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/* zeta.c
*
* Riemann zeta function of two arguments
*
*
*
* SYNOPSIS:
*
* double x, q, y, zeta();
*
* y = zeta( x, q );
*
*
*
* DESCRIPTION:
*
*
*
* inf.
* - -x
* zeta(x,q) = > (k+q)
* -
* k=0
*
* where x > 1 and q is not a negative integer or zero.
* The Euler-Maclaurin summation formula is used to obtain
* the expansion
*
* n
* - -x
* zeta(x,q) = > (k+q)
* -
* k=1
*
* 1-x inf. B x(x+1)...(x+2j)
* (n+q) 1 - 2j
* + --------- - ------- + > --------------------
* x-1 x - x+2j+1
* 2(n+q) j=1 (2j)! (n+q)
*
* where the B2j are Bernoulli numbers. Note that (see zetac.c)
* zeta(x,1) = zetac(x) + 1.
*
*
*
* ACCURACY:
*
*
*
* REFERENCE:
*
* Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals,
* Series, and Products, p. 1073; Academic Press, 1980.
*
*/
/*
* Cephes Math Library Release 2.0: April, 1987
* Copyright 1984, 1987 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#![allow(clippy::excessive_precision)]
use crate::cephes64::consts::MACHEP;
/* Expansion coefficients
* for Euler-Maclaurin summation formula
* (2k)! / B2k
* where B2k are Bernoulli numbers
*/
const A: [f64; 12] = [
12.0,
-720.0,
30240.0,
-1209600.0,
47900160.0,
-1.8924375803183791606e9, /*1.307674368e12/691 */
7.47242496e10,
-2.950130727918164224e12, /*1.067062284288e16/3617 */
1.1646782814350067249e14, /*5.109094217170944e18/43867 */
-4.5979787224074726105e15, /*8.028576626982912e20/174611 */
1.8152105401943546773e17, /*1.5511210043330985984e23/854513 */
-7.1661652561756670113e18 /*1.6938241367317436694528e27/236364091 */
];
/* 30 Nov 86 -- error in third coefficient fixed */
// $$\mathrm{zeta}(x, q) = \sum_{k = 0}^{\infty}{(k + q)^{-x}}$$
//
// $$\mathrm{zeta}(x, q) = \sum_{k = 1}^{n}{(k + q)^{-x}} +
// \frac{(n + q)^{1 - x}}{x - 1}-\frac{1}{2\,(n + q)^x} +
// \sum_{j=1}^{\infty}{\frac{B_{2j}\,x\,(x + 1)\,...\,(x + 2\,j)}
// {(2\,j)!\,(n + q)^{x + 2\,j + 1}}}$$
pub fn zeta(x: f64, q: f64) -> f64 {
//! Riemann zeta function of two arguments
//!
//! ## DESCRIPTION:
//!
#![doc=include_str!("zeta.svg")]
//!
//! where x > 1 and q is not a negative integer or zero.
//! The Euler-Maclaurin summation formula is used to obtain
//! the expansion
//!
#![doc=include_str!("zeta2.svg")]
//!
//! where the B2j are Bernoulli numbers. Note that (see zetac.c)
//! zeta(x,1) = zetac(x) + 1.
//!
//! ## ACCURACY:
//!
//! ## REFERENCE:
//!
//! Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals,
//! Series, and Products, p. 1073; Academic Press, 1980.
if x == 1.0 {
return f64::INFINITY;
} else if x < 1.0 {
//sf_error("zeta", SF_ERROR_DOMAIN, NULL);
return f64::NAN;
} else if q <= 0.0 {
if q == q.floor() {
//sf_error("zeta", SF_ERROR_SINGULAR, NULL);
return f64::INFINITY;
} else if x != x.floor() {
return f64::NAN; /* because q^-x not defined */
}
}
/* Asymptotic expansion
* https://dlmf.nist.gov/25.11#E43
*/
if q > 1e8 {
return (1.0 / (x - 1.0) + 1.0 / (2.0 * q)) * q.powf(1.0 - x);
}
/* Euler-Maclaurin summation formula */
/* Permit negative q but continue sum until n+q > +9 .
* This case should be handled by a reflection formula.
* If q<0 and x is an integer, there is a relation to
* the polyGamma function.
*/
let mut s = q.powf(-x);
let mut a = q;
let mut i = 0;
let mut b = 0.0;
while (i < 9) || (a <= 9.0) {
i += 1;
a += 1.0;
b = a.powf(-x);
s += b;
if (b / s).abs() < MACHEP {
return s;
}
}
let w = a;
s += b * w / (x - 1.0);
s -= 0.5 * b;
a = 1.0;
let mut k = 0.0;
for a_i in &A {
a *= x + k;
b /= w;
let t = a * b / a_i;
s += t;
if (t / s).abs() < MACHEP {
return s;
}
k += 1.0;
a *= x + k;
b /= w;
k += 1.0;
}
// TODO: Find a way to test this case
s
}
#[cfg(test)]
mod zeta_tests {
use super::*;
#[test]
fn zeta_trivials() {
assert_eq!(zeta(1.0, 1.0), f64::INFINITY);
assert_eq!(zeta(1.0 - 1e-10, 1.0).is_nan(), true);
assert_eq!(zeta(3.0, 0.0), f64::INFINITY);
assert_eq!(zeta(3.0, -10.0), f64::INFINITY);
assert_eq!(zeta(3.1, -10.5).is_nan(), true);
}
#[test]
fn zeta_large_q() {
assert_eq!(zeta(1.5, 1.1e8), 0.00019069251828251056);
assert_eq!(zeta(10.0, 1.1e8), 4.712195952465921e-74);
assert_eq!(zeta(1e10, 1.1e10), 0.0);
assert_eq!(zeta(10.0, 1.1e15), 4.712195759694296e-137);
assert_eq!(zeta(10.0, 1.1e25), 4.712195759694275e-227);
assert_eq!(zeta(10.0, 1.1e35), 4.7121956e-317);
}
#[test]
fn zeta_neg_q() {
assert_eq!(zeta(25.0, -0.9), 1.0000000000000055e+25);
assert_eq!(zeta(20.0, -1.5), 2097152.0006014686);
assert_eq!(zeta(100.0, -20.1), 9.99999999998579e+99);
assert_eq!(zeta(10.0, -1e5 + 0.1), 9999999997.435017);
assert_eq!(zeta(100.0, -1e5 + 0.1), 9.999999941792339e+99);
}
#[test]
fn zeta_nominal() {
assert_eq!(zeta(1.0 + 1e-15, 1e-20), 1.0000000000000511e+20);
assert_eq!(zeta(1.001, 0.002), 1503.6909814191445);
assert_eq!(zeta(1.0 + 1e-10, 1e-15), 1000000003453880.9);
assert_eq!(zeta(1.01, 0.01), 205.27439873109253);
assert_eq!(zeta(1.01, 1e5), 89.12509826963665);
assert_eq!(zeta(1.5, 0.9e7), 0.0006666666851851857);
assert_eq!(zeta(25.0, 0.9e7), 5.2235903489005884e-169);
assert_eq!(zeta(25.0, 0.01), 9.999999999999995e+49);
assert_eq!(zeta(100.0, 0.01), 9.99999999999998e+199);
}
}