adic 0.5.1

Arithmetic and rootfinding for p-adic numbers
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
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247

use num::Zero;

use crate::mapping::{Differentiable, IndexedMapping, Mapping};
use super::{euclid, Polynomial};


#[test]
fn new() {
    Polynomial::new(vec![1]);
    Polynomial::new(vec![1.0, 2.0]);
    assert_eq!(Polynomial::new(vec![0, 0, 4]), Polynomial::monomial(2, 4));
}

#[test]
fn degree() {
    assert_eq!(Polynomial::new(vec![-2, 0, 1]).degree(), Some(2));
    assert_eq!(Polynomial::new(vec![-1, 1]).degree(), Some(1));
    assert_eq!(Polynomial::new(vec![1]).degree(), Some(0));
    assert_eq!(Polynomial::<u32>::new(vec![]).degree(), None);
}

#[test]
fn display() {
    let expected = "1x^2 + 0x^1 + -1x^0".to_string();
    let actual = Polynomial::new(vec![-1, 0, 1]).to_string();
    assert_eq!(expected, actual);
}

#[test]
fn compose() {
    let f = Polynomial::new(vec![1, 1]);
    let g = Polynomial::new(vec![0, 0, 1]);
    assert_eq!(f.clone().compose(g.clone()).eval(2), Polynomial::new(vec![1, 0, 1]).eval(2));
    assert_eq!(g.compose(f.clone()).eval(2), Polynomial::new(vec![1, 2, 1]).eval(2));
    let f =|x| 1.0 / x;
    let g = Polynomial::new(vec![0.0, 0.0, 1.0]);
    assert_eq!(f.clone().compose(g.clone()).eval(2.0), Ok(0.25));
}

#[test]
fn eval_finite() {
    let f = Polynomial::new(vec![1, 1, 1, 1]);
    assert_eq!(f.eval_finite(3, 1), Ok(1));
    assert_eq!(f.eval_finite(3, 2), Ok(4));
    assert_eq!(f.eval_finite(3, 3), Ok(13));
    assert_eq!(f.eval_finite(3, 4), Ok(40));
    assert_eq!(f.eval_finite(3, 4), f.eval(3));
}

#[test]
fn ops() {

    let f0 = Polynomial::new(vec![6, 7, 1]);
    let f1 = Polynomial::new(vec![-6, -7, 1]);
    assert_eq!(f0.clone()+f1.clone(), Polynomial::new(vec![0, 0, 2]));
    assert_eq!(f0.clone()-f1.clone(), Polynomial::new(vec![12, 14]));
    assert_eq!(-f0.clone(), Polynomial::new(vec![-6, -7, -1]));
    assert_eq!(-f1.clone(), Polynomial::new(vec![6, 7, -1]));
    assert_eq!(f0.clone()+f1.clone(), -((-f0.clone()) + (-f1.clone())));
    assert_eq!(f0.clone()*f1.clone(), Polynomial::new(vec![-36, -84, -49, 0, 1]));

    assert_eq!(f0.derivative(), Polynomial::new(vec![7, 2]));
    assert_eq!(f1.derivative(), Polynomial::new(vec![-7, 2]));
    assert_eq!((f0.clone()+f1.clone()).derivative(), f0.derivative() + f1.derivative());
    assert_eq!((f0.clone()*f1.clone()).derivative(), f0.clone()*f1.derivative() + f0.derivative()*f1.clone());

}

#[test]
fn gcd() {

    // gcd(x^2 + 7 x + 6, x^2 - 5 x - 6) = 12 x + 12
    let f0 = Polynomial::new(vec![6, 7, 1]);
    let f1 = Polynomial::new(vec![-6, -5, 1]);
    let g = Polynomial::new(vec![1, 1]);
    assert_eq!(g, f0.gcd(&f1));
    assert_eq!(g, -f1.gcd(&f0));

    // gcd(a, 0) = a, gcd(0, b) = b
    let zero = Polynomial::zero();
    assert_eq!(f1, zero.gcd(&f1));
    assert_eq!(f0, f0.gcd(&zero));
    assert_eq!(zero, zero.gcd(&zero));

    // gcd(x^4 - 1, x - 1) = x - 1
    let f0 = Polynomial::new(vec![-1, 0, 0, 0, 1]);
    let f1 = Polynomial::new(vec![-1, 1]);
    assert_eq!(f1, f0.gcd(&f1));
    assert_eq!(f1, f1.gcd(&f0));

    // gcd(x^8 + x^6 - 3 x^4 - 3 x^3 + 8 x^2 + 2 x - 5, 3 x^6 + 5 x^4 - 4 x^2 - 9 x + 21) = const
    let f0 = Polynomial::new(vec![-5, 2, 8, -3, -3, 0, 1, 0, 1]);
    let f1 = Polynomial::new(vec![21, -9, -4, 0, 5, 0, 3]);
    let g = Polynomial::new(vec![1]);
    assert_eq!(g, -f0.gcd(&f1));
    assert_eq!(g, -f1.gcd(&f0));

    // gcd(81 x^4 - 16, 3 x^2 + 2 x) ~= 3 x + 2
    let f0 = Polynomial::new(vec![-16, 0, 0, 0, 81]);
    let f1 = Polynomial::new(vec![0, 2, 3]);
    let g = Polynomial::new(vec![2, 3]);
    assert_eq!(g, -f0.gcd(&f1));
    assert_eq!(g, -f1.gcd(&f0));
    let g = Polynomial::new(vec![2, 3]);
    assert_eq!(g, f0.gcd(&f1).primitive_monic());
    // gcd(81 x^4 - 16, 3 x + 2) ~= 3 x + 2
    let f0 = Polynomial::new(vec![-16, 0, 0, 0, 81]);
    let f1 = Polynomial::new(vec![2, 3]);
    let g = Polynomial::new(vec![2, 3]);
    assert_eq!(g, f0.gcd(&f1));
    assert_eq!(g, f1.gcd(&f0));
    // gcd(81 x^4 - 16, 3 x^2 - 2 x) ~= 3 x - 2
    let f1 = Polynomial::new(vec![0, -2, 3]);
    let g = Polynomial::new(vec![-2, 3]);
    assert_eq!(g, f0.gcd(&f1));
    assert_eq!(g, f1.gcd(&f0));
    let g = Polynomial::new(vec![-2, 3]);
    assert_eq!(g, f0.gcd(&f1).primitive_monic());
    // gcd(81 x^4 - 16, 3 x - 2) ~= 3 x - 2
    let f0 = Polynomial::new(vec![-16, 0, 0, 0, 81]);
    let f1 = Polynomial::new(vec![-2, 3]);
    let g = Polynomial::new(vec![-2, 3]);
    assert_eq!(g, f0.gcd(&f1));
    assert_eq!(g, f1.gcd(&f0));
    // gcd(81 x^4 - 16, 9 x^4 + 4 x^2) ~= 9 x^2 + 4
    let f1 = Polynomial::new(vec![0, 0, 4, 0, 9]);
    let g = Polynomial::new(vec![4, 0, 9]);
    assert_eq!(g, -f0.gcd(&f1));
    assert_eq!(g, f1.gcd(&f0));
    let g = Polynomial::new(vec![4, 0, 9]);
    assert_eq!(g, f0.gcd(&f1).primitive_monic());
    // gcd(81 x^4 - 16, 9 x^3 + 4 x) ~= 9 x^2 + 4
    let f1 = Polynomial::new(vec![0, 4, 0, 9]);
    let g = Polynomial::new(vec![4, 0, 9]);
    assert_eq!(g, -f0.gcd(&f1));
    assert_eq!(g, -f1.gcd(&f0));
    let g = Polynomial::new(vec![4, 0, 9]);
    assert_eq!(g, f0.gcd(&f1).primitive_monic());
    // gcd(81 x^4 - 16, 9 x^2 + 4) ~= 9 x^2 + 4
    let f1 = Polynomial::new(vec![4, 0, 9]);
    let g = Polynomial::new(vec![4, 0, 9]);
    assert_eq!(g, f0.gcd(&f1));
    assert_eq!(g, f1.gcd(&f0));

}

#[test]
fn square_free() {

    // (x + 1) (x + 2)^2 (x + 3)^3
    let f = Polynomial::new_big(vec![108, 324, 387, 238, 80, 14, 1]);
    let sff = f.raw_square_free_decomposition();
    assert_eq!(sff.len(), 3);
    assert_eq!(sff[0], Polynomial::new_big(vec![-1, -1]));
    assert_eq!(sff[1], Polynomial::new_big(vec![2, 1]));
    assert_eq!(sff[2], Polynomial::new_big(vec![3, 1]));

    // (x + 1) (x - 1) (x + 2)
    let f = Polynomial::new_big(vec![-2, -1, 1, 1]);
    let sff = f.raw_square_free_decomposition();
    assert_eq!(sff.len(), 1);
    assert_eq!(sff[0], f);

    // (x + 1) (x + 2)^3
    let f = Polynomial::new_big(vec![8, 20, 18, 7, 1]);
    let sff = f.raw_square_free_decomposition();
    assert_eq!(sff.len(), 3);
    assert_eq!(sff[0], Polynomial::new_big(vec![1, 1]));
    assert!(sff[1].is_constant());
    assert_eq!(sff[1], Polynomial::new_big(vec![1]));
    assert_eq!(sff[2], Polynomial::new_big(vec![2, 1]));

    // (x + 1) (x^2 + 1)^2
    let f = Polynomial::new_big(vec![1, 1, 2, 2, 1, 1]);
    let sff = f.raw_square_free_decomposition();
    assert_eq!(sff.len(), 2);
    assert_eq!(sff[0], Polynomial::new_big(vec![1, 1]));
    assert_eq!(sff[1], Polynomial::new_big(vec![1, 0, 1]));

    // (x - 1)^2 (x - 26)^2 = (x^2 - 27 x + 26)^2 = x^4 - 54 x^3 + 781 x^2 - 1404 x + 676
    let f = Polynomial::new_big(vec![676, -1404, 781, -54, 1]);
    let sff = f.raw_square_free_decomposition();
    assert_eq!(sff.len(), 2);
    assert!(sff[0].is_constant());
    assert_eq!(sff[1], Polynomial::new_big(vec![26, -27, 1]));

}

#[test]
fn subresultant() {

    use euclid::PolynomialSubresultant;

    // x^8 + x^6 - 3x4 - 3x^3 + 8x^2 + 2x - 5
    let a = Polynomial::new(vec![-5, 2, 8, -3, -3, 0, 1, 0, 1]);
    // 3x^6 + 5x^4 - 4x^2 - 9x + 21
    let b = Polynomial::new(vec![21, -9, -4, 0, 5, 0, 3]);

    // Non-primitive subresultants
    let mut subresultant = PolynomialSubresultant::new(a.clone(), b.clone());
    assert_eq!(Polynomial::new(vec![21, -9, -4, 0, 5, 0, 3]), subresultant.poly1);
    subresultant = subresultant.next();
    assert_eq!(Polynomial::new(vec![-3, 0, 1, 0, -5]), subresultant.poly1);
    subresultant = subresultant.next();
    assert_eq!(Polynomial::new(vec![441, -225, -117]), subresultant.poly1);
    subresultant = subresultant.next();
    assert_eq!(Polynomial::new(vec![224167500, -169966350]), subresultant.poly1);
    // The remaining steps cannot work because the coefficients break u32
    // (-117x^2 + -225x^1 + 441x^0) * -169966350
    // subresultant = subresultant.next();
    // assert_eq!(Polynomial::new(vec![1]), subresultant.poly1);
    // subresultant = subresultant.next();
    // assert!(subresultant.is_trivial());

    // Primitive subresultants
    let mut subresultant = PolynomialSubresultant::new(a.clone(), b.clone());
    assert_eq!(Polynomial::new(vec![21, -9, -4, 0, 5, 0, 3]), subresultant.poly1);
    subresultant = subresultant.next_primitive();
    assert_eq!(Polynomial::new(vec![3, 0, -1, 0, 5]), subresultant.poly1);
    subresultant = subresultant.next_primitive();
    assert_eq!(Polynomial::new(vec![-49, 25, 13]), subresultant.poly1);
    subresultant = subresultant.next_primitive();
    assert_eq!(Polynomial::new(vec![-6150, 4663]), subresultant.poly1);
    subresultant = subresultant.next_primitive();
    assert_eq!(Polynomial::new(vec![1]), subresultant.poly1);
    subresultant = subresultant.next_primitive();
    assert!(subresultant.is_trivial());

    // Primitive gcd
    let subresultant = PolynomialSubresultant::new(a.clone(), b.clone());
    assert_eq!(Polynomial::new(vec![1]), subresultant.into_primitive_gcd());

    // (x - 1)^2 (x - 26)^2 = (x^2 - 27 x + 26)^2 = x^4 - 54 x^3 + 781 x^2 - 1404 x + 676
    let a = Polynomial::new_big(vec![676, -1404, 781, -54, 1]);
    // f' = 4 x^3 - 162 x^2 + 1562 x - 1404
    let b = Polynomial::new_big(vec![-1404, 1562, -162, 4]);
    let mut subresultant = PolynomialSubresultant::new(a.clone(), b.clone());
    assert_eq!(Polynomial::new_big(vec![-65000, 67500, -2500]), subresultant.clone().into_gcd());
    assert_eq!(Polynomial::new_big(vec![-26, 27, -1]), subresultant.clone().into_factored_gcd());
    assert_eq!(Polynomial::new_big(vec![26, -27, 1]), subresultant.clone().into_primitive_gcd());
    assert_eq!(Polynomial::new_big(vec![-1404, 1562, -162, 4]), subresultant.poly1);
    subresultant = subresultant.next();
    assert_eq!(Polynomial::new_big(vec![-65000, 67500, -2500]), subresultant.poly1);
    subresultant = subresultant.next();
    assert!(subresultant.is_trivial());

}