MATLAB: Help with while loop

taylor serieswhile loop

Hello,

I have to create a program that will find the taylor series expansion at an arbitrary value x. I also have to find the relevant error of the series expansion from the true exp.

Create a MATLAB program to compute this series and stop at %error < 3%. Display the value of the series approximation and how many terms that you used to calculate this value.

x=1.5;
y=0;
n=0;
error=0;
while(error>3)
  if n<2
    f=1;
  elseif n>=2
    f=f*n;
  end
  y=y+(x^n/(f));
  error=((exp(x)-y)/exp(x))*100;
  n=n+1;
end
disp(n-1)
disp(error)

This is currently what I have but my program will only compute for the first n. I can not seem to see what is going wrong and would appreciate any hints.

Best Answer

You were pretty close, but there are a couple of mistakes.

You start out the error at zero, then ask the loop to run only when the error is greater than 3....
You are not actually calculating the factorial for each new value of n.
You need to use the absolute value when calculating the error.

x=1.5;
y=0;
n=0;
ER=100;  % Don't name a variable 'error'.  Start >3 so loop runs!
while(ER>3)
    f = 1;
    for ii=2:n
      f=f*ii;
    end
    y = y+(x^n/(f));
    ER = (abs(exp(x)-y)/exp(x))*100;
    n=n+1;
  end
  disp(n-1)
  disp(ER)

Related Solutions

MATLAB: HELP!! Error: In an assignment A(I) = B, the number of elements in B and I must be the same.

In the Homework function, there are two Taylors - the nx1 vector local variable

 Taylor = zeros(n,1);

and the function

 function y = Taylor(x,n)

This causes a conflict at the line where the function is called and the result is assigned to the vector

 Taylor(i) = Taylor(x,i)

Variables and functions cannot have the same name. I recommend changing the local variable to

 taylorVctr = zeros(n,1);  % the nx1 vector

or removing it altogether since it doesn't seem to be used outside of the for loop.

Note that the Taylor function has a return value of y but never initializes this value, but there is an f that is summed at each iteration so this should be replaced with y (or vice versa) else there will be an error. Also the second input parameter n is unused. It may be that it should be used in the for loop

 for i=1:n

as this would give results that show that the error is reduced between the exact function and the Taylor series approximation as the number of terms in the latter is increased. Further, y (or f) should be initialized as

 y = f_exact(0);

rather than as 1 (even though it results in the same answer) just to be clear that our a or c is zero (is this true - that a is zero in this Taylor series expansion and so is the Maclaurin series?). And…I think that the exponent is incorrect on the first term. What we should have is the nth derivative evaluated at a (or c which is zero) which, because the function is exp(-x/4), is just f_exact(a) multiplied by (-1.4)^i. That product is then multiplied by (x-a)^i which is then divided by i!. What I think you really want for this code (based on http://en.wikipedia.org/wiki/Taylor_series) is:

 function [y] = Taylor(x,n)
  a = 0;
  y = f_exact(a);
for i=1:n
  % note that the ith derivative is f_exact(a)*(-1/4)^i
  y = y + f_exact(a)*(-1/4)^i*(x-a)^i/factorial(i); 
end

The function f_exact takes as an input parameter x and then immediately redefines it to a row vector with three elements. Why? I think that this line of code and the for loop should be removed as the output is a 10x3 matrix with each row being identical. If the idea is to have f_exact represent exp(-x/4), then that is all it should do:

function y=f_exact(x)
   y = exp(-x/4);
end

unless there is some reason why x is replaced with a row vector and the for loop is there. Note that if this isn't changed, then there will be the error In an assignment A(I) = B, the number of elements in B and I must be the same. when assigning taylorVctr(i) = Taylor(x,i);.

MATLAB: For and while loops

function F = getFacWhile(n)
    F=1;
    while n>1
        F=F*n;
        n=n-1;
    end
end
function F = getFacWhile(n)
    F=1;
    for i=1:n
        F=F*i;
    end
end

Best Answer

Related Solutions

MATLAB: HELP!! Error: In an assignment A(I) = B, the number of elements in B and I must be the same.

MATLAB: For and while loops

Related Question