Designare per fvnctiones

Expolitio gradatim

Ab envntiato ad main pavcorvm versvvm

Programmata partis 4 tota intra main scripta svnt. Dvm in vno visorio continentvr, id toleratvr; sed programma dvcentorvm versvvm sic scriptvm postvlat vt omnia simvl in capite habeantvr, et solvm tentamen totivs admittit. Expolitio gradatim, sive designatio descendens, ordinem laboris invertit: primvm scribitvr qvid programma faciat, lingva qvaestionis, vt series vocationvm fvnctionvm qvae nondvm exsistvnt; deinde singvlae earvm fvnctionvm vt nova qvaestio, minor, svmvntvr, et ratio repetitvr donec ad fvnctiones tam simplices perveniatvr vt earvm corpvs sine vlla decompositione scribatvr.

Qvaestio seqvens exemplo inservit. Integer $n \geq 1$ est perfectvs si svmma divisorvm eivs propriorvm (positivorvm qvi a $n$ differvnt) aeqvalis est $n$, deficiens si minor est et abvndans si maior: $28 = 1 + 2 + 4 + 7 + 14$ perfectvs est; 8, cvm $1 + 2 + 4 = 7$, deficiens est; 12, cvm $1 + 2 + 3 + 4 + 6 = 16$, abvndans. Programma qvaeritvr qvod $n$ legat et svmmam divisorvm eivs propriorvm speciemqve eivs scribat. Primvs gradvs designationis est main, et ante omnia scribitvr:

#include <cstdint>#include <iostream>#include <print>#include <string>​enum class Kind { deficient, perfect, abundant };​// Svmma divisorvm positivorvm ipsivs n.// Praecondicio: n >= 1.// Postcondicio: reddit svmmam omnivm divisorvm positivorvm ipsivs n, ipso n inclvso.std::int64_t divisorSum(int n);​// Species ipsivs n secvndvm svmmam divisorvm eivs propriorvm.// Praecondicio: n >= 1.// Postcondicio: reddit deficient, perfect vel abundant provt svmma minor, aeqvalis vel maior qvam n est.Kind classify(int n);​// Nomen speciei, ad id scribendvm.// Praecondicio: nvlla.// Postcondicio: reddit «deficiens», «perfectvs» vel «abvndans».std::string kindName(Kind kind);​int main(){    int n{};    std::cin >> n;    std::println("Svmma divisorvm propriorvm nvmeri {}: {}", n, divisorSum(n) - n);    std::println("{} est {}", n, kindName(classify(n)));}

/usr/bin/x86_64-linux-gnu-ld.bfd: main.o: in function `main':main.cpp:27:(.text+0x22a): undefined reference to `divisorSum(int)'/usr/bin/x86_64-linux-gnu-ld.bfd: main.cpp:28:(.text+0x3fd): undefined reference to `classify(int)'/usr/bin/x86_64-linux-gnu-ld.bfd: main.cpp:28:(.text+0x40d): undefined reference to `kindName[abi:cxx11](Kind)'collect2: error: ld returned 1 exit status

Programma sine vllo monito compilatvr, qvia qvaeqve vocatio cvm declaratione sva congrvit, qvod solvm compilator inspicit, vt lectio Fvnctiones docvit. Qvod deficit est conivnctio: conivnctor definitiones qvaerit neqve invenit (undefined reference, referentia indefinita; additamentvm [abi:cxx11] est nota interna GCC pro fvnctionibvs qvae std::string reddvnt). Hic statvs est prorsvs is designationis hoc loco: interfacies decretae svnt, corpora non. Si haec ita se habent, main iam vt specificatio programmatis legitvr, et tres declarationes svnt mvnera reliqva, singvla qvaestione originali minora.

Arbor decompositionis

Secvndvs gradvs in singvlas declarationes descendit. kindName est decisio trivm casvvm. classify nititvr divisorSum: nvmervs perfectvs est cvm svmma omnivm eivs divisorvm est $2n$. Et divisorSum est circvitvs divisorvm per paria $(d, n/d)$ lectionis Algorithmi nvmerici, qvi in $\sqrt{n}$ consistit. Eventvs vt arbor delineatvr: radix est qvaestio tota; filii cvivsqve nodi, qvaestiones minores qvae evm solvvnt; folia, operationes qvas lingva iam praebet.

Arbore perfecta, definitiones ab imo svrsvm scribvntvr, ita vt qvaeqve fvnctio aliis iam scriptis nitatvr:

#include <cstdint>#include <iostream>#include <print>#include <string>​enum class Kind { deficient, perfect, abundant };​// Svmma divisorvm positivorvm ipsivs n.// Praecondicio: n >= 1.// Postcondicio: reddit svmmam omnivm divisorvm positivorvm ipsivs n, ipso n inclvso.std::int64_t divisorSum(int n){    std::int64_t sum{0};    for (int d{1}; d <= n / d; ++d) {        if (n % d == 0) {            sum += d;            if (d != n / d) {                sum += n / d;            }        }    }    return sum;}​// Species ipsivs n secvndvm svmmam divisorvm eivs propriorvm.// Praecondicio: n >= 1.// Postcondicio: reddit deficient, perfect vel abundant provt svmma minor, aeqvalis vel maior qvam n est.Kind classify(int n){    const std::int64_t proper{divisorSum(n) - n};    if (proper < n) {        return Kind::deficient;    }    if (proper == n) {        return Kind::perfect;    }    return Kind::abundant;}​// Nomen speciei, ad id scribendvm.// Praecondicio: nvlla.// Postcondicio: reddit «deficiens», «perfectvs» vel «abvndans».std::string kindName(Kind kind){    if (kind == Kind::deficient) {        return "deficiens";    }    if (kind == Kind::perfect) {        return "perfectvs";    }    return "abvndans";}​int main(){    int n{};    std::cin >> n;    std::println("Svmma divisorvm propriorvm nvmeri {}: {}", n, divisorSum(n) - n);    std::println("{} est {}", n, kindName(classify(n)));}

28

Svmma divisorvm propriorvm nvmeri 28: 2828 est perfectvs

Tres notae eventvs methodo debentvr. Primo, main non mvtatvm est cvm ea qvae infra svnt scriberentvr. Secvndo, svmma vt std::int64_t redditvr (ex <cstdint>, integer 64 bitvvm qvo etiam lectio Recvrsio vtitvr), qvia pro $n$ prope maximvm ipsivs int svmma divisorvm eivs id excedere potest, et id in interfacie decernere omnes vocantes tvetvr. Tertio, qvaeqve fvnctio seorsvm confirmatvr: divisorSum(28) 56 dare debet, divisorSum(1) 1 dare debet et divisorSum(36) 91 dare debet, vbi par $(6, 6)$ semel tantvm nvmeratvr; vltima sectio lectionis has confirmationes in programma convertit.