What's wrong with this code?

function sincomp(x,n)
i = 1;
tru = sin(x);
ser = 0;
fprintf('\n');
fprintf('order true value     approximation     error\n');
while (1)
    if i > n, break, end
    ser = ser + (-1)^(i - 1) * x^(2*i-1) / factorial(2*i-1);
    err = (tru - ser) / tru * 100;
    fprintf('%3d %14.10f %14.10f %12.8f\n',i,tru,ser,err);
    i = i + 1;
end

Answer 1

[Stephen23]

Your basic idea is okay, but a few tweaks would make the code simpler and more robust. Because there is a clear practical limit to the number of iterations that are possible (limited by factorial) I would recommend using a for loop: this avoids any chance of an unintentional infinite loop (which is an easy trap to fall into with while loops), and means that you do not need to keep track of the loop variable yourself. So you would get:

N = 0.9;
Z = 0;
T = sin(N);
for k = 0:7 % 8 terms
  p = k*2+1;
  Z = Z + (-1)^k * (N^p / factorial(p));
  fprintf('val =%+19.15f  err =%+20.15f%%\n',Z,100*(T-Z)/T)
end

giving:

val = +0.900000000000000  err = -14.894559211300594%
val = +0.778500000000000  err =  +0.616206282224993%
val = +0.783420750000000  err =  -0.011979720262797%
val = +0.783325849821429  err =  +0.000135295499462%
val = +0.783326917448438  err =  -0.000000998427861%
val = +0.783326909586820  err =  +0.000000005191053%
val = +0.783326909627640  err =  -0.000000000020041%
val = +0.783326909627483  err =  +0.000000000000057%

Addendum: It would really make more sense mathematically to calculate the error the other way around (i.e. Z-T, not T-Z):

fprintf('val =%+19.15f  err =%+20.15f%%\n',Z,100*(Z-T)/T)

giving

val = +0.900000000000000  err = +14.894559211300594%
val = +0.778500000000000  err =  -0.616206282224993%
val = +0.783420750000000  err =  +0.011979720262797%
val = +0.783325849821429  err =  -0.000135295499462%
val = +0.783326917448438  err =  +0.000000998427861%
val = +0.783326909586820  err =  -0.000000005191053%
val = +0.783326909627640  err =  +0.000000000020041%
val = +0.783326909627483  err =  -0.000000000000057%

because then V_est = V_true + V_true*err, e.g.:

>> sin(0.9) * (1+0.14894559211300594) % 1st term
ans =  0.900000000000000
>> sin(0.9) * (1-0.00616206282224993) % 2nd term
ans =  0.778500000000000
... etc

whereas as the question is written it makes no sense any way you look at it:

>> sin(0.9) * (1-0.14894559211300594)
ans =    0.666653819254967
>> 0.9 * (1-0.14894559211300594)
ans =    0.765948967098295

Whoever wrote the question did not consider the effect of removing the abs from the commonly used formula for relative error. Of course if they had kept the abs then this addendum would become moot.