1// Ceres Solver - A fast non-linear least squares minimizer
2// Copyright 2010, 2011, 2012 Google Inc. All rights reserved.
3// http://code.google.com/p/ceres-solver/
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29// Author: sameeragarwal@google.com (Sameer Agarwal)
30
31#include "ceres/implicit_schur_complement.h"
32
33#include "Eigen/Dense"
34#include "ceres/block_sparse_matrix.h"
35#include "ceres/block_structure.h"
36#include "ceres/internal/eigen.h"
37#include "ceres/internal/scoped_ptr.h"
38#include "ceres/types.h"
39#include "glog/logging.h"
40
41namespace ceres {
42namespace internal {
43
44ImplicitSchurComplement::ImplicitSchurComplement(int num_eliminate_blocks,
45                                                 bool preconditioner)
46    : num_eliminate_blocks_(num_eliminate_blocks),
47      preconditioner_(preconditioner),
48      A_(NULL),
49      D_(NULL),
50      b_(NULL),
51      block_diagonal_EtE_inverse_(NULL),
52      block_diagonal_FtF_inverse_(NULL) {
53}
54
55ImplicitSchurComplement::~ImplicitSchurComplement() {
56}
57
58void ImplicitSchurComplement::Init(const BlockSparseMatrix& A,
59                                   const double* D,
60                                   const double* b) {
61  // Since initialization is reasonably heavy, perhaps we can save on
62  // constructing a new object everytime.
63  if (A_ == NULL) {
64    A_.reset(new PartitionedMatrixView(A, num_eliminate_blocks_));
65  }
66
67  D_ = D;
68  b_ = b;
69
70  // Initialize temporary storage and compute the block diagonals of
71  // E'E and F'E.
72  if (block_diagonal_EtE_inverse_ == NULL) {
73    block_diagonal_EtE_inverse_.reset(A_->CreateBlockDiagonalEtE());
74    if (preconditioner_) {
75      block_diagonal_FtF_inverse_.reset(A_->CreateBlockDiagonalFtF());
76    }
77    rhs_.resize(A_->num_cols_f());
78    rhs_.setZero();
79    tmp_rows_.resize(A_->num_rows());
80    tmp_e_cols_.resize(A_->num_cols_e());
81    tmp_e_cols_2_.resize(A_->num_cols_e());
82    tmp_f_cols_.resize(A_->num_cols_f());
83  } else {
84    A_->UpdateBlockDiagonalEtE(block_diagonal_EtE_inverse_.get());
85    if (preconditioner_) {
86      A_->UpdateBlockDiagonalFtF(block_diagonal_FtF_inverse_.get());
87    }
88  }
89
90  // The block diagonals of the augmented linear system contain
91  // contributions from the diagonal D if it is non-null. Add that to
92  // the block diagonals and invert them.
93  AddDiagonalAndInvert(D_, block_diagonal_EtE_inverse_.get());
94  if (preconditioner_)  {
95    AddDiagonalAndInvert((D_ ==  NULL) ? NULL : D_ + A_->num_cols_e(),
96                         block_diagonal_FtF_inverse_.get());
97  }
98
99  // Compute the RHS of the Schur complement system.
100  UpdateRhs();
101}
102
103// Evaluate the product
104//
105//   Sx = [F'F - F'E (E'E)^-1 E'F]x
106//
107// By breaking it down into individual matrix vector products
108// involving the matrices E and F. This is implemented using a
109// PartitionedMatrixView of the input matrix A.
110void ImplicitSchurComplement::RightMultiply(const double* x, double* y) const {
111  // y1 = F x
112  tmp_rows_.setZero();
113  A_->RightMultiplyF(x, tmp_rows_.data());
114
115  // y2 = E' y1
116  tmp_e_cols_.setZero();
117  A_->LeftMultiplyE(tmp_rows_.data(), tmp_e_cols_.data());
118
119  // y3 = -(E'E)^-1 y2
120  tmp_e_cols_2_.setZero();
121  block_diagonal_EtE_inverse_->RightMultiply(tmp_e_cols_.data(),
122                                             tmp_e_cols_2_.data());
123  tmp_e_cols_2_ *= -1.0;
124
125  // y1 = y1 + E y3
126  A_->RightMultiplyE(tmp_e_cols_2_.data(), tmp_rows_.data());
127
128  // y5 = D * x
129  if (D_ != NULL) {
130    ConstVectorRef Dref(D_ + A_->num_cols_e(), num_cols());
131    VectorRef(y, num_cols()) =
132        (Dref.array().square() *
133         ConstVectorRef(x, num_cols()).array()).matrix();
134  } else {
135    VectorRef(y, num_cols()).setZero();
136  }
137
138  // y = y5 + F' y1
139  A_->LeftMultiplyF(tmp_rows_.data(), y);
140}
141
142// Given a block diagonal matrix and an optional array of diagonal
143// entries D, add them to the diagonal of the matrix and compute the
144// inverse of each diagonal block.
145void ImplicitSchurComplement::AddDiagonalAndInvert(
146    const double* D,
147    BlockSparseMatrix* block_diagonal) {
148  const CompressedRowBlockStructure* block_diagonal_structure =
149      block_diagonal->block_structure();
150  for (int r = 0; r < block_diagonal_structure->rows.size(); ++r) {
151    const int row_block_pos = block_diagonal_structure->rows[r].block.position;
152    const int row_block_size = block_diagonal_structure->rows[r].block.size;
153    const Cell& cell = block_diagonal_structure->rows[r].cells[0];
154    MatrixRef m(block_diagonal->mutable_values() + cell.position,
155                row_block_size, row_block_size);
156
157    if (D != NULL) {
158      ConstVectorRef d(D + row_block_pos, row_block_size);
159      m += d.array().square().matrix().asDiagonal();
160    }
161
162    m = m
163        .selfadjointView<Eigen::Upper>()
164        .llt()
165        .solve(Matrix::Identity(row_block_size, row_block_size));
166  }
167}
168
169// Similar to RightMultiply, use the block structure of the matrix A
170// to compute y = (E'E)^-1 (E'b - E'F x).
171void ImplicitSchurComplement::BackSubstitute(const double* x, double* y) {
172  const int num_cols_e = A_->num_cols_e();
173  const int num_cols_f = A_->num_cols_f();
174  const int num_cols =  A_->num_cols();
175  const int num_rows = A_->num_rows();
176
177  // y1 = F x
178  tmp_rows_.setZero();
179  A_->RightMultiplyF(x, tmp_rows_.data());
180
181  // y2 = b - y1
182  tmp_rows_ = ConstVectorRef(b_, num_rows) - tmp_rows_;
183
184  // y3 = E' y2
185  tmp_e_cols_.setZero();
186  A_->LeftMultiplyE(tmp_rows_.data(), tmp_e_cols_.data());
187
188  // y = (E'E)^-1 y3
189  VectorRef(y, num_cols).setZero();
190  block_diagonal_EtE_inverse_->RightMultiply(tmp_e_cols_.data(), y);
191
192  // The full solution vector y has two blocks. The first block of
193  // variables corresponds to the eliminated variables, which we just
194  // computed via back substitution. The second block of variables
195  // corresponds to the Schur complement system, so we just copy those
196  // values from the solution to the Schur complement.
197  VectorRef(y + num_cols_e, num_cols_f) =  ConstVectorRef(x, num_cols_f);
198}
199
200// Compute the RHS of the Schur complement system.
201//
202// rhs = F'b - F'E (E'E)^-1 E'b
203//
204// Like BackSubstitute, we use the block structure of A to implement
205// this using a series of matrix vector products.
206void ImplicitSchurComplement::UpdateRhs() {
207  // y1 = E'b
208  tmp_e_cols_.setZero();
209  A_->LeftMultiplyE(b_, tmp_e_cols_.data());
210
211  // y2 = (E'E)^-1 y1
212  Vector y2 = Vector::Zero(A_->num_cols_e());
213  block_diagonal_EtE_inverse_->RightMultiply(tmp_e_cols_.data(), y2.data());
214
215  // y3 = E y2
216  tmp_rows_.setZero();
217  A_->RightMultiplyE(y2.data(), tmp_rows_.data());
218
219  // y3 = b - y3
220  tmp_rows_ = ConstVectorRef(b_, A_->num_rows()) - tmp_rows_;
221
222  // rhs = F' y3
223  rhs_.setZero();
224  A_->LeftMultiplyF(tmp_rows_.data(), rhs_.data());
225}
226
227}  // namespace internal
228}  // namespace ceres
229