ConditionEstimator.h
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1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2016 Rasmus Munk Larsen (rmlarsen@google.com)
5 //
6 // This Source Code Form is subject to the terms of the Mozilla
7 // Public License v. 2.0. If a copy of the MPL was not distributed
8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9 
10 #ifndef EIGEN_CONDITIONESTIMATOR_H
11 #define EIGEN_CONDITIONESTIMATOR_H
12 
13 namespace Eigen {
14 
15 namespace internal {
16 
17 template <typename Vector, typename RealVector, bool IsComplex>
19  static inline Vector run(const Vector& v) {
20  const RealVector v_abs = v.cwiseAbs();
21  return (v_abs.array() == static_cast<typename Vector::RealScalar>(0))
22  .select(Vector::Ones(v.size()), v.cwiseQuotient(v_abs));
23  }
24 };
25 
26 // Partial specialization to avoid elementwise division for real vectors.
27 template <typename Vector>
28 struct rcond_compute_sign<Vector, Vector, false> {
29  static inline Vector run(const Vector& v) {
30  return (v.array() < static_cast<typename Vector::RealScalar>(0))
31  .select(-Vector::Ones(v.size()), Vector::Ones(v.size()));
32  }
33 };
34 
55 template <typename Decomposition>
57 {
58  typedef typename Decomposition::MatrixType MatrixType;
59  typedef typename Decomposition::Scalar Scalar;
60  typedef typename Decomposition::RealScalar RealScalar;
63  const bool is_complex = (NumTraits<Scalar>::IsComplex != 0);
64 
65  eigen_assert(dec.rows() == dec.cols());
66  const Index n = dec.rows();
67  if (n == 0)
68  return 0;
69 
70  // Disable Index to float conversion warning
71 #ifdef __INTEL_COMPILER
72  #pragma warning push
73  #pragma warning ( disable : 2259 )
74 #endif
75  Vector v = dec.solve(Vector::Ones(n) / Scalar(n));
76 #ifdef __INTEL_COMPILER
77  #pragma warning pop
78 #endif
79 
80  // lower_bound is a lower bound on
81  // ||inv(matrix)||_1 = sup_v ||inv(matrix) v||_1 / ||v||_1
82  // and is the objective maximized by the ("super-") gradient ascent
83  // algorithm below.
84  RealScalar lower_bound = v.template lpNorm<1>();
85  if (n == 1)
86  return lower_bound;
87 
88  // Gradient ascent algorithm follows: We know that the optimum is achieved at
89  // one of the simplices v = e_i, so in each iteration we follow a
90  // super-gradient to move towards the optimal one.
91  RealScalar old_lower_bound = lower_bound;
92  Vector sign_vector(n);
93  Vector old_sign_vector;
94  Index v_max_abs_index = -1;
95  Index old_v_max_abs_index = v_max_abs_index;
96  for (int k = 0; k < 4; ++k)
97  {
99  if (k > 0 && !is_complex && sign_vector == old_sign_vector) {
100  // Break if the solution stagnated.
101  break;
102  }
103  // v_max_abs_index = argmax |real( inv(matrix)^T * sign_vector )|
104  v = dec.adjoint().solve(sign_vector);
105  v.real().cwiseAbs().maxCoeff(&v_max_abs_index);
106  if (v_max_abs_index == old_v_max_abs_index) {
107  // Break if the solution stagnated.
108  break;
109  }
110  // Move to the new simplex e_j, where j = v_max_abs_index.
111  v = dec.solve(Vector::Unit(n, v_max_abs_index)); // v = inv(matrix) * e_j.
112  lower_bound = v.template lpNorm<1>();
113  if (lower_bound <= old_lower_bound) {
114  // Break if the gradient step did not increase the lower_bound.
115  break;
116  }
117  if (!is_complex) {
118  old_sign_vector = sign_vector;
119  }
120  old_v_max_abs_index = v_max_abs_index;
121  old_lower_bound = lower_bound;
122  }
123  // The following calculates an independent estimate of ||matrix||_1 by
124  // multiplying matrix by a vector with entries of slowly increasing
125  // magnitude and alternating sign:
126  // v_i = (-1)^{i} (1 + (i / (dim-1))), i = 0,...,dim-1.
127  // This improvement to Hager's algorithm above is due to Higham. It was
128  // added to make the algorithm more robust in certain corner cases where
129  // large elements in the matrix might otherwise escape detection due to
130  // exact cancellation (especially when op and op_adjoint correspond to a
131  // sequence of backsubstitutions and permutations), which could cause
132  // Hager's algorithm to vastly underestimate ||matrix||_1.
133  Scalar alternating_sign(RealScalar(1));
134  for (Index i = 0; i < n; ++i) {
135  // The static_cast is needed when Scalar is a complex and RealScalar implements expression templates
136  v[i] = alternating_sign * static_cast<RealScalar>(RealScalar(1) + (RealScalar(i) / (RealScalar(n - 1))));
137  alternating_sign = -alternating_sign;
138  }
139  v = dec.solve(v);
140  const RealScalar alternate_lower_bound = (2 * v.template lpNorm<1>()) / (3 * RealScalar(n));
141  return numext::maxi(lower_bound, alternate_lower_bound);
142 }
143 
157 template <typename Decomposition>
159 rcond_estimate_helper(typename Decomposition::RealScalar matrix_norm, const Decomposition& dec)
160 {
161  typedef typename Decomposition::RealScalar RealScalar;
162  eigen_assert(dec.rows() == dec.cols());
163  if (dec.rows() == 0) return NumTraits<RealScalar>::infinity();
164  if (matrix_norm == RealScalar(0)) return RealScalar(0);
165  if (dec.rows() == 1) return RealScalar(1);
166  const RealScalar inverse_matrix_norm = rcond_invmatrix_L1_norm_estimate(dec);
167  return (inverse_matrix_norm == RealScalar(0) ? RealScalar(0)
168  : (RealScalar(1) / inverse_matrix_norm) / matrix_norm);
169 }
170 
171 } // namespace internal
172 
173 } // namespace Eigen
174 
175 #endif
Eigen
Namespace containing all symbols from the Eigen library.
Definition: jet.h:637
MatrixType
MatrixXf MatrixType
Definition: benchmark-blocking-sizes.cpp:52
eigen_assert
#define eigen_assert(x)
Definition: Macros.h:1037
is_complex
Definition: numpy.h:418
Eigen::internal::rcond_estimate_helper
Decomposition::RealScalar rcond_estimate_helper(typename Decomposition::RealScalar matrix_norm, const Decomposition &dec)
Reciprocal condition number estimator.
Definition: ConditionEstimator.h:159
n
int n
Definition: BiCGSTAB_simple.cpp:1
Eigen::internal::true_type
Definition: Meta.h:96
Eigen::internal::rcond_compute_sign::run
static Vector run(const Vector &v)
Definition: ConditionEstimator.h:19
RealScalar
NumTraits< Scalar >::Real RealScalar
Definition: bench_gemm.cpp:47
Eigen::internal::rcond_compute_sign< Vector, Vector, false >::run
static Vector run(const Vector &v)
Definition: ConditionEstimator.h:29
v
Array< int, Dynamic, 1 > v
Definition: Array_initializer_list_vector_cxx11.cpp:1
Eigen::internal::rcond_compute_sign
Definition: ConditionEstimator.h:18
Eigen::internal::rcond_invmatrix_L1_norm_estimate
Decomposition::RealScalar rcond_invmatrix_L1_norm_estimate(const Decomposition &dec)
Definition: ConditionEstimator.h:56
internal
Definition: BandTriangularSolver.h:13
Eigen::numext::maxi
EIGEN_DEVICE_FUNC EIGEN_ALWAYS_INLINE T maxi(const T &x, const T &y)
Definition: Eigen/src/Core/MathFunctions.h:1093
ceres::Vector
Eigen::Matrix< double, Eigen::Dynamic, 1 > Vector
Definition: gtsam/3rdparty/ceres/eigen.h:38
Eigen::NumTraits
Holds information about the various numeric (i.e. scalar) types allowed by Eigen.
Definition: NumTraits.h:232
i
int i
Definition: BiCGSTAB_step_by_step.cpp:9
Scalar
SCALAR Scalar
Definition: bench_gemm.cpp:46
Eigen::Index
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
The Index type as used for the API.
Definition: Meta.h:74


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