static ScalarValue Psi0asymptotic(ScalarValue z)
{
/* coefficients in the asymptotic expansion for large z;
* let w = z^(-2) and write the expression in the form
*
* ln(z) - 1/(2z) - 1/12 w (1 + c1 w + c2 w + c3 w + ... )
*/
const double c1 = -0.1;
const double c2 = 1.0 / 21.0;
const double c3 = -0.05;
const double c4 = -1.0 / 12.0;
var zi = 1.0 / z;
var w = zi * zi;
/* Horner method evaluation of term in parentheses */
var sum = (w * (c3 / c2)) + 1.0;
sum = sum * (c2 / c1);
sum = ((sum * w) + 1.0) * c1 * w;
sum += 1.0;
/* correction added to log(z) */
var cs = sum * w;
cs = cs * c4;
cs = cs + (zi * (-0.5));
return(z.Log() + cs);
}
public static ScalarValue HurwitzZeta(ScalarValue s, ScalarValue q)
{
if (s.Re <= 1.0)
throw new YAMPArgumentRangeException("s", 1.0);
if (q.Re <= 0.0)
throw new YAMPArgumentRangeException("q", 0.0);
var max_bits = 54.0;
var ln_term0 = -s * q.Log();
var qabs = q.Abs();
var sabs = s.Abs();
var ss = s;
if ((sabs > max_bits && qabs < 1.0) || (sabs > 0.5 * max_bits && qabs < 0.25))
{
return q.Pow(-ss);
}
else if (sabs > 0.5 * max_bits && qabs < 1.0)
{
var p1 = q.Pow(-ss);
var p2 = (q / (1.0 + q)).Pow(ss);
var p3 = (q / (2.0 + q)).Pow(ss);
return p1 * (1.0 + p2 + p3);
}
/* Euler-Maclaurin summation formula
* [Moshier, p. 400, with several typo corrections]
*/
const int jmax = 12;
const int kmax = 10;
var pmax = (kmax + q).Pow(-ss);
var scp = s;
var pcp = pmax / (kmax + q);
var ans = pmax * ((kmax + q) / (s - 1.0) + 0.5);
for (var k = 0; k < kmax; k++)
{
ans += (k + q).Pow(-ss);
}
for (var j = 0; j <= jmax; j++)
{
var delta = COEFFICIENTS[j + 1] * scp * pcp;
ans += delta;
if ((delta / ans).Abs() < 0.5 * Double.Epsilon)
{
break;
}
scp *= (s + 2 * j + 1) * (s + 2 * j + 2);
pcp /= (kmax + q) * (kmax + q);
}
return ans;
}
public static ScalarValue Polylog(int n, ScalarValue z)
{
if (n == 2)
{
return(SpenceFunction.DiLog(z));
}
if (n == -1)
{
return(PolylogNeg1(z));
}
if (n == 0)
{
return(PolylogZero(z));
}
if (n == 1)
{
return(PolylogPos1(z));
}
if (n == -2)
{
return(PolylogNeg2(z));
}
if (n == -3)
{
return(PolylogNeg3(z));
}
if (n == -4)
{
return(PolylogNeg4(z));
}
if (n < -4)
{
return(PolylogNegative(n, z));
}
if (z == ScalarValue.One && n > 1)
{
return(PolylogZetaPositive(n));
}
if (-z == ScalarValue.One && n > 1)
{
return(PolylogZetaNegative(n));
}
/* This recurrence provides formulas for n < 2.
*
* d 1
* -- Li (x) = --- Li (x) .
* dx n x n-1
*
*/
var s = ScalarValue.Zero;
var ah = Math.Abs(z.Re) + Math.Abs(z.Im);
int i, j;
if (ah > 3.0)
{
return(PolylogInversion(n, z));
}
else if (ah >= 0.75)
{
var ad = 0.0;
var x = z.Log();
var h = -((-x).Log());
for (i = 1; i < n; i++)
{
h += 1.0 / i;
}
var p = ScalarValue.One;
s = PolylogZetaPositive(n);
for (j = 1; j <= n + 1; j++)
{
p = p * x / j;
if (j == n - 1)
{
s += h * p;
}
else
{
s += PolylogZetaPositive(n - j) * p;
}
}
j = n + 3;
x = x * x;
for (; ;)
{
p = p * x / ((j - 1) * j);
h = PolylogZetaPositive(n - j);
h = h * p;
s += h;
ah = Math.Abs(h.Re) + Math.Abs(h.Im);
ad = Math.Abs(s.Re) + Math.Abs(s.Im);
if (ah < ad * double.Epsilon)
{
break;
}
j += 2;
}
return(s);
}
else if (ah >= 1e-6)
{
var p = z * z * z;
var ad = 0.0;
var k = 3.0;
var h = ScalarValue.Zero;
do
{
p = p * z;
k += 1.0;
h = p / Math.Pow(k, n);
s += h;
ah = Math.Abs(h.Re) + Math.Abs(h.Im);
ad = Math.Abs(s.Re) + Math.Abs(s.Im);
}while (ah > ad * 1.1e-16);
}
s += z * z * z / Math.Pow(3.0, n);
s += z * z / Math.Pow(2.0, n);
return(s + z);
}
static ScalarValue Psi0asymptotic(ScalarValue z)
{
/* coefficients in the asymptotic expansion for large z;
* let w = z^(-2) and write the expression in the form
*
* ln(z) - 1/(2z) - 1/12 w (1 + c1 w + c2 w + c3 w + ... )
*/
const double c1 = -0.1;
const double c2 = 1.0 / 21.0;
const double c3 = -0.05;
const double c4 = -1.0 / 12.0;
var zi = 1.0 / z;
var w = zi * zi;
/* Horner method evaluation of term in parentheses */
var sum = (w * (c3 / c2)) + 1.0;
sum = sum * (c2 / c1);
sum = ((sum * w) + 1.0) * c1 * w;
sum += 1.0;
/* correction added to log(z) */
var cs = sum * w;
cs = cs * c4;
cs = cs + (zi * (-0.5));
return z.Log() + cs;
}
public static ScalarValue HurwitzZeta(ScalarValue s, ScalarValue q)
{
if (s.Re <= 1.0)
{
throw new YAMPArgumentRangeException("s", 1.0);
}
if (q.Re <= 0.0)
{
throw new YAMPArgumentRangeException("q", 0.0);
}
var max_bits = 54.0;
var ln_term0 = -s *q.Log();
var qabs = q.Abs();
var sabs = s.Abs();
var ss = s;
if ((sabs > max_bits && qabs < 1.0) || (sabs > 0.5 * max_bits && qabs < 0.25))
{
return(q.Pow(-ss));
}
else if (sabs > 0.5 * max_bits && qabs < 1.0)
{
var p1 = q.Pow(-ss);
var p2 = (q / (1.0 + q)).Pow(ss);
var p3 = (q / (2.0 + q)).Pow(ss);
return(p1 * (1.0 + p2 + p3));
}
/* Euler-Maclaurin summation formula
* [Moshier, p. 400, with several typo corrections]
*/
const int jmax = 12;
const int kmax = 10;
var pmax = (kmax + q).Pow(-ss);
var scp = s;
var pcp = pmax / (kmax + q);
var ans = pmax * ((kmax + q) / (s - 1.0) + 0.5);
for (var k = 0; k < kmax; k++)
{
ans += (k + q).Pow(-ss);
}
for (var j = 0; j <= jmax; j++)
{
var delta = COEFFICIENTS[j + 1] * scp * pcp;
ans += delta;
if ((delta / ans).Abs() < 0.5 * Double.Epsilon)
{
break;
}
scp *= (s + 2 * j + 1) * (s + 2 * j + 2);
pcp /= (kmax + q) * (kmax + q);
}
return(ans);
}
protected override ScalarValue GetValue(ScalarValue value)
{
return value.Log(2.0);
}
protected override ScalarValue GetValue(ScalarValue value)
{
return(value.Log(2.0));
}