public static BigDecimal MultiplyRound(BigDecimal x, Rational f) { if (f.CompareTo(BigInteger.Zero) == 0) return BigDecimal.Zero; /* Convert the rational value with two digits of extra precision */ var mc = new MathContext(2 + x.Precision); BigDecimal fbd = f.ToBigDecimal(mc); /* and the precision of the product is then dominated by the precision in x */ return MultiplyRound(x, fbd); }
public static BigDecimal Log(Rational r, MathContext mc) { /* the value is undefined if x is negative. */ if (r.CompareTo(Rational.Zero) <= 0) throw new ArithmeticException("Cannot take log of negative " + r); if (r.CompareTo(Rational.One) == 0) return BigDecimal.Zero; /* log(r+epsr) = log(r)+epsr/r. Convert the precision to an absolute error in the result. * eps contains the required absolute error of the result, epsr/r. */ double eps = PrecisionToError(System.Math.Log(r.ToDouble()), mc.Precision); /* Convert this further into a requirement of the relative precision in r, given that * epsr/r is also the relative precision of r. Add one safety digit. */ var mcloc = new MathContext(1 + ErrorToPrecision(eps)); BigDecimal resul = Log(r.ToBigDecimal(mcloc)); return resul.Round(mc); }
public static BigDecimal Zeta(int n, MathContext mc) { if (n <= 0) throw new NotSupportedException("Zeta at negative argument " + n + " not supported"); if (n == 1) throw new ArithmeticException("Pole at zeta(1) "); if (n%2 == 0) { /* Even indices. Abramowitz-Stegun 23.2.16. Start with 2^(n-1)*B(n)/n! */ Rational b = Bernoulli.Default[n].Abs(); b = b.Divide(Factorial.Default[n]); b = b.Multiply(BigInteger.One.ShiftLeft(n - 1)); /* to be multiplied by pi^n. Absolute error in the result of pi^n is n times * error in pi times pi^(n-1). Relative error is n*error(pi)/pi, requested by mc. * Need one more digit in pi if n=10, two digits if n=100 etc, and add one extra digit. */ var mcpi = new MathContext(mc.Precision + (int) (System.Math.Log10(10.0*n))); BigDecimal piton = PiRound(mcpi).Pow(n, mc); return MultiplyRound(piton, b); } if (n == 3) { /* Broadhurst BBP <a href="http://arxiv.org/abs/math/9803067">arXiv:math/9803067</a> * Error propagation: S31 is roughly 0.087, S33 roughly 0.131 */ int[] a31 = {1, -7, -1, 10, -1, -7, 1, 0}; int[] a33 = {1, 1, -1, -2, -1, 1, 1, 0}; BigDecimal S31 = BroadhurstBbp(3, 1, a31, mc); BigDecimal S33 = BroadhurstBbp(3, 3, a33, mc); S31 = S31.Multiply(new BigDecimal(48)); S33 = S33.Multiply(new BigDecimal(32)); return S31.Add(S33).Divide(new BigDecimal(7), mc); } if (n == 5) { /* Broadhurst BBP <a href=http://arxiv.org/abs/math/9803067">arXiv:math/9803067</a> * Error propagation: S51 is roughly -11.15, S53 roughly 22.165, S55 is roughly 0.031 * 9*2048*S51/6265 = -3.28. 7*2038*S53/61651= 5.07. 738*2048*S55/61651= 0.747. * The result is of the order 1.03, so we add 2 digits to S51 and S52 and one digit to S55. */ int[] a51 = {31, -1614, -31, -6212, -31, -1614, 31, 74552}; int[] a53 = {173, 284, -173, -457, -173, 284, 173, -111}; int[] a55 = {1, 0, -1, -1, -1, 0, 1, 1}; BigDecimal S51 = BroadhurstBbp(5, 1, a51, new MathContext(2 + mc.Precision)); BigDecimal S53 = BroadhurstBbp(5, 3, a53, new MathContext(2 + mc.Precision)); BigDecimal S55 = BroadhurstBbp(5, 5, a55, new MathContext(1 + mc.Precision)); S51 = S51.Multiply(new BigDecimal(18432)); S53 = S53.Multiply(new BigDecimal(14336)); S55 = S55.Multiply(new BigDecimal(1511424)); return S51.Add(S53).Subtract(S55).Divide(new BigDecimal(62651), mc); } /* Cohen et al Exp Math 1 (1) (1992) 25 */ var betsum = new Rational(); var bern = new Bernoulli(); var fact = new Factorial(); for (int npr = 0; npr <= (n + 1)/2; npr++) { Rational b = bern[2*npr].Multiply(bern[n + 1 - 2*npr]); b = b.Divide(fact[2*npr]).Divide(fact[n + 1 - 2*npr]); b = b.Multiply(1 - 2*npr); if (npr%2 == 0) betsum = betsum.Add(b); else betsum = betsum.Subtract(b); } betsum = betsum.Divide(n - 1); /* The first term, including the facor (2pi)^n, is essentially most * of the result, near one. The second term below is roughly in the range 0.003 to 0.009. * So the precision here is matching the precisionn requested by mc, and the precision * requested for 2*pi is in absolute terms adjusted. */ var mcloc = new MathContext(2 + mc.Precision + (int) (System.Math.Log10(n))); BigDecimal ftrm = PiRound(mcloc).Multiply(new BigDecimal(2)); ftrm = ftrm.Pow(n); ftrm = MultiplyRound(ftrm, betsum.ToBigDecimal(mcloc)); var exps = new BigDecimal(0); /* the basic accuracy of the accumulated terms before multiplication with 2 */ double eps = System.Math.Pow(10d, -mc.Precision); if (n%4 == 3) { /* since the argument n is at least 7 here, the drop * of the terms is at rather constant pace at least 10^-3, for example * 0.0018, 0.2e-7, 0.29e-11, 0.74e-15 etc for npr=1,2,3.... We want 2 times these terms * fall below eps/10. */ int kmax = mc.Precision/3; eps /= kmax; /* need an error of eps for 2/(exp(2pi)-1) = 0.0037 * The absolute error is 4*exp(2pi)*err(pi)/(exp(2pi)-1)^2=0.0075*err(pi) */ BigDecimal exp2p = PiRound(new MathContext(3 + ErrorToPrecision(3.14, eps/0.0075))); exp2p = Exp(exp2p.Multiply(new BigDecimal(2))); BigDecimal c = exp2p.Subtract(BigDecimal.One); exps = DivideRound(1, c); for (int npr = 2; npr <= kmax; npr++) { /* the error estimate above for npr=1 is the worst case of * the absolute error created by an error in 2pi. So we can * safely re-use the exp2p value computed above without * reassessment of its error. */ c = PowRound(exp2p, npr).Subtract(BigDecimal.One); c = MultiplyRound(c, (BigInteger.ValueOf(npr)).Pow(n)); c = DivideRound(1, c); exps = exps.Add(c); } } else { /* since the argument n is at least 9 here, the drop * of the terms is at rather constant pace at least 10^-3, for example * 0.0096, 0.5e-7, 0.3e-11, 0.6e-15 etc. We want these terms * fall below eps/10. */ int kmax = (1 + mc.Precision)/3; eps /= kmax; /* need an error of eps for 2/(exp(2pi)-1)*(1+4*Pi/8/(1-exp(-2pi)) = 0.0096 * at k=7 or = 0.00766 at k=13 for example. * The absolute error is 0.017*err(pi) at k=9, 0.013*err(pi) at k=13, 0.012 at k=17 */ BigDecimal twop = PiRound(new MathContext(3 + ErrorToPrecision(3.14, eps/0.017))); twop = twop.Multiply(new BigDecimal(2)); BigDecimal exp2p = Exp(twop); BigDecimal c = exp2p.Subtract(BigDecimal.One); exps = DivideRound(1, c); c = BigDecimal.One.Subtract(DivideRound(1, exp2p)); c = DivideRound(twop, c).Multiply(new BigDecimal(2)); c = DivideRound(c, n - 1).Add(BigDecimal.One); exps = MultiplyRound(exps, c); for (int npr = 2; npr <= kmax; npr++) { c = PowRound(exp2p, npr).Subtract(BigDecimal.One); c = MultiplyRound(c, (BigInteger.ValueOf(npr)).Pow(n)); BigDecimal d = DivideRound(1, exp2p.Pow(npr)); d = BigDecimal.One.Subtract(d); d = DivideRound(twop, d).Multiply(new BigDecimal(2*npr)); d = DivideRound(d, n - 1).Add(BigDecimal.One); d = DivideRound(d, c); exps = exps.Add(d); } } exps = exps.Multiply(new BigDecimal(2)); return ftrm.Subtract(exps, mc); }
private static BigDecimal BroadhurstBbp(int n, int p, int[] a, MathContext mc) { /* Explore the actual magnitude of the result first with a quick estimate. */ double x = 0.0; for (int k = 1; k < 10; k++) x += a[(k - 1)%8]/System.Math.Pow(2d, p*(k + 1)/2d)/System.Math.Pow(k, n); /* Convert the relative precision and estimate of the result into an absolute precision. */ double eps = PrecisionToError(x, mc.Precision); /* Divide this through the number of terms in the sum to account for error accumulation * The divisor 2^(p(k+1)/2) means that on the average each 8th term in k has shrunk by * relative to the 8th predecessor by 1/2^(4p). 1/2^(4pc) = 10^(-precision) with c the 8term * cycles yields c=log_2( 10^precision)/4p = 3.3*precision/4p with k=8c */ var kmax = (int) (6.6*mc.Precision/p); /* Now eps is the absolute error in each term */ eps /= kmax; BigDecimal res = BigDecimal.Zero; for (int c = 0;; c++) { var r = new Rational(); for (int k = 0; k < 8; k++) { var tmp = new Rational(BigInteger.ValueOf(a[k]), (BigInteger.ValueOf((1 + 8*c + k))).Pow(n)); /* floor( (pk+p)/2) */ int pk1h = p*(2 + 8*c + k)/2; tmp = tmp.Divide(BigInteger.One.ShiftLeft(pk1h)); r = r.Add(tmp); } if (System.Math.Abs(r.ToDouble()) < eps) break; var mcloc = new MathContext(1 + ErrorToPrecision(r.ToDouble(), eps)); res = res.Add(r.ToBigDecimal(mcloc)); } return res.Round(mc); }
public static BigDecimal PowRound(BigDecimal x, Rational q) { /** Special cases: x^1=x and x^0 = 1 */ if (q.CompareTo(BigInteger.One) == 0) return x; if (q.Sign == 0) return BigDecimal.One; if (q.IsInteger) { /* We are sure that the denominator is positive here, because normalize() has been * called during constrution etc. */ return PowRound(x, q.Numerator); } /* Refuse to operate on the general negative basis. The integer q have already been handled above. */ if (x.CompareTo(BigDecimal.Zero) < 0) throw new ArithmeticException("Cannot power negative " + x); if (q.IsIntegerFraction) { /* Newton method with first estimate in double precision. * The disadvantage of this first line here is that the result must fit in the * standard range of double precision numbers exponents. */ double estim = System.Math.Pow(x.ToDouble(), q.ToDouble()); var res = new BigDecimal(estim); /* The error in x^q is q*x^(q-1)*Delta(x). * The relative error is q*Delta(x)/x, q times the relative error of x. */ var reserr = new BigDecimal(0.5*q.Abs().ToDouble() *x.Ulp().Divide(x.Abs(), MathContext.Decimal64).ToDouble()); /* The main point in branching the cases above is that this conversion * will succeed for numerator and denominator of q. */ int qa = q.Numerator.ToInt32(); int qb = q.Denominator.ToInt32(); /* Newton iterations. */ BigDecimal xpowa = PowRound(x, qa); for (;;) { /* numerator and denominator of the Newton term. The major * disadvantage of this implementation is that the updates of the powers * of the new estimate are done in full precision calling BigDecimal.pow(), * which becomes slow if the denominator of q is large. */ BigDecimal nu = res.Pow(qb).Subtract(xpowa); BigDecimal de = MultiplyRound(res.Pow(qb - 1), q.Denominator); /* estimated correction */ BigDecimal eps = nu.Divide(de, MathContext.Decimal64); BigDecimal err = res.Multiply(reserr, MathContext.Decimal64); int precDiv = 2 + ErrorToPrecision(eps, err); if (precDiv <= 0) { /* The case when the precision is already reached and any precision * will do. */ eps = nu.Divide(de, MathContext.Decimal32); } else { eps = nu.Divide(de, new MathContext(precDiv)); } res = SubtractRound(res, eps); /* reached final precision if the relative error fell below reserr, * |eps/res| < reserr */ if (eps.Divide(res, MathContext.Decimal64).Abs().CompareTo(reserr) < 0) { /* delete the bits of extra precision kept in this * working copy. */ return res.Round(new MathContext(ErrorToPrecision(reserr.ToDouble()))); } } } /* The error in x^q is q*x^(q-1)*Delta(x) + Delta(q)*x^q*log(x). * The relative error is q/x*Delta(x) + Delta(q)*log(x). Convert q to a floating point * number such that its relative error becomes negligible: Delta(q)/q << Delta(x)/x/log(x) . */ int precq = 3 + ErrorToPrecision((x.Ulp().Divide(x, MathContext.Decimal64)).ToDouble() /System.Math.Log(x.ToDouble())); /* Perform the actual calculation as exponentiation of two floating point numbers. */ return Pow(x, q.ToBigDecimal(new MathContext(precq))); }