Why do we use absolute value to find the cosine series?-CodePudding

Why do we take the absolute value of term to find the cos series when will the term even be less than 0.0001 please explain if possible with example?

X = (22.0/7.0) * (Theta/180.0);    
CosX = 1.0;
Term = 1.0;
i = 1;

while(fabs(Term)>=0.0001){

    Term = (-1) * Term * X*X/((2*i-1)*2*i);
    CosX = CosX   Term;
    i  ;
}

CodePudding user response：

The sign of the terms alternate: the Taylor series of cosine is

 1 - x^2/2!   x^4/4! - x^6/6!   ...
// ^        ^        ^

Then, the first non-trivial term for |x| > 0 is negative and clearly < 0.0001.

Another way to handle this is to unroll twice using only positive terms.

do {
    Term = Term * X*X/((2*i-1)*2*i);
    CosX = CosX - Term;
    i  ;
    Term = Term * X*X/((2*i-1)*2*i);
    CosX = CosX   Term;
    i  ;
} while (Term > 0.0001);

CodePudding user response：

The idea of the loop condition is to continue approximation until the last change is very small, indicating being close to the precise value. Of course such a condition, one way or another, is needed, because the correct value is unknown and because arriving at it (even if known) would be taking inefficient time.

However "small" for a change, a delta, is the kind of small where -10 is larger than 2. I.e. if the last change was by a big but negative step, the loop should continue. Only when the change does not move a lot anymore in either direction (positive or negative) a reasonable approximation can be assumed.

Wether that is done via taking the absolute value of the change for comparison, or do two checks (one against a small positive change and one against a large negative one - where the largest one is of course very close to 0) does not matter.

CodePudding user response：

We do not have to use fabs in the while loop

We can check both conditions (negative && positive) instead:

while( wTerm >= 0.0001 || wTerm <= -0.0001){