Question

Use bisection and RK4 write matlab code

in Fnd v such that ycoo)

Answer 1

(1) Bisection Method in MATLAB :

a=input('Enter function with right hand side zero:','s');

f=inline(a);

xl=input('Enter the first value of guess interval:') ;

xu=input('Enter the end value of guess interval:');

tol=input('Enter the allowed error:');

if f(xu)*f(xl)<0

else

    fprintf('The guess is incorrect! Enter new guesses\n');

    xl=input('Enter the first value of guess interval:\n') ;

    xu=input('Enter the end value of guess interval:\n');

end

for i=2:1000

xr=(xu+xl)/2;

if f(xu)*f(xr)<0

    xl=xr;

else

    xu=xr;

end

if f(xl)*f(xr)<0

    xu=xr;

else

    xl=xr;

end

xnew(1)=0;

xnew(i)=xr;

if abs((xnew(i)-xnew(i-1))/xnew(i))<tol,break,end

end

str = ['The required root of the equation is: ', num2str(xr), '']

(2) RK4 Method in MATLAB :

function a = runge_kutta(df)

% asking initial conditions

x0 = input('Enter initial value of x : ');

y0 = input ('Enter initial value of y : ');

x1 = input( 'Enter value of x at which y is to be calculated : ');

tol = input( 'Enter desired level of accuracy in the final result : ');

%choose the order of Runge-Kutta method

r = menu ( ' Which order of Runge Kutta u want to use', ...

' 2nd order ' , ' 3rd order ' , ' 4th order ');

switch r

case 1

% calculating the value of h

n =ceil( (x1-x0)/sqrt(tol));

h = ( x1 - x0)/n;

for i = 1 : n

X(1,1) = x0; Y (1,1) = y0;

k1 = h*feval( df , X(1,i), Y(1,i));

k2 = h*feval( df , X(1,i) + h , Y(1,i) + k1);

k = 1/2* ( k1+ k2);

X( 1, i+1) = X(1,i) + h;

Y( 1 ,i+1) = Y(1,i) + k;

end

case 2

% calculating the value of h

n =ceil( (x1-x0)/nthroot(tol,3));

h = ( x1 - x0)/n;

for i = 1 : n

X(1,1) = x0; Y (1,1) = y0;

k1 = h*feval( df , X(1,i), Y(1,i));

k2 = h*feval( df , X(1,i) + h/2, Y(1,i) + k1);

k3 = h*feval( df , X(1,i) + h, Y(1,i) + k2);

k = 1/6* ( k1+ 4*k2 + k3);

X( 1, i+1) = X(1,i) + h;

Y( 1 ,i+1) = Y(1,i) + k;

end

case 3

% calculating the value of h

n =ceil( (x1-x0)/nthroot(tol,3));

h = ( x1 - x0)/n;

for i = 1 : n

X(1,1) = x0; Y (1,1) = y0;

k1 = h*feval( df , X(1,i), Y(1,i));

k2 = h*feval( df , X(1,i) + h/2, Y(1,i) + k1);

k3 = h*feval( df , X(1,i) + h/2, Y(1,i) + k2);

k4 = h*feval( df , X(1,i) + h, Y(1,i) + k3);

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

X( 1, i+1) = X(1,i) + h;

Y( 1 ,i+1) = Y(1,i) + k;

end

end

%displaying results

fprintf( 'for x = %g \n y = %g \n' , x1,Y(1,n+1))

%displaying graph

x = 1:n+1;

y = Y(1,n+1)*ones(1,n+1) - Y(1,:);

plot(x,y,'r')

xlabel = (' no of interval ');

ylabel = ( ' Error ');