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)));
}
