I have written this function to return the factorial of a given number
func factorial(_ n: Int) -> Int {
if n == 0 {
return 1
}
else {
return n * factorial(n - 1)
}
}
print( factorial(20) ) // 2432902008176640000
Works as it should, as long the given number does not exceed 20, because then the result becomes too high!
How can I circumvent this limit and thus calculate the factorial of higher numbers?
I have searched around and found some bignum libraries for Swift. I'm doing this to learn and be familiar with Swift, therefore I want to figure this out on my own.
The last factorial is 170! “Did you know? The number 170 is the highest possible number you can calculate a factorial for? Any higher than 170, and the mathematical answer is infinity.” - visualfractions.com/calculator/fac…
Most of the time when you need factorials, you'll need many of them, so the best idea is to build a table. Then you can compute each one in constant time on average by using n! = n*(n-1)! .
Here's an approach that will let you find very large factorials.
Represent large numbers as an array of digits. For instance 987
would be [9, 8, 7]
. Multiplying that number by an integer n
would require two steps.
n
.For example 987 * 2
:
let arr = [9, 8, 7]
let arr2 = arr.map { $0 * 2 }
print(arr2) // [18, 16, 14]
Now, perform the carry operation. Starting at the one's digit, 14
is too big, so keep the 4
and carry the 1
. Add the 1
to 16
to get 17
.
[18, 17, 4]
Repeat with the ten's place:
[19, 7, 4]
And then with the hundred's place:
[1, 9, 7, 4]
Finally, for printing, you could convert this back to a string:
let arr = [1, 9, 7, 4]
print(arr.map(String.init).joined())
1974
Applying that technique, here is a carryAll
function that performs the carry operation, and a factorial
that uses it to calculate very large factorials:
func carryAll(_ arr: [Int]) -> [Int] {
var result = [Int]()
var carry = 0
for val in arr.reversed() {
let total = val + carry
let digit = total % 10
carry = total / 10
result.append(digit)
}
while carry > 0 {
let digit = carry % 10
carry = carry / 10
result.append(digit)
}
return result.reversed()
}
func factorial(_ n: Int) -> String {
var result = [1]
for i in 2...n {
result = result.map { $0 * i }
result = carryAll(result)
}
return result.map(String.init).joined()
}
print(factorial(1000))
402387260077093773543702433923003985719374864210714632543799910429938512398629020592044208486969404800479988610197196058631666872994808558901323829669944590997424504087073759918823627727188732519779505950995276120874975462497043601418278094646496291056393887437886487337119181045825783647849977012476632889835955735432513185323958463075557409114262417474349347553428646576611667797396668820291207379143853719588249808126867838374559731746136085379534524221586593201928090878297308431392844403281231558611036976801357304216168747609675871348312025478589320767169132448426236131412508780208000261683151027341827977704784635868170164365024153691398281264810213092761244896359928705114964975419909342221566832572080821333186116811553615836546984046708975602900950537616475847728421889679646244945160765353408198901385442487984959953319101723355556602139450399736280750137837615307127761926849034352625200015888535147331611702103968175921510907788019393178114194545257223865541461062892187960223838971476088506276862967146674697562911234082439208160153780889893964518263243671616762179168909779911903754031274622289988005195444414282012187361745992642956581746628302955570299024324153181617210465832036786906117260158783520751516284225540265170483304226143974286933061690897968482590125458327168226458066526769958652682272807075781391858178889652208164348344825993266043367660176999612831860788386150279465955131156552036093988180612138558600301435694527224206344631797460594682573103790084024432438465657245014402821885252470935190620929023136493273497565513958720559654228749774011413346962715422845862377387538230483865688976461927383814900140767310446640259899490222221765904339901886018566526485061799702356193897017860040811889729918311021171229845901641921068884387121855646124960798722908519296819372388642614839657382291123125024186649353143970137428531926649875337218940694281434118520158014123344828015051399694290153483077644569099073152433278288269864602789864321139083506217095002597389863554277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994871701244516461260379029309120889086942028510640182154399457156805941872748998094254742173582401063677404595741785160829230135358081840096996372524230560855903700624271243416909004153690105933983835777939410970027753472000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
You can use this library: BigInt
Install it using CocoaPods:
pod 'BigInt'
Then you can use it like this:
import BigInt
func factorial(_ n: Int) -> BigInt {
if n == 0 {
return 1
}
else {
return BigInt(n) * factorial(n - 1)
}
}
print( factorial(50) ) // 30414093201713378043612608166064768844377641568960512000000000000
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