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
|
// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2015 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
// * Neither the name of Google Inc. nor the names of its contributors may be
// used to endorse or promote products derived from this software without
// specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
//
// Author: moll.markus@arcor.de (Markus Moll)
// sameeragarwal@google.com (Sameer Agarwal)
#ifndef CERES_INTERNAL_POLYNOMIAL_SOLVER_H_
#define CERES_INTERNAL_POLYNOMIAL_SOLVER_H_
#include <string>
#include <vector>
#include "ceres/internal/eigen.h"
#include "ceres/internal/port.h"
namespace ceres {
namespace internal {
// All polynomials are assumed to be the form
//
// sum_{i=0}^N polynomial(i) x^{N-i}.
//
// and are given by a vector of coefficients of size N + 1.
// Evaluate the polynomial at x using the Horner scheme.
inline double EvaluatePolynomial(const Vector& polynomial, double x) {
double v = 0.0;
for (int i = 0; i < polynomial.size(); ++i) {
v = v * x + polynomial(i);
}
return v;
}
// Use the companion matrix eigenvalues to determine the roots of the
// polynomial.
//
// This function returns true on success, false otherwise.
// Failure indicates that the polynomial is invalid (of size 0) or
// that the eigenvalues of the companion matrix could not be computed.
// On failure, a more detailed message will be written to LOG(ERROR).
// If real is not NULL, the real parts of the roots will be returned in it.
// Likewise, if imaginary is not NULL, imaginary parts will be returned in it.
bool FindPolynomialRoots(const Vector& polynomial,
Vector* real,
Vector* imaginary);
// Return the derivative of the given polynomial. It is assumed that
// the input polynomial is at least of degree zero.
Vector DifferentiatePolynomial(const Vector& polynomial);
// Find the minimum value of the polynomial in the interval [x_min,
// x_max]. The minimum is obtained by computing all the roots of the
// derivative of the input polynomial. All real roots within the
// interval [x_min, x_max] are considered as well as the end points
// x_min and x_max. Since polynomials are differentiable functions,
// this ensures that the true minimum is found.
void MinimizePolynomial(const Vector& polynomial,
double x_min,
double x_max,
double* optimal_x,
double* optimal_value);
// Structure for storing sample values of a function.
//
// Clients can use this struct to communicate the value of the
// function and or its gradient at a given point x.
struct FunctionSample {
FunctionSample()
: x(0.0),
value(0.0),
value_is_valid(false),
gradient(0.0),
gradient_is_valid(false) {
}
std::string ToDebugString() const;
double x;
double value; // value = f(x)
bool value_is_valid;
double gradient; // gradient = f'(x)
bool gradient_is_valid;
};
// Given a set of function value and/or gradient samples, find a
// polynomial whose value and gradients are exactly equal to the ones
// in samples.
//
// Generally speaking,
//
// degree = # values + # gradients - 1
//
// Of course its possible to sample a polynomial any number of times,
// in which case, generally speaking the spurious higher order
// coefficients will be zero.
Vector FindInterpolatingPolynomial(const std::vector<FunctionSample>& samples);
// Interpolate the function described by samples with a polynomial,
// and minimize it on the interval [x_min, x_max]. Depending on the
// input samples, it is possible that the interpolation or the root
// finding algorithms may fail due to numerical difficulties. But the
// function is guaranteed to return its best guess of an answer, by
// considering the samples and the end points as possible solutions.
void MinimizeInterpolatingPolynomial(const std::vector<FunctionSample>& samples,
double x_min,
double x_max,
double* optimal_x,
double* optimal_value);
} // namespace internal
} // namespace ceres
#endif // CERES_INTERNAL_POLYNOMIAL_SOLVER_H_
|
