clausen 1.0.1

Rust implementation of Clausen functions.
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227

use crate::general::{co, si};
use crate::range_reduction::{range_reduce_even, range_reduce_odd};
use num::Complex;
use polylog::Li;

/// Standard Clausen function Cl_n(x) for a real argument
pub fn cln(n: i32, x: f64) -> f64 {
    if n < 1 {
        cln_neg(n, x)
    } else {
        cln_pos(n, x)
    }
}


/// Standard Clausen function Cl_n(x) for a real argument and n < 1
fn cln_neg(n: i32, x: f64) -> f64 {
    let eix = Complex::cis(x); // e^(i*x)

    if is_even(n) {
        if x == 0.0 {
            x
        } else {
            eix.li(n).im
        }
    } else {
        eix.li(n).re
    }
}


/// Standard Clausen function Cl_n(x) for a real argument and n > 0
fn cln_pos(n: i32, x: f64) -> f64 {
    let (r, sgn) = range_reduce(n, x);

    if is_even(n) && r == 0.0 {
        r
    } else if is_even(n) && r == core::f64::consts::PI {
        0.0
    } else if n < 10 {
        sgn*cln_pos_zeta(n, r)
    } else {
        sgn*cln_pos_series(n, r)
    }
}


/// Series expansion of Cl_n(x) in terms of the zeta function from
/// [Jiming Wu, Xiaoping Zhang, Dongjie Liu, "An efficient calculation
/// of the Clausen functions Cl_n(θ)(n >= 2)", Bit Numer Math 50,
/// 193-206 (2010), https://doi.org/10.1007/s10543-009-0246-8].
fn cln_pos_zeta(n: i32, x: f64) -> f64 {
    // first line in Eq.(2.13)
    let term1 = if x == 0.0 {
        0.0
    } else {
        let sgn = if is_even((n + 1)/2) { 1.0 } else { -1.0 };
        sgn*x.powi(n - 1)*inv_fac(n - 1)*(2.0*(0.5*x).sin()).ln()
    };

    // second line in Eq.(2.13)
    let sgn = if is_even(n/2) { 1.0 } else { -1.0 };
    let term2 = pcal(n, x) - sgn*inv_fac(n - 2)*nsum(n, x);

    term1 + term2
}


/// returns 1/n!
fn inv_fac(n: i32) -> f64 {
    [1.0, 1.0, 0.5, 1.0/6.0, 1.0/24.0, 1.0/120.0, 1.0/720.0, 1.0/5040.0, 1.0/40320.0][n as usize]
}


/// returns P_n(x)
fn pcal(n: i32, x: f64) -> f64 {
    let mut sum = 0.0;
    let x2 = x*x;
    let fl = (n - 1)/2;

    for i in (3..=n).step_by(2) {
        let sgn = if is_even(fl + (i - 1)/2) { 1.0 } else { -1.0 };
        sum = x2*sum + sgn*zeta(i)*inv_fac(n - i);
    }

    if is_even(n) {
        sum *= x;
    }

    sum
}


/// returns zeta(n) for n = 2,...,9
fn zeta(n: i32) -> f64 {
    [1.6449340668482264, 1.2020569031595943, 1.0823232337111382,
     1.0369277551433699, 1.0173430619844491, 1.0083492773819228,
     1.0040773561979443, 1.0020083928260822][(n - 2) as usize]
}


/// returns sum in Eq.(2.13), n > 1
fn nsum(n: i32, x: f64) -> f64 {
    let mut sum = 0.0;
    let mut xn = 1.0;

    for i in 0..=(n - 3) {
        sum += binomial(n - 2, i)*xn*ncal(n - 2 - i, x);
        xn *= -x;
    }

    sum + xn*ncal(0, x)
}


/// returns N_n(x) from Eq.(2.11)
fn ncal(n: i32, x: f64) -> f64 {
    let mut sum = 0.0;
    let xn1 = x.powi(n + 1);
    let x2 = x*x;
    let mut xn = xn1*x2; // x^(n + 3)

    for (k, bn) in B.iter().enumerate() {
        let old_sum = sum;
        sum += bn*xn/((2*(k as i32) + n + 3) as f64);
        if sum == old_sum { break; }
        xn *= x2;
    }

    (xn1/((n + 1) as f64) + sum)/((n + 1) as f64)
}


/// returns the binomial coefficient (n, k)
fn binomial(n: i32, k: i32) -> f64 {
    [[1.0, 0.0,  0.0,  0.0,   0.0,   0.0,  0.0,  0.0],
     [1.0, 1.0,  0.0,  0.0,   0.0,   0.0,  0.0,  0.0],
     [1.0, 2.0,  1.0,  0.0,   0.0,   0.0,  0.0,  0.0],
     [1.0, 3.0,  3.0,  1.0,   0.0,   0.0,  0.0,  0.0],
     [1.0, 4.0,  6.0,  4.0,   1.0,   0.0,  0.0,  0.0],
     [1.0, 5.0, 10.0, 10.0,   5.0,   1.0,  0.0,  0.0],
     [1.0, 6.0, 15.0, 20.0,  15.0,   6.0,  1.0,  0.0],
     [1.0, 7.0, 21.0, 35.0,  35.0,  21.0,  7.0,  1.0]
    ][n as usize][k as usize]
}


/// Series expansion of Cl_n(x) in terms of sin(x) and cos(x)
fn cln_pos_series(n: i32, x: f64) -> f64 {
    if is_even(n) {
        si(n, x)
    } else {
        co(n, x)
    }
}


/// map x to [0,pi] using the symmetries of Cl_n(x)
fn range_reduce(n: i32, x: f64) -> (f64, f64) {
    if is_even(n) {
        range_reduce_even(x)
    } else {
        (range_reduce_odd(x), 1.0)
    }
}


fn is_even(n: i32) -> bool {
    n % 2 == 0
}


// (-1)^k B_{2k}/(2k)! where B_{2k} are the even Bernoulli numbers
// B(k) = 2*(-1)^(2*k + 1)*SpecialFunctions.zeta(2*k)/(2*pi)^(2*k)
const B: [f64; 202] = [
    -8.3333333333333333e-002,-1.3888888888888889e-003,-3.3068783068783069e-005,-8.2671957671957672e-007,
    -2.0876756987868099e-008,-5.2841901386874932e-010,-1.3382536530684679e-011,-3.3896802963225829e-013,
    -8.5860620562778446e-015,-2.1748686985580619e-016,-5.5090028283602295e-018,-1.3954464685812523e-019,
    -3.5347070396294675e-021,-8.9535174270375469e-023,-2.2679524523376831e-024,-5.7447906688722024e-026,
    -1.4551724756148649e-027,-3.6859949406653102e-029,-9.3367342570950447e-031,-2.3650224157006299e-032,
    -5.9906717624821343e-034,-1.5174548844682903e-035,-3.8437581254541882e-037,-9.7363530726466910e-039,
    -2.4662470442006810e-040,-6.2470767418207437e-042,-1.5824030244644914e-043,-4.0082736859489360e-045,
    -1.0153075855569556e-046,-2.5718041582418717e-048,-6.5144560352338149e-050,-1.6501309906896525e-051,
    -4.1798306285394759e-053,-1.0587634667702909e-054,-2.6818791912607707e-056,-6.7932793511074212e-058,
    -1.7207577616681405e-059,-4.3587303293488938e-061,-1.1040792903684667e-062,-2.7966655133781345e-064,
    -7.0840365016794702e-066,-1.7944074082892241e-067,-4.5452870636110961e-069,-1.1513346631982052e-070,
    -2.9163647710923614e-072,-7.3872382634973376e-074,-1.8712093117637953e-075,-4.7398285577617994e-077,
    -1.2006125993354507e-078,-3.0411872415142924e-080,-7.7034172747051063e-082,-1.9512983909098831e-083,
    -4.9426965651594615e-085,-1.2519996659171848e-086,-3.1713522017635155e-088,-8.0331289707353345e-090,
    -2.0348153391661466e-091,-5.1542474664474739e-093,-1.3055861352149467e-094,-3.3070883141750912e-096,
    -8.3769525600490913e-098,-2.1219068717497138e-099,-5.3748528956122803e-101,-1.3614661432172069e-102,
    -3.4486340279933990e-104,-8.7354920416383551e-106,-2.2127259833925497e-107,-5.6049003928372242e-109,
    -1.4197378549991788e-110,-3.5962379982587627e-112,-9.1093772660782319e-114,-2.3074322171091233e-115,
    -5.8447940852990020e-117,-1.4805036371705745e-118,-3.7501595226227197e-120,-9.4992650419929583e-122,
    -2.4061919444675199e-123,-6.0949553971026848e-125,-1.5438702377042471e-126,-3.9106689968592923e-128,
    -9.9058402898794298e-130,-2.5091786578563553e-131,-6.3558237896024598e-133,-1.6099489734616250e-134,
    -4.0780483898724622e-136,-1.0329817245315190e-137,-2.6165732752609202e-139,-6.6278575334086228e-141,
    -1.6788559257443673e-142,-4.2525917390343006e-144,-1.0771940713664573e-145,-2.7285644580839580e-147,
    -6.9115345134370137e-149,-1.7507121442157682e-150,-4.4346056667239196e-152,-1.1232987378487150e-153,
    -2.8453489425693971e-155,-7.2073530684151368e-157,-1.8256438595501418e-158,-4.6244099189746555e-160,
    -1.1713767165946985e-161,-2.9671318854112523e-163,-7.5158328663197353e-165,-1.9037827051837494e-166,
    -4.8223379271757316e-168,-1.2215124667619569e-169,-3.0941272241548313e-171,-7.8375158172837117e-173,
    -1.9852659485568249e-174,-5.0287373938151486e-176,-1.2737940624195893e-177,-3.2265580530233667e-179,
    -8.1729670255761088e-181,-2.0702367322529201e-182,-5.2439709032927841e-184,-1.3283133472690102e-185,
    -3.3646570148302941e-187,-8.5227757823275058e-189,-2.1588443254591854e-190,-5.4684165588767235e-192,
    -1.3851660959868732e-193,-3.5086667096656512e-195,-8.8875566007447618e-197,-2.2512443861893281e-198,
    -5.7024686469217722e-200,-1.4444521824735857e-201,-3.6588401210745441e-203,-9.2679502956336812e-205,
    -2.3475992347298971e-206,-5.9465383295169881e-208,-1.5062757553029781e-209,-3.8154410604763518e-211,
    -9.6646251091260105e-213,-2.4480781387902631e-214,-6.2010543667790175e-216,-1.5707454206813435e-217,
    -3.9787446306053875e-219,-1.0078277884588346e-220,-2.5528576108572180e-222,-6.4664638700600961e-224,
    -1.6379744332372530e-225,-4.1490377087871461e-227,-1.0509635290775169e-228,-2.6621217182765621e-230,
    -6.7432330873938832e-232,-1.7080808949773099e-233,-4.3266194508991167e-235,-1.0959455098376500e-236,
    -2.7760624066064009e-238,-7.0318482225589324e-240,-1.7811879627573501e-241,-4.5118018169014740e-243,
    -1.1428527511202687e-244,-2.8948798368101921e-246,-7.3328162891986569e-248,-1.8574240646335573e-249,
    -4.7049101188608530e-251,-1.1917676554344843e-252,-3.0187827368818927e-254,-7.6466660014982318e-256,
    -1.9369231254735576e-257,-4.9062835924299293e-259,-1.2427761521749539e-260,-3.1479887685209103e-262,
    -7.9739487029830971e-264,-2.0198248022238287e-265,-5.1162759927867278e-267,-1.2959678485750701e-268,
    -3.2827249095010017e-270,-8.3152393350706909e-272,-2.1062747292467202e-273,-5.3352562160805552e-275,
    -1.3514361871210552e-276,-3.4232278524048296e-278,-8.6711374470768819e-280,-2.1964247741580717e-281,
    -5.5636089474762561e-283,-1.4092786097034883e-284,-3.5697444204246398e-286,-9.0422682494513886e-288,
    -2.2904333046148606e-289,-5.8017353369352214e-291,-1.4695967287946349e-292,-3.7225320009595016e-294,
    -9.4292837120924187e-296,-2.3884654665215509e-297,-6.0500537039203004e-299,-1.5324965059522865e-300,
    -3.8818589977708149e-302,-9.8328637096699497e-304,-2.4906934741438690e-305,-6.3090002722625804e-307,
    -1.5980884379636898e-308,-4.0480053024903931e-310,-1.0253717215969654e-311,-2.5972969126396544e-313,
    -6.5790299364809836e-315,-1.6664877509565689e-316,-4.2212627863094242e-318,-1.0692583549355590e-319,
    -2.7084630535382438e-321,-6.8606170609008831e-323
];