parallelization sum of list

Question

Is there a simple way to parallelize sum of list elements ? when I use:

n = 500;
T1[z_] = Table[RandomReal[]/(RandomReal[] + RandomReal[] z), {i, n}];
Timing@Parallelize@Total@T1[1.]

error

Parallelize::nopar1: "T1[1.] cannot be parallelized; proceeding with sequential evaluation."

It is only an examle. I my calculation I have a huge expression with elements p/(q+r z) which are a result of using Apart[]

---

Thanks for your help , but the problem is something else f[z_]- very big formula (rational function) which should be integrated

Af[z_]:=Apart@f[z]

after that the Af[z] = p1/(q1+r1 z)+p2/(q2+ r2 z) +.....

thus integral is simply defined:

F[z_]:=Af[z]/. p_/(q_ + r_ z) -> p/r Log[q + r z]

The problem is: F[z] has a lot of simple elements, and I should compute F[z] for many complex points as quick as possible

regards, Olaf

Answer 1

tl; dr Do not try to parallelize this. Use vector arithmetic (vectorization) instead.

Total cannot be parallelized because it already makes use of all your CPU cores in a way that is much more effective than Mathematica's parallel tools. The same applies to vector arithmetic. If a and b are two large arrays of equal size, then operations such as a^2, a*b, a.b, Total[a], etc. will run very efficiently and will make use of all your CPU cores. In fact these will often be faster than a naïve for loop one might write in C.

Answer 2

n = 10^7;
p = RandomReal[1, n];
q = RandomReal[1, n];
r = RandomReal[1, n];

T1[z_] := p/(q + z r)

AbsoluteTiming[
 Total[T1[1.]]
]

{0.143513, 6.92922*10^6}