This document serves as an overview for solving problems common in Computational Mathematics. Of note, `primeSieve`

and `primeCount`

are based on the excellent work by Kim Walisch.

`primeSieve`

The primeSieve function is based on the Segmented Sieve of Eratosthenes. As stated in the linked article, the sieve itself is already very efficient. The problem from an efficiency standpoint, is due to the memory requirements. The segmented version overcomes this by only sieving small sections at a time, which greatly facilitates use of the cache.

```
library(RcppAlgos)
library(microbenchmark)
microbenchmark(primeSieve(1e6))
Unit: milliseconds
expr min lq mean median uq max neval
primeSieve(1e+06) 1.151969 1.175597 1.28727 1.264687 1.316879 1.687415 100
## Single threaded primes under a billion!!!
system.time(a <- primeSieve(10^9))
user system elapsed
1.161 0.091 1.253
## Using 8 threads we can get under 0.5 seconds!!!
system.time(primeSieve(10^9, nThreads = 8))
user system elapsed
2.033 0.045 0.374
## Quickly generate large primes over small interval. N.B. The
## order for the bounds does not matter.
options(scipen = 50)
system.time(myPs <- primeSieve(10^13 + 10^3, 10^13))
user system elapsed
0.016 0.005 0.021
myPs
[1] 10000000000037 10000000000051 10000000000099 10000000000129
[5] 10000000000183 10000000000259 10000000000267 10000000000273
[9] 10000000000279 10000000000283 10000000000313 10000000000343
[13] 10000000000391 10000000000411 10000000000433 10000000000453
[17] 10000000000591 10000000000609 10000000000643 10000000000649
[21] 10000000000657 10000000000687 10000000000691 10000000000717
[25] 10000000000729 10000000000751 10000000000759 10000000000777
[29] 10000000000853 10000000000883 10000000000943 10000000000957
[33] 10000000000987 10000000000993
## Object created is small
object.size(myPs)
320 bytes
```

Since version `2.3.0`

, we are implementing the cache-friendly improvements for larger primes originally developed by Tomás Oliveira.

```
## Version <= 2.2.0.. i.e. older versions
system.time(old <- RcppAlgos2.2::primeSieve(1e15, 1e15 + 1e9))
user system elapsed
7.615 0.140 7.792
## v2.3.0 is over 3x faster!
system.time(a <- primeSieve(1e15, 1e15 + 1e9))
user system elapsed
2.237 0.183 2.420
## And using nThreads we are ~8x faster
system.time(b <- primeSieve(1e15, 1e15 + 1e9, nThreads = 8))
user system elapsed
4.872 0.807 0.917
identical(a, b)
[1] TRUE
identical(a, old)
[1] TRUE
```

`primeCount`

The library by Kim Walisch relies on OpenMP for parallel computation with Legendre’s Formula. Currently, the default compiler on `macOS`

is `clang`

, which does not support `OpenMP`

. James Balamuta (a.k.a. TheCoatlessProfessor… well at least we think so) has written a great article on this topic, which you can find here: https://thecoatlessprofessor.com/programming/openmp-in-r-on-os-x/. One of the goals of `RcppAlgos`

is to be accessible by all users. With this in mind, we set out to count primes in parallel *without* `OpenMP`

.

At first glance, this seems trivial as we have a function in `Primes.cpp`

called `phiWorker`

that counts the primes up to `x`

. If you look in phi.cpp in the `primecount`

library by Kim Walisch, we see that `OpenMP`

does its magic on a for loop that makes repeated calls to `phi`

(which is what `phiWorker`

is based on). All we need to do is break this loop into *n* intervals where *n* is the number of threads. Simple, right?

We can certainly do this, but what you will find is that *n - 1* threads will complete very quickly and the *n ^{th}* thread will be left with a heavy computation. In order to alleviate this unbalanced load, we take advantage of thread pooling provided by

`RcppThread`

which allows us to reuse threads efficiently as well as breaking up the loop mentioned above into smaller intervals. The idea is to completely calculate `phi`

up to a limit With this is mind, here are some results:

```
## Enumerate the number of primes below trillion
system.time(underOneTrillion <- primeCount(10^12))
user system elapsed
0.478 0.000 0.480
underOneTrillion
[1] 37607912018
## Enumerate the number of primes below a billion in 2 milliseconds
library(microbenchmark)
microbenchmark(primeCount(10^9))
Unit: milliseconds
expr min lq mean median uq max neval
primeCount(10^9) 1.93462 1.938516 2.044785 1.957809 2.090828 3.584245 100
system.time(underOneHundredTrillion <- primeCount(1e14, nThreads = 8))
user system elapsed
49.894 0.102 6.774
underOneHundredTrillion
[1] 3204941750802
## From Kim Walisch's primecount library:
## Josephs-MBP:primecount-4 josephwood$ ./primecount 1e14 --legendre --time
## 3204941750802
## Seconds: 4.441
```

`RcppAlgos`

comes equipped with several functions for quickly generating essential components for problems common in computational mathematics. All functions below can be executed in parallel by using the argument `nThreads`

.

The following sieving functions (`primeFactorizeSieve`

, `divisorsSieve`

, `numDivisorSieve`

, & `eulerPhiSieve`

) are very useful and flexible. Generate components up to a number or between two bounds.

```
## get the number of divisors for every number from 1 to n
numDivisorSieve(20)
[1] 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 5 2 6 2 6
## If you want the complete factorization from 1 to n, use divisorsList
system.time(allFacs <- divisorsSieve(10^5, namedList = TRUE))
user system elapsed
0.040 0.003 0.043
allFacs[c(4339, 15613, 22080)]
$`4339`
[1] 1 4339
$`15613`
[1] 1 13 1201 15613
$`22080`
[1] 1 2 3 4 5 6 8 10 12 15
[11] 16 20 23 24 30 32 40 46 48 60
[21] 64 69 80 92 96 115 120 138 160 184
[31] 192 230 240 276 320 345 368 460 480 552
[41] 690 736 920 960 1104 1380 1472 1840 2208 2760
[51] 3680 4416 5520 7360 11040 22080
## Between two bounds
primeFactorizeSieve(10^12, 10^12 + 5)
[[1]]
[1] 2 2 2 2 2 2 2 2 2 2 2 2 5 5 5 5 5 5 5 5 5 5 5 5
[[2]]
[1] 73 137 99990001
[[3]]
[1] 2 3 166666666667
[[4]]
[1] 61 14221 1152763
[[5]]
[1] 2 2 17 149 197 501001
[[6]]
[1] 3 5 66666666667
## Creating a named object
eulerPhiSieve(20, namedVector = TRUE)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 2 2 4 2 6 4 6 4 10 4 12 6 8 8 16 6 18 8
system.time(a <- eulerPhiSieve(1e12, 1e12 + 1e7))
user system elapsed
0.998 0.042 1.041
## Using nThreads for greater efficiency
system.time(b <- eulerPhiSieve(1e12, 1e12 + 1e7, nThreads = 8))
user system elapsed
3.576 0.014 0.485
identical(a, b)
[1] TRUE
```

There are three very fast vectorized functions for general factoring (e.g. all divisors of number), primality testing, as well as prime factoring (`divisorsRcpp`

, `isPrimeRcpp`

, `primeFactorize`

).

```
## get result for individual numbers
primeFactorize(123456789)
[1] 3 3 3607 3803
## or for an entire vector
set.seed(100)
myVec <- sample(-100000000:100000000, 5)
divisorsRcpp(myVec, namedList = TRUE)
$`-38446778`
[1] -38446778 -19223389 -2 -1 1
[6] 2 19223389 38446778
$`-48465500`
[1] -48465500 -24232750 -12116375 -9693100 -4846550
[6] -2423275 -1938620 -969310 -484655 -387724
[11] -193862 -96931 -500 -250 -125
[16] -100 -50 -25 -20 -10
[21] -5 -4 -2 -1 1
[26] 2 4 5 10 20
[31] 25 50 100 125 250
[36] 500 96931 193862 387724 484655
[41] 969310 1938620 2423275 4846550 9693100
[46] 12116375 24232750 48465500
$`10464487`
[1] 1 11 317 3001 3487 33011
[7] 951317 10464487
$`-88723370`
[1] -88723370 -44361685 -17744674 -8872337 -10
[6] -5 -2 -1 1 2
[11] 5 10 8872337 17744674 44361685
[16] 88723370
$`-6290143`
[1] -6290143 -1 1 6290143
## Creating a named object
isPrimeRcpp(995:1000, namedVector = TRUE)
995 996 997 998 999 1000
FALSE FALSE TRUE FALSE FALSE FALSE
system.time(a <- primeFactorize(1e12:(1e12 + 1e5)))
user system elapsed
1.721 0.004 1.725
## Using nThreads for greater efficiency
system.time(b <- primeFactorize(1e12:(1e12 + 1e5), nThreads = 8))
user system elapsed
3.155 0.002 0.410
identical(a, b)
[1] TRUE
```