How would you code a method to evaluate a n-th degree polynomial? I once needed to code this type of routine for a project I was working on. At that time I used to program fast FPU processors, so I never paid attention to performance. I went *careless*ly coding the straight forward approach:

**void** **polynomial::eval**(**const** vector<**double**>& coef, **double** x, **double** *res)
{
double res=0;
**for** (**int** i=0; i<coef.size(); i++)
res += coef[i] * pow(x, i);
**return** res;
}

The library ran smooth on our P-IV intel processor. It was used for a couple of years without problems. As time went by and product cost was pushed down, we sooner decided to move to a XScale intel processor. It was then when we noticed quite an impact on the performance (from 0.2 to 12 sec overhead). It turns out that my *harmless *function was called 9 million times! A mathematician friend of mine pointed me out to a more efficient algorithm: Horner’s scheme. I admire it for the elegance and simplicity of its solution: just a rewrite of the commonly used polynomial expression:

The new routine runs now 10 times faster!

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Nice post and nice weblog!

In fact, it was a physicist coworker of mine who reminded me of Horner’s.

Keep up the good work 😉

10 times faster? Not exactly.

While your old algorithm is of complexity O(n^2), where n is the degree, the Horner’s method is just O(n). Hence for the same degree, your old code would run about n times *slower* then the new one. If the degree is 10, then it is correct to say the new code will run in about 1/10th of the time.

Depending on your polynomials and how they are evaluated, there could

be even better approaches.

Now you might have been talking about overall system performance in which case it would be difficult to figure out the correct gains without knowing the system particulars.

In any case, enjoy the performance! 😉

The evaluation of pow(x,i) in C/C++ costs something close to 100-200 multiplications, as opposed to the assumption of (i-1) multiplications in Waldeck’s analysis above.