PKK

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Printing String …………………….. .MODEL SMALL .STACK 100H .DATA STRING DB 'My name is Md. Mehedi', '$' .CODE MAIN PROC FAR MOV AX,@DATA MOV DS,AX ; load address of the string LEA DX,STRING ;output the string ;loaded in dx MOV AH,09H INT 21H ;interrupt to exit MOV AH,4CH INT 21H MAIN ENDP END MAIN




Upper to Lower Case
………………………………..

.MODEL SMALL
.STACK 100H 
.CODE
MAIN PROC       
    MOV AH,1     ;INPUT
    INT 21H                  
    MOV BL,AL    
    
    ADD BL,20H   ;Upper to Lower     
    
    MOV AH,2
    MOV DL,BL
    INT 21H        
  
    EXIT:
    MOV AH,4CH
    INT 21H
    MAIN ENDP
END MAIN
























Bisection


ORG 100H
.MODEL SMALL
.STACK 100H
.DATA
R DB 'ENTER RANGE: $'
F DB 0
S DB 1
SUM DB ?
N DW ?
FAB DB 13, 10, 13, 10, '**FIBONACCI SERIES**$'
.CODE
MAIN PROC
   
    MOV DX, OFFSET R
    MOV AH, 9
    INT 21H
   
    MOV N, 0
    MOV BL, 10
    INPUT:
    MOV AH, 1
    INT 21H
    CMP AL, 13
    JE NEXT
    SUB AL, 30H
    MOV AH, 0
    MOV CX, AX
    MOV AX, N
    MUL BL
    ADD AX, CX
    MOV N, AX
    JMP INPUT
   
    NEXT:
    LEA DX, FAB
    MOV AH, 9
    INT 21H
   
    MOV CX, N
    L:
   
    PUSH CX
   
    MOV DL, 10
    MOV AH, 2
    INT 21H
   
    MOV DL, 13
    MOV AH, 2
    INT 21H
   
    MOV BL, F
    ADD BL, S
    MOV SUM, BL
   
    MOV AH, 0
    MOV AL, SUM
   
    MOV DX, 0
    MOV BX, 10
    MOV CX, 0
    L1:
    DIV BX
    PUSH DX
    MOV DX, 0
    MOV AH, 0
    INC CX
    CMP AX, 0
    JNE L1
    MOV AH, 2
    L2:
    POP DX
    ADD DX, 48
    INT 21H
    LOOP L2
   
    MOV BL, S
    MOV F, BL
   
    MOV BL, SUM
    MOV S, BL
   
    POP CX
    LOOP L
    RET
   
    MAIN ENDP



Euler




% Prompt the user to input initial values and step size
xo = input("Enter the value of xo: ");
yo = input("Enter the value of yo: ");
h = input("Enter the value of h: ");

% Prompt the user to input the function as a string
func_str = input("Enter the function in terms of x and y: ", 's');

% Convert the function string to a function handle
f = str2func(['@(x,y)', func_str]);

% Prompt the user to input the number of iterations
n = (yo-xo)/h;

% Implement Euler's Method
for i = 1:n
    y = yo + h * f(xo, yo);
    fprintf('y%i = %f\n', i, y);
    yo = y;
    xo = xo + h;
end


False position


%function = @(argument)(expression)
f =input ('Enter the EXP: ');

%Get user input for initial value/intervals
a=input('Enter the lower bound, a: ');
b=input('Enter the upper bound, b: ');  

%check if there is a root in the interval
while f(a)*f(b)>=0
    fprintf('There is no root in the given interval.\n');
    a=input('Enter the lower bound, a: ');
    b=input('Enter the upper bound, b: ');
end

acc=input('Enter the values of accuracy: ');
iteration= 0;
matched_roots = 0;

while abs(b-a)>acc
    c=(a*f(b) - b*f(a)) / (f(b) - f(a));
    fc = f(c);
   
    %plot the interval
    x_interval = [a,b];
    y_interval = [f(a), f(b)];
    plot(x_interval, y_interval, 'b-o');
    hold on;
   
    %plot root
    plot(c, fc, 'ro', 'MarkerSize', 8);
   
    %check if the root is matched twice
    if f(a)*fc<=0
        b=c;
    else
        a=c;
    end
   
    iteration = iteration+1;
    fprintf('Iteration %d: Interval[%f, %f], Root = %f\n', iteration,a,b,c);
   
    %check if the root matched twice
    if abs(f(c))<acc
        matched_roots = 0;
    end
   
    %break if the root is matched twice
    if matched_roots>=2
        break;
    end
end
fprintf('The root of the equation according to the false Position Method is: %f\n',c);




Fixed Point


% fixedpointfun
function xr = FixedPointIteartion(fun,x0)
pmin = 0; % x limits of the plot. Change for different functions if needed.
pmax = 1;

% First plot g(x) and y=x to see the behaviour
figure
hold on
fplot(@(x) fun(x),[pmin,pmax],'LineWidth',2)
fplot(@(x) x,[pmin,pmax],'LineWidth',2)
fplot(0,[pmin,pmax],'-k')
ylim([0,1]) % y limits of the plot. Change for different functions if needed.
xlabel('x')
ylabel('g(x)')
title('Fixed-Point Iteration Demo')
grid on

% Initialize variables
es = 0.001;
maxit = 100;

% prepare the output header
fprintf('\n  #         xi          er\n');

count = 0;  % iteration counter
er = 1;     % initial error value to enter loop
xi = x0;     % initialize xi and xold
xold = xi;
fxold = fun(xold); % will be used for plotting

fprintf('%3g %10.6g %10.4g\n',count,xi,er);
plot([xold,xold],[xold,fxold],'--k');  % allows us to visualize the approximations

erplot = zeros(1,50);

% main loop
while (abs(er) > es && count<maxit)
    count = count +1;
    xold = xi;  % keep the previous estimate
    fxold = fun(xold); % will be used for plotting
   
    xi = fun(xi);   % calculate the new estimate
    fxi = fun(xi);
   
    plot([xold,xi,xi],[fxold,xi,fxi],'--k');  % allows us to visualize the approximations
       
    er = (xi-xold)/xi;  % relative error
    erplot(count) = er;
   
    fprintf('%3g %10.6g %10.4g\n',count,xi,er);
    % pause  % Wait for key press
end

if count >= maxit
    disp(' ')
    disp(['Failed to converge in ',num2str(maxit),' iterations'])
    disp(' ')
    disp('--------------------------------------------')
    disp(' ')
    xr = NaN;
else
    disp(' ')
    disp(['Root has been found at x = ',num2str(xi)])
    disp(' ')
    disp('--------------------------------------------')
    disp(' ')
    xr = xi;
end

figure
hold on
fplot(0,[0,count],'-k')
plot(erplot(1:count),'LineWidth',2)
title('Relative Error')
xlabel('Iteration Number')
ylabel('Relative Error (er)')
hold off
end

% use it command line "FixedPointIteartion(@(x)cos(x),0.1)"
% use it command line "FixedPointIteartion(@(x)(cos(x)+3)/2,0.1)"




Gauss Elimination


function x = gausselimination(A,b)
[rA,cA] = size(A);  % Determine the size of A
[rb,cb] = size(b);  % Determine the size of b
% Input checks
if rA ~= cA
    disp('Error: A is not a square matrix')
    return
else
    n = rA;
end
if cb ~= 1
    disp('Error: b is not a column vector')
    return
end
if cA ~= rb
    disp('Error: Matrix and vector dimensions are not compatible')
    return
end
dA = det(A);
if dA == 0
    disp('Error: Matrix is singular')
    return
end
% Initialize the result x
x = zeros(rA,1); % initialize x as a column vector
% Step 1: Form the combined matrix [A b]
a = [A b];
disp(' ')
disp('Combined Matrix [A|b]')
disp('-----------------------------')
disp(a)
disp(' ')
% Step 2: Forward elimination
disp('-----------------------------')
disp('Forward Elimination')
disp('-----------------------------')
disp(' ')
for k = 1:n
    disp(['Pivot Row ',num2str(k)])
    disp('-----------------------------')
    for i = k+1:n
        m = a(i,k)/a(k,k);
        for j = k:n+1            
            a(i,j)=a(i,j)-m*a(k,j); % eliminate elements in the rows below
        end        
    end
    disp(a)
    disp(' ')  
end
disp('-----------------------------')
disp('Obtain the Result')
disp('-----------------------------')
disp(' ')
% Back substitution
x(n) = a(n,n+1)/a(n,n);
for i = n-1:-1:1
    sum = a(i,n+1);
    for j = i+1:n
        sum = sum - a(i,j)*x(j);
    end
    x(i) = sum/a(i,i);
end
disp(x)
end

Example-
% defines matrix A A = [1, 1, 1; 2, 3,-1; 1, 3, 2]; % defines vector b b = [ 2; 9; 5]; x = gausselimination(A,b)



Gauss Jacobi


function x = gaussjacobi(A,b,x0)
[rA,cA] = size(A);  % Determine the size of A
[rb,cb] = size(b);  % Determine the size of b
% Input checks
if rA ~= cA
    disp('Error: A is not a square matrix')
    return
else
    n = rA;
end
if cb ~= 1
    disp('Error: b is not a column vector')
    return
end
if cA ~= rb
    disp('Error: Matrix and vector dimensions are not compatible')
    return
end
dA = det(A);
if dA == 0
    disp('Error: Matrix is singular')
    return
end
% Initialize the variables
maxit = 100;
es = 0.0001;
er = 1;
xnew = x0;
count = 0;
% Prepare R and D-inverse matrices
R = A;
Dinv = zeros(n);
for i=1:n
    R(i,i)=0;
    Dinv(i,i)=1/A(i,i);
end
% For display only
outmat = zeros(1,n+2);
outmat(1,:) = [count transpose(xnew) er];
% main loop
while(er>es && count<maxit)
    count = count + 1;          % Increment the counter
    xold = xnew;                % keep the old estimate
    xnew = Dinv*(b - R*xold);   % update the new estimate
    er = norm(xnew-xold)/norm(xnew);    % approximate relative error
    outmat =[outmat; count transpose(xnew) er]; % For display
end
% Display the results
if count<maxit
    disp('Gauss-Jacobi Method')
    disp('-------------------------------------------------')
    disp('Output format:')
    disp('First column = iteration number')
    disp('Next columns = estimates of the components of x')
    disp('Last column = er')
    disp(' ')
    disp(outmat);
    disp(' ');
    x = xnew;
    disp(['In ',num2str(count),' iterations, x was found to be: '])
    disp(x);
    disp(' ');
else
    disp(['Failed to converge in ',num2str(count),' iterations'])
    x = NAN;
end


Example-
A= [26 2 2;3 27 1;2 3 17]; b= [12.6;-14.3;6.0]; x0= [0;0;0]; gaussjacobi(A,b,x0);


Gauss Jordan


function x = gaussjordan(A,b)
[rA,cA] = size(A);  % Determine the size of A
[rb,cb] = size(b);  % Determine the size of b
% Input checks
if rA ~= cA
    disp('Error: A is not a square matrix')
    return
else
    n = rA;
end
if cb ~= 1
    disp('Error: b is not a column vector')
    return
end
if cA ~= rb
    disp('Error: Matrix and vector dimensions are not compatible')
    return
end
dA = det(A);
if dA == 0
    disp('Error: Matrix is singular')
    return
end
% Initialize the result x
x = zeros(rA,1); % initialize x as a col
% Step 1: Form the combined matrix [A b]
a = [A b];
disp(' ')
disp('Combined Matrix [A|b]')
disp('-----------------------------')
disp(a)
disp(' ')
% Step 2: Forward elimination
disp('-----------------------------')
disp('Forward Elimination')
disp('-----------------------------')
disp(' ')
for k = 1:n
    disp(['Pivot Row ',num2str(k)])
    disp('-----------------------------')
    a(k,:) = a(k,:)/a(k,k);  % normalize the pivot row
    for i = k+1:n
        m = a(i,k)/a(k,k);
        for j = k:n+1            
            a(i,j)=a(i,j)-m*a(k,j); % eliminate elements in the rows below
        end        
    end
    for i = 1:k-1
        m = a(i,k)/a(k,k);
        for j = k:n+1            
            a(i,j)=a(i,j)-m*a(k,j); % eliminate elements in the rows above
        end      
    end
    disp(a)
    disp(' ')  
end
disp('-----------------------------')
disp('Obtain the Result')
disp('-----------------------------')
disp(' ')
x = a(:,n+1);   % obtain x directly as the right hand column
disp(x)
end

Example-
% defines matrix A A = [1, 1, 1; 2, 3,-1; 1, 3, 2]; % defines vector b b = [ 2; 9; 5]; x = gaussjordan(A,b)




Gauss Seidel



function x = gaussseidel(A,b,x0)
[rA,cA] = size(A);  % Determine the size of A
[rb,cb] = size(b);  % Determine the size of b
% Input checks
if rA ~= cA
    disp('Error: A is not a square matrix')
    return
else
    n = rA;
end
if cb ~= 1
    disp('Error: b is not a column vector')
    return
end
if cA ~= rb
    disp('Error: Matrix and vector dimensions are not compatible')
    return
end
dA = det(A);
if dA == 0
    disp('Error: Matrix is singular')
    return
end
% Initialize the variables
maxit = 100;
es = 0.0001;
er = 1;
xnew = x0;
count = 0;
% Prepare U and L matrices
U = zeros(n);
for i=1:n
    for j = i+1:n
        U(i,j)=A(i,j);
    end
end
L = A - U;
% For display only
outmat = zeros(1,n+2);
outmat(1,:) = [count transpose(xnew) er];
% main loop
while(er>es && count<maxit)
    count = count + 1;              % increment the counter
    xold = xnew;                    % keep the old estimate
    % Calculate the new estimate by forward substitution
    xnew(1) = (b(1) - U(1,:)*xnew)/L(1,1);
    for i = 2:n
        sum = 0;
        for j = 1:i-1
            sum = sum + L(i,j)*xnew(j);
        end
    xnew(i) = (b(i)-U(i,:)*xnew-sum)/L(i,i);
    end
    er = norm(xnew-xold)/norm(xnew);    % absolute relative error
    outmat =[outmat; count transpose(xnew) er]; % for display
end
% Display the results
if count<maxit
    disp('Gauss-Seidel Method')
    disp('-------------------------------------------------')
    disp('Output format:')
    disp('First column = iteration number')
    disp('Next columns = estimates of the components of x')
    disp('Last column = er')
    disp(' ')
    disp(outmat);
    disp(' ');
    x = xnew;
    disp(['In ',num2str(count),' iterations, x was found to be: '])
    disp(x);
    fprintf('\n er = ');
    fprintf('%10.6g\n\n',er);
else
    disp(['Failed to converge in ',num2str(count),' iterations'])
    x = 'NAN';
end


Example =
A= [26 2 2;3 27 1;2 3 17]; b= [12.6;-14.3;6.0]; x0= [0;0;0]; gaussseidel(A,b,x0);



Newton Rapson Method


% NewtonRaphsonMethod
function xr = NewtonRaphsonMethod(fun, dfun, x0)
% Set limits for the plot
pmin = 0;
pmax = 3;

% Plot the function and y=x to visualize behavior
figure
hold on
fplot(@(x) fun(x), [pmin, pmax], 'LineWidth', 2)
fplot(@(x) x, [pmin, pmax], 'LineWidth', 2)
fplot(0, [pmin, pmax], '-k')
xlabel('x')
ylabel('f(x)')
title('Newton-Raphson Method Demo')
grid on

% Initialize variables
es = 0.001;
maxit = 100;

% Print output header
fprintf('\n  #         xi          er\n');

count = 0;  % Iteration counter
er = 1;     % Initial error value
xi = x0;    % Initial guess
erplot = zeros (1,50);
fprintf('%3g %10.6g %10.4g\n', count, xi, er);

% Main loop
while (abs(er) > es && count < maxit)
    count = count + 1;
    xold = xi;  % Keep the previous estimate
    fxold = fun(xold);
   
    % Calculate the new estimate using Newton-Raphson formula
    xi = xold - fun(xold) / dfun(xold);
   
    plot([xold, xi], [fxold, 0], '--k');  % Visualize the approximation
   
    er = (xi - xold) / xi;  % Relative error
    erplot (count) = er;
    fprintf('%3g %10.6g %10.4g\n', count, xi, er);
end

if count >= maxit
    disp(' ')
    disp(['Failed to converge in ', num2str(maxit), ' iterations'])
    disp(' ')
    disp('--------------------------------------------')
    disp(' ')
    xr = NaN;
else
    disp(' ')
    disp(['Root has been found at x = ', num2str(xi)])
    disp(' ')
    disp('--------------------------------------------')
    disp(' ')
    xr = xi;
end

% Plot the relative error versus iteration number
figure
hold on
fplot(0, [0, count], '-k')
plot(erplot(1:count),'LineWidth',2)
title('Relative Error')
xlabel('Iteration Number')
ylabel('Relative Error (er)')
hold off
end


Example:
% Define your trigonometric function and its derivative fun = @(x) x*sin(x)+cos(x); dfun = @(x) x*cos(x); % Initial guess x0 = 1; % Call the NewtonRaphsonMethod function with your function, its derivative, and initial guess root = NewtonRaphsonMethod(fun, dfun, x0); % Define your trigonometric function and its derivative fun = @(x) sin(x) - x/2; dfun = @(x) cos(x) - 0.5; % Initial guess x0 = 1; % Call the NewtonRaphsonMethod function with your function, its derivative, and initial guess root = NewtonRaphsonMethod(fun, dfun, x0); % Define your function and its derivative fun = @(x) x^3 - 2*x - 5; dfun = @(x) 3*x^2 - 2; % Initial guess x0 = 2; % Call the NewtonRaphsonMethod function with your function, its derivative, and initial guess root = NewtonRaphsonMethod(fun, dfun, x0);



Runge Kutta


% Prompt the user to input initial values and step size
xo = input("Enter the value of xo: ");
yo = input("Enter the value of yo: ");
h = input("Enter the value of h: ");
% Prompt the user to input the function as a string
func_str = input("Enter the function in terms of x and y: ", 's');
% Convert the function string to a function handle
f = str2func(['@(x,y)', func_str]);
% Prompt the user to input the number of iterations
n = (yo-xo)/h;
% Implement Runge-Kutta Method (4th Order)
for i = 1:n
    k1 = h * f(xo, yo);
    k2 = h * f(xo + h/2, yo + k1/2);
    k3 = h * f(xo + h/2, yo + k2/2);
    k4 = h * f(xo + h, yo + k3);
 
    y = yo + (k1 + 2*k2 + 2*k3 + k4)/6;
   
    fprintf('y%i = %f\n', i, y);
   
    yo = y;
    xo = xo + h;
end





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