Faulhaber's formula: Difference between revisions
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=={{header|zkl}}== |
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{{libheader|GMP}} GNU Multiple Precision Arithmetic Library |
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Uses |
Uses code from the Bernoulli numbers task (copied here). |
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<lang zkl>var [const] BN=Import("zklBigNum"); // libGMP (GNU MP Bignum Library) |
<lang zkl>var [const] BN=Import("zklBigNum"); // libGMP (GNU MP Bignum Library) |
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Revision as of 11:22, 6 February 2019
In mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers as a (p + 1)th-degree polynomial function of n, the coefficients involving Bernoulli numbers.
- Task
Generate the first 10 closed-form expressions, starting with p = 0.
- See also
- Bernoulli numbers and binomial coefficients on Wikipedia.
C
<lang c>#include <stdbool.h>
- include <stdio.h>
- include <stdlib.h>
int binomial(int n, int k) {
int num, denom, i;
if (n < 0 || k < 0 || n < k) return -1; if (n == 0 || k == 0) return 1;
num = 1; for (i = k + 1; i <= n; ++i) { num = num * i; }
denom = 1; for (i = 2; i <= n - k; ++i) { denom *= i; }
return num / denom;
}
int gcd(int a, int b) {
int temp; while (b != 0) { temp = a % b; a = b; b = temp; } return a;
}
typedef struct tFrac {
int num, denom;
} Frac;
Frac makeFrac(int n, int d) {
Frac result; int g;
if (d == 0) { result.num = 0; result.denom = 0; return result; }
if (n == 0) { d = 1; } else if (d < 0) { n = -n; d = -d; }
g = abs(gcd(n, d)); if (g > 1) { n = n / g; d = d / g; }
result.num = n; result.denom = d; return result;
}
Frac negateFrac(Frac f) {
return makeFrac(-f.num, f.denom);
}
Frac subFrac(Frac lhs, Frac rhs) {
return makeFrac(lhs.num * rhs.denom - lhs.denom * rhs.num, rhs.denom * lhs.denom);
}
Frac multFrac(Frac lhs, Frac rhs) {
return makeFrac(lhs.num * rhs.num, lhs.denom * rhs.denom);
}
bool equalFrac(Frac lhs, Frac rhs) {
return (lhs.num == rhs.num) && (lhs.denom == rhs.denom);
}
bool lessFrac(Frac lhs, Frac rhs) {
return (lhs.num * rhs.denom) < (rhs.num * lhs.denom);
}
void printFrac(Frac f) {
printf("%d", f.num); if (f.denom != 1) { printf("/%d", f.denom); }
}
Frac bernoulli(int n) {
Frac a[16]; int j, m;
if (n < 0) { a[0].num = 0; a[0].denom = 0; return a[0]; }
for (m = 0; m <= n; ++m) { a[m] = makeFrac(1, m + 1); for (j = m; j >= 1; --j) { a[j - 1] = multFrac(subFrac(a[j - 1], a[j]), makeFrac(j, 1)); } }
if (n != 1) { return a[0]; }
return negateFrac(a[0]);
}
void faulhaber(int p) {
Frac coeff, q; int j, pwr, sign;
printf("%d : ", p); q = makeFrac(1, p + 1); sign = -1; for (j = 0; j <= p; ++j) { sign = -1 * sign; coeff = multFrac(multFrac(multFrac(q, makeFrac(sign, 1)), makeFrac(binomial(p + 1, j), 1)), bernoulli(j)); if (equalFrac(coeff, makeFrac(0, 1))) { continue; } if (j == 0) { if (!equalFrac(coeff, makeFrac(1, 1))) { if (equalFrac(coeff, makeFrac(-1, 1))) { printf("-"); } else { printFrac(coeff); } } } else { if (equalFrac(coeff, makeFrac(1, 1))) { printf(" + "); } else if (equalFrac(coeff, makeFrac(-1, 1))) { printf(" - "); } else if (lessFrac(makeFrac(0, 1), coeff)) { printf(" + "); printFrac(coeff); } else { printf(" - "); printFrac(negateFrac(coeff)); } } pwr = p + 1 - j; if (pwr > 1) { printf("n^%d", pwr); } else { printf("n"); } } printf("\n");
}
int main() {
int i;
for (i = 0; i < 10; ++i) { faulhaber(i); }
return 0;
}</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
C++
Uses C++17 <lang cpp>#include <iostream>
- include <numeric>
- include <sstream>
- include <vector>
class Frac { public: Frac(long n, long d) { if (d == 0) { throw new std::runtime_error("d must not be zero"); }
long nn = n; long dd = d; if (nn == 0) { dd = 1; } else if (dd < 0) { nn = -nn; dd = -dd; }
long g = abs(std::gcd(nn, dd)); if (g > 1) { nn /= g; dd /= g; }
num = nn; denom = dd; }
Frac operator-() const { return Frac(-num, denom); }
Frac operator+(const Frac& rhs) const { return Frac(num*rhs.denom + denom * rhs.num, rhs.denom*denom); }
Frac operator-(const Frac& rhs) const { return Frac(num*rhs.denom - denom * rhs.num, rhs.denom*denom); }
Frac operator*(const Frac& rhs) const { return Frac(num*rhs.num, denom*rhs.denom); }
bool operator==(const Frac& rhs) const { return num == rhs.num && denom == rhs.denom; }
bool operator!=(const Frac& rhs) const { return num != rhs.num || denom != rhs.denom; }
bool operator<(const Frac& rhs) const { if (denom == rhs.denom) { return num < rhs.num; } return num * rhs.denom < rhs.num * denom; }
friend std::ostream& operator<<(std::ostream&, const Frac&);
static Frac ZERO() { return Frac(0, 1); }
static Frac ONE() { return Frac(1, 1); }
private: long num; long denom; };
std::ostream & operator<<(std::ostream & os, const Frac &f) { if (f.num == 0 || f.denom == 1) { return os << f.num; }
std::stringstream ss; ss << f.num << "/" << f.denom; return os << ss.str(); }
Frac bernoulli(int n) { if (n < 0) { throw new std::runtime_error("n may not be negative or zero"); }
std::vector<Frac> a; for (int m = 0; m <= n; m++) { a.push_back(Frac(1, m + 1)); for (int j = m; j >= 1; j--) { a[j - 1] = (a[j - 1] - a[j]) * Frac(j, 1); } }
// returns 'first' Bernoulli number if (n != 1) return a[0]; return -a[0]; }
int binomial(int n, int k) { if (n < 0 || k < 0 || n < k) { throw new std::runtime_error("parameters are invalid"); } if (n == 0 || k == 0) return 1;
int num = 1; for (int i = k + 1; i <= n; i++) { num *= i; }
int denom = 1; for (int i = 2; i <= n - k; i++) { denom *= i; }
return num / denom; }
void faulhaber(int p) { using namespace std; cout << p << " : ";
auto q = Frac(1, p + 1); int sign = -1; for (int j = 0; j < p + 1; j++) { sign *= -1; auto coeff = q * Frac(sign, 1) * Frac(binomial(p + 1, j), 1) * bernoulli(j); if (coeff == Frac::ZERO()) { continue; } if (j == 0) { if (coeff == -Frac::ONE()) { cout << "-"; } else if (coeff != Frac::ONE()) { cout << coeff; } } else { if (coeff == Frac::ONE()) { cout << " + "; } else if (coeff == -Frac::ONE()) { cout << " - "; } else if (coeff < Frac::ZERO()) { cout << " - " << -coeff; } else { cout << " + " << coeff; } } int pwr = p + 1 - j; if (pwr > 1) { cout << "n^" << pwr; } else { cout << "n"; } } cout << endl; }
int main() { for (int i = 0; i < 10; i++) { faulhaber(i); }
return 0; }</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
C#
<lang csharp>using System;
namespace FaulhabersFormula {
internal class Frac { private long num; private long denom;
public static readonly Frac ZERO = new Frac(0, 1); public static readonly Frac ONE = new Frac(1, 1);
public Frac(long n, long d) { if (d == 0) { throw new ArgumentException("d must not be zero"); } long nn = n; long dd = d; if (nn == 0) { dd = 1; } else if (dd < 0) { nn = -nn; dd = -dd; } long g = Math.Abs(Gcd(nn, dd)); if (g > 1) { nn /= g; dd /= g; } num = nn; denom = dd; }
private static long Gcd(long a, long b) { if (b == 0) { return a; } return Gcd(b, a % b); }
public static Frac operator -(Frac self) { return new Frac(-self.num, self.denom); }
public static Frac operator +(Frac lhs, Frac rhs) { return new Frac(lhs.num * rhs.denom + lhs.denom * rhs.num, rhs.denom * lhs.denom); }
public static Frac operator -(Frac lhs, Frac rhs) { return lhs + -rhs; }
public static Frac operator *(Frac lhs, Frac rhs) { return new Frac(lhs.num * rhs.num, lhs.denom * rhs.denom); }
public static bool operator <(Frac lhs, Frac rhs) { double x = (double)lhs.num / lhs.denom; double y = (double)rhs.num / rhs.denom; return x < y; }
public static bool operator >(Frac lhs, Frac rhs) { double x = (double)lhs.num / lhs.denom; double y = (double)rhs.num / rhs.denom; return x > y; }
public static bool operator ==(Frac lhs, Frac rhs) { return lhs.num == rhs.num && lhs.denom == rhs.denom; }
public static bool operator !=(Frac lhs, Frac rhs) { return lhs.num != rhs.num || lhs.denom != rhs.denom; }
public override string ToString() { if (denom == 1) { return num.ToString(); } return string.Format("{0}/{1}", num, denom); }
public override bool Equals(object obj) { var frac = obj as Frac; return frac != null && num == frac.num && denom == frac.denom; }
public override int GetHashCode() { var hashCode = 1317992671; hashCode = hashCode * -1521134295 + num.GetHashCode(); hashCode = hashCode * -1521134295 + denom.GetHashCode(); return hashCode; } }
class Program { static Frac Bernoulli(int n) { if (n < 0) { throw new ArgumentException("n may not be negative or zero"); } Frac[] a = new Frac[n + 1]; for (int m = 0; m <= n; m++) { a[m] = new Frac(1, m + 1); for (int j = m; j >= 1; j--) { a[j - 1] = (a[j - 1] - a[j]) * new Frac(j, 1); } } // returns 'first' Bernoulli number if (n != 1) return a[0]; return -a[0]; }
static int Binomial(int n, int k) { if (n < 0 || k < 0 || n < k) { throw new ArgumentException(); } if (n == 0 || k == 0) return 1; int num = 1; for (int i = k + 1; i <= n; i++) { num = num * i; } int denom = 1; for (int i = 2; i <= n - k; i++) { denom = denom * i; } return num / denom; }
static void Faulhaber(int p) { Console.Write("{0} : ", p); Frac q = new Frac(1, p + 1); int sign = -1; for (int j = 0; j <= p; j++) { sign *= -1; Frac coeff = q * new Frac(sign, 1) * new Frac(Binomial(p + 1, j), 1) * Bernoulli(j); if (Frac.ZERO == coeff) continue; if (j == 0) { if (Frac.ONE != coeff) { if (-Frac.ONE == coeff) { Console.Write("-"); } else { Console.Write(coeff); } } } else { if (Frac.ONE == coeff) { Console.Write(" + "); } else if (-Frac.ONE == coeff) { Console.Write(" - "); } else if (Frac.ZERO < coeff) { Console.Write(" + {0}", coeff); } else { Console.Write(" - {0}", -coeff); } } int pwr = p + 1 - j; if (pwr > 1) { Console.Write("n^{0}", pwr); } else { Console.Write("n"); } } Console.WriteLine(); }
static void Main(string[] args) { for (int i = 0; i < 10; i++) { Faulhaber(i); } } }
}</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
D
<lang D>import std.algorithm : fold; import std.exception : enforce; import std.format : formattedWrite; import std.numeric : cmp, gcd; import std.range : iota; import std.stdio; import std.traits;
auto abs(T)(T val) if (isNumeric!T) {
if (val < 0) { return -val; } return val;
}
struct Frac {
long num; long denom;
enum ZERO = Frac(0, 1); enum ONE = Frac(1, 1);
this(long n, long d) in { enforce(d != 0, "Parameter d may not be zero."); } body { auto nn = n; auto dd = d; if (nn == 0) { dd = 1; } else if (dd < 0) { nn = -nn; dd = -dd; } auto g = gcd(abs(nn), abs(dd)); if (g > 1) { nn /= g; dd /= g; } num = nn; denom = dd; }
auto opBinary(string op)(Frac rhs) const { static if (op == "+" || op == "-") { return mixin("Frac(num*rhs.denom"~op~"denom*rhs.num, rhs.denom*denom)"); } else if (op == "*") { return Frac(num*rhs.num, denom*rhs.denom); } }
auto opUnary(string op : "-")() const { return Frac(-num, denom); }
int opCmp(Frac rhs) const { return cmp(cast(real) this, cast(real) rhs); }
bool opEquals(Frac rhs) const { return num == rhs.num && denom == rhs.denom; }
void toString(scope void delegate(const(char)[]) sink) const { if (denom == 1) { formattedWrite(sink, "%d", num); } else { formattedWrite(sink, "%d/%s", num, denom); } }
T opCast(T)() const if (isFloatingPoint!T) { return cast(T) num / denom; }
}
auto abs(Frac f) {
if (f.num >= 0) { return f; } return -f;
}
auto bernoulli(int n) in {
enforce(n >= 0, "Parameter n must not be negative.");
} body {
Frac[] a; a.length = n+1; a[0] = Frac.ZERO; foreach (m; 0..n+1) { a[m] = Frac(1, m+1); foreach_reverse (j; 1..m+1) { a[j-1] = (a[j-1] - a[j]) * Frac(j, 1); } } if (n != 1) { return a[0]; } return -a[0];
}
auto binomial(int n, int k) in {
enforce(n>=0 && k>=0 && n>=k);
} body {
if (n==0 || k==0) return 1; auto num = iota(k+1, n+1).fold!"a*b"(1); auto den = iota(2, n-k+1).fold!"a*b"(1); return num / den;
}
auto faulhaber(int p) {
write(p, " : "); auto q = Frac(1, p+1); auto sign = -1; foreach (j; 0..p+1) { sign *= -1; auto coeff = q * Frac(sign, 1) * Frac(binomial(p+1, j), 1) * bernoulli(j); if (coeff == Frac.ZERO) continue; if (j == 0) { if (coeff == -Frac.ONE) { write("-"); } else if (coeff != Frac.ONE) { write(coeff); } } else { if (coeff == Frac.ONE) { write(" + "); } else if (coeff == -Frac.ONE) { write(" - "); } else if (coeff > Frac.ZERO) { write(" + ", coeff); } else { write(" - ", -coeff); } } auto pwr = p + 1 - j; if (pwr > 1) { write("n^", pwr); } else { write("n"); } } writeln;
}
void main() {
foreach (i; 0..10) { faulhaber(i); }
}</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
EchoLisp
<lang scheme> (lib 'math) ;; for bernoulli numbers (string-delimiter "")
- returns list of polynomial coefficients
(define (Faulhaber p) (cons 0 (for/list ([k (in-range p -1 -1)]) (* (Cnp (1+ p) k) (bernoulli k)))))
- prints formal polynomial
(define (task (pmax 10))
(for ((p pmax)) (writeln p '→ (/ 1 (1+ p)) '* (poly->string 'n (Faulhaber p)))))
- extra credit - compute sums
(define (Faulcomp n p) (printf "Σ(1..%d) n^%d = %d" n p (/ (poly n (Faulhaber p)) (1+ p) ))) </lang>
- Output:
(task) 0 → 1 * n 1 → 1/2 * n^2 + n 2 → 1/3 * n^3 + 3/2 n^2 + 1/2 n 3 → 1/4 * n^4 + 2 n^3 + n^2 4 → 1/5 * n^5 + 5/2 n^4 + 5/3 n^3 -1/6 n 5 → 1/6 * n^6 + 3 n^5 + 5/2 n^4 -1/2 n^2 6 → 1/7 * n^7 + 7/2 n^6 + 7/2 n^5 -7/6 n^3 + 1/6 n 7 → 1/8 * n^8 + 4 n^7 + 14/3 n^6 -7/3 n^4 + 2/3 n^2 8 → 1/9 * n^9 + 9/2 n^8 + 6 n^7 -21/5 n^5 + 2 n^3 -3/10 n 9 → 1/10 * n^10 + 5 n^9 + 15/2 n^8 -7 n^6 + 5 n^4 -3/2 n^2 (Faulcomp 100 2) Σ(1..100) n^2 = 338350 (Faulcomp 100 1) Σ(1..100) n^1 = 5050 (lib 'bigint) (Faulcomp 100 9) Σ(1..100) n^9 = 10507499300049998000 ;; check it ... (for/sum ((n 101)) (expt n 9)) → 10507499300049998500
GAP
Straightforward implementation using GAP polynomials, and two different formulas: one based on Stirling numbers of the second kind (sum1, see Python implementation below in this page), and the usual Faulhaber formula (sum2). No optimization is made (one could compute Stirling numbers row by row, or the product in sum1 may be kept from one call to the other). Notice the Bernoulli term in the first formula is here only to correct the value of sum1(0), which is off by one because sum1 computes sums from 0 to n.
<lang gap>n := X(Rationals, "n"); sum1 := p -> Sum([0 .. p], k -> Stirling2(p, k) * Product([0 .. k], j -> n + 1 - j) / (k + 1)) + 2 * Bernoulli(2 * p + 1); sum2 := p -> Sum([0 .. p], j -> (-1)^j * Binomial(p + 1, j) * Bernoulli(j) * n^(p + 1 - j)) / (p + 1); ForAll([0 .. 20], k -> sum1(k) = sum2(k));
for p in [0 .. 9] do
Print(sum1(p), "\n");
od;
n 1/2*n^2+1/2*n 1/3*n^3+1/2*n^2+1/6*n 1/4*n^4+1/2*n^3+1/4*n^2 1/5*n^5+1/2*n^4+1/3*n^3-1/30*n 1/6*n^6+1/2*n^5+5/12*n^4-1/12*n^2 1/7*n^7+1/2*n^6+1/2*n^5-1/6*n^3+1/42*n 1/8*n^8+1/2*n^7+7/12*n^6-7/24*n^4+1/12*n^2 1/9*n^9+1/2*n^8+2/3*n^7-7/15*n^5+2/9*n^3-1/30*n 1/10*n^10+1/2*n^9+3/4*n^8-7/10*n^6+1/2*n^4-3/20*n^2</lang>
Go
<lang Go>package main
import ( "fmt" "math/big" )
func bernoulli(z *big.Rat, n int64) *big.Rat { if z == nil { z = new(big.Rat) } a := make([]big.Rat, n+1) for m := range a { a[m].SetFrac64(1, int64(m+1)) for j := m; j >= 1; j-- { d := &a[j-1] d.Mul(z.SetInt64(int64(j)), d.Sub(d, &a[j])) } } return z.Set(&a[0]) }
func main() { // allocate needed big.Rat's once q := new(big.Rat) c := new(big.Rat) // coefficients be := new(big.Rat) // for Bernoulli numbers bi := big.NewRat(1, 1) // for binomials
for p := int64(0); p < 10; p++ { fmt.Print(p, " : ") q.SetFrac64(1, p+1) neg := true for j := int64(0); j <= p; j++ { neg = !neg if neg { c.Neg(q) } else { c.Set(q) } bi.Num().Binomial(p+1, j) bernoulli(be, j) c.Mul(c, bi) c.Mul(c, be) if c.Num().BitLen() == 0 { continue } if j == 0 { fmt.Printf(" %4s", c.RatString()) } else { fmt.Printf(" %+2d/%-2d", c.Num(), c.Denom()) } fmt.Print("×n") if exp := p + 1 - j; exp > 1 { fmt.Printf("^%-2d", exp) } } fmt.Println() } }</lang>
- Output:
0 : 1×n 1 : 1/2×n^2 -1/2 ×n 2 : 1/3×n^3 -1/2 ×n^2 +1/6 ×n 3 : 1/4×n^4 -1/2 ×n^3 +1/4 ×n^2 4 : 1/5×n^5 -1/2 ×n^4 +1/3 ×n^3 -1/30×n 5 : 1/6×n^6 -1/2 ×n^5 +5/12×n^4 -1/12×n^2 6 : 1/7×n^7 -1/2 ×n^6 +1/2 ×n^5 -1/6 ×n^3 +1/42×n 7 : 1/8×n^8 -1/2 ×n^7 +7/12×n^6 -7/24×n^4 +1/12×n^2 8 : 1/9×n^9 -1/2 ×n^8 +2/3 ×n^7 -7/15×n^5 +2/9 ×n^3 -1/30×n 9 : 1/10×n^10 -1/2 ×n^9 +3/4 ×n^8 -7/10×n^6 +1/2 ×n^4 -3/20×n^2
Haskell
Bernouilli polynomials
<lang Haskell>import Data.Ratio ((%), numerator, denominator) import Data.List (intercalate, transpose) import Control.Arrow ((&&&), (***)) import Data.Char (isSpace) import Data.Monoid ((<>))
-- FAULHABER ------------------------------------------------------------------- faulhaber :: Rational faulhaber =
tail $ scanl (\rs n -> let xs = zipWith ((*) . (n %)) [2 ..] rs in 1 - sum xs : xs) [] [0 ..]
-- EXPRESSION STRINGS ---------------------------------------------------------- polynomials :: (String, String) polynomials = fmap ((ratioPower =<<) . reverse . flip zip [1 ..]) faulhaber
-- Rows of (Power string, Ratio string) tuples -> Printable lines expressionTable :: (String, String) -> [String] expressionTable ps =
let cols = transpose (fullTable ps) in expressionRow <$> zip [0 ..] (transpose $ zipWith (\(lw, rw) col -> (fmap (justifyLeft lw ' ' *** justifyLeft rw ' ') col)) (colWidths cols) cols)
-- Value pair -> String pair (lifted into list for use with >>=) ratioPower :: (Rational, Integer) -> [(String, String)] ratioPower (nd, j) =
let (num, den) = (numerator &&& denominator) nd sn | num == 0 = [] | (j /= 1) = ("n^" <> show j) | otherwise = "n" sr | num == 0 = [] | den == 1 && num == 1 = [] | den == 1 = show num <> "n" | otherwise = intercalate "/" [show num, show den] s = sr <> sn in if null s then [] else [(sn, sr)]
-- Rows of uneven length -> All rows padded to length of longest fullTable :: (String, String) -> (String, String) fullTable xs =
let lng = maximum $ length <$> xs in (<>) <*> (flip replicate ([], []) . (-) lng . length) <$> xs
justifyLeft :: Int -> Char -> String -> String justifyLeft n c s = take n (s <> replicate n c)
-- (Row index, Expression pairs) -> String joined by conjunctions expressionRow :: (Int, [(String, String)]) -> String expressionRow (i, row) =
concat [ show i , " -> " , foldr (\s a -> concat [ s , if blank a || head a == '-' then " " else " + " , a ]) "" (polyTerm <$> row) ]
-- (Power string, Ratio String) -> Combined string with possible '*' polyTerm :: (String, String) -> String polyTerm (l, r)
| blank l || blank r = l <> r | head r == '-' = concat ["- ", l, " * ", tail r] | otherwise = intercalate " * " [l, r]
blank :: String -> Bool blank = all isSpace
-- Maximum widths of power and ratio elements in each column colWidths :: (String, String) -> [(Int, Int)] colWidths =
fmap (foldr (\(ls, rs) (lMax, rMax) -> (max (length ls) lMax, max (length rs) rMax)) (0, 0))
-- Length of string excluding any leading '-' unsignedLength :: String -> Int unsignedLength xs =
let l = length xs in case l of 0 -> 0 _ -> case head xs of '-' -> l - 1 _ -> l
-- TEST ------------------------------------------------------------------------ main :: IO () main = (putStrLn . unlines . expressionTable . take 10) polynomials</lang>
- Output:
0 -> n 1 -> n^2 * 1/2 + n * 1/2 2 -> n^3 * 1/3 + n^2 * 1/2 + n * 1/6 3 -> n^4 * 1/4 + n^3 * 1/2 + n^2 * 1/4 4 -> n^5 * 1/5 + n^4 * 1/2 + n^3 * 1/3 - n * 1/30 5 -> n^6 * 1/6 + n^5 * 1/2 + n^4 * 5/12 - n^2 * 1/12 6 -> n^7 * 1/7 + n^6 * 1/2 + n^5 * 1/2 - n^3 * 1/6 + n * 1/42 7 -> n^8 * 1/8 + n^7 * 1/2 + n^6 * 7/12 - n^4 * 7/24 + n^2 * 1/12 8 -> n^9 * 1/9 + n^8 * 1/2 + n^7 * 2/3 - n^5 * 7/15 + n^3 * 2/9 - n * 1/30 9 -> n^10 * 1/10 + n^9 * 1/2 + n^8 * 3/4 - n^6 * 7/10 + n^4 * 1/2 - n^2 * 3/20
J
Implementation:
<lang J>Bsecond=:verb define"0
+/,(<:*(_1^[)*!*(y^~1+[)%1+])"0/~i.1x+y
)
Bfirst=: Bsecond - 1&=
Faul=:adverb define
(0,|.(%m+1x) * (_1x&^ * !&(m+1) * Bfirst) i.1+m)&p.
)</lang>
Task example:
<lang J> 0 Faul 0 1x&p.
1 Faul
0 1r2 1r2&p.
2 Faul
0 1r6 1r2 1r3&p.
3 Faul
0 0 1r4 1r2 1r4&p.
4 Faul
0 _1r30 0 1r3 1r2 1r5&p.
5 Faul
0 0 _1r12 0 5r12 1r2 1r6&p.
6 Faul
0 1r42 0 _1r6 0 1r2 1r2 1r7&p.
7 Faul
0 0 1r12 0 _7r24 0 7r12 1r2 1r8&p.
8 Faul
0 _1r30 0 2r9 0 _7r15 0 2r3 1r2 1r9&p.
9 Faul
0 0 _3r20 0 1r2 0 _7r10 0 3r4 1r2 1r10&p.</lang>
Double checking our work:
<lang J> Fcheck=: dyad def'+/(1+i.y)^x'"0
9 Faul i.5
0 1 513 20196 282340
9 Fcheck i.5
0 1 513 20196 282340
2 Faul i.5
0 1 5 14 30
2 Fcheck i.5
0 1 5 14 30</lang>
Java
<lang Java>import java.util.Arrays; import java.util.stream.IntStream;
public class FaulhabersFormula {
private static long gcd(long a, long b) { if (b == 0) { return a; } return gcd(b, a % b); }
private static class Frac implements Comparable<Frac> { private long num; private long denom;
public static final Frac ZERO = new Frac(0, 1); public static final Frac ONE = new Frac(1, 1);
public Frac(long n, long d) { if (d == 0) throw new IllegalArgumentException("d must not be zero"); long nn = n; long dd = d; if (nn == 0) { dd = 1; } else if (dd < 0) { nn = -nn; dd = -dd; } long g = Math.abs(gcd(nn, dd)); if (g > 1) { nn /= g; dd /= g; } num = nn; denom = dd; }
public Frac plus(Frac rhs) { return new Frac(num * rhs.denom + denom * rhs.num, rhs.denom * denom); }
public Frac unaryMinus() { return new Frac(-num, denom); }
public Frac minus(Frac rhs) { return this.plus(rhs.unaryMinus()); }
public Frac times(Frac rhs) { return new Frac(this.num * rhs.num, this.denom * rhs.denom); }
@Override public int compareTo(Frac o) { double diff = toDouble() - o.toDouble(); return Double.compare(diff, 0.0); }
@Override public boolean equals(Object obj) { return null != obj && obj instanceof Frac && this.compareTo((Frac) obj) == 0; }
@Override public String toString() { if (denom == 1) { return Long.toString(num); } return String.format("%d/%d", num, denom); }
private double toDouble() { return (double) num / denom; } }
private static Frac bernoulli(int n) { if (n < 0) throw new IllegalArgumentException("n may not be negative or zero"); Frac[] a = new Frac[n + 1]; Arrays.fill(a, Frac.ZERO); for (int m = 0; m <= n; ++m) { a[m] = new Frac(1, m + 1); for (int j = m; j >= 1; --j) { a[j - 1] = a[j - 1].minus(a[j]).times(new Frac(j, 1)); } } // returns 'first' Bernoulli number if (n != 1) return a[0]; return a[0].unaryMinus(); }
private static int binomial(int n, int k) { if (n < 0 || k < 0 || n < k) throw new IllegalArgumentException(); if (n == 0 || k == 0) return 1; int num = IntStream.rangeClosed(k + 1, n).reduce(1, (a, b) -> a * b); int den = IntStream.rangeClosed(2, n - k).reduce(1, (acc, i) -> acc * i); return num / den; }
private static void faulhaber(int p) { System.out.printf("%d : ", p); Frac q = new Frac(1, p + 1); int sign = -1; for (int j = 0; j <= p; ++j) { sign *= -1; Frac coeff = q.times(new Frac(sign, 1)).times(new Frac(binomial(p + 1, j), 1)).times(bernoulli(j)); if (Frac.ZERO.equals(coeff)) continue; if (j == 0) { if (!Frac.ONE.equals(coeff)) { if (Frac.ONE.unaryMinus().equals(coeff)) { System.out.print("-"); } else { System.out.print(coeff); } } } else { if (Frac.ONE.equals(coeff)) { System.out.print(" + "); } else if (Frac.ONE.unaryMinus().equals(coeff)) { System.out.print(" - "); } else if (coeff.compareTo(Frac.ZERO) > 0) { System.out.printf(" + %s", coeff); } else { System.out.printf(" - %s", coeff.unaryMinus()); } } int pwr = p + 1 - j; if (pwr > 1) System.out.printf("n^%d", pwr); else System.out.print("n"); } System.out.println(); }
public static void main(String[] args) { for (int i = 0; i <= 9; ++i) { faulhaber(i); } }
}</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
Julia
Module: <lang julia>module Faulhaber
function bernoulli(n::Integer)
n ≥ 0 || throw(DomainError(n, "n must be a positive-or-0 number")) a = fill(0 // 1, n + 1) for m in 1:n a[m] = 1 // (m + 1) for j in m:-1:2 a[j - 1] = (a[j - 1] - a[j]) * j end end return ifelse(n != 1, a[1], -a[1])
end
const _exponents = collect(Char, "⁰¹²³⁴⁵⁶⁷⁸⁹") toexponent(n) = join(_exponents[reverse(digits(n)) .+ 1])
function formula(p::Integer)
print(p, ": ") q = 1 // (p + 1) s = -1 for j in 0:p s *= -1 coeff = q * s * binomial(p + 1, j) * bernoulli(j) iszero(coeff) && continue if iszero(j) print(coeff == 1 ? "" : coeff == -1 ? "-" : "$coeff") else print(coeff == 1 ? " + " : coeff == -1 ? " - " : coeff > 0 ? " + $coeff " : " - $(-coeff) ") end pwr = p + 1 - j if pwr > 1 print("n", toexponent(pwr)) else print("n") end end println()
end
end # module Faulhaber</lang>
Main: <lang julia>Faulhaber.formula.(1:10)</lang>
- Output:
1: + 1//2 n 2: + 1//2 n² + 1//3 n 3: + 1//2 n³ + 1//2 n² - 1//6 n 4: + 1//2 n⁴ + 2//3 n³ - 1//3 n² + 1//30 n 5: + 1//2 n⁵ + 5//6 n⁴ - 5//9 n³ + 1//12 n² + 1//30 n 6: + 1//2 n⁶ + n⁵ - 5//6 n⁴ + 1//6 n³ + 1//10 n² - 1//42 n 7: + 1//2 n⁷ + 7//6 n⁶ - 7//6 n⁵ + 7//24 n⁴ + 7//30 n³ - 1//12 n² - 1//42 n 8: + 1//2 n⁸ + 4//3 n⁷ - 14//9 n⁶ + 7//15 n⁵ + 7//15 n⁴ - 2//9 n³ - 2//21 n² + 1//30 n 9: + 1//2 n⁹ + 3//2 n⁸ - 2//1 n⁷ + 7//10 n⁶ + 21//25 n⁵ - 1//2 n⁴ - 2//7 n³ + 3//20 n² + 1//30 n 10: + 1//2 n¹⁰ + 5//3 n⁹ - 5//2 n⁸ + n⁷ + 7//5 n⁶ - n⁵ - 5//7 n⁴ + 1//2 n³ + 1//6 n² - 5//66 n
Kotlin
As Kotlin doesn't have support for rational numbers built in, a cut-down version of the Frac class in the Arithmetic/Rational task has been used in order to express the polynomial coefficients as fractions. <lang scala>// version 1.1.2
fun gcd(a: Long, b: Long): Long = if (b == 0L) a else gcd(b, a % b)
class Frac : Comparable<Frac> {
val num: Long val denom: Long
companion object { val ZERO = Frac(0, 1) val ONE = Frac(1, 1) } constructor(n: Long, d: Long) { require(d != 0L) var nn = n var dd = d if (nn == 0L) { dd = 1 } else if (dd < 0) { nn = -nn dd = -dd } val g = Math.abs(gcd(nn, dd)) if (g > 1) { nn /= g dd /= g } num = nn denom = dd }
constructor(n: Int, d: Int) : this(n.toLong(), d.toLong()) operator fun plus(other: Frac) = Frac(num * other.denom + denom * other.num, other.denom * denom)
operator fun unaryMinus() = Frac(-num, denom)
operator fun minus(other: Frac) = this + (-other)
operator fun times(other: Frac) = Frac(this.num * other.num, this.denom * other.denom) fun abs() = if (num >= 0) this else -this
override fun compareTo(other: Frac): Int { val diff = this.toDouble() - other.toDouble() return when { diff < 0.0 -> -1 diff > 0.0 -> +1 else -> 0 } }
override fun equals(other: Any?): Boolean { if (other == null || other !is Frac) return false return this.compareTo(other) == 0 }
override fun toString() = if (denom == 1L) "$num" else "$num/$denom" fun toDouble() = num.toDouble() / denom
}
fun bernoulli(n: Int): Frac {
require(n >= 0) val a = Array<Frac>(n + 1) { Frac.ZERO } for (m in 0..n) { a[m] = Frac(1, m + 1) for (j in m downTo 1) a[j - 1] = (a[j - 1] - a[j]) * Frac(j, 1) } return if (n != 1) a[0] else -a[0] // returns 'first' Bernoulli number
}
fun binomial(n: Int, k: Int): Int {
require(n >= 0 && k >= 0 && n >= k) if (n == 0 || k == 0) return 1 val num = (k + 1..n).fold(1) { acc, i -> acc * i } val den = (2..n - k).fold(1) { acc, i -> acc * i } return num / den
}
fun faulhaber(p: Int) {
print("$p : ") val q = Frac(1, p + 1) var sign = -1 for (j in 0..p) { sign *= -1 val coeff = q * Frac(sign, 1) * Frac(binomial(p + 1, j), 1) * bernoulli(j) if (coeff == Frac.ZERO) continue if (j == 0) { print(when { coeff == Frac.ONE -> "" coeff == -Frac.ONE -> "-" else -> "$coeff" }) } else { print(when { coeff == Frac.ONE -> " + " coeff == -Frac.ONE -> " - " coeff > Frac.ZERO -> " + $coeff" else -> " - ${-coeff}" }) } val pwr = p + 1 - j if (pwr > 1) print("n^${p + 1 - j}") else print("n") } println()
}
fun main(args: Array<String>) {
for (i in 0..9) faulhaber(i)
}</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
Maxima
<lang maxima>sum1(p):=sum(stirling2(p,k)*pochhammer(n-k+1,k+1)/(k+1),k,0,p)$ sum2(p):=sum((-1)^j*binomial(p+1,j)*bern(j)*n^(p-j+1),j,0,p)/(p+1)$
makelist(expand(sum1(p)-sum2(p)),p,1,10); [0,0,0,0,0,0,0,0,0,0]
for p from 0 thru 9 do print(expand(sum2(p)));</lang>
- Output:
n n^2/2+n/2 n^3/3+n^2/2+n/6 n^4/4+n^3/2+n^2/4 n^5/5+n^4/2+n^3/3-n/30 n^6/6+n^5/2+(5*n^4)/12-n^2/12 n^7/7+n^6/2+n^5/2-n^3/6+n/42 n^8/8+n^7/2+(7*n^6)/12-(7*n^4)/24+n^2/12 n^9/9+n^8/2+(2*n^7)/3-(7*n^5)/15+(2*n^3)/9-n/30 n^10/10+n^9/2+(3*n^8)/4-(7*n^6)/10+n^4/2-(3*n^2)/20
Modula-2
<lang modula2>MODULE Faulhaber; FROM EXCEPTIONS IMPORT AllocateSource,ExceptionSource,GetMessage,RAISE; FROM FormatString IMPORT FormatString; FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
VAR TextWinExSrc : ExceptionSource;
(* Helper Functions *) PROCEDURE Abs(n : INTEGER) : INTEGER; BEGIN
IF n < 0 THEN RETURN -n END; RETURN n
END Abs;
PROCEDURE Binomial(n,k : INTEGER) : INTEGER; VAR i,num,denom : INTEGER; BEGIN
IF (n < 0) OR (k < 0) OR (n < k) THEN RAISE(TextWinExSrc, 0, "Argument Exception.") END; IF (n = 0) OR (k = 0) THEN RETURN 1 END; num := 1; FOR i:=k+1 TO n DO num := num * i END; denom := 1; FOR i:=2 TO n - k DO denom := denom * i END; RETURN num / denom
END Binomial;
PROCEDURE GCD(a,b : INTEGER) : INTEGER; BEGIN
IF b = 0 THEN RETURN a END; RETURN GCD(b, a MOD b)
END GCD;
PROCEDURE WriteInteger(n : INTEGER); VAR buf : ARRAY[0..15] OF CHAR; BEGIN
FormatString("%i", buf, n); WriteString(buf)
END WriteInteger;
(* Fraction Handling *) TYPE Frac = RECORD
num,denom : INTEGER;
END;
PROCEDURE InitFrac(n,d : INTEGER) : Frac; VAR nn,dd,g : INTEGER; BEGIN
IF d = 0 THEN RAISE(TextWinExSrc, 0, "The denominator must not be zero.") END; IF n = 0 THEN d := 1 ELSIF d < 0 THEN n := -n; d := -d END; g := Abs(GCD(n, d)); IF g > 1 THEN n := n / g; d := d / g END; RETURN Frac{n, d}
END InitFrac;
PROCEDURE EqualFrac(a,b : Frac) : BOOLEAN; BEGIN
RETURN (a.num = b.num) AND (a.denom = b.denom)
END EqualFrac;
PROCEDURE LessFrac(a,b : Frac) : BOOLEAN; BEGIN
RETURN a.num * b.denom < b.num * a.denom
END LessFrac;
PROCEDURE NegateFrac(f : Frac) : Frac; BEGIN
RETURN Frac{-f.num, f.denom}
END NegateFrac;
PROCEDURE SubFrac(lhs,rhs : Frac) : Frac; BEGIN
RETURN InitFrac(lhs.num * rhs.denom - lhs.denom * rhs.num, rhs.denom * lhs.denom)
END SubFrac;
PROCEDURE MultFrac(lhs,rhs : Frac) : Frac; BEGIN
RETURN InitFrac(lhs.num * rhs.num, lhs.denom * rhs.denom)
END MultFrac;
PROCEDURE Bernoulli(n : INTEGER) : Frac; VAR
a : ARRAY[0..15] OF Frac; i,j,m : INTEGER;
BEGIN
IF n < 0 THEN RAISE(TextWinExSrc, 0, "n may not be negative or zero.") END; FOR m:=0 TO n DO a[m] := Frac{1, m + 1}; FOR j:=m TO 1 BY -1 DO a[j-1] := MultFrac(SubFrac(a[j-1], a[j]), Frac{j, 1}) END END; IF n # 1 THEN RETURN a[0] END; RETURN NegateFrac(a[0])
END Bernoulli;
PROCEDURE WriteFrac(f : Frac); BEGIN
WriteInteger(f.num); IF f.denom # 1 THEN WriteString("/"); WriteInteger(f.denom) END
END WriteFrac;
(* Target *) PROCEDURE Faulhaber(p : INTEGER); VAR
j,pwr,sign : INTEGER; q,coeff : Frac;
BEGIN
WriteInteger(p); WriteString(" : "); q := InitFrac(1, p + 1); sign := -1; FOR j:=0 TO p DO sign := -1 * sign; coeff := MultFrac(MultFrac(MultFrac(q, Frac{sign, 1}), Frac{Binomial(p + 1, j), 1}), Bernoulli(j)); IF EqualFrac(coeff, Frac{0, 1}) THEN CONTINUE END; IF j = 0 THEN IF NOT EqualFrac(coeff, Frac{1, 1}) THEN IF EqualFrac(coeff, Frac{-1, 1}) THEN WriteString("-") ELSE WriteFrac(coeff) END END ELSE IF EqualFrac(coeff, Frac{1, 1}) THEN WriteString(" + ") ELSIF EqualFrac(coeff, Frac{-1, 1}) THEN WriteString(" - ") ELSIF LessFrac(Frac{0, 1}, coeff) THEN WriteString(" + "); WriteFrac(coeff) ELSE WriteString(" - "); WriteFrac(NegateFrac(coeff)) END END; pwr := p + 1 - j; IF pwr > 1 THEN WriteString("n^"); WriteInteger(pwr) ELSE WriteString("n") END END; WriteLn
END Faulhaber;
(* Main *) VAR i : INTEGER; BEGIN
FOR i:=0 TO 9 DO Faulhaber(i) END; ReadChar
END Faulhaber.</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
PARI/GP
PARI/GP has 2 built in functions: bernfrac(n) for Bernoulli numbers and bernpol(n) for Bernoulli polynomials. Using Bernoulli polynomials in PARI/GP is more simple, clear and much faster. (See version #2).
Version #1. Using Bernoulli numbers.
This version is using "Faulhaber's" formula based on Bernoulli numbers.
It's not worth using Bernoulli numbers in PARI/GP, because too much cleaning if you are avoiding "dirty" (but correct) result.
Note: Find ssubstr() function here on RC.
<lang parigp>
\\ Using "Faulhaber's" formula based on Bernoulli numbers. aev 2/7/17
\\ In str string replace all occurrences of the search string ssrch with the replacement string srepl. aev 3/8/16
sreplace(str,ssrch,srepl)={
my(sn=#str,ssn=#ssrch,srn=#srepl,sres="",Vi,Vs,Vz,vin,vin1,vi,L=List(),tok,zi,js=1); if(sn==0,return("")); if(ssn==0||ssn>sn,return(str)); \\ Vi - found ssrch indexes Vi=sfindalls(str,ssrch); vin=#Vi; if(vin==0,return(str)); vin1=vin+1; Vi=Vec(Vi,vin1); Vi[vin1]=sn+1; for(i=1,vin1, vi=Vi[i]; for(j=js,sn, \\print("ij:",i,"/",j,": ",sres); if(j!=vi, sres=concat(sres,ssubstr(str,j,1)), sres=concat(sres,srepl); js=j+ssn; break) ); \\fend j ); \\fend i return(sres);
} B(n)=(bernfrac(n)); Comb(n,k)={my(r=0); if(k<=n, r=n!/(n-k)!/k!); return(r)}; Faulhaber2(p)={
my(s="",s1="",s2="",c1=0,c2=0); for(j=0,p, c1=(-1)^j*Comb(p+1,j)*B(j); c2=(p+1-j); s2="*n"; if(c1==0, next); if(c2==1, s1="", s1=Str("^",c2)); s=Str(s,c1,s2,s1,"+") ); s=ssubstr(s,1,#s-1); s=sreplace(s,"1*n","n"); s=sreplace(s,"+-","-"); if(p==0, s="n", s=Str("(",s,")/",p+1)); print(p,": ",s);
} {\\ Testing: for(i=0,9, Faulhaber2(i))} </lang>
- Output:
0: n 1: (n^2+n)/2 2: (n^3+3/2*n^2+1/2*n)/3 3: (n^4+2*n^3+n^2)/4 4: (n^5+5/2*n^4+5/3*n^3-1/6*n)/5 5: (n^6+3*n^5+5/2*n^4-1/2*n^2)/6 6: (n^7+7/2*n^6+7/2*n^5-7/6*n^3+1/6*n)/7 7: (n^8+4*n^7+14/3*n^6-7/3*n^4+2/3*n^2)/8 8: (n^9+9/2*n^8+6*n^7-21/5*n^5+2*n^3-3/10*n)/9 9: (n^10+5*n^9+15/2*n^8-7*n^6+5*n^4-3/2*n^2)/10 time = 16 ms.
Version #2. Using Bernoulli polynomials.
This version is using the sums of pth powers formula from Bernoulli polynomials. It has small, simple and clear code, and produces instant result. <lang parigp> \\ Using a formula based on Bernoulli polynomials. aev 2/5/17 Faulhaber1(m)={
my(B,B1,B2,Bn); Bn=bernpol(m+1); x=n+1; B1=eval(Bn); x=0; B2=eval(Bn); Bn=(B1-B2)/(m+1); if(m==0, Bn=Bn-1); print(m,": ",Bn);
} {\\ Testing:
for(i=0,9, Faulhaber1(i))}
</lang>
- Output:
0: n 1: 1/2*n^2 + 1/2*n 2: 1/3*n^3 + 1/2*n^2 + 1/6*n 3: 1/4*n^4 + 1/2*n^3 + 1/4*n^2 4: 1/5*n^5 + 1/2*n^4 + 1/3*n^3 - 1/30*n 5: 1/6*n^6 + 1/2*n^5 + 5/12*n^4 - 1/12*n^2 6: 1/7*n^7 + 1/2*n^6 + 1/2*n^5 - 1/6*n^3 + 1/42*n 7: 1/8*n^8 + 1/2*n^7 + 7/12*n^6 - 7/24*n^4 + 1/12*n^2 8: 1/9*n^9 + 1/2*n^8 + 2/3*n^7 - 7/15*n^5 + 2/9*n^3 - 1/30*n 9: 1/10*n^10 + 1/2*n^9 + 3/4*n^8 - 7/10*n^6 + 1/2*n^4 - 3/20*n^2 > ## *** last result computed in 0 ms
Perl
<lang perl>use 5.014; use Math::Algebra::Symbols;
sub bernoulli_number {
my ($n) = @_;
return 0 if $n > 1 && $n % 2;
my @A; for my $m (0 .. $n) { $A[$m] = symbols(1) / ($m + 1);
for (my $j = $m ; $j > 0 ; $j--) { $A[$j - 1] = $j * ($A[$j - 1] - $A[$j]); } }
return $A[0];
}
sub binomial {
my ($n, $k) = @_; return 1 if $k == 0 || $n == $k; binomial($n - 1, $k - 1) + binomial($n - 1, $k);
}
sub faulhaber_s_formula {
my ($p) = @_;
my $formula = 0; for my $j (0 .. $p) { $formula += binomial($p + 1, $j) * bernoulli_number($j) * symbols('n')**($p + 1 - $j); }
(symbols(1) / ($p + 1) * $formula) =~ s/\$n/n/gr =~ s/\*\*/^/gr =~ s/\*/ /gr;
}
foreach my $i (0 .. 9) {
say "$i: ", faulhaber_s_formula($i);
}</lang>
- Output:
0: n 1: 1/2 n+1/2 n^2 2: 1/6 n+1/2 n^2+1/3 n^3 3: 1/4 n^2+1/2 n^3+1/4 n^4 4: -1/30 n+1/3 n^3+1/2 n^4+1/5 n^5 5: -1/12 n^2+5/12 n^4+1/2 n^5+1/6 n^6 6: 1/42 n-1/6 n^3+1/2 n^5+1/2 n^6+1/7 n^7 7: 1/12 n^2-7/24 n^4+7/12 n^6+1/2 n^7+1/8 n^8 8: -1/30 n+2/9 n^3-7/15 n^5+2/3 n^7+1/2 n^8+1/9 n^9 9: -3/20 n^2+1/2 n^4-7/10 n^6+3/4 n^8+1/2 n^9+1/10 n^10
Perl 6
<lang perl6>sub bernoulli_number($n) {
return 1/2 if $n == 1; return 0/1 if $n % 2;
my @A; for 0..$n -> $m { @A[$m] = 1 / ($m + 1); for $m, $m-1 ... 1 -> $j { @A[$j - 1] = $j * (@A[$j - 1] - @A[$j]); } }
return @A[0];
}
sub binomial($n, $k) {
$k == 0 || $n == $k ?? 1 !! binomial($n-1, $k-1) + binomial($n-1, $k);
}
sub faulhaber_s_formula($p) {
my @formula = gather for 0..$p -> $j { take '(' ~ join('/', (binomial($p+1, $j) * bernoulli_number($j)).Rat.nude) ~ ")*n^{$p+1 - $j}"; }
my $formula = join(' + ', @formula.grep({!m{'(0/1)*'}}));
$formula .= subst(rx{ '(1/1)*' }, , :g); $formula .= subst(rx{ '^1'» }, , :g);
"1/{$p+1} * ($formula)";
}
for 0..9 -> $p {
say "f($p) = ", faulhaber_s_formula($p);
}</lang>
- Output:
f(0) = 1/1 * (n) f(1) = 1/2 * (n^2 + n) f(2) = 1/3 * (n^3 + (3/2)*n^2 + (1/2)*n) f(3) = 1/4 * (n^4 + (2/1)*n^3 + n^2) f(4) = 1/5 * (n^5 + (5/2)*n^4 + (5/3)*n^3 + (-1/6)*n) f(5) = 1/6 * (n^6 + (3/1)*n^5 + (5/2)*n^4 + (-1/2)*n^2) f(6) = 1/7 * (n^7 + (7/2)*n^6 + (7/2)*n^5 + (-7/6)*n^3 + (1/6)*n) f(7) = 1/8 * (n^8 + (4/1)*n^7 + (14/3)*n^6 + (-7/3)*n^4 + (2/3)*n^2) f(8) = 1/9 * (n^9 + (9/2)*n^8 + (6/1)*n^7 + (-21/5)*n^5 + (2/1)*n^3 + (-3/10)*n) f(9) = 1/10 * (n^10 + (5/1)*n^9 + (15/2)*n^8 + (-7/1)*n^6 + (5/1)*n^4 + (-3/2)*n^2)
Phix
<lang Phix>include builtins\pfrac.e -- (0.8.0+)
function bernoulli(integer n)
sequence a = {} for m=0 to n do a = append(a,{1,m+1}) for j=m to 1 by -1 do a[j] = frac_mul({j,1},frac_sub(a[j+1],a[j])) end for end for if n!=1 then return a[1] end if return frac_uminus(a[1])
end function
function binomial(integer n, k)
if n<0 or k<0 or n<k then ?9/0 end if if n=0 or k=0 then return 1 end if integer num = 1, denom = 1 for i=k+1 to n do num *= i end for for i=2 to n-k do denom *= i end for return num / denom
end function
procedure faulhaber(integer p)
string res = sprintf("%d : ", p) frac q = {1, p+1} for j=0 to p do frac bj = bernoulli(j) if frac_ne(bj,frac_zero) then frac coeff = frac_mul({binomial(p+1,j),p+1},bj) string s = frac_sprint(coeff) if j=0 then if s="1" then s = "" end if else if s[1]='-' then s[1..1] = " - " else s[1..0] = " + " end if end if res &= s&"n" integer pwr = p+1-j if pwr>1 then res &= sprintf("^%d", pwr) end if end if end for printf(1,"%s\n",{res})
end procedure
for i=0 to 9 do
faulhaber(i)
end for</lang>
- Output:
0 : n 1 : 1/2n^2 + 1/2n 2 : 1/3n^3 + 1/2n^2 + 1/6n 3 : 1/4n^4 + 1/2n^3 + 1/4n^2 4 : 1/5n^5 + 1/2n^4 + 1/3n^3 - 1/30n 5 : 1/6n^6 + 1/2n^5 + 5/12n^4 - 1/12n^2 6 : 1/7n^7 + 1/2n^6 + 1/2n^5 - 1/6n^3 + 1/42n 7 : 1/8n^8 + 1/2n^7 + 7/12n^6 - 7/24n^4 + 1/12n^2 8 : 1/9n^9 + 1/2n^8 + 2/3n^7 - 7/15n^5 + 2/9n^3 - 1/30n 9 : 1/10n^10 + 1/2n^9 + 3/4n^8 - 7/10n^6 + 1/2n^4 - 3/20n^2
Python
The following implementation does not use Bernoulli numbers, but Stirling numbers of the second kind, based on the relation: .
Then summing: .
One has then to expand the product in order to get a polynomial in the variable . Also, for the sum of , the sum is too large by one (since we start at ), this has to be taken into account.
Note: a number of the formulae above are invisible to the majority of browsers, including Chrome, IE/Edge, Safari and Opera. They may (subject to the installation of necessary fronts) be visible to Firefox.
<lang python>from fractions import Fraction
def nextu(a):
n = len(a) a.append(1) for i in range(n - 1, 0, -1): a[i] = i * a[i] + a[i - 1] return a
def nextv(a):
n = len(a) - 1 b = [(1 - n) * x for x in a] b.append(1) for i in range(n): b[i + 1] += a[i] return b
def sumpol(n):
u = [0, 1] v = [[1], [1, 1]] yield [Fraction(0), Fraction(1)] for i in range(1, n): v.append(nextv(v[-1])) t = [0] * (i + 2) p = 1 for j, r in enumerate(u): r = Fraction(r, j + 1) for k, s in enumerate(v[j + 1]): t[k] += r * s yield t u = nextu(u)
def polstr(a):
s = "" q = False n = len(a) - 1 for i, x in enumerate(reversed(a)): i = n - i if i < 2: m = "n" if i == 1 else "" else: m = "n^%d" % i c = str(abs(x)) if i > 0: if c == "1": c = "" else: m = " " + m if x != 0: if q: t = " + " if x > 0 else " - " s += "%s%s%s" % (t, c, m) else: t = "" if x > 0 else "-" s = "%s%s%s" % (t, c, m) q = True if q: return s else: return "0"
for i, p in enumerate(sumpol(10)):
print(i, ":", polstr(p))</lang>
- Output:
0 : n 1 : 1/2 n^2 + 1/2 n 2 : 1/3 n^3 + 1/2 n^2 + 1/6 n 3 : 1/4 n^4 + 1/2 n^3 + 1/4 n^2 4 : 1/5 n^5 + 1/2 n^4 + 1/3 n^3 - 1/30 n 5 : 1/6 n^6 + 1/2 n^5 + 5/12 n^4 - 1/12 n^2 6 : 1/7 n^7 + 1/2 n^6 + 1/2 n^5 - 1/6 n^3 + 1/42 n 7 : 1/8 n^8 + 1/2 n^7 + 7/12 n^6 - 7/24 n^4 + 1/12 n^2 8 : 1/9 n^9 + 1/2 n^8 + 2/3 n^7 - 7/15 n^5 + 2/9 n^3 - 1/30 n 9 : 1/10 n^10 + 1/2 n^9 + 3/4 n^8 - 7/10 n^6 + 1/2 n^4 - 3/20 n^2
Racket
Racket will simplify rational numbers; if this code simplifies the expressions too much for your tastes (e.g. you like 1/1 * (n)
) then tweak the simplify... clauses to taste.
<lang racket>#lang racket/base
(require racket/match
racket/string math/number-theory)
(define simplify-arithmetic-expression
(letrec ((s-a-e (match-lambda [(list (and op '+) l ... (list '+ m ...) r ...) (s-a-e `(,op ,@l ,@m ,@r))] [(list (and op '+) l ... (? number? n1) m ... (? number? n2) r ...) (s-a-e `(,op ,@l ,(+ n1 n2) ,@m ,@r))] [(list (and op '+) (app s-a-e l _) ... 0 (app s-a-e r _) ...) (s-a-e `(,op ,@l ,@r))] [(list (and op '+) (app s-a-e x _)) (values x #t)] [(list (and op '*) l ... (list '* m ...) r ...) (s-a-e `(,op ,@l ,@m ,@r))] [(list (and op '*) l ... (? number? n1) m ... (? number? n2) r ...) (s-a-e `(,op ,@l ,(* n1 n2) ,@m ,@r))] [(list (and op '*) (app s-a-e l _) ... 1 (app s-a-e r _) ...) (s-a-e `(,op ,@l ,@r))] [(list (and op '*) (app s-a-e l _) ... 0 (app s-a-e r _) ...) (values 0 #t)] [(list (and op '*) (app s-a-e x _)) (values x #t)] [(list 'expt (app s-a-e x x-simplified?) 1) (values x x-simplified?)] [(list op (app s-a-e a #f) ...) (values `(,op ,@a) #f)] [(list op (app s-a-e a _) ...) (s-a-e `(,op ,@a))] [e (values e #f)]))) s-a-e))
(define (expression->infix-string e)
(define (parenthesise-maybe s p?) (if p? (string-append "(" s ")") s)) (letrec ((e->is (lambda (paren?) (match-lambda [(list (and op (or '+ '- '* '*)) (app (e->is #t) a p?) ...) (define bits (map parenthesise-maybe a p?)) (define compound (string-join bits (format " ~a " op))) (values (if paren? (string-append "(" compound ")") compound) #f)] [(list 'expt (app (e->is #t) x xp?) (app (e->is #t) n np?)) (values (format "~a^~a" (parenthesise-maybe x xp?) (parenthesise-maybe n np?)) #f)] [(? number? (app number->string s)) (values s #f)] [(? symbol? (app symbol->string s)) (values s #f)])))) (define-values (str needs-parens?) ((e->is #f) e)) str))
(define (faulhaber p)
(define p+1 (add1 p)) (define-values (simpler simplified?) (simplify-arithmetic-expression `(* ,(/ 1 p+1) (+ ,@(for/list ((j (in-range p+1))) `(* ,(* (expt -1 j) (binomial p+1 j)) (* ,(bernoulli-number j) (expt n ,(- p+1 j))))))))) simpler)
(for ((p (in-range 0 (add1 9))))
(printf "f(~a) = ~a~%" p (expression->infix-string (faulhaber p))))
</lang>
- Output:
f(0) = n f(1) = 1/2 * (n^2 + n) f(2) = 1/3 * (n^3 + (3/2 * n^2) + (1/2 * n)) f(3) = 1/4 * (n^4 + (2 * n^3) + n^2) f(4) = 1/5 * (n^5 + (5/2 * n^4) + (5/3 * n^3) + (-1/6 * n)) f(5) = 1/6 * (n^6 + (3 * n^5) + (5/2 * n^4) + (-1/2 * n^2)) f(6) = 1/7 * (n^7 + (7/2 * n^6) + (7/2 * n^5) + (-7/6 * n^3) + (1/6 * n)) f(7) = 1/8 * (n^8 + (4 * n^7) + (14/3 * n^6) + (-7/3 * n^4) + (2/3 * n^2)) f(8) = 1/9 * (n^9 + (9/2 * n^8) + (6 * n^7) + (-21/5 * n^5) + (2 * n^3) + (-3/10 * n)) f(9) = 1/10 * (n^10 + (5 * n^9) + (15/2 * n^8) + (-7 * n^6) + (5 * n^4) + (-3/2 * n^2))
Sidef
<lang ruby>func faulhaber_s_formula(p) {
var formula = gather { { |j| take "(#{binomial(p+1, j) * j.bernfrac -> as_rat})*n^#{p+1 - j}" } << 0..p }
formula.grep! { !.contains('(0)*') }.join!(' + ')
formula -= /\(1\)\*/g formula -= /\^1\b/g formula.gsub!(/\(([^+]*?)\)/, { _ })
"1/#{p + 1} * (#{formula})"
}
{ |p|
printf("%2d: %s\n", p, faulhaber_s_formula(p))
} << ^10</lang>
- Output:
0: 1/1 * (n) 1: 1/2 * (n^2 + n) 2: 1/3 * (n^3 + 3/2*n^2 + 1/2*n) 3: 1/4 * (n^4 + 2*n^3 + n^2) 4: 1/5 * (n^5 + 5/2*n^4 + 5/3*n^3 + -1/6*n) 5: 1/6 * (n^6 + 3*n^5 + 5/2*n^4 + -1/2*n^2) 6: 1/7 * (n^7 + 7/2*n^6 + 7/2*n^5 + -7/6*n^3 + 1/6*n) 7: 1/8 * (n^8 + 4*n^7 + 14/3*n^6 + -7/3*n^4 + 2/3*n^2) 8: 1/9 * (n^9 + 9/2*n^8 + 6*n^7 + -21/5*n^5 + 2*n^3 + -3/10*n) 9: 1/10 * (n^10 + 5*n^9 + 15/2*n^8 + -7*n^6 + 5*n^4 + -3/2*n^2)
By not simplifying the formulas, we can have a much cleaner code: <lang ruby>func faulhaber_s_formula(p) {
"1/#{p + 1} * (" + gather { { |j| take "#{binomial(p+1, j) * j.bernfrac -> as_rat}*n^#{p+1 - j}" } << 0..p }.join(' + ') + ")"
}
{ |p|
printf("%2d: %s\n", p, faulhaber_s_formula(p))
} << ^10</lang>
- Output:
0: 1/1 * (1*n^1) 1: 1/2 * (1*n^2 + 1*n^1) 2: 1/3 * (1*n^3 + 3/2*n^2 + 1/2*n^1) 3: 1/4 * (1*n^4 + 2*n^3 + 1*n^2 + 0*n^1) 4: 1/5 * (1*n^5 + 5/2*n^4 + 5/3*n^3 + 0*n^2 + -1/6*n^1) 5: 1/6 * (1*n^6 + 3*n^5 + 5/2*n^4 + 0*n^3 + -1/2*n^2 + 0*n^1) 6: 1/7 * (1*n^7 + 7/2*n^6 + 7/2*n^5 + 0*n^4 + -7/6*n^3 + 0*n^2 + 1/6*n^1) 7: 1/8 * (1*n^8 + 4*n^7 + 14/3*n^6 + 0*n^5 + -7/3*n^4 + 0*n^3 + 2/3*n^2 + 0*n^1) 8: 1/9 * (1*n^9 + 9/2*n^8 + 6*n^7 + 0*n^6 + -21/5*n^5 + 0*n^4 + 2*n^3 + 0*n^2 + -3/10*n^1) 9: 1/10 * (1*n^10 + 5*n^9 + 15/2*n^8 + 0*n^7 + -7*n^6 + 0*n^5 + 5*n^4 + 0*n^3 + -3/2*n^2 + 0*n^1)
zkl
GNU Multiple Precision Arithmetic Library
Uses code from the Bernoulli numbers task (copied here). <lang zkl>var [const] BN=Import("zklBigNum"); // libGMP (GNU MP Bignum Library)
fcn faulhaberFormula(p){ //-->(Rational,Rational...)
[p..0,-1].pump(List(),'wrap(k){ B(k)*BN(p+1).binomial(k) }) .apply('*(Rational(1,p+1)))
}</lang> <lang zkl>foreach p in (10){
println("F(%d) --> %s".fmt(p,polyRatString(faulhaberFormula(p))))
}</lang> <lang zkl>class Rational{ // Weenie Rational class, can handle BigInts
fcn init(_a,_b){ var a=_a, b=_b; normalize(); } fcn toString{ if(b==1) a.toString() else "%d/%d".fmt(a,b) } var [proxy] isZero=fcn{ a==0 }; fcn normalize{ // divide a and b by gcd g:= a.gcd(b); a/=g; b/=g; if(b<0){ a=-a; b=-b; } // denominator > 0 self } fcn __opAdd(n){ if(Rational.isChildOf(n)) self(a*n.b + b*n.a, b*n.b); // Rat + Rat else self(b*n + a, b); // Rat + Int } fcn __opSub(n){ self(a*n.b - b*n.a, b*n.b) } // Rat - Rat fcn __opMul(n){ if(Rational.isChildOf(n)) self(a*n.a, b*n.b); // Rat * Rat else self(a*n, b); // Rat * Int } fcn __opDiv(n){ self(a*n.b,b*n.a) } // Rat / Rat
}</lang> <lang zkl>fcn B(N){ // calculate Bernoulli(n) --> Rational
var A=List.createLong(100,0); // aka static aka not thread safe foreach m in (N+1){ A[m]=Rational(BN(1),BN(m+1)); foreach j in ([m..1, -1]){ A[j-1]= (A[j-1] - A[j])*j; } } A[0]
} fcn polyRatString(terms){ // (a1,a2...)-->"a1n + a2n^2 ..."
str:=[1..].zipWith('wrap(n,a){ if(a.isZero) "" else "+ %sn^%s ".fmt(a,n) }, terms) .pump(String) .replace(" 1n"," n").replace("n^1 ","n ").replace("+ -","- "); if(not str) return(" "); // all zeros if(str[0]=="+") str[1,*]; // leave leading space else String("-",str[2,*]);
}</lang>
- Output:
F(0) --> n F(1) --> 1/2n + 1/2n^2 F(2) --> 1/6n + 1/2n^2 + 1/3n^3 F(3) --> 1/4n^2 + 1/2n^3 + 1/4n^4 F(4) --> -1/30n + 1/3n^3 + 1/2n^4 + 1/5n^5 F(5) --> -1/12n^2 + 5/12n^4 + 1/2n^5 + 1/6n^6 F(6) --> 1/42n - 1/6n^3 + 1/2n^5 + 1/2n^6 + 1/7n^7 F(7) --> 1/12n^2 - 7/24n^4 + 7/12n^6 + 1/2n^7 + 1/8n^8 F(8) --> -1/30n + 2/9n^3 - 7/15n^5 + 2/3n^7 + 1/2n^8 + 1/9n^9 F(9) --> -3/20n^2 + 1/2n^4 - 7/10n^6 + 3/4n^8 + 1/2n^9 + 1/10n^10