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
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
|
/**
* \file
* \brief Polynomial in canonical (monomial) basis
*//*
* Authors:
* MenTaLguY <mental@rydia.net>
* Krzysztof Kosiński <tweenk.pl@gmail.com>
*
* Copyright 2007-2015 Authors
*
* This library is free software; you can redistribute it and/or
* modify it either under the terms of the GNU Lesser General Public
* License version 2.1 as published by the Free Software Foundation
* (the "LGPL") or, at your option, under the terms of the Mozilla
* Public License Version 1.1 (the "MPL"). If you do not alter this
* notice, a recipient may use your version of this file under either
* the MPL or the LGPL.
*
* You should have received a copy of the LGPL along with this library
* in the file COPYING-LGPL-2.1; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
* You should have received a copy of the MPL along with this library
* in the file COPYING-MPL-1.1
*
* The contents of this file are subject to the Mozilla Public License
* Version 1.1 (the "License"); you may not use this file except in
* compliance with the License. You may obtain a copy of the License at
* http://www.mozilla.org/MPL/
*
* This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
* OF ANY KIND, either express or implied. See the LGPL or the MPL for
* the specific language governing rights and limitations.
*/
#ifndef LIB2GEOM_SEEN_POLY_H
#define LIB2GEOM_SEEN_POLY_H
#include <assert.h>
#include <vector>
#include <iostream>
#include <algorithm>
#include <complex>
#include <2geom/coord.h>
#include <2geom/utils.h>
namespace Geom {
/** @brief Polynomial in canonical (monomial) basis.
* @ingroup Fragments */
class Poly : public std::vector<double>{
public:
// coeff; // sum x^i*coeff[i]
//unsigned size() const { return coeff.size();}
unsigned degree() const { return size()-1;}
//double operator[](const int i) const { return (*this)[i];}
//double& operator[](const int i) { return (*this)[i];}
Poly operator+(const Poly& p) const {
Poly result;
const unsigned out_size = std::max(size(), p.size());
const unsigned min_size = std::min(size(), p.size());
result.reserve(out_size);
for(unsigned i = 0; i < min_size; i++) {
result.push_back((*this)[i] + p[i]);
}
for(unsigned i = min_size; i < size(); i++)
result.push_back((*this)[i]);
for(unsigned i = min_size; i < p.size(); i++)
result.push_back(p[i]);
assert(result.size() == out_size);
return result;
}
Poly operator-(const Poly& p) const {
Poly result;
const unsigned out_size = std::max(size(), p.size());
const unsigned min_size = std::min(size(), p.size());
result.reserve(out_size);
for(unsigned i = 0; i < min_size; i++) {
result.push_back((*this)[i] - p[i]);
}
for(unsigned i = min_size; i < size(); i++)
result.push_back((*this)[i]);
for(unsigned i = min_size; i < p.size(); i++)
result.push_back(-p[i]);
assert(result.size() == out_size);
return result;
}
Poly operator-=(const Poly& p) {
const unsigned out_size = std::max(size(), p.size());
const unsigned min_size = std::min(size(), p.size());
resize(out_size);
for(unsigned i = 0; i < min_size; i++) {
(*this)[i] -= p[i];
}
for(unsigned i = min_size; i < out_size; i++)
(*this)[i] = -p[i];
return *this;
}
Poly operator-(const double k) const {
Poly result;
const unsigned out_size = size();
result.reserve(out_size);
for(unsigned i = 0; i < out_size; i++) {
result.push_back((*this)[i]);
}
result[0] -= k;
return result;
}
Poly operator-() const {
Poly result;
result.resize(size());
for(unsigned i = 0; i < size(); i++) {
result[i] = -(*this)[i];
}
return result;
}
Poly operator*(const double p) const {
Poly result;
const unsigned out_size = size();
result.reserve(out_size);
for(unsigned i = 0; i < out_size; i++) {
result.push_back((*this)[i]*p);
}
assert(result.size() == out_size);
return result;
}
// equivalent to multiply by x^terms, negative terms are disallowed
Poly shifted(unsigned const terms) const {
Poly result;
size_type const out_size = size() + terms;
result.reserve(out_size);
result.resize(terms, 0.0);
result.insert(result.end(), this->begin(), this->end());
assert(result.size() == out_size);
return result;
}
Poly operator*(const Poly& p) const;
template <typename T>
T eval(T x) const {
T r = 0;
for(int k = size()-1; k >= 0; k--) {
r = r*x + T((*this)[k]);
}
return r;
}
template <typename T>
T operator()(T t) const { return (T)eval(t);}
void normalize();
void monicify();
Poly() {}
Poly(const Poly& p) : std::vector<double>(p) {}
Poly(const double a) {push_back(a);}
public:
template <class T, class U>
void val_and_deriv(T x, U &pd) const {
pd[0] = back();
int nc = size() - 1;
int nd = pd.size() - 1;
for(unsigned j = 1; j < pd.size(); j++)
pd[j] = 0.0;
for(int i = nc -1; i >= 0; i--) {
int nnd = std::min(nd, nc-i);
for(int j = nnd; j >= 1; j--)
pd[j] = pd[j]*x + operator[](i);
pd[0] = pd[0]*x + operator[](i);
}
double cnst = 1;
for(int i = 2; i <= nd; i++) {
cnst *= i;
pd[i] *= cnst;
}
}
static Poly linear(double ax, double b) {
Poly p;
p.push_back(b);
p.push_back(ax);
return p;
}
};
inline Poly operator*(double a, Poly const & b) { return b * a;}
Poly integral(Poly const & p);
Poly derivative(Poly const & p);
Poly divide_out_root(Poly const & p, double x);
Poly compose(Poly const & a, Poly const & b);
Poly divide(Poly const &a, Poly const &b, Poly &r);
Poly gcd(Poly const &a, Poly const &b, const double tol=1e-10);
/*** solve(Poly p)
* find all p.degree() roots of p.
* This function can take a long time with suitably crafted polynomials, but in practice it should be fast. Should we provide special forms for degree() <= 4?
*/
std::vector<std::complex<double> > solve(const Poly & p);
#ifdef HAVE_GSL
/*** solve_reals(Poly p)
* find all real solutions to Poly p.
* currently we just use solve and pick out the suitably real looking values, there may be a better algorithm.
*/
std::vector<double> solve_reals(const Poly & p);
#endif
double polish_root(Poly const & p, double guess, double tol);
/** @brief Analytically solve quadratic equation.
* The equation is given in the standard form: ax^2 + bx + c = 0.
* Only real roots are returned. */
std::vector<Coord> solve_quadratic(Coord a, Coord b, Coord c);
/** @brief Analytically solve cubic equation.
* The equation is given in the standard form: ax^3 + bx^2 + cx + d = 0.
* Only real roots are returned. */
std::vector<Coord> solve_cubic(Coord a, Coord b, Coord c, Coord d);
inline std::ostream &operator<< (std::ostream &out_file, const Poly &in_poly) {
if(in_poly.size() == 0)
out_file << "0";
else {
for(int i = (int)in_poly.size()-1; i >= 0; --i) {
if(i == 1) {
out_file << "" << in_poly[i] << "*x";
out_file << " + ";
} else if(i) {
out_file << "" << in_poly[i] << "*x^" << i;
out_file << " + ";
} else
out_file << in_poly[i];
}
}
return out_file;
}
} // namespace Geom
#endif //LIB2GEOM_SEEN_POLY_H
/*
Local Variables:
mode:c++
c-file-style:"stroustrup"
c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
indent-tabs-mode:nil
fill-column:99
End:
*/
// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:fileencoding=utf-8:textwidth=99 :
|
