在Swift 3中，当结果过高时，如何计算阶乘? [英] In Swift 3, how to calculate the factorial when the result becomes too high?

问题描述

我已经编写了此函数以返回给定数字的阶乘

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

只要给定的数字不超过20，就可以正常工作，因为这样结果会变得太高！

Works as it should, as long the given number does not exceed 20, because then the result becomes too high!

我该如何规避此限制，从而计算出较高数字的阶乘?

How can I circumvent this limit and thus calculate the factorial of higher numbers?

我到处搜索并找到了一些Swift的bignum库.我这样做是为了学习和熟悉Swift，因此我想自己解决这个问题.

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.

推荐答案

以下是一种可以让您找到非常大的阶乘的方法.

Here's an approach that will let you find very large factorials.

将大数字表示为数字数组.例如，987将为[9, 8, 7].将该数字乘以整数n将需要两个步骤.

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.

1. 将该数组中的每个值乘以n.
2. 执行进位运算以返回再次为个位数的结果.

例如987 * 2:

let arr = [9, 8, 7]
let arr2 = arr.map { $0 * 2 }
print(arr2)  // [18, 16, 14]

现在，执行进位操作.从一个数字开始，14太大，因此请保留4并随身携带1.将1添加到16以获得17.

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]

以十位代替:

[19, 7, 4]

然后是百位:

[1, 9, 7, 4]

最后，要进行打印，您可以将其转换回字符串:

Finally, for printing, you could convert this back to a string:

let arr = [1, 9, 7, 4]
print(arr.map(String.init).joined())

1974

1974

---

应用该技术，这里是执行进位运算的carryAll函数，以及使用它计算非常大的阶乘的factorial:

---

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))

4023872600770937735437024339230039857193748642107146325437999104299385123986290205920442084869694048004799886101971960586316668729948085589013238296699445909974245040870737599188236277271887325197795059509952761208749754624970436014182780946464962910563938874378864873371191810458257836478499770124766328898359557354325131853239584630755574091142624174743493475534286465766116677973966688202912073791438537195882498081268678383745597317461360853795345242215865932019280908782973084313928444032812315586110369768013573042161687476096758713483120254785893207671691324484262361314125087802080002616831510273418279777047846358681701643650241536913982812648102130927612448963599287051149649754199093422215668325720808213331861168115536​​158365469840467089756029009505376164758477284218896796462449451607653534081989013854424879849599533191017233555566021394503997362807501378376153071277619268490343526252000158885351473316117021039681759215109077880193931781141945452572238655414610628921879602238389714 7608850627686296714667469756291123408243920816015378088989396451826324367161676217916890977991190375403127462228998800519544441428201218736174599264295658174662830295557029902432415318161721046583203678690611726015878352075151628422554026517048330422614397428693306169089796848259012545832716822645806652676995865268227280707578139185817888965220816434834482599326604336766017699961283186078838615027946595513115655203609398818061213855860030143569452722420634463179746059468257310379008402443243846565724501440282188525247093519062092902313649327349756551395872055965422874977401141334696271542284586237738753823048386568897646192738381490014076731044664025989949022222176590433990188601856652648506179970235619389701786004081188972991831102117122984590164192106888438712185564612496079872290851929681937238864261483965738229112312502418664935314397013742853192664987533721894069428143411852015801412334482801505139969429015348307764456909907315243327828826986460278986432113908350621709500259738986 3554277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994871701244516461260379029309120889086942028510640182154399457156805941872748998094254742173582401063677404595741785160829230135358081840096996372524230560855903700624271243416909004153690105933983835777939410970027753472000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

402387260077093773543702433923003985719374864210714632543799910429938512398629020592044208486969404800479988610197196058631666872994808558901323829669944590997424504087073759918823627727188732519779505950995276120874975462497043601418278094646496291056393887437886487337119181045825783647849977012476632889835955735432513185323958463075557409114262417474349347553428646576611667797396668820291207379143853719588249808126867838374559731746136085379534524221586593201928090878297308431392844403281231558611036976801357304216168747609675871348312025478589320767169132448426236131412508780208000261683151027341827977704784635868170164365024153691398281264810213092761244896359928705114964975419909342221566832572080821333186116811553615836546984046708975602900950537616475847728421889679646244945160765353408198901385442487984959953319101723355556602139450399736280750137837615307127761926849034352625200015888535147331611702103968175921510907788019393178114194545257223865541461062892187960223838971476088506276862967146674697562911234082439208160153780889893964518263243671616762179168909779911903754031274622289988005195444414282012187361745992642956581746628302955570299024324153181617210465832036786906117260158783520751516284225540265170483304226143974286933061690897968482590125458327168226458066526769958652682272807075781391858178889652208164348344825993266043367660176999612831860788386150279465955131156552036093988180612138558600301435694527224206344631797460594682573103790084024432438465657245014402821885252470935190620929023136493273497565513958720559654228749774011413346962715422845862377387538230483865688976461927383814900140767310446640259899490222221765904339901886018566526485061799702356193897017860040811889729918311021171229845901641921068884387121855646124960798722908519296819372388642614839657382291123125024186649353143970137428531926649875337218940694281434118520158014123344828015051399694290153483077644569099073152433278288269864602789864321139083506217095002597389863554277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994871701244516461260379029309120889086942028510640182154399457156805941872748998094254742173582401063677404595741785160829230135358081840096996372524230560855903700624271243416909004153690105933983835777939410970027753472000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

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