taylor expanssion calculation result is wrong

function [SN] = sintaylorfunction(A, tol)

k = 0; 

error = 0 ;

sin = 0;

SN = sin; 

sinplus1 = 0;

SM = sinplus1;

  while E <= tol
       
      sin = (-1)^k * A^(2*k+1)/factorial(2*k+1);
      
      SN = SN + sin;
      
     sinplus1 = (-1)^(k+1) * A^(2*(k+1)+1)/factorial(2*(k+1)+1);
     
     SM = SM + sinplus1;
     
     error= abs( max(SM - SN));
     
  end
end

I wrote the sin function in this way. When I calculate sin(pi/2), I get sin(pi/2) = 1.5708. However, we know that sin(pi/2) = 1. Where is my mistake in my function?

Answer 1

[Ameer Hamza]

kou are not incrementing 'k' inside the while-loop. Also, the while-condition is incorrect. Try the following code

sintaylorfunction(pi/2, 0.01)

function [SN] = sintaylorfunction(A, tol)

k = 0;
error = inf;
sin = 0;
SN = sin;
sinplus1 = 0;
SM = sinplus1;

while error >= tol
    sin = (-1)^k * A^(2*k+1)/factorial(2*k+1);
    SN = SN + sin;
    sinplus1 = (-1)^(k+1) * A^(2*(k+1)+1)/factorial(2*(k+1)+1);
    SM = SM + sinplus1;
    error= abs( max(SM - SN));
    k = k+1;
end
end

[Zeynep Toprak]

thank very much!! I didnt know the increment term. It helps very much. Well, my another question is that will I use the same things for cos expansion in the same while loop except for its fomula ?

[Ameer Hamza]

Yes, the formula for taylor series of cos is a bit different as shown in your question too. The method will be same.

[Zeynep Toprak]

sintaylorfunction(pi/2, 0.01)

function [SN CN] = sintaylorfunction(A, tol)

k = 0;
error = inf;
sin = 0;
SN = sin;
sinplus1 = 0;
SM = sinplus1;


while error >= tol
    sin = (-1)^k * A^(2*k+1)/factorial(2*k+1);
    SN = SN + sin;
    sinplus1 = (-1)^(k+1) * A^(2*(k+1)+1)/factorial(2*(k+1)+1);
    SM = SM + sinplus1;
    error= abs( max(SM - SN));
    k = k+1;
end

n=0;
error_cos=inf;
cos=1;
CN=cos;
cosplus1 = 1;
CM = cosplus1;

while error_cos >=tol
    cos = (-1)^n * A^(2*n)/factorial(2*n);
    CN = CN + cos;
    cosplus1 = (-1)^(n+1) * A^(2*(n+1))/factorial(2*(n+1));
    CM = CM + cosplus1;
    error= abs( max(CM - CN));
    n
 = n+1; 
end

end

again I get wrong result for cos. I cannot find my mistake.

[Ameer Hamza]

I have simplified your code a bit. Following will work

[q, w] = sintaylorfunction(pi/2, 0.01)

function [SN CN] = sintaylorfunction(A, tol)

k = 0;
error = inf;
SN = 0;

while error >= tol
    sin = (-1)^k * A^(2*k+1)/factorial(2*k+1);
    SN = SN + sin;
    sinplus1 = (-1)^(k+1) * A^(2*(k+1)+1)/factorial(2*(k+1)+1);
    error= abs(sinplus1-sin);
    k = k+1;
end

n=0;
error_cos=inf;
CN=0;

while error_cos >=tol
    cos = (-1)^n * A^(2*n)/factorial(2*n);
    CN = CN + cos;
    cosplus1 = (-1)^(n+1) * A^(2*(n+1))/factorial(2*(n+1));
    error_cos= abs(cosplus1-cos);
    n = n+1; 
end

end

[Zeynep Toprak]

many thanks dear Hamza. I see my missings.

[Ameer Hamza]

I am glad to be of help!

Answer 2

[Steven Lord]

I want to walk through your code and comment on a few places.

function [SN] = sintaylorfunction(A, tol)

k = 0; 

error = 0 ;

Don't create a variable named error. That identifier already has a meaning in MATLAB.

sin = 0;

The identifier sin also already has a meaning in MATLAB, so I recommend you choose a different variable name.

SN = sin; 

sinplus1 = 0;

SM = sinplus1;

  while E <= tol

The variable E doesn't exist.

       
      sin = (-1)^k * A^(2*k+1)/factorial(2*k+1);

From the role this plays, a better name might be term.

      
      SN = SN + sin;
      
     sinplus1 = (-1)^(k+1) * A^(2*(k+1)+1)/factorial(2*(k+1)+1);
     
     SM = SM + sinplus1;
     
     error= abs( max(SM - SN));

In this loop you change neither the (nonexistent) variable E nor the variable tol. So if this were to enter the loop, you'd never exit.

FYI, you might want to check your answer using the funm function to compute the matrix cosine and matrix sine.