics. This notes are constructed for students of CHARUSAT studying in courses: First Semester B Pharm, First Semester B. Sc. as per prescribed syllabus and teaching scheme.

I have referred several text book to construct these notes. In many places I have compiled the material available in textbooks that discuss the theories (without proof!) and concepts described by author/s. Interested students can read in more details about the topic/s from the Book mentioned here. Also students can download the PDF document of topics which I have not posted here. At the end I am collecting the exercise problems,some Quiz questions from these resources for practice purpose. 

Send your comments and suggestions for better improvement in these posts if any!!!

Thank You all

Few Books that are used:

1. "Naive Set Theory": By Paul R Halmos, Van Nostrand Reinhold Comp
2. "Sets, Functions, and Logic An Introduction to Abstract Mathematics": By Keith Devlin, Chapman & Hall/CRC
3. "The Mathematics of Matrices" A First Book of Matrix Theory and Linear Algebra, by Philip J. Davis,John Wiley and Sons, Inc. Second Edition
4. "Theory and Problems of Set Theory and Related Topics" by Seymour Lipschutz, Schaum's Outline Series,Second Edition
5. "Calculus and Analytical Geometry"  by Thomas G B and R. L. Finney, Addison Wesley $9^{th}$ Edition
6. "Advanced  Engineering Mathematics" by Erwin Kreyszig, John Wiley and Sons, India $8^{th}$ Edition

Follow the link for

1. Numbers and Set Theory Introduction
2. Relations and Functions Introduction only  (Contains Some Quiz Questions)
3. Introduction to Matrices Some Basics
4. Introduction to Determinants (Upto Order 3 only)

Students can download the material not included in above posts here

Select from Below Mathematics Software Exercise Quiz Questions

Solution to the Questions will be displayed at the end of topic discussion.